From aacdddbf007134591bc2123fe1e12b3477afcff3 Mon Sep 17 00:00:00 2001
From: "J. Derek Tucker" <jdtuck@sandia.gov>
Date: Thu, 18 Nov 2021 07:59:44 -0700
Subject: [PATCH 01/12] stub files

---
 fdasrsf/image.py           | 21 +++++++++++++++++++++
 fdasrsf/image_functions.py |  6 ++++++
 2 files changed, 27 insertions(+)
 create mode 100644 fdasrsf/image.py
 create mode 100644 fdasrsf/image_functions.py

diff --git a/fdasrsf/image.py b/fdasrsf/image.py
new file mode 100644
index 00000000..648109da
--- /dev/null
+++ b/fdasrsf/image.py
@@ -0,0 +1,21 @@
+"""
+image warping using SRVF framework
+
+moduleauthor:: J. Derek Tucker <jdtuck@sandia.gov>
+
+"""
+
+
+def reparam_image(It, Im, gam, b, stepsize=1e-4, itermax=20):
+    """
+    This function warps an image to another using SRVF framework
+
+    :param Im: numpy ndarray of shape (N,N) respresenting a NxN image
+    :param Im: numpy ndarray of shape (N,N) respresenting a NxN image
+    :param gam: numpy ndarray of shape (N,N) respresenting an initial warping function
+    :param b: numpy ndarray of 
+
+    :rtype: numpy ndarray
+    :return f: smoothed functions functions
+
+    """
\ No newline at end of file
diff --git a/fdasrsf/image_functions.py b/fdasrsf/image_functions.py
new file mode 100644
index 00000000..79066931
--- /dev/null
+++ b/fdasrsf/image_functions.py
@@ -0,0 +1,6 @@
+"""
+functions for SRVF image manipulations
+
+moduleauthor:: J. Derek Tucker <jdtuck@sandia.gov>
+
+"""
\ No newline at end of file

From 1d69af1ef20cfd0de99837174c2826117d81ae40 Mon Sep 17 00:00:00 2001
From: Derek Tucker <jdtuck@sandia.gov>
Date: Sun, 12 Feb 2023 21:25:38 -0700
Subject: [PATCH 02/12] add c++ files

---
 src/ImageRegister.h     | 623 +++++++++++++++++++++++++++++++++++
 src/UnitSquareImage.cpp | 708 ++++++++++++++++++++++++++++++++++++++++
 src/UnitSquareImage.h   |  32 ++
 src/myVector.cpp        | 125 +++++++
 src/myVector.h          |  20 ++
 5 files changed, 1508 insertions(+)
 create mode 100644 src/ImageRegister.h
 create mode 100644 src/UnitSquareImage.cpp
 create mode 100644 src/UnitSquareImage.h
 create mode 100644 src/myVector.cpp
 create mode 100644 src/myVector.h

diff --git a/src/ImageRegister.h b/src/ImageRegister.h
new file mode 100644
index 00000000..22a4728b
--- /dev/null
+++ b/src/ImageRegister.h
@@ -0,0 +1,623 @@
+#ifndef SURFACECHRISTOFFEL_H
+#define SURFACECHRISTOFFEL_H
+
+#include <cstring>
+#include <cmath>
+using namespace std;
+
+#define SQRT5	2.23606797749978969641
+
+// Prototypes: -----------------------------------------------------------------
+template<typename T>
+void cpyArray(T *x, const T *y, int n);
+
+template<typename T>
+T innerProd(const T *x, const T *y, int N, int d);
+
+template<typename T>
+T det2(const T *A);
+
+template<typename T>
+T det3(const T *A);
+
+template<typename T>
+inline void jacob(T z, const T x[2], const T y[2]);
+
+template<typename T>
+void jacobMat(T *z, const T *x, const T *y, int n, int t);
+
+template<typename T>
+void cross(T z[3], const T x[3], const T y[3]);
+
+template<typename T>
+void cross2(T z, const T x[2], const T y[2]);
+
+template<typename T>
+void matmul3(T *C, const T *A, const T *B);
+
+template<typename T>
+void matvec3(T y[3], const T *A, const T x[3]);
+
+template<typename T>
+void aat3(T *AA, const T *A);
+
+template<typename T>
+void ata3(T *AA, const T *A);
+
+template<typename T>
+void div3(T x[3], T div);
+
+template<typename T>
+void mult3(T x[3], T mult);
+
+template<typename T>
+void assign3(T *y, const T x[3], int s = 1);
+
+template<typename T>
+T L2norm3(T x[3]);
+
+template<typename T>
+T L2norm(T x, int s);
+
+template<typename T>
+T cdiff(const T *F, T dx, int s);
+
+template<typename T>
+T fdiff(const T *F, T dx, int s);
+
+template<typename T>
+T bdiff(const T *F, T dx, int s);
+
+template<typename T>
+void cdiff2(T df[2], const T *F, T dx, int s, int S);
+
+template<typename T>
+void fdiff2(T df[2], const T *F, T dx, int s, int S);
+
+template<typename T>
+void bdiff2(T df[2], const T *F, T dx, int s, int S);
+
+template<typename T>
+void cdiff3(T df[3], const T *F, T dx, int s, int S);
+
+template<typename T>
+void fdiff3(T df[3], const T *F, T dx, int s, int S);
+
+template<typename T>
+void bdiff3(T df[3], const T *F, T dx, int s, int S);
+
+template<typename T>
+void sort3(T x[3]);
+
+template<typename T>
+void solvecubic(T c[3]);
+
+template<typename T>
+void ldu3(T *A, int P[3]);
+
+template<typename T>
+void ldubsolve3(T x[3], const T y[3], const T *LDU, const int P[3]);
+
+template<typename T>
+void unit3(T x[3]);
+
+template<typename T>
+void svd3(T *U, T S[3], T *V, const T *A);
+
+template<typename T>
+void thomas(T *x, const T *a, const T *b, T *c, int n);
+
+template<typename T>
+void spline(T *D, const T *y, int n);
+
+template<typename T>
+void lookupspline(T &t, int &k, T dist, T len, int n);
+
+template<typename T>
+T evalspline(T t, const T D[2], const T y[2]);
+
+template<typename T>
+void interp2(T *Dy, const T *Z, int nx, int ny);
+
+template<typename T>
+void evalinterp2(T u, T *Dx, T *zx, const T *Dy, const T *Z, int nx, int ny);
+
+
+// Subroutines ============================================================
+template<typename T>
+void cpyArray(T *x, const T *y, int n) {
+    for (int i=0; i<n; ++i) {
+        x[i] = y[i];
+    }
+}
+
+template<typename T>
+inline T innerProd(const T *x, const T *y, int N, int d) {
+    T s = 0;
+    for (int i=0; i<d; i++) {
+        s += x[N*i]*y[N*i];
+    }
+    return s;
+}
+
+template<typename T>
+inline void jacob(T z, const T x[2], const T y[2]) {
+	z = x[1]*y[2]-x[2]*y[1];
+}
+
+template<typename T>
+inline void jacobMat(T *z, const T *x, const T *y, int n, int t) {
+    int i, N = n*t, d = 2;
+    if (d == 2) {
+        for (i=0; i<N; ++i) {
+            z[i] = x[N*0 + i]*y[N*1 + i]-x[N*1 + i]*y[N*0 + i];
+        }
+    }
+}
+
+template<typename T>
+inline void cross(T z[3], const T x[3], const T y[3]) {
+	z[0] = x[1]*y[2]-x[2]*y[1];
+	z[1] = -(x[0]*y[2]-x[2]*y[0]);
+	z[2] = x[0]*y[1]-x[1]*y[0];
+}
+
+template<typename T>
+inline void cross2(T z, const T x[2], const T y[2]) {
+	z = x[1]*y[2]-x[2]*y[1];
+}
+
+template<typename T>
+inline T cdiff(const T *F, T dx, int s) {
+	return (F[s]-F[-s])/(2*dx);
+}
+
+template<typename T>
+inline T fdiff(const T *F, T dx, int s) {
+	return (F[s]-F[0])/dx;
+}
+
+template<typename T>
+inline T bdiff(const T *F, T dx, int s) {
+	return (F[0]-F[-s])/dx;
+}
+
+template<typename T>
+inline void cdiff2(T df[2], const T *F, T dx, int s, int S) {
+    int i;
+    for (i=0; i<2; ++i){
+        df[i] = (F[i*S+s]-F[i*S-s])/(2*dx);
+    }
+}
+
+template<typename T>
+inline void fdiff2(T df[2], const T *F, T dx, int s, int S) {
+    int i;
+    for (i=0; i<2; ++i){
+        df[i] = (F[i*S+s]-F[i*S])/(2*dx);
+    }
+}
+
+template<typename T>
+inline void bdiff2(T df[2], const T *F, T dx, int s, int S) {
+    int i;
+    for (i=0; i<2; ++i){
+        df[i] = (F[i*S]-F[i*S-s])/(2*dx);
+    }
+}
+
+template<typename T>
+inline void cdiff3(T df[3], const T *F, T dx, int s, int S) {
+	df[0] = (F[s]-F[-s])/(2*dx);
+	df[1] = (F[S+s]-F[S-s])/(2*dx);
+	df[2] = (F[2*S+s]-F[2*S-s])/(2*dx);
+}
+
+template<typename T>
+inline void fdiff3(T df[3], const T *F, T dx, int s, int S) {
+	df[0] = (F[s]-F[0])/dx;
+	df[1] = (F[S+s]-F[S])/dx;
+	df[2] = (F[2*S+s]-F[2*S])/dx;
+}
+
+template<typename T>
+inline void bdiff3(T df[3], const T *F, T dx, int s, int S) {
+	df[0] = (F[0]-F[-s])/dx;
+	df[1] = (F[S]-F[S-s])/dx;
+	df[2] = (F[2*S]-F[2*S-s])/dx;
+}
+
+template<typename T>
+inline void matmul3(T *C, const T *A, const T *B) {
+	C[3*0+0] = A[3*0+0]*B[3*0+0] + A[3*1+0]*B[3*0+1] + A[3*2+0]*B[3*0+2];
+	C[3*1+0] = A[3*0+0]*B[3*1+0] + A[3*1+0]*B[3*1+1] + A[3*2+0]*B[3*1+2];
+	C[3*2+0] = A[3*0+0]*B[3*2+0] + A[3*1+0]*B[3*2+1] + A[3*2+0]*B[3*2+2];
+
+	C[3*0+1] = A[3*0+1]*B[3*0+0] + A[3*1+1]*B[3*0+1] + A[3*2+1]*B[3*0+2];
+	C[3*1+1] = A[3*0+1]*B[3*1+0] + A[3*1+1]*B[3*1+1] + A[3*2+1]*B[3*1+2];
+	C[3*2+1] = A[3*0+1]*B[3*2+0] + A[3*1+1]*B[3*2+1] + A[3*2+1]*B[3*2+2];
+
+	C[3*0+2] = A[3*0+2]*B[3*0+0] + A[3*1+2]*B[3*0+1] + A[3*2+2]*B[3*0+2];
+	C[3*1+2] = A[3*0+2]*B[3*1+0] + A[3*1+2]*B[3*1+1] + A[3*2+2]*B[3*1+2];
+	C[3*2+2] = A[3*0+2]*B[3*2+0] + A[3*1+2]*B[3*2+1] + A[3*2+2]*B[3*2+2];
+}
+
+template<typename T>
+inline void matvec3(T y[3], const T *A, const T x[3]) {
+	y[0] = A[3*0+0]*x[0] + A[3*1+0]*x[1] + A[3*2+0]*x[2];
+	y[1] = A[3*0+1]*x[0] + A[3*1+1]*x[1] + A[3*2+1]*x[2];
+	y[2] = A[3*0+2]*x[0] + A[3*1+2]*x[1] + A[3*2+2]*x[2];
+}
+
+template<typename T>
+inline void aat3(T *AA, const T *A) {
+	AA[3*0+0] = A[3*0+0]*A[3*0+0] + A[3*1+0]*A[3*1+0] + A[3*2+0]*A[3*2+0];
+	AA[3*1+0] = A[3*0+0]*A[3*0+1] + A[3*1+0]*A[3*1+1] + A[3*2+0]*A[3*2+1];
+	AA[3*2+0] = A[3*0+0]*A[3*0+2] + A[3*1+0]*A[3*1+2] + A[3*2+0]*A[3*2+2];
+
+	AA[3*0+1] = AA[3*1+0];
+	AA[3*1+1] = A[3*0+1]*A[3*0+1] + A[3*1+1]*A[3*1+1] + A[3*2+1]*A[3*2+1];
+	AA[3*2+1] = A[3*0+1]*A[3*0+2] + A[3*1+1]*A[3*1+2] + A[3*2+1]*A[3*2+2];
+
+	AA[3*0+2] = AA[3*2+0];
+	AA[3*1+2] = AA[3*2+1];
+	AA[3*2+2] = A[3*0+2]*A[3*0+2] + A[3*1+2]*A[3*1+2] + A[3*2+2]*A[3*2+2];
+}
+
+template<typename T>
+inline void ata3(T *AA, const T *A) {
+	AA[3*0+0] = A[3*0+0]*A[3*0+0] + A[3*0+1]*A[3*0+1] + A[3*0+2]*A[3*0+2];
+	AA[3*1+0] = A[3*0+0]*A[3*1+0] + A[3*0+1]*A[3*1+1] + A[3*0+2]*A[3*1+2];
+	AA[3*2+0] = A[3*0+0]*A[3*2+0] + A[3*0+1]*A[3*2+1] + A[3*0+2]*A[3*2+2];
+
+	AA[3*0+1] = AA[3*1+0];
+	AA[3*1+1] = A[3*1+0]*A[3*1+0] + A[3*1+1]*A[3*1+1] + A[3*1+2]*A[3*1+2];
+	AA[3*2+1] = A[3*1+0]*A[3*2+0] + A[3*1+1]*A[3*2+1] + A[3*1+2]*A[3*2+2];
+
+	AA[3*0+2] = AA[3*2+0];
+	AA[3*1+2] = AA[3*2+1];
+	AA[3*2+2] = A[3*2+0]*A[3*2+0] + A[3*2+1]*A[3*2+1] + A[3*2+2]*A[3*2+2];
+
+}
+
+template<typename T>
+inline void div3(T x[3], T div) {
+	x[0] /= div;
+	x[1] /= div;
+	x[2] /= div;
+}
+
+template<typename T>
+inline void mult3(T x[3], T mult) {
+	x[0] *= mult;
+	x[1] *= mult;
+	x[2] *= mult;
+}
+
+template<typename T>
+inline T det2(const T *A) {
+	return A[2*0+0]*A[2*1+1] - A[2*0+1]*A[2*1+0];
+}
+
+template<typename T>
+inline T det3(const T *A) {
+	return A[3*0+0]*(A[3*1+1]*A[3*2+2] - A[3*2+1]*A[3*1+2]) -
+		A[3*1+0]*(A[3*0+1]*A[3*2+2] - A[3*2+1]*A[3*0+2]) +
+		A[3*2+0]*(A[3*0+1]*A[3*1+2] - A[3*1+1]*A[3*0+2]);
+}
+
+template<typename T>
+inline void assign3(T *y, const T x[3], int s) {
+	y[0] = x[0];
+	y[s] = x[1];
+	y[2*s] = x[2];
+}
+
+template<typename T>
+inline T L2norm3(T x[3]) {
+	return (T)sqrt(x[0]*x[0] + x[1]*x[1] + x[2]*x[2]);
+}
+
+template<typename T>
+inline T L2norm(T x, int s) {
+    double temp = 0;
+    for (int i = 0; i < s; ++i){
+        temp += x[i]*x[i];
+    }
+	return (T)sqrt(temp);
+}
+
+template<typename T>
+inline void sort3(T x[3]) {
+	T tmp;
+	int k;
+
+	k = (x[1] <= x[0] || x[2] <= x[0]) + (x[2] <= x[0] && x[2] <= x[1]);
+	tmp = x[2];
+	x[2] = x[k];
+	x[k] = tmp;
+
+	k = (x[1] <= x[0]);
+	tmp = x[1];
+	x[1] = x[k];
+	x[k] = tmp;
+}
+
+template<typename T>
+void solvecubic(T c[3]) {
+	const double sq3d2 = 0.86602540378443864676, c2d3 = c[2]/3.0,
+		c2sq = ((double)c[2])*c[2], Q = (3.0*c[1]-c2sq)/9.0,
+		R = (c[2]*(9.0*c[1]-2.0*c2sq)-27.0*c[0])/54.0;
+	double tmp, t, sint, cost;
+
+	if (Q < 0) {
+		tmp = 2*sqrt(-Q);
+		t = acos(R/sqrt(-Q*Q*Q))/3;
+		cost = tmp*cos(t);
+		sint = tmp*sin(t);
+
+		c[0] = (T)(cost - c2d3);
+
+		cost = -0.5*cost - c2d3;
+		sint = sq3d2*sint;
+
+		c[1] = (T)(cost - sint);
+		c[2] = (T)(cost + sint);
+	}
+	else {
+#if __STDC_VERSION__ >= 199901L
+		tmp = cbrt(R);
+#else
+		tmp = pow(R,1.0/3);
+#endif
+		c[0] = (T)(-c2d3 + 2*tmp);
+		c[1] = c[2] = (T)(-c2d3 - tmp);
+	}
+}
+
+template<typename T>
+void ldu3(T *A, int P[3]) {
+	int tmp, k;
+	P[1] = 1;
+	P[2] = 2;
+
+	P[0] = ((fabs(A[3*0+0]) < fabs(A[3*1+0])) | (fabs(A[3*0+0]) < fabs(A[3*2+0]))) +
+		((fabs(A[3*0+0]) < fabs(A[3*2+0])) & (fabs(A[3*1+0]) < fabs(A[3*2+0])));
+	P[P[0]] = 0;
+
+	k = 1 + (fabs(A[3*P[1]+1]) < fabs(A[3*P[2]+1]));
+	tmp = P[1];
+	P[1] = P[k];
+	P[k] = tmp;
+
+
+	if (A[3*P[0]+0] != 0) {
+		A[3*P[1]+0] = A[3*P[1]+0]/A[3*P[0]+0];
+		A[3*P[2]+0] = A[3*P[2]+0]/A[3*P[0]+0];
+		A[3*P[0]+1] = A[3*P[0]+1]/A[3*P[0]+0];
+		A[3*P[0]+2] = A[3*P[0]+2]/A[3*P[0]+0];
+	}
+
+	A[3*P[1]+1] = A[3*P[1]+1] - A[3*P[0]+1]*A[3*P[1]+0]*A[3*P[0]+0];
+
+	if (A[3*P[1]+1] != 0) {
+		A[3*P[2]+1] = (A[3*P[2]+1] - A[3*P[0]+1]*A[3*P[2]+0]*A[3*P[0]+0])/A[3*P[1]+1];
+		A[3*P[1]+2] = (A[3*P[1]+2] - A[3*P[0]+2]*A[3*P[1]+0]*A[3*P[0]+0])/A[3*P[1]+1];
+	}
+
+	A[3*P[2]+2] = A[3*P[2]+2] - A[3*P[0]+2]*A[3*P[2]+0]*A[3*P[0]+0] - A[3*P[1]+2]*A[3*P[2]+1]*A[3*P[1]+1];
+}
+
+template<typename T>
+inline void ldubsolve3(T x[3], const T y[3], const T *LDU, const int P[3]) {
+	x[P[2]] = y[2];
+	x[P[1]] = y[1] - LDU[3*P[2]+1]*x[P[2]];
+	x[P[0]] = y[0] - LDU[3*P[2]+0]*x[P[2]] - LDU[3*P[1]+0]*x[P[1]];
+}
+
+template<typename T>
+inline void unit3(T x[3]) {
+	double tmp = sqrt(((double)x[0])*x[0]+((double)x[1])*x[1]+((double)x[2])*x[2]);
+	x[0] = (T)(x[0]/tmp);
+	x[1] = (T)(x[1]/tmp);
+	x[2] = (T)(x[2]/tmp);
+}
+
+template<typename T>
+void svd3(T *U, T S[3], T *V, const T *A) {
+	const T thr = (T)(1e-10);
+	int P[3], k;
+	T y[3], AA[3][3], LDU[3][3];
+
+	ata3((double *)AA, A);
+
+	S[2] = -AA[0][0] - AA[1][1] - AA[2][2];
+	S[1] = AA[0][0]*AA[1][1] + AA[2][2]*AA[0][0] + AA[2][2]*AA[1][1] -
+		AA[2][1]*AA[1][2] - AA[2][0]*AA[0][2] - AA[1][0]*AA[0][1];
+	S[0] = AA[2][1]*AA[1][2]*AA[0][0] + AA[2][0]*AA[0][2]*AA[1][1] + AA[1][0]*AA[0][1]*AA[2][2] -
+		AA[0][0]*AA[1][1]*AA[2][2] - AA[1][0]*AA[2][1]*AA[0][2] - AA[2][0]*AA[0][1]*AA[1][2];
+
+	solvecubic(S);
+
+	if (S[0] < 0)
+		S[0] = 0;
+	if (S[1] < 0)
+		S[1] = 0;
+	if (S[2] < 0)
+		S[2] = 0;
+
+	sort3(S);
+
+	memcpy(LDU,AA,sizeof(LDU));
+	LDU[0][0] -= S[0];
+	LDU[1][1] -= S[0];
+	LDU[2][2] -= S[0];
+
+	ldu3((T *)LDU, P);
+
+	y[0] = y[1] = y[2] = 0;
+	k = (fabs(LDU[P[1]][1]) < fabs(LDU[P[0]][0]) || fabs(LDU[P[2]][2]) < fabs(LDU[P[0]][0])) +
+		(fabs(LDU[P[2]][2]) < fabs(LDU[P[0]][0]) && fabs(LDU[P[2]][2]) < fabs(LDU[P[1]][1]));
+	y[k] = 1;
+
+	ldubsolve3(V+(3*0+0), y, (T *)LDU, P);
+
+	memcpy(LDU,AA,sizeof(LDU));
+	LDU[0][0] -= S[2];
+	LDU[1][1] -= S[2];
+	LDU[2][2] -= S[2];
+
+	ldu3((T *)LDU, P);
+
+	y[0] = y[1] = y[2] = 0;
+	k = (fabs(LDU[P[1]][1]) <= fabs(LDU[P[0]][0]) || fabs(LDU[P[2]][2]) <= fabs(LDU[P[0]][0])) +
+		(fabs(LDU[P[2]][2]) <= fabs(LDU[P[0]][0]) && fabs(LDU[P[2]][2]) <= fabs(LDU[P[1]][1]));
+	y[k] = 1;
+
+	ldubsolve3(V+(3*2+0), y, (T *)LDU, P);
+
+	cross(V+(3*1+0), V+(3*2+0), V+(3*0+0));
+
+	k = (S[0] > thr) + (S[1] > thr) + (S[2] > thr);
+
+	switch (k) {
+		case 0:
+			memcpy(U,V,9*sizeof(T));
+			break;
+		case 1:
+			matvec3(U+(3*0+0), A, V+(3*0+0));
+
+			y[0] = y[1] = y[2] = 0;
+			k = (fabs(U[3*0+1]) <= fabs(U[3*0+0]) || fabs(U[3*0+2]) <= fabs(U[3*0+0])) +
+				(fabs(U[3*0+2]) <= fabs(U[3*0+0]) && fabs(U[3*0+2]) <= fabs(U[3*0+1]));
+			y[k] = 1;
+
+			cross(U+(3*1+0), y, U+(3*0+0));
+			cross(U+(3*2+0), U+(3*0+0), U+(3*1+0));
+			break;
+		case 2:
+			matvec3(U+(3*0+0), A, V+(3*0+0));
+			matvec3(U+(3*1+0), A, V+(3*1+0));
+
+			cross(U+(3*2+0), U+(3*0+0), U+(3*1+0));
+			break;
+		case 3:
+			matmul3(U, A, V);
+			break;
+	}
+
+	unit3(V+(3*0+0));
+	unit3(V+(3*1+0));
+	unit3(V+(3*2+0));
+
+	unit3(U+(3*0+0));
+	unit3(U+(3*1+0));
+	unit3(U+(3*2+0));
+
+	S[0] = (T)sqrt(S[0]);
+	S[1] = (T)sqrt(S[1]);
+	S[2] = (T)sqrt(S[2]);
+}
+
+template<typename T>
+void thomas(T *x, const T *a, const T *b, T *c, int n) {
+	T tmp;
+	int i;
+
+	c[0] /= b[0];
+	x[0] /= b[0];
+
+	for (i = 1; i < n; ++i) {
+		tmp = 1/(b[i] - c[i-1] * a[i]);
+		c[i] *= tmp;
+		x[i] = (x[i] - x[i-1] * a[i])*tmp;
+	}
+
+	for (i = n-2; i >= 0; --i) {
+		x[i] -= c[i]*x[i+1];
+	}
+}
+
+template<typename T>
+void spline(T *D, const T *y, int n) {
+	int i;
+	T *a, *b, *c;
+
+	a = new T[n];
+	b = new T[n];
+	c = new T[n];
+
+	if (n < 4) {
+		a[0] = 0;
+		b[0] = 2;
+		c[0] = 1;
+		D[0] = 3*(y[1]-y[0]);
+
+		a[n-1] = 1;
+		b[n-1] = 2;
+		c[n-1] = 0;
+		D[n-1] = 3*(y[n-1]-y[n-2]);
+	}
+	else {
+		a[0] = 0;
+		b[0] = 2;
+		c[0] = 4;
+		D[0] = -5*y[0] + 4*y[1] + y[2];
+
+		a[n-1] = 4;
+		b[n-1] = 2;
+		c[n-1] = 0;
+		D[n-1] = 5*y[n-1] - 4*y[n-2] - y[n-3];
+	}
+
+	for (i = 1; i < n-1; ++i) {
+		a[i] = 1;
+		b[i] = 4;
+		c[i] = 1;
+		D[i] = 3*(y[i+1]-y[i-1]);
+	}
+
+	thomas(D, a, b, c, n);
+
+	delete [] a;
+	delete [] b;
+	delete [] c;
+}
+
+template<typename T>
+inline void lookupspline(T &t, int &k, T dist, T len, int n) {
+	t = (n-1)*dist/len;
+	k = (int)floor(t);
+
+	k = (k > 0)*k;
+	k += (k > n-2)*(n-2-k);
+
+	t -= k;
+}
+
+template<typename T>
+inline T evalspline(T t, const T D[2], const T y[2]) {
+	const T c[4] = { y[0], D[0], 3*(y[1]-y[0])-2*D[0]-D[1], 2*(y[0]-y[1])+D[0]+D[1] };
+
+	return t*(t*(t*c[3] + c[2]) + c[1]) + c[0];
+}
+
+template<typename T>
+void interp2(T *Dy, const T *Z, int nx, int ny) {
+
+	for (int i = 0; i < ny; ++i) {
+		spline(Dy+(nx*i+0), Z+(nx*i+0), nx);
+	}
+}
+
+template<typename T>
+void evalinterp2(T u, T *Dx, T *zx, const T *Dy, const T *Z, int nx, int ny) {
+
+	for (int i = 0; i < ny; ++i) {
+		zx[i] = evalspline(u, Dy+(nx*i+0), Z+(nx*i+0));
+	}
+
+	spline(Dx, zx, ny);
+}
+
+#endif // !SURFACE_H
diff --git a/src/UnitSquareImage.cpp b/src/UnitSquareImage.cpp
new file mode 100644
index 00000000..a06086d7
--- /dev/null
+++ b/src/UnitSquareImage.cpp
@@ -0,0 +1,708 @@
+#include <cstring>
+#include <cmath>
+#include <iostream>
+#include <stdio.h>
+#include <Rcpp.h>
+#include "UnitSquareImage.h"
+#include "ImageRegister.h"
+
+using namespace std;
+using namespace Rcpp;
+
+RcppExport SEXP find_grad_2D(SEXP dfdui, SEXP dfdvi, SEXP fi, SEXP ni, SEXP ti, SEXP di){
+    int n = as<int>(ni);
+    int t = as<int>(ti);
+    int d = as<int>(di);
+    NumericVector f(fi);
+    NumericVector dfdu(dfdui);
+    NumericVector dfdv(dfdvi);
+
+    findgrad2D(dfdu.begin(), dfdv.begin(), f.begin(), n, t, d);
+    List ret;
+    ret["dfdu"] = dfdu;
+    ret["dfdv"] = dfdv;
+    return(ret);
+}
+
+RcppExport SEXP check_cross(SEXP fi, SEXP ni, SEXP ti, SEXP Di){
+    int temp;
+    int n = as<int>(ni);
+    int t = as<int>(ti);
+    int D = as<int>(Di);
+    NumericVector f(fi);
+
+    temp = check_crossing(f.begin(), n, t, D);
+
+    return wrap(temp);
+}
+
+RcppExport SEXP find_phistar(SEXP wi, SEXP qi, SEXP bi, SEXP ni, SEXP ti, SEXP di, SEXP Ki){
+    int n = as<int>(ni);
+    int t = as<int>(ti);
+    int d = as<int>(di);
+    int K = as<int>(Ki);
+    NumericVector w(wi);
+    NumericVector q(qi);
+    NumericVector b(bi);
+
+    findphistar(w.begin(), q.begin(), b.begin(), n, t, d, K);
+
+    return(w);
+}
+
+void findgrad(double *dfdu, double *dfdv, const double *f, int n, int t) {
+    int i, j, k, N = n*t;
+    double du, dv;
+
+    du = 1.0/(t-1);
+    dv = 1.0/(n-1);
+
+    for (i = 0; i < 1; ++i) {
+        // j = 0, k = 0
+        dfdu[n*(t*i + 0) + 0] = fdiff(f + n*(t*i + 0) + 0, du, n);
+        dfdv[n*(t*i + 0) + 0] = fdiff(f + n*(t*i + 0) + 0, dv, 1);
+
+        // k = 0
+        for (j = 1; j < n-1; ++j) {
+            dfdu[n*(t*i + 0) + j] = fdiff(f + n*(t*i + 0) + j, du, n);
+            dfdv[n*(t*i + 0) + j] = cdiff(f + n*(t*i + 0) + j, dv, 1);
+        }
+
+        // j = n-1, k = 0
+        dfdu[n*(t*i + 0) + n-1] = fdiff(f + n*(t*i + 0) + n-1, du, n);
+        dfdv[n*(t*i + 0) + n-1] = bdiff(f + n*(t*i + 0) + n-1, dv, 1);
+
+        for (k = 1; k < t-1; ++k) {
+            // j = 0
+            dfdu[n*(t*i + k) + 0] = cdiff(f + n*(t*i + k) + 0, du, n);
+            dfdv[n*(t*i + k) + 0] = fdiff(f + n*(t*i + k) + 0, dv, 1);
+
+            for (j = 1; j < n-1; ++j) {
+                dfdu[n*(t*i + k) + j] = cdiff(f + n*(t*i + k) + j, du, n);
+                dfdv[n*(t*i + k) + j] = cdiff(f + n*(t*i + k) + j, dv, 1);
+            }
+
+            // j = n-1
+            dfdu[n*(t*i + k) + n-1] = cdiff(f + n*(t*i + k) + n-1, du, n);
+            dfdv[n*(t*i + k) + n-1] = bdiff(f + n*(t*i + k) + n-1, dv, 1);
+        }
+
+        // j = 0, k = t-1
+        dfdu[n*(t*i + t-1) + 0] = bdiff(f + n*(t*i + t-1) + 0, du, n);
+        dfdv[n*(t*i + t-1) + 0] = fdiff(f + n*(t*i + t-1) + 0, dv, 1);
+
+        // k = t-1
+        for (j = 1; j < n-1; ++j) {
+            dfdu[n*(t*i + t-1) + j] = bdiff(f + n*(t*i + t-1) + j, du, n);
+            dfdv[n*(t*i + t-1) + j] = cdiff(f + n*(t*i + t-1) + j, dv, 1);
+        }
+
+        // j = n-1, k = t-1
+        dfdu[n*(t*i + t-1) + n-1] = bdiff(f + n*(t*i + t-1) + n-1, du, n);
+        dfdv[n*(t*i + t-1) + n-1] = bdiff(f + n*(t*i + t-1) + n-1, dv, 1);
+    }
+
+}
+
+// ------------------------------------------------------------------------
+void findgrad2D(double *dfdu, double *dfdv, double *f, int n, int t, int d) {
+    int i, j, k, N = n*t;
+    double du, dv;
+
+    du = 1.0/(t-1);
+    dv = 1.0/(n-1);
+
+    for (i = 0; i < d; ++i) {
+        // j = 0, k = 0
+        dfdu[n*(t*i + 0) + 0] = fdiff(f + n*(t*i + 0) + 0, du, n);
+        dfdv[n*(t*i + 0) + 0] = fdiff(f + n*(t*i + 0) + 0, dv, 1);
+
+        // k = 0
+        for (j = 1; j < n-1; ++j) {
+            dfdu[n*(t*i + 0) + j] = fdiff(f + n*(t*i + 0) + j, du, n);
+            dfdv[n*(t*i + 0) + j] = cdiff(f + n*(t*i + 0) + j, dv, 1);
+        }
+
+        // j = n-1, k = 0
+        dfdu[n*(t*i + 0) + n-1] = fdiff(f + n*(t*i + 0) + n-1, du, n);
+        dfdv[n*(t*i + 0) + n-1] = bdiff(f + n*(t*i + 0) + n-1, dv, 1);
+
+        for (k = 1; k < t-1; ++k) {
+            // j = 0
+            dfdu[n*(t*i + k) + 0] = cdiff(f + n*(t*i + k) + 0, du, n);
+            dfdv[n*(t*i + k) + 0] = fdiff(f + n*(t*i + k) + 0, dv, 1);
+
+            for (j = 1; j < n-1; ++j) {
+                dfdu[n*(t*i + k) + j] = cdiff(f + n*(t*i + k) + j, du, n);
+                dfdv[n*(t*i + k) + j] = cdiff(f + n*(t*i + k) + j, dv, 1);
+            }
+
+            // j = n-1
+            dfdu[n*(t*i + k) + n-1] = cdiff(f + n*(t*i + k) + n-1, du, n);
+            dfdv[n*(t*i + k) + n-1] = bdiff(f + n*(t*i + k) + n-1, dv, 1);
+        }
+
+        // j = 0, k = t-1
+        dfdu[n*(t*i + t-1) + 0] = bdiff(f + n*(t*i + t-1) + 0, du, n);
+        dfdv[n*(t*i + t-1) + 0] = fdiff(f + n*(t*i + t-1) + 0, dv, 1);
+
+        // k = t-1
+        for (j = 1; j < n-1; ++j) {
+            dfdu[n*(t*i + t-1) + j] = bdiff(f + n*(t*i + t-1) + j, du, n);
+            dfdv[n*(t*i + t-1) + j] = cdiff(f + n*(t*i + t-1) + j, dv, 1);
+        }
+
+        // j = n-1, k = t-1
+        dfdu[n*(t*i + t-1) + n-1] = bdiff(f + n*(t*i + t-1) + n-1, du, n);
+        dfdv[n*(t*i + t-1) + n-1] = bdiff(f + n*(t*i + t-1) + n-1, dv, 1);
+    }
+
+}
+
+// ------------------------------------------------------------------------
+void multfact_image(double *multfact, const double *dfdu, const double *dfdv, int n, int t,int d) {
+    int N = n*t;
+
+    //     multfact = new double[N];
+
+    if (d < 3) {
+        for (int i = 0; i < N; ++i) {
+            multfact[i] = abs(dfdu[N*0+i]*dfdv[N*1+i] - dfdu[N*1+i]*dfdv[N*0+i]);
+        }
+    }
+    else if (d < 4) {
+        for (int i = 0; i < N; ++i) {
+            multfact[i] = pow(pow(dfdu[N*0+i]*dfdv[N*1+i] - dfdu[N*1+i]*dfdv[N*0+i],2) +
+                    pow(dfdu[N*0+i]*dfdv[N*2+i] - dfdu[N*2+i]*dfdv[N*0+i],2) +
+                    pow(dfdu[N*1+i]*dfdv[N*2+i] - dfdu[N*2+i]*dfdv[N*1+i],2), 1/2);
+        }
+    }
+    else {
+        for (int i = 0; i < N; ++i) {
+            multfact[i] = pow(pow(dfdu[N*0+i]*dfdv[N*1+i] - dfdu[N*1+i]*dfdv[N*0+i],2) +
+                    pow(dfdu[N*0+i]*dfdv[N*2+i] - dfdu[N*2+i]*dfdv[N*0+i],2) +
+                    pow(dfdu[N*0+i]*dfdv[N*3+i] - dfdu[N*3+i]*dfdv[N*0+i],2) +
+                    pow(dfdu[N*1+i]*dfdv[N*2+i] - dfdu[N*2+i]*dfdv[N*1+i],2) +
+                    pow(dfdu[N*1+i]*dfdv[N*3+i] - dfdu[N*3+i]*dfdv[N*1+i],2) +
+                    pow(dfdu[N*2+i]*dfdv[N*3+i] - dfdu[N*3+i]*dfdv[N*2+i],2), 1/2);
+        }
+    }
+}
+
+// ------------------------------------------------------------------------
+void surface_to_q(double *q, const double *f, const double *multfact, int n, int t, int d) {
+    int N = n*t;
+
+    for (int k = 0; k < d; ++k) {
+        for (int i = 0; i < N; ++i) {
+            q[N*k + i] = sqrt(multfact[i])*f[N*k + i];
+        }
+    }
+}
+
+// ------------------------------------------------------------------------
+void Calculate_Distance(double *H, const double *q1, const double *q2, int n, int t, int d) {
+    int N = n*t*d;
+    double tmp, du, dv;
+
+    du = 1.0/(n-1);
+    dv = 1.0/(t-1);
+
+    *H = 0;
+
+    for (int i = 0; i < N; ++i) {
+        tmp = q1[i] - q2[i];
+        *H += tmp*tmp;
+    }
+
+    *H *= du*dv;
+}
+
+// ------------------------------------------------------------------------
+void findphistar(double *w, double *q, double *b, int n, int t, int d, int K) {
+    int D = 2;
+    double du, dv, dbxdu, dbydv, divb, *dqdu, *dqdv, *expr1, *expr2;
+
+    du = 1.0/(t-1);
+    dv = 1.0/(n-1);
+
+    expr1 = new double[d];
+    expr2 = new double[d];
+
+    dqdu = new double[n*t*d];
+    dqdv = new double[n*t*d];
+
+    findgrad2D(dqdu, dqdv, q, n, t, d);
+
+    memset(w,0,n*t*d*K*sizeof(double));
+
+    for (int j = 0; j < K; ++j) {
+        // k = 0, i = 0
+        dbxdu = fdiff(b + n*(t*(D*j + 0) + 0) + 0, du, n);
+        dbydv = fdiff(b + n*(t*(D*j + 1) + 0) + 0, dv, 1);
+        divb = 0.5*(dbxdu + dbydv);
+
+        for (int ii = 0; ii < d; ++ii) {
+            expr1[ii] = divb*q[n*(t*ii + 0) + 0];
+            expr2[ii] = dqdu[n*(t*ii + 0) + 0]*b[n*(t*(D*j + 0) + 0) + 0] + dqdv[n*(t*ii + 0) + 0]*b[n*(t*(D*j + 1) + 0) + 0];
+            w[n*(t*(d*j + ii) + 0) + 0] = expr1[ii] + expr2[ii];
+        }
+
+        // k = 0
+        for (int i = 1; i < n-1; ++i) {
+            dbxdu = fdiff(b + n*(t*(D*j + 0) + 0) + i, du, n);
+            dbydv = cdiff(b + n*(t*(D*j + 1) + 0) + i, dv, 1);
+            divb = 0.5*(dbxdu + dbydv);
+
+            for (int ii = 0; ii < d; ++ii) {
+                expr1[ii] = divb*q[n*(t*ii + 0) + i];
+                expr2[ii] = dqdu[n*(t*ii + 0) + i]*b[n*(t*(D*j + 0) + 0) + i] + dqdv[n*(t*ii + 0) + i]*b[n*(t*(D*j + 1) + 0) + i];
+                w[n*(t*(d*j + ii) + 0) + i] = expr1[ii] + expr2[ii];
+            }
+        }
+
+        // i = n-1, k = 0
+        dbxdu = fdiff(b + n*(t*(D*j + 0) + 0) + n-1, du, n);
+        dbydv = bdiff(b + n*(t*(D*j + 1) + 0) + n-1, dv, 1);
+        divb = 0.5*(dbxdu + dbydv);
+
+        for (int ii = 0; ii < d; ++ii) {
+            expr1[ii] = divb*q[n*(t*ii + 0) + n-1];
+            expr2[ii] = dqdu[n*(t*ii + 0) + n-1]*b[n*(t*(D*j + 0) + 0) + n-1] + dqdv[n*(t*ii + 0) + n-1]*b[n*(t*(D*j + 1) + 0) + n-1];
+            w[n*(t*(d*j + ii) + 0) + n-1] = expr1[ii] + expr2[ii];
+        }
+
+        for (int k = 1; k < t-1; ++k) {
+            // i = 0
+            dbxdu = cdiff(b + n*(t*(D*j + 0) + k) + 0, du, n);
+            dbydv = fdiff(b + n*(t*(D*j + 1) + k) + 0, dv, 1);
+            divb = 0.5*(dbxdu + dbydv);
+
+            for (int ii = 0; ii < d; ++ii) {
+                expr1[ii] = divb*q[n*(t*ii + k) + 0];
+                expr2[ii] = dqdu[n*(t*ii + k) + 0]*b[n*(t*(D*j + 0) + k) + 0] + dqdv[n*(t*ii + k) + 0]*b[n*(t*(D*j + 1) + k) + 0];
+                w[n*(t*(d*j + ii) + k) + 0] = expr1[ii] + expr2[ii];
+            }
+
+            for (int i = 1; i < n-1; ++i) {
+                dbxdu = cdiff(b + n*(t*(D*j + 0) + k) + i, du, n);
+                dbydv = cdiff(b + n*(t*(D*j + 1) + k) + i, dv, 1);
+                divb = 0.5*(dbxdu + dbydv);
+
+                for (int ii = 0; ii < d; ++ii) {
+                    expr1[ii] = divb*q[n*(t*ii + k) + i];
+                    expr2[ii] = dqdu[n*(t*ii + k) + i]*b[n*(t*(D*j + 0) + k) + i] + dqdv[n*(t*ii + k) + i]*b[n*(t*(D*j + 1) + k) + i];
+                    w[n*(t*(d*j + ii) + k) + i] = expr1[ii] + expr2[ii];
+                }
+            }
+
+            // i = n-1
+            dbxdu = cdiff(b + n*(t*(D*j + 0) + k) + n-1, du, n);
+            dbydv = bdiff(b + n*(t*(D*j + 1) + k) + n-1, dv, 1);
+            divb = 0.5*(dbxdu + dbydv);
+
+            for (int ii = 0; ii < d; ++ii) {
+                expr1[ii] = divb*q[n*(t*ii + k) + n-1];
+                expr2[ii] = dqdu[n*(t*ii + k) + n-1]*b[n*(t*(D*j + 0) + k) + n-1] + dqdv[n*(t*ii + k) + n-1]*b[n*(t*(D*j + 1) + k) + n-1];
+                w[n*(t*(d*j + ii) + k) + n-1] = expr1[ii] + expr2[ii];
+            }
+        }
+
+        // i = 0, k = t-1
+        dbxdu = bdiff(b + n*(t*(2*j + 0) + t-1) + 0, du, n);
+        dbydv = fdiff(b + n*(t*(2*j + 1) + t-1) + 0, dv, 1);
+        divb = 0.5*(dbxdu + dbydv);
+
+        for (int ii = 0; ii < d; ++ii) {
+            expr1[ii] = divb*q[n*(t*ii + t-1) + 0];
+            expr2[ii] = dqdu[n*(t*ii + t-1) + 0]*b[n*(t*(D*j + 0) + t-1) + 0] + dqdv[n*(t*ii + t-1) + 0]*b[n*(t*(D*j + 1) + t-1) + 0];
+            w[n*(t*(d*j + ii) + t-1) + 0] = expr1[ii] + expr2[ii];
+        }
+
+        // k = t-1
+        for (int i = 1; i < n-1; ++i) {
+            dbxdu = bdiff(b + n*(t*(D*j + 0) + t-1) + i, du, n);
+            dbydv = cdiff(b + n*(t*(D*j + 1) + t-1) + i, dv, 1);
+            divb = 0.5*(dbxdu + dbydv);
+
+            for (int ii = 0; ii < d; ++ii) {
+                expr1[ii] = divb*q[n*(t*ii + t-1) + i];
+                expr2[ii] = dqdu[n*(t*ii + t-1) + i]*b[n*(t*(D*j + 0) + t-1) + i] + dqdv[n*(t*ii + t-1) + i]*b[n*(t*(D*j + 1) + t-1) + i];
+                w[n*(t*(d*j + ii) + t-1) + i] = expr1[ii] + expr2[ii];
+            }
+        }
+
+        // i = n-1, k = t-1
+        dbxdu = bdiff(b + n*(t*(D*j + 0) + t-1) + n-1, du, n);
+        dbydv = bdiff(b + n*(t*(D*j + 1) + t-1) + n-1, dv, 1);
+        divb = 0.5*(dbxdu + dbydv);
+
+        for (int ii = 0; ii < d; ++ii) {
+            expr1[ii] = divb*q[n*(t*ii + t-1) + n-1];
+            expr2[ii] = dqdu[n*(t*ii + t-1) + n-1]*b[n*(t*(D*j + 0) + t-1) + n-1] + dqdv[n*(t*ii + t-1) + n-1]*b[n*(t*(D*j + 1) + t-1) + n-1];
+            w[n*(t*(d*j + ii) + t-1) + n-1] = expr1[ii] + expr2[ii];
+        }
+    }
+
+    delete [] dqdu;
+    delete [] dqdv;
+    delete [] expr1;
+    delete [] expr2;
+
+    return;
+}
+
+
+// ------------------------------------------------------------------------
+void findupdategam(double *gamupdate, const double *v, const double *w, const double *b, int n, int t, int d, int K) {
+    int i, k, N = n*t, D = 2;
+    double innp, du, dv;
+
+    du = 1.0/(n-1);
+    dv = 1.0/(t-1);
+
+    memset(gamupdate,0,n*t*D*sizeof(double));
+
+    for (k = 0; k < K; ++k) {
+        innp = 0;
+        for (i = 0; i < N*d; ++i) {
+            innp += v[i]*w[N*d*k + i];
+        }
+
+        innp *= du*dv;
+
+        for (i = 0; i < N; ++i) {
+            gamupdate[N*0 + i] += innp*b[N*D*k + N*0 + i];
+            gamupdate[N*1 + i] += innp*b[N*D*k + N*1 + i];
+        }
+    }
+
+    return;
+}
+
+// ------------------------------------------------------------------------
+void updategam(double *gamnew, const double *gamupdate, const double *gamid, double eps, int n, int t, int D) {
+
+    for (int i = 0; i < n*t*D; ++i) {
+        gamnew[i] = gamid[i] + eps*gamupdate[i];
+    }
+
+    return;
+}
+
+// ------------------------------------------------------------------------
+void Apply_gam_gamid(double *gamcum, const double *gamid, const double *gaminc, int m, int n) {
+    int k;
+    double u0, v0, ndu, ndv, t, *y, *D1, *D2;
+
+    u0 = 0;
+    v0 = 0;
+    ndu = 1;
+    ndv = 1;
+
+    y = new double[n];
+    D1 = new double[n];
+    D2 = new double[m];
+
+    for (int i = 0; i < m; ++i) { // for each row (i-th)
+        for (int j = 0; j < n; ++j) {
+            y[j] = gamid[m*(n*0+j)+i];
+        } // y = (i-th) row of gamid[N*0]
+
+        spline(D1, y, n);
+
+        for (int j = 0; j < n; ++j) {
+            lookupspline(t, k, gaminc[m*(n*0+j)+i]-u0, ndu, n);
+            gamcum[m*(n*0+j)+i] = evalspline(t, D1+k, y+k);
+        }
+    }
+
+    for (int j = 0; j < n; ++j) {
+        spline(D2, gamid + m*(n*1+j)+0, m); // for each column (j-th)
+
+        for (int i = 0; i < m; ++i) {
+            lookupspline(t, k, gaminc[m*(n*1+j)+i]-v0, ndv, m);
+            gamcum[m*(n*1+j)+i] = evalspline(t, D2+k, gamid + m*(n*1+j)+k);
+        }
+    }
+
+    delete [] y;
+    delete [] D1;
+    delete [] D2;
+
+    return;
+}
+
+//-------------------------------------------------------------------------
+void Apply_Gamma_Surf(double *Fnew, const double *F, const double *gam, int m, int n, int d) {
+    int j, N = m*n;
+    double *Du, *Dv, *zu, u0, v0, ndu, ndv, u, v;
+
+    Dv = new double[N];
+    Du = new double[n];
+    zu = new double[n];
+
+    ndu = 1;
+    ndv = 1;
+    u0 = 0;
+    v0 = 0;
+
+    for (int k = 0; k < d; ++k) {
+        interp2(Dv, F+(N*k+0), m, n); //col
+
+        for (int i = 0; i < N; ++i) {
+            lookupspline(v, j, gam[N*1+i]-v0, ndv, m); // col
+            evalinterp2(v, Du, zu, Dv+j, F+(N*k+j), m, n); // row
+
+            lookupspline(u, j, gam[N*0+i]-u0, ndu, n); // row
+            Fnew[N*k + i] = evalspline(u, Du+j, zu+j); // row
+        }
+    }
+
+    delete [] Dv;
+    delete [] Du;
+    delete [] zu;
+
+    return;
+}
+
+// ------------------------------------------------------------------------
+int check_crossing(double *f, int n, int t, int D) {
+    int is_diffeo = 1;
+    int N = n*t, nC = 0;
+    double c, *dfdu, *dfdv;
+
+    dfdu = new double[n*t*D];
+    dfdv = new double[n*t*D];
+
+    findgrad2D(dfdu,dfdv,f,n,t,D);
+
+    for (int i=0; i<N; ++i) {
+        c = dfdu[N*0 + i]*dfdv[N*1 + i] - dfdu[N*1 + i]*dfdv[N*0 + i];
+
+        if (c < 0) {
+            nC += 1;
+        }
+    }
+
+    if (nC > 0)
+        is_diffeo = 0;
+
+    delete [] dfdu;
+    delete [] dfdv;
+
+    return is_diffeo;
+}
+
+//-------------------------------------------------------------------------
+int ReparamSurf(double *Fnew, double *gamnew, double *H,
+        double *Ft, const double *Fm, const double *gam,
+        double *b, const double *gamid,
+        const int n, const int t, const int d, const int D, const int K,
+        double eps=0.1, const double tol=1e-8, const int itermax=100)
+{
+    int iter = 0, N = n*t, is_diffeo;
+    double *qt, *qm, *v, *w, *gamupdate, *gaminc, *gamold, *Fu, *Fv, *multfact, Hdiff;
+
+    qt = new double[n*t*d];
+    qm = new double[n*t*d];
+    v = new double[n*t*d];
+    w = new double[n*t*d*K];
+    gamupdate = new double[n*t*D];
+    gaminc = new double[n*t*D];
+    gamold = new double[n*t*D];
+    Fu = new double[n*t*d];
+    Fv = new double[n*t*d];
+    multfact = new double[n*t];
+
+    // initial gampold and Fold
+    cpyArray(gamold,gam,n*t*D);
+    Apply_Gamma_Surf(Fnew,Fm,gamold,n,t,d);
+
+    // check point
+    cpyArray(gamnew,gamold,n*t*D);
+
+    // initial q-functions for Ft and Fnew
+    findgrad2D(Fu,Fv,Ft,n,t,d);
+    multfact_image(multfact,Fu,Fv,n,t,d);
+    surface_to_q(qt,Ft,multfact,n,t,d);
+
+    findgrad2D(Fu,Fv,Fnew,n,t,d);
+    multfact_image(multfact,Fu,Fv,n,t,d);
+    surface_to_q(qm,Fnew,multfact,n,t,d);
+
+    // compute initial energy
+    Hdiff = 100;
+    Calculate_Distance(H+iter,qt,qm,n,t,d);
+    Rprintf("iter %d, energy %f \n", iter, *(H+iter));
+
+    // main iteration
+    for (iter = 1; iter < itermax && Hdiff > tol; ++iter) {
+        // basis to tangent space
+        findphistar(w,qm,b,n,t,d,K);
+
+        // find v = q1-q2
+        for (int i = 0; i < N*d; ++i) {
+            v[i] = qt[i]-qm[i];
+        }
+
+        findupdategam(gamupdate,v,w,b,n,t,d,K);
+
+update:
+        updategam(gaminc,gamupdate,gamid,eps,n,t,D);
+
+        // compute update gampnew
+        Apply_gam_gamid(gamnew,gamold,gaminc,n,t);
+        is_diffeo = check_crossing(gamnew,n,t,D);
+        if (!is_diffeo) {
+            eps *= 0.5;
+            goto update;
+        }
+
+        Apply_Gamma_Surf(Fnew,Fm,gamnew,n,t,d); // apply cumulative deformation to original object
+
+        findgrad2D(Fu,Fv,Fnew,n,t,d);
+        multfact_image(multfact,Fu,Fv,n,t,d);
+        surface_to_q(qm,Fnew,multfact,n,t,d);
+
+        Calculate_Distance(H+iter,qt,qm,n,t,d);
+        Hdiff = (H[iter-1]-H[iter])/H[iter-1];
+
+        Rprintf("iter %d, energy %f \n", iter, *(H+iter));
+
+        // update iteration or break out
+        if (H[iter] <= H[iter-1]) {
+            cpyArray(gamold,gamnew,n*t*D);
+        }
+        else {
+            eps *= 0.5;
+            goto update;
+        }
+    }
+
+    delete [] qt;
+    delete [] qm;
+    delete [] v;
+    delete [] w;
+    delete [] gamupdate;
+    delete [] gaminc;
+    delete [] gamold;
+    delete [] Fu;
+    delete [] Fv;
+    delete [] multfact;
+
+    return iter;
+}
+
+// inverse transform ------------------------------------------------------
+void InvtGamma(double *gaminv, const double *gam, const double *gamid, int n) {
+    int i, j, k;
+    double ndu, ndv, t, *y, *D;
+
+    ndu = 1;
+    ndv = 1;
+
+    y = new double[n];
+    D = new double[n];
+
+    for (i = 0; i < n; ++i) {
+        for (j = 0; j < n; ++j) {
+            y[j] = gam[n*(n*0+j)+i];
+        }
+
+        spline(D, y, n);
+
+        for (j = 0; j < n; ++j) {
+            lookupspline(t, k, gamid[n*(n*0+j)+i], ndu, n);
+            gaminv[n*(n*0+j)+i] = evalspline(t, D+k, y+k);
+        }
+    }
+
+    for (j = 0; j < n; ++j) {
+        spline(D, gam + n*(n*1+j)+0, n);
+
+        for (i = 0; i < n; ++i) {
+            lookupspline(t, k, gamid[n*(n*1+j)+i], ndv, n);
+            gaminv[n*(n*1+j)+i] = evalspline(t, D+k, gam + n*(n*1+j)+k);
+        }
+    }
+
+    delete [] y;
+    delete [] D;
+
+    return;
+}
+
+// ------------------------------------------------------------------------
+void jacob_image(double *A, const double *F, int n, int t) {
+    int j, k, N = n*t;
+    double dfdu[2], dfdv[2], du, dv;
+    double c=0.0;
+    
+    du = 1.0/(n-1);
+    dv = 1.0/(t-1);
+
+    // j = 0, k = 0
+    fdiff2(dfdu, F + n*(t*0 + 0) + 0, du, n, N);
+    fdiff2(dfdv, F + n*(t*0 + 0) + 0, dv, 1, N);
+    jacob(c,dfdu,dfdv);
+    A[n*0 + 0] = sqrt(abs(c));
+
+    // k = 0
+    for (j = 1; j < n-1; ++j) {
+        fdiff2(dfdu, F + n*(t*0 + 0) + j, du, n, N);
+        cdiff2(dfdv, F + n*(t*0 + 0) + j, dv, 1, N);
+        jacob(c,dfdu,dfdv);
+        A[n*0 + j] = sqrt(abs(c));
+    }
+
+    // j = n-1, k = 0
+    fdiff2(dfdu, F + n*(t*0 + 0) + n-1, du, n, N);
+    bdiff2(dfdv, F + n*(t*0 + 0) + n-1, dv, 1, N);
+    jacob(c,dfdu,dfdv);
+    A[n*0 + n-1] = sqrt(abs(c));
+
+    for (k = 1; k < t-1; ++k) {
+        // j = 0
+        cdiff2(dfdu, F + n*(t*0 + k) + 0, du, n, N);
+        fdiff2(dfdv, F + n*(t*0 + k) + 0, dv, 1, N);
+        jacob(c,dfdu,dfdv);
+        A[n*k + 0] = sqrt(abs(c));
+
+        for (j = 1; j < n-1; ++j) {
+            cdiff2(dfdu, F + n*(t*0 + k) + j, du, n, N);
+            cdiff2(dfdv, F + n*(t*0 + k) + j, dv, 1, N);
+            jacob(c,dfdu,dfdv);
+            A[n*k + j] = sqrt(abs(c));
+        }
+
+        // j = n-1
+        cdiff2(dfdu, F + n*(t*0 + k) + n-1, du, n, N);
+        bdiff2(dfdv, F + n*(t*0 + k) + n-1, dv, 1, N);
+        jacob(c,dfdu,dfdv);
+        A[n*k + n-1] = sqrt(abs(c));
+    }
+
+    // j = 0, k = t-1
+    bdiff2(dfdu, F + n*(t*0 + t-1) + 0, du, n, N);
+    fdiff2(dfdv, F + n*(t*0 + t-1) + 0, dv, 1, N);
+    jacob(c,dfdu,dfdv);
+    A[n*(t-1) + 0] = sqrt(abs(c));
+
+    // k = t-1
+    for (j = 1; j < n-1; ++j) {
+        bdiff2(dfdu, F + n*(t*0 + t-1) + j, du, n, N);
+        cdiff2(dfdv, F + n*(t*0 + t-1) + j, dv, 1, N);
+        jacob(c,dfdu,dfdv);
+        A[n*(t-1) + j] = sqrt(abs(c));
+    }
+
+    // j = n-1, k = t-1
+    bdiff2(dfdu, F + n*(t*0 + t-1) + n-1, du, n, N);
+    bdiff2(dfdv, F + n*(t*0 + t-1) + n-1, dv, 1, N);
+    jacob(c,dfdu,dfdv);
+    A[n*(t-1) + n-1] = sqrt(abs(c));
+
+    return;
+
+}
diff --git a/src/UnitSquareImage.h b/src/UnitSquareImage.h
new file mode 100644
index 00000000..f4e150b8
--- /dev/null
+++ b/src/UnitSquareImage.h
@@ -0,0 +1,32 @@
+#ifndef UNITSQUAREIMAGE_H
+#define UNITSQUAREIMAGE_H
+
+#include <cstring>
+#include <cmath>
+#include <iostream>
+#include "ImageRegister.h"
+
+using namespace std;
+
+
+void findgrad(double *dfdu, double *dfdv, const double *f, int n, int t);
+void findgrad2D(double *dfdu, double *dfdv, double *f, int n, int t, int d);
+void multfact_image(double *multfact, const double *dfdu, const double *dfdv, int n, int t, int d);
+void surface_to_q(double *q, const double *f, const double *multfact, int n, int t, int d);
+void Calculate_Distance(double *H, const double *q1, const double *q2, int n, int t, int d);
+void findphistar(double *w, double *q, double *b, int n, int t, int d, int K);
+void findupdategam(double *gamupdate, const double *v, const double *w, const double *b, int n, int t, int d, int K);
+void updategam(double *gamnew, const double *gamupdate, const double *gamid, double eps, int n, int t, int D);
+void Apply_gam_gamid(double *gamcum, const double *gamid, const double *gaminc, int m, int n);
+void Apply_Gamma_Surf(double *Fnew, const double *F, const double *gam, int n, int d);
+int check_crossing(double *f, int n, int t, int D);
+int ReparamSurf(double *Fnew, double *gamnew, double *H,
+        double *Ft, const double *Fm, const double *gam,
+        double *b, const double *gamid,
+        const int n, const int t, const int d, const int D, const int K,
+        const double eps, const double tol, const int itermax);
+
+void InvtGamma(double *gaminv, const double *gam, const double *gamid, int n);
+void jacob_image(double *A, const double *F, int n, int t);
+
+#endif
diff --git a/src/myVector.cpp b/src/myVector.cpp
new file mode 100644
index 00000000..51676ccb
--- /dev/null
+++ b/src/myVector.cpp
@@ -0,0 +1,125 @@
+#include <iostream>
+#include <cstring>
+#include <cmath>
+#include <Rcpp.h>
+#include "myVector.h"
+
+using namespace std;
+using namespace Rcpp;
+
+double InProd(const double *u, const double *v, int d)
+{
+    double innp = 0.0;
+
+    for (int i = 0; i < d; i++) {
+        innp += u[i]*v[i];
+    }
+
+    return innp;
+}
+
+void GramSchmitd(double *x, int &n, int d)
+{
+    // n=# of basis; d= dimension for each basis element
+    double innp;
+    int cnt = 0, k;
+
+    cnt = n; //initial # of vectors
+
+    // 1st vector
+    k = 0;
+    innp = sqrt(InProd(x,x,d));
+    for (int i=0; i<d; ++i) {
+        x[d*k + i] /= innp;
+    }
+
+    // 2nd to the last
+    k = 1;
+    do{
+        // kth vector
+        for (int j=0; j<k; ++j) {
+            innp = InProd(x+d*k, x+d*j,d);
+
+            for (int i=0; i<d; ++i) {
+                x[d*k+i] -= innp*x[d*j+i];
+            }
+        }
+
+        innp = sqrt(InProd(x+d*k,x+d*k,d));
+        if (innp>EPS) {
+            for (int i=0; i<d; ++i) {
+                x[d*k+i] /= innp;
+            }
+            k += 1;
+        }
+        else {
+            for (int i=0; i<d; ++i) {
+                x[d*k+i] = x[d*cnt+i];
+            }
+            cnt -= 1;
+        }
+    } while (k < (cnt-1));
+
+    n = cnt;
+}
+
+double innerSquare(const double *u, const double *v, int n1, int n2, int d)
+{
+    int N = n1*n2;
+    double innp = 0.0, du, dv;
+
+    du = 1.0/(n1-1);
+    dv = 1.0/(n2-1);
+
+    for (int i = 0; i < N*d; i++) {
+        innp += u[i]*v[i];
+    }
+
+    innp *= du*dv;
+    return innp;
+}
+
+void GramSchmitdSquare(double *x, int &n, int n1, int n2, int d)
+{
+    // n=# of basis; d= dimension for each basis element
+    double innp;
+    int cnt = 0, k, N = n1*n2*d, i, j;
+
+    cnt = n-1;
+
+    // 1st vector
+    k = 0;
+    innp = sqrt(innerSquare(x,x,n1,n2,d));
+    for (i=0; i<N; i++) {
+        x[N*k + i] /= innp;
+    }
+
+    // 2nd to the last
+    k = 1;
+    do{
+        // kth vector
+        for (j=0; j<k; j++) {
+            innp = innerSquare(x+N*k, x+N*j,n1,n2,d);
+
+            for (i=0; i<N; i++) {
+                x[N*k+i] -= innp*x[N*j + i];
+            }
+        }
+
+        innp = sqrt(innerSquare(x+N*k,x+N*k,n1,n2,d));
+        if (innp>EPS) {
+            for (i=0; i<N; i++) {
+                x[N*k + i] /= innp;
+            }
+            k += 1;
+        }
+        else {
+            for (i=0; i<N; i++) {
+                x[N*k + i] = x[N*cnt + i];
+            }
+            cnt -= 1;
+        }
+    } while (k <= cnt);
+
+    n = k;
+}
diff --git a/src/myVector.h b/src/myVector.h
new file mode 100644
index 00000000..0148c09f
--- /dev/null
+++ b/src/myVector.h
@@ -0,0 +1,20 @@
+#ifndef MYVECTOR_H
+#define MYVECTOR_H
+
+#include <iostream>
+#include <cstring>
+#include <cmath>
+
+using namespace std;
+
+#define SQRT5	2.23606797749978969641
+#define EPS	0.00000000001
+
+// Prototypes: -----------------------------------------------------------------
+double InProd(const double *u, const double *v, int d);
+void GramSchmitd(double *x, int &n, int d);
+
+double innerSquare(const double *u, const double *v, int n, int t, int d);
+void GramSchmitdSquare(double *x, int &n, int n1, int n2, int d);
+
+#endif

From 460707023fdc557d1e2ea8130bc4933e6d889bb1 Mon Sep 17 00:00:00 2001
From: Derek Tucker <jdtuck@sandia.gov>
Date: Sun, 12 Feb 2023 21:34:32 -0700
Subject: [PATCH 03/12] add docs

---
 README.md            | 2 ++
 doc/source/image.rst | 8 ++++++++
 doc/source/index.rst | 3 +++
 fdasrsf/__init__.py  | 3 ++-
 4 files changed, 15 insertions(+), 1 deletion(-)
 create mode 100644 doc/source/image.rst

diff --git a/README.md b/README.md
index 8fb670ef..13c459c8 100644
--- a/README.md
+++ b/README.md
@@ -108,3 +108,5 @@ T. Harris, J. D. Tucker, B. Li, and L. Shand, "Elastic depths for detecting shap
 M. K. Ahn, J. D. Tucker, W. Wu, and A. Srivastava. “Regression Models Using Shapes of Functions as Predictors” Computational Statistics and Data Analysis, 10.1016/j.csda.2020.107017, 2020. 
 
 J. D. Tucker, L. Shand, and K. Chowdhary. “Multimodal Bayesian Registration of Noisy Functions using Hamiltonian Monte Carlo”, Computational Statistics and Data Analysis, accepted, 2021.
+
+Q. Xie, S. Kurtek, E. Klassen, G. E. Christensen and A. Srivastava. Metric-based pairwise and multiple image registration. IEEE European Conference on Computer Vision (ECCV), September, 2014
diff --git a/doc/source/image.rst b/doc/source/image.rst
new file mode 100644
index 00000000..9b021d65
--- /dev/null
+++ b/doc/source/image.rst
@@ -0,0 +1,8 @@
+.. image:: ../artwork/logo.png
+  :width: 200px
+  :align: right
+  
+Elastic Image Warping
+=========================
+.. automodule:: image
+   :members:
diff --git a/doc/source/index.rst b/doc/source/index.rst
index b9db6b1e..3e155564 100644
--- a/doc/source/index.rst
+++ b/doc/source/index.rst
@@ -28,6 +28,7 @@ using functional component analysis and regression.
    pcr_regression.rst
    elastic_glm_regression.rst
    tolerance.rst
+   image.rst
    curve_statistics.rst
    geodesic.rst
    utility_functions.rst
@@ -101,6 +102,8 @@ References
 
    J. D. Tucker, L. Shand, and K. Chowdhary. “Multimodal Bayesian Registration of Noisy Functions using Hamiltonian Monte Carlo”, Computational Statistics and Data Analysis, accepted, 2021.
 
+   Q. Xie, S. Kurtek, E. Klassen, G. E. Christensen and A. Srivastava. Metric-based pairwise and multiple image registration. IEEE European Conference on Computer Vision (ECCV), September, 2014
+
 
 Indices and tables
 ==================
diff --git a/fdasrsf/__init__.py b/fdasrsf/__init__.py
index cc902889..d811217e 100644
--- a/fdasrsf/__init__.py
+++ b/fdasrsf/__init__.py
@@ -7,7 +7,7 @@
 """
 __all__ = ["time_warping", "utility_functions", "bayesian_functions", "curve_stats", "geodesic", "curve_functions", 
            "geometry", "pcr_regression", "tolerance", "boxplots", "curve_regression", "regression", "fPCA", 
-           "elastic_glm_regression", "curve_pcr_regression"]
+           "elastic_glm_regression", "curve_pcr_regression", "image"]
 
 __version__ = "2.3.12"
 
@@ -35,6 +35,7 @@
 from .curve_pcr_regression import elastic_curve_pcr_regression
 from .boxplots import ampbox, phbox
 from .tolerance import bootTB, pcaTB
+from .image import reparam_image
 from .curve_functions import resamplecurve, calculatecentroid, curve_to_q, optimum_reparam_curve, find_best_rotation, elastic_distance_curve
 from .curve_functions import q_to_curve, find_rotation_and_seed_coord
 from .curve_stats import fdacurve

From 2ccf0095074826cd3bf498f4a366bfb014dde33d Mon Sep 17 00:00:00 2001
From: Derek Tucker <jdtuck@sandia.gov>
Date: Sun, 12 Feb 2023 22:11:31 -0700
Subject: [PATCH 04/12] continue function development

---
 fdasrsf/image.py           | 31 +++++++++++---
 fdasrsf/image_functions.py | 82 +++++++++++++++++++++++++++++++++++++-
 src/UnitSquareImage.cpp    |  1 -
 3 files changed, 106 insertions(+), 8 deletions(-)

diff --git a/fdasrsf/image.py b/fdasrsf/image.py
index 648109da..98b720a8 100644
--- a/fdasrsf/image.py
+++ b/fdasrsf/image.py
@@ -5,17 +5,36 @@
 
 """
 
+import numpy as np
+import fdasrsf.image_functions as fif
 
-def reparam_image(It, Im, gam, b, stepsize=1e-4, itermax=20):
+def reparam_image(It, Im, gam=None, b=None, stepsize=1e-4, itermax=20):
     """
     This function warps an image to another using SRVF framework
 
-    :param Im: numpy ndarray of shape (N,N) respresenting a NxN image
-    :param Im: numpy ndarray of shape (N,N) respresenting a NxN image
-    :param gam: numpy ndarray of shape (N,N) respresenting an initial warping function
-    :param b: numpy ndarray of 
+    :param Im: numpy ndarray of shape (N,N) representing a NxN image
+    :param Im: numpy ndarray of shape (N,N) representing a NxN image
+    :param gam: numpy ndarray of shape (N,N) representing an initial warping function
+    :param b: numpy ndarray representing basis matrix
 
     :rtype: numpy ndarray
     :return f: smoothed functions functions
 
-    """
\ No newline at end of file
+    """
+
+    m = It.shape[0]
+    n = It.shape[1]
+
+    gamid = fif.makediffeoid(m,n)
+
+    if gam is None:
+        gam = gamid.copy()
+
+    # main loop
+    H = np.zeros(itermax+1)
+    Iold = fif.apply_gam_imag(Im, gam)
+    Iold -= Iold.min()
+    Iold /= Iold.max()
+    qt = fif.image_to_q(It)
+    qm = fif.image_to_q(Iold)
+    gamold = gam.copy()
\ No newline at end of file
diff --git a/fdasrsf/image_functions.py b/fdasrsf/image_functions.py
index 79066931..2559df18 100644
--- a/fdasrsf/image_functions.py
+++ b/fdasrsf/image_functions.py
@@ -3,4 +3,84 @@
 
 moduleauthor:: J. Derek Tucker <jdtuck@sandia.gov>
 
-"""
\ No newline at end of file
+"""
+
+import numpy as np
+from scipy.interpolate import griddata
+
+
+def apply_gam_to_gam(gamnew, gam):
+    m = gam.shape[0]
+    n = gam.shape[1]
+    D = gam.shape[2]
+    md = 8
+    mt = md*m
+    nt = md*n
+
+    U = np.linspace(0,1,m)
+    V = np.linspace(0,1,n)
+
+    return
+
+
+def apply_gam_imag(F, gam):
+    (m,n,d) = F.shape
+
+    Fnew = np.zeros((m,n,d))
+
+    U = np.linspace(0,1,m)
+    V = np.linspace(0,1,n)
+
+    for j in range(d):
+        Fnew[:,:,j] = griddata((U,V), F[:,:,j], (gam[:,:,0], gam[:,:,1]), method='cubic')
+    
+    return Fnew
+
+
+def makediffeoid(nrow,ncol):
+    D = 2
+    gamid = np.zeros((nrow,ncol,D))
+
+    U, V = np.meshgrid(np.linspace(0,1,ncol), np.linspace(0,1,nrow), indexing='xy')
+
+    gamid[:,:,0] = U
+    gamid[:,:,1] = V
+
+    return gamid
+
+
+def image_to_q(F):
+    d = F.shape[2]
+
+    if d < 2:
+        raise NameError('Data dimension is wrong!')
+    
+    q = F.copy()
+
+    sqrtmultfact = np.sqrt(Jacob_imag(F))
+    for i in range(d):
+        q[:,:,i] = sqrtmultfact*F[:,:,i]
+    
+    return q
+
+
+def Jacob_imag(F):
+    (m,n,d) = F.shape
+
+    if d < 2:
+        raise NameError('Data dimension is wrong!')
+    
+    dfdu, dfdv = compgrad2D(F)
+
+    multFactor = np.zeros((m,n))
+
+    if d==2:
+        multFactor = dfdu[:,:,0]@dfdv[:,:,1] - dfdu[:,:,1]@dfdv[:,:,0]
+        multFactor = np.abs(multFactor)
+    elif d==3:
+        multFactor = (dfdu[:,:,1]@dfdv[:,:,2] - dfdu[:,:,2]@dfdv[:,:,1])**2 + (dfdu[:,:,0]@dfdv[:,:,2] - dfdu[:,:,2]@dfdv[:,:,0])**2 + (dfdu[:,:,0]@dfdv[:,:,1] - dfdu[:,:,1]@dfdv[:,:,0])**2
+        multFactor = np.sqrt(multFactor)
+    
+    return multFactor
+
+
diff --git a/src/UnitSquareImage.cpp b/src/UnitSquareImage.cpp
index a06086d7..c4e18605 100644
--- a/src/UnitSquareImage.cpp
+++ b/src/UnitSquareImage.cpp
@@ -2,7 +2,6 @@
 #include <cmath>
 #include <iostream>
 #include <stdio.h>
-#include <Rcpp.h>
 #include "UnitSquareImage.h"
 #include "ImageRegister.h"
 

From 4ccb47a52c78dac965ac42da442ca80cdf39dab6 Mon Sep 17 00:00:00 2001
From: "J. Derek Tucker" <jdtuck@sandia.gov>
Date: Mon, 13 Feb 2023 11:33:37 -0700
Subject: [PATCH 05/12] finish first pass of image warping code (basic)

---
 fdasrsf/image.py           |   73 +-
 fdasrsf/image_functions.py |  155 +-
 setup.py                   |    5 +
 src/ImageRegister.h        |  929 +++--
 src/UnitSquareImage.cpp    |   47 +-
 src/UnitSquareImage.h      |    2 +-
 src/cUnitSquareImage.pxd   |    6 +
 src/imagecpp.cpp           | 7761 ++++++++++++++++++++++++++++++++++++
 src/imagecpp.pyx           |   59 +
 src/myVector.cpp           |    2 -
 src/myVector.h             |    5 -
 11 files changed, 8495 insertions(+), 549 deletions(-)
 create mode 100644 src/cUnitSquareImage.pxd
 create mode 100644 src/imagecpp.cpp
 create mode 100644 src/imagecpp.pyx

diff --git a/fdasrsf/image.py b/fdasrsf/image.py
index 98b720a8..11e1eb2e 100644
--- a/fdasrsf/image.py
+++ b/fdasrsf/image.py
@@ -7,6 +7,8 @@
 
 import numpy as np
 import fdasrsf.image_functions as fif
+import imagecpp as im
+
 
 def reparam_image(It, Im, gam=None, b=None, stepsize=1e-4, itermax=20):
     """
@@ -18,7 +20,10 @@ def reparam_image(It, Im, gam=None, b=None, stepsize=1e-4, itermax=20):
     :param b: numpy ndarray representing basis matrix
 
     :rtype: numpy ndarray
-    :return f: smoothed functions functions
+    :return gamnew: diffeomorphism
+    :return Inew: warped image
+    :return H: energy
+    :return stepsize: final stepsize
 
     """
 
@@ -29,6 +34,11 @@ def reparam_image(It, Im, gam=None, b=None, stepsize=1e-4, itermax=20):
 
     if gam is None:
         gam = gamid.copy()
+    
+    if b is None: 
+        M = 10
+        basetype = 't'
+        b = fif.formbasisTid(M, m, n, basetype)
 
     # main loop
     H = np.zeros(itermax+1)
@@ -37,4 +47,63 @@ def reparam_image(It, Im, gam=None, b=None, stepsize=1e-4, itermax=20):
     Iold /= Iold.max()
     qt = fif.image_to_q(It)
     qm = fif.image_to_q(Iold)
-    gamold = gam.copy()
\ No newline at end of file
+    gamold = gam.copy()
+
+    gamnew = gamold.copy()
+    Inew = Iold.copy()
+    iter = 0
+    H[iter] = fif.compEnergy(qt,qm)
+
+    print('Iteration %d, energy %f\n'% (iter-1,H(iter)))
+
+    gamupdate = fif.updateGam(qt,qm,b)
+
+    cutoff = 1e-3
+
+    for iter in range(1,itermax+1):
+        gaminc = gamid + stepsize*gamupdate
+
+        G = im.check_crossing(gamnew)
+
+        if G == 0:
+            print('Possible crossing!\n')
+            gamnew = gamold.copy()
+            stepsize *= 0.67
+            H[iter] = H[iter-1]
+            continue
+        
+        else:
+            gamnew = fif.apply_gam_gamid(gamnew,gaminc)
+        
+        Inew = fif.apply_gam_imag(Im, gamnew)
+        Inew -= Inew.min()
+        Inew /= Inew.max()
+        qm = fif.image_to_q(Inew)
+        H[iter] = fif.compEnergy(qt,qm)
+        print('Iteration %d, energy %f\n'% (iter-1,H(iter)))
+
+        if (iter > 4):
+            hstop = 1
+            for i in range(4):
+                hstop *= (H[iter] >= H[iter-i])
+            
+            if hstop != 0:
+                print('Warning: energy constantly increasing\n')
+                break
+        
+        if (iter > 4) and (np.abs(H[iter] <= H[iter-1])<cutoff and np.abs(H[iter-1] - H[iter-2])<cutoff and np.abs(H[iter-2] <= H[iter-3])<cutoff):
+            print('Warning: energy is not changing\n')
+            break
+
+        if ((iter > 1) and (H[iter] > H[iter-1])) or ((iter > 3) and ((H[iter-1] <= H[iter-2]) and (H[iter-2] > H[iter-3]))):
+            stepsize *= 0.9
+            gamnew = gamold.copy()
+            H[iter] = H[iter-1]
+            continue
+        
+        gamold = gamnew.copy()
+        gamupdate = fif.updateGam(qt,qm,b)
+    
+    H = H[0:iter]
+
+    return gamnew,Inew,H,stepsize
\ No newline at end of file
diff --git a/fdasrsf/image_functions.py b/fdasrsf/image_functions.py
index 2559df18..b2220160 100644
--- a/fdasrsf/image_functions.py
+++ b/fdasrsf/image_functions.py
@@ -7,20 +7,7 @@
 
 import numpy as np
 from scipy.interpolate import griddata
-
-
-def apply_gam_to_gam(gamnew, gam):
-    m = gam.shape[0]
-    n = gam.shape[1]
-    D = gam.shape[2]
-    md = 8
-    mt = md*m
-    nt = md*n
-
-    U = np.linspace(0,1,m)
-    V = np.linspace(0,1,n)
-
-    return
+import imagecpp as im
 
 
 def apply_gam_imag(F, gam):
@@ -70,7 +57,7 @@ def Jacob_imag(F):
     if d < 2:
         raise NameError('Data dimension is wrong!')
     
-    dfdu, dfdv = compgrad2D(F)
+    dfdu, dfdv = im.compgrad2D(F)
 
     multFactor = np.zeros((m,n))
 
@@ -84,3 +71,141 @@ def Jacob_imag(F):
     return multFactor
 
 
+def compEnergy(q1,q2):
+    (m,n,d) = q1.shape
+    ds = 1/(m-1)/(n-1)
+
+    tmp = q1-q2
+
+    H = np.dot(tmp.ravel(),tmp.ravel())*ds
+
+    return H
+
+
+def updateGam(qt,qm,b):
+    v = qt - qm
+    w = findphistar(qt,b)
+    gamupdate = findupdategam(v,w,b)
+
+    return gamupdate
+
+
+def findphistar(q,b):
+    (m,n,D,K) = b.shape
+
+    d = q.shape[2]
+    dbxdu = np.zeros((m,n,K))
+    dbydv = np.zeros((m,n,K))
+    expr1 = np.zeros((m,n,d,K))
+    expr2 = np.zeros((m,n,d,K))
+
+    for k in range(K):
+        dbxdu[:,:,k], tmp = im.compgrad2D(b[:,:,0,k])
+        tmp, dbydv[:,:,k] = im.compgrad2D(b[:,:,1,k])
+    
+    divb = dbxdu + dbydv
+
+    dqdu, dqdv = im.compgrad2D(q)
+
+    for k in range(K):
+        for j in range(d):
+            expr1[:,:,j,k] = divb[:,:,k]*q[:,:,j]
+            expr2[:,:,j,k] = dqdu[:,:,j]*b[:,:,0,k] + dqdv[:,:,j]*b[:,:,1,k]
+    
+    w = 0.5 * expr1 + expr2
+
+    return w
+
+
+def findupdategam(v,w,b):
+    (m,n,D,K) = b.shape
+    ds = 1/((m-1)*(n-1))
+
+    innp = np.zeros(K)
+
+    gamupdate = np.zeros((m,n,D))
+
+    for k in range(K):
+        vt = w[:,:,:,k]
+        innp[k] = np.dot(v.ravel(),vt.ravel())*ds
+
+        gamupdate = gamupdate + innp[k]*b[:,:,:,k]
+    
+    return gamupdate
+
+
+def apply_gam_gamid(gamid, gaminc):
+    (m, n, d) = gamid.shape
+
+    U = np.linspace(0,1,m)
+    V = np.linspace(0,1,n)
+
+    gamcum = np.zeros((m,n,d))
+    for j in range(d):
+        gamcum[:,:,j] = griddata((U,V), gamid[:,:,j], (gaminc[:,:,0], gaminc[:,:,1]), method='cubic')
+    
+    return gamcum
+
+
+def gen_basis(m, n, M=10, N=10, baseType = 't', ortho=False):
+
+    b = formbasisTid(M, m, n, baseType)
+
+    if ortho:
+        b = GramSchmidt(b)
+    
+    return b
+
+
+def formbasisTid(M, m, n, base_type):
+    U, V = np.meshgrid(np.linspace(0,1,n), np.linspace(0,1,m), indexing='xy')
+
+    idx = 0
+
+    if base_type == 't':
+        b = np.zeros((m,n,2,2*M))
+        for s in range(M):
+            const = np.sqrt(2)*np.pi*s
+            sPI2 = 2*np.pi*s
+
+            b[:,:,0,idx] = np.zeros((m,n))
+            b[:,:,1,idx] = (np.cos(sPI2*V)-1)/const
+
+            b[:,:,0,idx+1] = (np.cos(sPI2*U)-1)/const
+            b[:,:,1,idx+1] = np.zeros((m,n))
+
+            idx += 2
+
+    return b
+
+
+def GramSchmidt(b):
+    (m,n,D,N) = b.shape
+
+    ds = 1/((m-1)*(n-1))
+
+    cnt = 0
+    G = np.zeros((m,n,D,N))
+    G[:,:,:,cnt] = b[:,:,:,cnt]
+
+    dvx, dvy = im.compgrad2D(G[:,:,:,cnt])
+    l = np.dot(dvx.ravel(),dvx.ravel())*ds + np.dot(dvy.ravel(),dvy.ravel())*ds
+    G[:,:,:,cnt] = G[:,:,:,cnt]/np.sqrt(l)
+
+    for i in range(1,N):
+        G[:,:,:,i] = b[:,:,:,i]
+        dv1x, dv1y = im.compgrad2D(G[:,:,:,i])
+
+        for j in range(0,i-1):
+            dv2x, dv2y = im.compgrad2D(G[:,:,:,j])
+            t = np.dot(dv1x.ravel(),dv2x.ravel())*ds + np.dot(dv1y.ravel(),dv2y.ravel())*ds
+            G[:,:,:,i] = G[:,:,:,i]-t*G[:,:,:,j]
+        
+        v = G[:,:,:,i]
+        l = np.dot(v.ravel(),v.ravel())*ds
+
+        if l>0:
+            cnt += 1
+            G[:,:,:,cnt] = G[:,:,:,i]/np.sqrt(l)
+    
+    return G
diff --git a/setup.py b/setup.py
index e2275825..cd89b717 100644
--- a/setup.py
+++ b/setup.py
@@ -84,6 +84,11 @@ def run(self):
         include_dirs=[numpy.get_include()],
         language="c++"
     ),
+    Extension(name="cimage",
+        sources=["src/imagecpp.pyx", "src/UnitSquareImage.cpp"],
+        include_dirs=[numpy.get_include()],
+        language="c++"
+    ),
     dp_build.ffibuilder.distutils_extension(),
 ]
 
diff --git a/src/ImageRegister.h b/src/ImageRegister.h
index 22a4728b..c9835511 100644
--- a/src/ImageRegister.h
+++ b/src/ImageRegister.h
@@ -1,623 +1,596 @@
 #ifndef SURFACECHRISTOFFEL_H
 #define SURFACECHRISTOFFEL_H
 
-#include <cstring>
 #include <cmath>
+#include <cstring>
 using namespace std;
 
-#define SQRT5	2.23606797749978969641
+#define SQRT5 2.23606797749978969641
 
 // Prototypes: -----------------------------------------------------------------
-template<typename T>
-void cpyArray(T *x, const T *y, int n);
+template <typename T> void cpyArray(T *x, const T *y, int n);
 
-template<typename T>
-T innerProd(const T *x, const T *y, int N, int d);
+template <typename T> T innerProd(const T *x, const T *y, int N, int d);
 
-template<typename T>
-T det2(const T *A);
+template <typename T> T det2(const T *A);
 
-template<typename T>
-T det3(const T *A);
+template <typename T> T det3(const T *A);
 
-template<typename T>
-inline void jacob(T z, const T x[2], const T y[2]);
+template <typename T> inline void jacob(T z, const T x[2], const T y[2]);
 
-template<typename T>
-void jacobMat(T *z, const T *x, const T *y, int n, int t);
+template <typename T> void jacobMat(T *z, const T *x, const T *y, int n, int t);
 
-template<typename T>
-void cross(T z[3], const T x[3], const T y[3]);
+template <typename T> void cross(T z[3], const T x[3], const T y[3]);
 
-template<typename T>
-void cross2(T z, const T x[2], const T y[2]);
+template <typename T> void cross2(T z, const T x[2], const T y[2]);
 
-template<typename T>
-void matmul3(T *C, const T *A, const T *B);
+template <typename T> void matmul3(T *C, const T *A, const T *B);
 
-template<typename T>
-void matvec3(T y[3], const T *A, const T x[3]);
+template <typename T> void matvec3(T y[3], const T *A, const T x[3]);
 
-template<typename T>
-void aat3(T *AA, const T *A);
+template <typename T> void aat3(T *AA, const T *A);
 
-template<typename T>
-void ata3(T *AA, const T *A);
+template <typename T> void ata3(T *AA, const T *A);
 
-template<typename T>
-void div3(T x[3], T div);
+template <typename T> void div3(T x[3], T div);
 
-template<typename T>
-void mult3(T x[3], T mult);
+template <typename T> void mult3(T x[3], T mult);
 
-template<typename T>
-void assign3(T *y, const T x[3], int s = 1);
+template <typename T> void assign3(T *y, const T x[3], int s = 1);
 
-template<typename T>
-T L2norm3(T x[3]);
+template <typename T> T L2norm3(T x[3]);
 
-template<typename T>
-T L2norm(T x, int s);
+template <typename T> T L2norm(T x, int s);
 
-template<typename T>
-T cdiff(const T *F, T dx, int s);
+template <typename T> T cdiff(const T *F, T dx, int s);
 
-template<typename T>
-T fdiff(const T *F, T dx, int s);
+template <typename T> T fdiff(const T *F, T dx, int s);
 
-template<typename T>
-T bdiff(const T *F, T dx, int s);
+template <typename T> T bdiff(const T *F, T dx, int s);
 
-template<typename T>
-void cdiff2(T df[2], const T *F, T dx, int s, int S);
+template <typename T> void cdiff2(T df[2], const T *F, T dx, int s, int S);
 
-template<typename T>
-void fdiff2(T df[2], const T *F, T dx, int s, int S);
+template <typename T> void fdiff2(T df[2], const T *F, T dx, int s, int S);
 
-template<typename T>
-void bdiff2(T df[2], const T *F, T dx, int s, int S);
+template <typename T> void bdiff2(T df[2], const T *F, T dx, int s, int S);
 
-template<typename T>
-void cdiff3(T df[3], const T *F, T dx, int s, int S);
+template <typename T> void cdiff3(T df[3], const T *F, T dx, int s, int S);
 
-template<typename T>
-void fdiff3(T df[3], const T *F, T dx, int s, int S);
+template <typename T> void fdiff3(T df[3], const T *F, T dx, int s, int S);
 
-template<typename T>
-void bdiff3(T df[3], const T *F, T dx, int s, int S);
+template <typename T> void bdiff3(T df[3], const T *F, T dx, int s, int S);
 
-template<typename T>
-void sort3(T x[3]);
+template <typename T> void sort3(T x[3]);
 
-template<typename T>
-void solvecubic(T c[3]);
+template <typename T> void solvecubic(T c[3]);
 
-template<typename T>
-void ldu3(T *A, int P[3]);
+template <typename T> void ldu3(T *A, int P[3]);
 
-template<typename T>
+template <typename T>
 void ldubsolve3(T x[3], const T y[3], const T *LDU, const int P[3]);
 
-template<typename T>
-void unit3(T x[3]);
+template <typename T> void unit3(T x[3]);
 
-template<typename T>
-void svd3(T *U, T S[3], T *V, const T *A);
+template <typename T> void svd3(T *U, T S[3], T *V, const T *A);
 
-template<typename T>
-void thomas(T *x, const T *a, const T *b, T *c, int n);
+template <typename T> void thomas(T *x, const T *a, const T *b, T *c, int n);
 
-template<typename T>
-void spline(T *D, const T *y, int n);
+template <typename T> void spline(T *D, const T *y, int n);
 
-template<typename T>
-void lookupspline(T &t, int &k, T dist, T len, int n);
+template <typename T> void lookupspline(T &t, int &k, T dist, T len, int n);
 
-template<typename T>
-T evalspline(T t, const T D[2], const T y[2]);
+template <typename T> T evalspline(T t, const T D[2], const T y[2]);
 
-template<typename T>
-void interp2(T *Dy, const T *Z, int nx, int ny);
+template <typename T> void interp2(T *Dy, const T *Z, int nx, int ny);
 
-template<typename T>
+template <typename T>
 void evalinterp2(T u, T *Dx, T *zx, const T *Dy, const T *Z, int nx, int ny);
 
-
 // Subroutines ============================================================
-template<typename T>
-void cpyArray(T *x, const T *y, int n) {
-    for (int i=0; i<n; ++i) {
-        x[i] = y[i];
-    }
+template <typename T> void cpyArray(T *x, const T *y, int n) {
+  for (int i = 0; i < n; ++i) {
+    x[i] = y[i];
+  }
 }
 
-template<typename T>
-inline T innerProd(const T *x, const T *y, int N, int d) {
-    T s = 0;
-    for (int i=0; i<d; i++) {
-        s += x[N*i]*y[N*i];
-    }
-    return s;
+template <typename T> inline T innerProd(const T *x, const T *y, int N, int d) {
+  T s = 0;
+  for (int i = 0; i < d; i++) {
+    s += x[N * i] * y[N * i];
+  }
+  return s;
 }
 
-template<typename T>
-inline void jacob(T z, const T x[2], const T y[2]) {
-	z = x[1]*y[2]-x[2]*y[1];
+template <typename T> inline void jacob(T z, const T x[2], const T y[2]) {
+  z = x[1] * y[2] - x[2] * y[1];
 }
 
-template<typename T>
+template <typename T>
 inline void jacobMat(T *z, const T *x, const T *y, int n, int t) {
-    int i, N = n*t, d = 2;
-    if (d == 2) {
-        for (i=0; i<N; ++i) {
-            z[i] = x[N*0 + i]*y[N*1 + i]-x[N*1 + i]*y[N*0 + i];
-        }
+  int i, N = n * t, d = 2;
+  if (d == 2) {
+    for (i = 0; i < N; ++i) {
+      z[i] = x[N * 0 + i] * y[N * 1 + i] - x[N * 1 + i] * y[N * 0 + i];
     }
+  }
 }
 
-template<typename T>
-inline void cross(T z[3], const T x[3], const T y[3]) {
-	z[0] = x[1]*y[2]-x[2]*y[1];
-	z[1] = -(x[0]*y[2]-x[2]*y[0]);
-	z[2] = x[0]*y[1]-x[1]*y[0];
+template <typename T> inline void cross(T z[3], const T x[3], const T y[3]) {
+  z[0] = x[1] * y[2] - x[2] * y[1];
+  z[1] = -(x[0] * y[2] - x[2] * y[0]);
+  z[2] = x[0] * y[1] - x[1] * y[0];
 }
 
-template<typename T>
-inline void cross2(T z, const T x[2], const T y[2]) {
-	z = x[1]*y[2]-x[2]*y[1];
+template <typename T> inline void cross2(T z, const T x[2], const T y[2]) {
+  z = x[1] * y[2] - x[2] * y[1];
 }
 
-template<typename T>
-inline T cdiff(const T *F, T dx, int s) {
-	return (F[s]-F[-s])/(2*dx);
+template <typename T> inline T cdiff(const T *F, T dx, int s) {
+  return (F[s] - F[-s]) / (2 * dx);
 }
 
-template<typename T>
-inline T fdiff(const T *F, T dx, int s) {
-	return (F[s]-F[0])/dx;
+template <typename T> inline T fdiff(const T *F, T dx, int s) {
+  return (F[s] - F[0]) / dx;
 }
 
-template<typename T>
-inline T bdiff(const T *F, T dx, int s) {
-	return (F[0]-F[-s])/dx;
+template <typename T> inline T bdiff(const T *F, T dx, int s) {
+  return (F[0] - F[-s]) / dx;
 }
 
-template<typename T>
+template <typename T>
 inline void cdiff2(T df[2], const T *F, T dx, int s, int S) {
-    int i;
-    for (i=0; i<2; ++i){
-        df[i] = (F[i*S+s]-F[i*S-s])/(2*dx);
-    }
+  int i;
+  for (i = 0; i < 2; ++i) {
+    df[i] = (F[i * S + s] - F[i * S - s]) / (2 * dx);
+  }
 }
 
-template<typename T>
+template <typename T>
 inline void fdiff2(T df[2], const T *F, T dx, int s, int S) {
-    int i;
-    for (i=0; i<2; ++i){
-        df[i] = (F[i*S+s]-F[i*S])/(2*dx);
-    }
+  int i;
+  for (i = 0; i < 2; ++i) {
+    df[i] = (F[i * S + s] - F[i * S]) / (2 * dx);
+  }
 }
 
-template<typename T>
+template <typename T>
 inline void bdiff2(T df[2], const T *F, T dx, int s, int S) {
-    int i;
-    for (i=0; i<2; ++i){
-        df[i] = (F[i*S]-F[i*S-s])/(2*dx);
-    }
+  int i;
+  for (i = 0; i < 2; ++i) {
+    df[i] = (F[i * S] - F[i * S - s]) / (2 * dx);
+  }
 }
 
-template<typename T>
+template <typename T>
 inline void cdiff3(T df[3], const T *F, T dx, int s, int S) {
-	df[0] = (F[s]-F[-s])/(2*dx);
-	df[1] = (F[S+s]-F[S-s])/(2*dx);
-	df[2] = (F[2*S+s]-F[2*S-s])/(2*dx);
+  df[0] = (F[s] - F[-s]) / (2 * dx);
+  df[1] = (F[S + s] - F[S - s]) / (2 * dx);
+  df[2] = (F[2 * S + s] - F[2 * S - s]) / (2 * dx);
 }
 
-template<typename T>
+template <typename T>
 inline void fdiff3(T df[3], const T *F, T dx, int s, int S) {
-	df[0] = (F[s]-F[0])/dx;
-	df[1] = (F[S+s]-F[S])/dx;
-	df[2] = (F[2*S+s]-F[2*S])/dx;
+  df[0] = (F[s] - F[0]) / dx;
+  df[1] = (F[S + s] - F[S]) / dx;
+  df[2] = (F[2 * S + s] - F[2 * S]) / dx;
 }
 
-template<typename T>
+template <typename T>
 inline void bdiff3(T df[3], const T *F, T dx, int s, int S) {
-	df[0] = (F[0]-F[-s])/dx;
-	df[1] = (F[S]-F[S-s])/dx;
-	df[2] = (F[2*S]-F[2*S-s])/dx;
-}
+  df[0] = (F[0] - F[-s]) / dx;
+  df[1] = (F[S] - F[S - s]) / dx;
+  df[2] = (F[2 * S] - F[2 * S - s]) / dx;
+}
 
-template<typename T>
-inline void matmul3(T *C, const T *A, const T *B) {
-	C[3*0+0] = A[3*0+0]*B[3*0+0] + A[3*1+0]*B[3*0+1] + A[3*2+0]*B[3*0+2];
-	C[3*1+0] = A[3*0+0]*B[3*1+0] + A[3*1+0]*B[3*1+1] + A[3*2+0]*B[3*1+2];
-	C[3*2+0] = A[3*0+0]*B[3*2+0] + A[3*1+0]*B[3*2+1] + A[3*2+0]*B[3*2+2];
+template <typename T> inline void matmul3(T *C, const T *A, const T *B) {
+  C[3 * 0 + 0] = A[3 * 0 + 0] * B[3 * 0 + 0] + A[3 * 1 + 0] * B[3 * 0 + 1] +
+                 A[3 * 2 + 0] * B[3 * 0 + 2];
+  C[3 * 1 + 0] = A[3 * 0 + 0] * B[3 * 1 + 0] + A[3 * 1 + 0] * B[3 * 1 + 1] +
+                 A[3 * 2 + 0] * B[3 * 1 + 2];
+  C[3 * 2 + 0] = A[3 * 0 + 0] * B[3 * 2 + 0] + A[3 * 1 + 0] * B[3 * 2 + 1] +
+                 A[3 * 2 + 0] * B[3 * 2 + 2];
 
-	C[3*0+1] = A[3*0+1]*B[3*0+0] + A[3*1+1]*B[3*0+1] + A[3*2+1]*B[3*0+2];
-	C[3*1+1] = A[3*0+1]*B[3*1+0] + A[3*1+1]*B[3*1+1] + A[3*2+1]*B[3*1+2];
-	C[3*2+1] = A[3*0+1]*B[3*2+0] + A[3*1+1]*B[3*2+1] + A[3*2+1]*B[3*2+2];
+  C[3 * 0 + 1] = A[3 * 0 + 1] * B[3 * 0 + 0] + A[3 * 1 + 1] * B[3 * 0 + 1] +
+                 A[3 * 2 + 1] * B[3 * 0 + 2];
+  C[3 * 1 + 1] = A[3 * 0 + 1] * B[3 * 1 + 0] + A[3 * 1 + 1] * B[3 * 1 + 1] +
+                 A[3 * 2 + 1] * B[3 * 1 + 2];
+  C[3 * 2 + 1] = A[3 * 0 + 1] * B[3 * 2 + 0] + A[3 * 1 + 1] * B[3 * 2 + 1] +
+                 A[3 * 2 + 1] * B[3 * 2 + 2];
 
-	C[3*0+2] = A[3*0+2]*B[3*0+0] + A[3*1+2]*B[3*0+1] + A[3*2+2]*B[3*0+2];
-	C[3*1+2] = A[3*0+2]*B[3*1+0] + A[3*1+2]*B[3*1+1] + A[3*2+2]*B[3*1+2];
-	C[3*2+2] = A[3*0+2]*B[3*2+0] + A[3*1+2]*B[3*2+1] + A[3*2+2]*B[3*2+2];
+  C[3 * 0 + 2] = A[3 * 0 + 2] * B[3 * 0 + 0] + A[3 * 1 + 2] * B[3 * 0 + 1] +
+                 A[3 * 2 + 2] * B[3 * 0 + 2];
+  C[3 * 1 + 2] = A[3 * 0 + 2] * B[3 * 1 + 0] + A[3 * 1 + 2] * B[3 * 1 + 1] +
+                 A[3 * 2 + 2] * B[3 * 1 + 2];
+  C[3 * 2 + 2] = A[3 * 0 + 2] * B[3 * 2 + 0] + A[3 * 1 + 2] * B[3 * 2 + 1] +
+                 A[3 * 2 + 2] * B[3 * 2 + 2];
 }
 
-template<typename T>
-inline void matvec3(T y[3], const T *A, const T x[3]) {
-	y[0] = A[3*0+0]*x[0] + A[3*1+0]*x[1] + A[3*2+0]*x[2];
-	y[1] = A[3*0+1]*x[0] + A[3*1+1]*x[1] + A[3*2+1]*x[2];
-	y[2] = A[3*0+2]*x[0] + A[3*1+2]*x[1] + A[3*2+2]*x[2];
+template <typename T> inline void matvec3(T y[3], const T *A, const T x[3]) {
+  y[0] = A[3 * 0 + 0] * x[0] + A[3 * 1 + 0] * x[1] + A[3 * 2 + 0] * x[2];
+  y[1] = A[3 * 0 + 1] * x[0] + A[3 * 1 + 1] * x[1] + A[3 * 2 + 1] * x[2];
+  y[2] = A[3 * 0 + 2] * x[0] + A[3 * 1 + 2] * x[1] + A[3 * 2 + 2] * x[2];
 }
 
-template<typename T>
-inline void aat3(T *AA, const T *A) {
-	AA[3*0+0] = A[3*0+0]*A[3*0+0] + A[3*1+0]*A[3*1+0] + A[3*2+0]*A[3*2+0];
-	AA[3*1+0] = A[3*0+0]*A[3*0+1] + A[3*1+0]*A[3*1+1] + A[3*2+0]*A[3*2+1];
-	AA[3*2+0] = A[3*0+0]*A[3*0+2] + A[3*1+0]*A[3*1+2] + A[3*2+0]*A[3*2+2];
+template <typename T> inline void aat3(T *AA, const T *A) {
+  AA[3 * 0 + 0] = A[3 * 0 + 0] * A[3 * 0 + 0] + A[3 * 1 + 0] * A[3 * 1 + 0] +
+                  A[3 * 2 + 0] * A[3 * 2 + 0];
+  AA[3 * 1 + 0] = A[3 * 0 + 0] * A[3 * 0 + 1] + A[3 * 1 + 0] * A[3 * 1 + 1] +
+                  A[3 * 2 + 0] * A[3 * 2 + 1];
+  AA[3 * 2 + 0] = A[3 * 0 + 0] * A[3 * 0 + 2] + A[3 * 1 + 0] * A[3 * 1 + 2] +
+                  A[3 * 2 + 0] * A[3 * 2 + 2];
 
-	AA[3*0+1] = AA[3*1+0];
-	AA[3*1+1] = A[3*0+1]*A[3*0+1] + A[3*1+1]*A[3*1+1] + A[3*2+1]*A[3*2+1];
-	AA[3*2+1] = A[3*0+1]*A[3*0+2] + A[3*1+1]*A[3*1+2] + A[3*2+1]*A[3*2+2];
+  AA[3 * 0 + 1] = AA[3 * 1 + 0];
+  AA[3 * 1 + 1] = A[3 * 0 + 1] * A[3 * 0 + 1] + A[3 * 1 + 1] * A[3 * 1 + 1] +
+                  A[3 * 2 + 1] * A[3 * 2 + 1];
+  AA[3 * 2 + 1] = A[3 * 0 + 1] * A[3 * 0 + 2] + A[3 * 1 + 1] * A[3 * 1 + 2] +
+                  A[3 * 2 + 1] * A[3 * 2 + 2];
 
-	AA[3*0+2] = AA[3*2+0];
-	AA[3*1+2] = AA[3*2+1];
-	AA[3*2+2] = A[3*0+2]*A[3*0+2] + A[3*1+2]*A[3*1+2] + A[3*2+2]*A[3*2+2];
+  AA[3 * 0 + 2] = AA[3 * 2 + 0];
+  AA[3 * 1 + 2] = AA[3 * 2 + 1];
+  AA[3 * 2 + 2] = A[3 * 0 + 2] * A[3 * 0 + 2] + A[3 * 1 + 2] * A[3 * 1 + 2] +
+                  A[3 * 2 + 2] * A[3 * 2 + 2];
 }
 
-template<typename T>
-inline void ata3(T *AA, const T *A) {
-	AA[3*0+0] = A[3*0+0]*A[3*0+0] + A[3*0+1]*A[3*0+1] + A[3*0+2]*A[3*0+2];
-	AA[3*1+0] = A[3*0+0]*A[3*1+0] + A[3*0+1]*A[3*1+1] + A[3*0+2]*A[3*1+2];
-	AA[3*2+0] = A[3*0+0]*A[3*2+0] + A[3*0+1]*A[3*2+1] + A[3*0+2]*A[3*2+2];
-
-	AA[3*0+1] = AA[3*1+0];
-	AA[3*1+1] = A[3*1+0]*A[3*1+0] + A[3*1+1]*A[3*1+1] + A[3*1+2]*A[3*1+2];
-	AA[3*2+1] = A[3*1+0]*A[3*2+0] + A[3*1+1]*A[3*2+1] + A[3*1+2]*A[3*2+2];
+template <typename T> inline void ata3(T *AA, const T *A) {
+  AA[3 * 0 + 0] = A[3 * 0 + 0] * A[3 * 0 + 0] + A[3 * 0 + 1] * A[3 * 0 + 1] +
+                  A[3 * 0 + 2] * A[3 * 0 + 2];
+  AA[3 * 1 + 0] = A[3 * 0 + 0] * A[3 * 1 + 0] + A[3 * 0 + 1] * A[3 * 1 + 1] +
+                  A[3 * 0 + 2] * A[3 * 1 + 2];
+  AA[3 * 2 + 0] = A[3 * 0 + 0] * A[3 * 2 + 0] + A[3 * 0 + 1] * A[3 * 2 + 1] +
+                  A[3 * 0 + 2] * A[3 * 2 + 2];
 
-	AA[3*0+2] = AA[3*2+0];
-	AA[3*1+2] = AA[3*2+1];
-	AA[3*2+2] = A[3*2+0]*A[3*2+0] + A[3*2+1]*A[3*2+1] + A[3*2+2]*A[3*2+2];
+  AA[3 * 0 + 1] = AA[3 * 1 + 0];
+  AA[3 * 1 + 1] = A[3 * 1 + 0] * A[3 * 1 + 0] + A[3 * 1 + 1] * A[3 * 1 + 1] +
+                  A[3 * 1 + 2] * A[3 * 1 + 2];
+  AA[3 * 2 + 1] = A[3 * 1 + 0] * A[3 * 2 + 0] + A[3 * 1 + 1] * A[3 * 2 + 1] +
+                  A[3 * 1 + 2] * A[3 * 2 + 2];
 
+  AA[3 * 0 + 2] = AA[3 * 2 + 0];
+  AA[3 * 1 + 2] = AA[3 * 2 + 1];
+  AA[3 * 2 + 2] = A[3 * 2 + 0] * A[3 * 2 + 0] + A[3 * 2 + 1] * A[3 * 2 + 1] +
+                  A[3 * 2 + 2] * A[3 * 2 + 2];
 }
 
-template<typename T>
-inline void div3(T x[3], T div) {
-	x[0] /= div;
-	x[1] /= div;
-	x[2] /= div;
+template <typename T> inline void div3(T x[3], T div) {
+  x[0] /= div;
+  x[1] /= div;
+  x[2] /= div;
 }
 
-template<typename T>
-inline void mult3(T x[3], T mult) {
-	x[0] *= mult;
-	x[1] *= mult;
-	x[2] *= mult;
+template <typename T> inline void mult3(T x[3], T mult) {
+  x[0] *= mult;
+  x[1] *= mult;
+  x[2] *= mult;
 }
 
-template<typename T>
-inline T det2(const T *A) {
-	return A[2*0+0]*A[2*1+1] - A[2*0+1]*A[2*1+0];
+template <typename T> inline T det2(const T *A) {
+  return A[2 * 0 + 0] * A[2 * 1 + 1] - A[2 * 0 + 1] * A[2 * 1 + 0];
 }
 
-template<typename T>
-inline T det3(const T *A) {
-	return A[3*0+0]*(A[3*1+1]*A[3*2+2] - A[3*2+1]*A[3*1+2]) -
-		A[3*1+0]*(A[3*0+1]*A[3*2+2] - A[3*2+1]*A[3*0+2]) +
-		A[3*2+0]*(A[3*0+1]*A[3*1+2] - A[3*1+1]*A[3*0+2]);
+template <typename T> inline T det3(const T *A) {
+  return A[3 * 0 + 0] *
+             (A[3 * 1 + 1] * A[3 * 2 + 2] - A[3 * 2 + 1] * A[3 * 1 + 2]) -
+         A[3 * 1 + 0] *
+             (A[3 * 0 + 1] * A[3 * 2 + 2] - A[3 * 2 + 1] * A[3 * 0 + 2]) +
+         A[3 * 2 + 0] *
+             (A[3 * 0 + 1] * A[3 * 1 + 2] - A[3 * 1 + 1] * A[3 * 0 + 2]);
 }
 
-template<typename T>
-inline void assign3(T *y, const T x[3], int s) {
-	y[0] = x[0];
-	y[s] = x[1];
-	y[2*s] = x[2];
+template <typename T> inline void assign3(T *y, const T x[3], int s) {
+  y[0] = x[0];
+  y[s] = x[1];
+  y[2 * s] = x[2];
 }
 
-template<typename T>
-inline T L2norm3(T x[3]) {
-	return (T)sqrt(x[0]*x[0] + x[1]*x[1] + x[2]*x[2]);
+template <typename T> inline T L2norm3(T x[3]) {
+  return (T)sqrt(x[0] * x[0] + x[1] * x[1] + x[2] * x[2]);
 }
 
-template<typename T>
-inline T L2norm(T x, int s) {
-    double temp = 0;
-    for (int i = 0; i < s; ++i){
-        temp += x[i]*x[i];
-    }
-	return (T)sqrt(temp);
+template <typename T> inline T L2norm(T x, int s) {
+  double temp = 0;
+  for (int i = 0; i < s; ++i) {
+    temp += x[i] * x[i];
+  }
+  return (T)sqrt(temp);
 }
 
-template<typename T>
-inline void sort3(T x[3]) {
-	T tmp;
-	int k;
+template <typename T> inline void sort3(T x[3]) {
+  T tmp;
+  int k;
 
-	k = (x[1] <= x[0] || x[2] <= x[0]) + (x[2] <= x[0] && x[2] <= x[1]);
-	tmp = x[2];
-	x[2] = x[k];
-	x[k] = tmp;
+  k = (x[1] <= x[0] || x[2] <= x[0]) + (x[2] <= x[0] && x[2] <= x[1]);
+  tmp = x[2];
+  x[2] = x[k];
+  x[k] = tmp;
 
-	k = (x[1] <= x[0]);
-	tmp = x[1];
-	x[1] = x[k];
-	x[k] = tmp;
+  k = (x[1] <= x[0]);
+  tmp = x[1];
+  x[1] = x[k];
+  x[k] = tmp;
 }
 
-template<typename T>
-void solvecubic(T c[3]) {
-	const double sq3d2 = 0.86602540378443864676, c2d3 = c[2]/3.0,
-		c2sq = ((double)c[2])*c[2], Q = (3.0*c[1]-c2sq)/9.0,
-		R = (c[2]*(9.0*c[1]-2.0*c2sq)-27.0*c[0])/54.0;
-	double tmp, t, sint, cost;
+template <typename T> void solvecubic(T c[3]) {
+  const double sq3d2 = 0.86602540378443864676, c2d3 = c[2] / 3.0,
+               c2sq = ((double)c[2]) * c[2], Q = (3.0 * c[1] - c2sq) / 9.0,
+               R = (c[2] * (9.0 * c[1] - 2.0 * c2sq) - 27.0 * c[0]) / 54.0;
+  double tmp, t, sint, cost;
 
-	if (Q < 0) {
-		tmp = 2*sqrt(-Q);
-		t = acos(R/sqrt(-Q*Q*Q))/3;
-		cost = tmp*cos(t);
-		sint = tmp*sin(t);
+  if (Q < 0) {
+    tmp = 2 * sqrt(-Q);
+    t = acos(R / sqrt(-Q * Q * Q)) / 3;
+    cost = tmp * cos(t);
+    sint = tmp * sin(t);
 
-		c[0] = (T)(cost - c2d3);
+    c[0] = (T)(cost - c2d3);
 
-		cost = -0.5*cost - c2d3;
-		sint = sq3d2*sint;
+    cost = -0.5 * cost - c2d3;
+    sint = sq3d2 * sint;
 
-		c[1] = (T)(cost - sint);
-		c[2] = (T)(cost + sint);
-	}
-	else {
+    c[1] = (T)(cost - sint);
+    c[2] = (T)(cost + sint);
+  } else {
 #if __STDC_VERSION__ >= 199901L
-		tmp = cbrt(R);
+    tmp = cbrt(R);
 #else
-		tmp = pow(R,1.0/3);
+    tmp = pow(R, 1.0 / 3);
 #endif
-		c[0] = (T)(-c2d3 + 2*tmp);
-		c[1] = c[2] = (T)(-c2d3 - tmp);
-	}
-}
-
-template<typename T>
-void ldu3(T *A, int P[3]) {
-	int tmp, k;
-	P[1] = 1;
-	P[2] = 2;
-
-	P[0] = ((fabs(A[3*0+0]) < fabs(A[3*1+0])) | (fabs(A[3*0+0]) < fabs(A[3*2+0]))) +
-		((fabs(A[3*0+0]) < fabs(A[3*2+0])) & (fabs(A[3*1+0]) < fabs(A[3*2+0])));
-	P[P[0]] = 0;
-
-	k = 1 + (fabs(A[3*P[1]+1]) < fabs(A[3*P[2]+1]));
-	tmp = P[1];
-	P[1] = P[k];
-	P[k] = tmp;
-
-
-	if (A[3*P[0]+0] != 0) {
-		A[3*P[1]+0] = A[3*P[1]+0]/A[3*P[0]+0];
-		A[3*P[2]+0] = A[3*P[2]+0]/A[3*P[0]+0];
-		A[3*P[0]+1] = A[3*P[0]+1]/A[3*P[0]+0];
-		A[3*P[0]+2] = A[3*P[0]+2]/A[3*P[0]+0];
-	}
-
-	A[3*P[1]+1] = A[3*P[1]+1] - A[3*P[0]+1]*A[3*P[1]+0]*A[3*P[0]+0];
-
-	if (A[3*P[1]+1] != 0) {
-		A[3*P[2]+1] = (A[3*P[2]+1] - A[3*P[0]+1]*A[3*P[2]+0]*A[3*P[0]+0])/A[3*P[1]+1];
-		A[3*P[1]+2] = (A[3*P[1]+2] - A[3*P[0]+2]*A[3*P[1]+0]*A[3*P[0]+0])/A[3*P[1]+1];
-	}
-
-	A[3*P[2]+2] = A[3*P[2]+2] - A[3*P[0]+2]*A[3*P[2]+0]*A[3*P[0]+0] - A[3*P[1]+2]*A[3*P[2]+1]*A[3*P[1]+1];
-}
-
-template<typename T>
+    c[0] = (T)(-c2d3 + 2 * tmp);
+    c[1] = c[2] = (T)(-c2d3 - tmp);
+  }
+}
+
+template <typename T> void ldu3(T *A, int P[3]) {
+  int tmp, k;
+  P[1] = 1;
+  P[2] = 2;
+
+  P[0] = ((fabs(A[3 * 0 + 0]) < fabs(A[3 * 1 + 0])) |
+          (fabs(A[3 * 0 + 0]) < fabs(A[3 * 2 + 0]))) +
+         ((fabs(A[3 * 0 + 0]) < fabs(A[3 * 2 + 0])) &
+          (fabs(A[3 * 1 + 0]) < fabs(A[3 * 2 + 0])));
+  P[P[0]] = 0;
+
+  k = 1 + (fabs(A[3 * P[1] + 1]) < fabs(A[3 * P[2] + 1]));
+  tmp = P[1];
+  P[1] = P[k];
+  P[k] = tmp;
+
+  if (A[3 * P[0] + 0] != 0) {
+    A[3 * P[1] + 0] = A[3 * P[1] + 0] / A[3 * P[0] + 0];
+    A[3 * P[2] + 0] = A[3 * P[2] + 0] / A[3 * P[0] + 0];
+    A[3 * P[0] + 1] = A[3 * P[0] + 1] / A[3 * P[0] + 0];
+    A[3 * P[0] + 2] = A[3 * P[0] + 2] / A[3 * P[0] + 0];
+  }
+
+  A[3 * P[1] + 1] =
+      A[3 * P[1] + 1] - A[3 * P[0] + 1] * A[3 * P[1] + 0] * A[3 * P[0] + 0];
+
+  if (A[3 * P[1] + 1] != 0) {
+    A[3 * P[2] + 1] = (A[3 * P[2] + 1] -
+                       A[3 * P[0] + 1] * A[3 * P[2] + 0] * A[3 * P[0] + 0]) /
+                      A[3 * P[1] + 1];
+    A[3 * P[1] + 2] = (A[3 * P[1] + 2] -
+                       A[3 * P[0] + 2] * A[3 * P[1] + 0] * A[3 * P[0] + 0]) /
+                      A[3 * P[1] + 1];
+  }
+
+  A[3 * P[2] + 2] = A[3 * P[2] + 2] -
+                    A[3 * P[0] + 2] * A[3 * P[2] + 0] * A[3 * P[0] + 0] -
+                    A[3 * P[1] + 2] * A[3 * P[2] + 1] * A[3 * P[1] + 1];
+}
+
+template <typename T>
 inline void ldubsolve3(T x[3], const T y[3], const T *LDU, const int P[3]) {
-	x[P[2]] = y[2];
-	x[P[1]] = y[1] - LDU[3*P[2]+1]*x[P[2]];
-	x[P[0]] = y[0] - LDU[3*P[2]+0]*x[P[2]] - LDU[3*P[1]+0]*x[P[1]];
-}
-
-template<typename T>
-inline void unit3(T x[3]) {
-	double tmp = sqrt(((double)x[0])*x[0]+((double)x[1])*x[1]+((double)x[2])*x[2]);
-	x[0] = (T)(x[0]/tmp);
-	x[1] = (T)(x[1]/tmp);
-	x[2] = (T)(x[2]/tmp);
-}
-
-template<typename T>
-void svd3(T *U, T S[3], T *V, const T *A) {
-	const T thr = (T)(1e-10);
-	int P[3], k;
-	T y[3], AA[3][3], LDU[3][3];
-
-	ata3((double *)AA, A);
-
-	S[2] = -AA[0][0] - AA[1][1] - AA[2][2];
-	S[1] = AA[0][0]*AA[1][1] + AA[2][2]*AA[0][0] + AA[2][2]*AA[1][1] -
-		AA[2][1]*AA[1][2] - AA[2][0]*AA[0][2] - AA[1][0]*AA[0][1];
-	S[0] = AA[2][1]*AA[1][2]*AA[0][0] + AA[2][0]*AA[0][2]*AA[1][1] + AA[1][0]*AA[0][1]*AA[2][2] -
-		AA[0][0]*AA[1][1]*AA[2][2] - AA[1][0]*AA[2][1]*AA[0][2] - AA[2][0]*AA[0][1]*AA[1][2];
-
-	solvecubic(S);
-
-	if (S[0] < 0)
-		S[0] = 0;
-	if (S[1] < 0)
-		S[1] = 0;
-	if (S[2] < 0)
-		S[2] = 0;
-
-	sort3(S);
-
-	memcpy(LDU,AA,sizeof(LDU));
-	LDU[0][0] -= S[0];
-	LDU[1][1] -= S[0];
-	LDU[2][2] -= S[0];
+  x[P[2]] = y[2];
+  x[P[1]] = y[1] - LDU[3 * P[2] + 1] * x[P[2]];
+  x[P[0]] = y[0] - LDU[3 * P[2] + 0] * x[P[2]] - LDU[3 * P[1] + 0] * x[P[1]];
+}
+
+template <typename T> inline void unit3(T x[3]) {
+  double tmp = sqrt(((double)x[0]) * x[0] + ((double)x[1]) * x[1] +
+                    ((double)x[2]) * x[2]);
+  x[0] = (T)(x[0] / tmp);
+  x[1] = (T)(x[1] / tmp);
+  x[2] = (T)(x[2] / tmp);
+}
+
+template <typename T> void svd3(T *U, T S[3], T *V, const T *A) {
+  const T thr = (T)(1e-10);
+  int P[3], k;
+  T y[3], AA[3][3], LDU[3][3];
+
+  ata3((double *)AA, A);
+
+  S[2] = -AA[0][0] - AA[1][1] - AA[2][2];
+  S[1] = AA[0][0] * AA[1][1] + AA[2][2] * AA[0][0] + AA[2][2] * AA[1][1] -
+         AA[2][1] * AA[1][2] - AA[2][0] * AA[0][2] - AA[1][0] * AA[0][1];
+  S[0] = AA[2][1] * AA[1][2] * AA[0][0] + AA[2][0] * AA[0][2] * AA[1][1] +
+         AA[1][0] * AA[0][1] * AA[2][2] - AA[0][0] * AA[1][1] * AA[2][2] -
+         AA[1][0] * AA[2][1] * AA[0][2] - AA[2][0] * AA[0][1] * AA[1][2];
+
+  solvecubic(S);
+
+  if (S[0] < 0)
+    S[0] = 0;
+  if (S[1] < 0)
+    S[1] = 0;
+  if (S[2] < 0)
+    S[2] = 0;
+
+  sort3(S);
 
-	ldu3((T *)LDU, P);
+  memcpy(LDU, AA, sizeof(LDU));
+  LDU[0][0] -= S[0];
+  LDU[1][1] -= S[0];
+  LDU[2][2] -= S[0];
 
-	y[0] = y[1] = y[2] = 0;
-	k = (fabs(LDU[P[1]][1]) < fabs(LDU[P[0]][0]) || fabs(LDU[P[2]][2]) < fabs(LDU[P[0]][0])) +
-		(fabs(LDU[P[2]][2]) < fabs(LDU[P[0]][0]) && fabs(LDU[P[2]][2]) < fabs(LDU[P[1]][1]));
-	y[k] = 1;
-
-	ldubsolve3(V+(3*0+0), y, (T *)LDU, P);
-
-	memcpy(LDU,AA,sizeof(LDU));
-	LDU[0][0] -= S[2];
-	LDU[1][1] -= S[2];
-	LDU[2][2] -= S[2];
-
-	ldu3((T *)LDU, P);
-
-	y[0] = y[1] = y[2] = 0;
-	k = (fabs(LDU[P[1]][1]) <= fabs(LDU[P[0]][0]) || fabs(LDU[P[2]][2]) <= fabs(LDU[P[0]][0])) +
-		(fabs(LDU[P[2]][2]) <= fabs(LDU[P[0]][0]) && fabs(LDU[P[2]][2]) <= fabs(LDU[P[1]][1]));
-	y[k] = 1;
-
-	ldubsolve3(V+(3*2+0), y, (T *)LDU, P);
-
-	cross(V+(3*1+0), V+(3*2+0), V+(3*0+0));
-
-	k = (S[0] > thr) + (S[1] > thr) + (S[2] > thr);
-
-	switch (k) {
-		case 0:
-			memcpy(U,V,9*sizeof(T));
-			break;
-		case 1:
-			matvec3(U+(3*0+0), A, V+(3*0+0));
-
-			y[0] = y[1] = y[2] = 0;
-			k = (fabs(U[3*0+1]) <= fabs(U[3*0+0]) || fabs(U[3*0+2]) <= fabs(U[3*0+0])) +
-				(fabs(U[3*0+2]) <= fabs(U[3*0+0]) && fabs(U[3*0+2]) <= fabs(U[3*0+1]));
-			y[k] = 1;
-
-			cross(U+(3*1+0), y, U+(3*0+0));
-			cross(U+(3*2+0), U+(3*0+0), U+(3*1+0));
-			break;
-		case 2:
-			matvec3(U+(3*0+0), A, V+(3*0+0));
-			matvec3(U+(3*1+0), A, V+(3*1+0));
-
-			cross(U+(3*2+0), U+(3*0+0), U+(3*1+0));
-			break;
-		case 3:
-			matmul3(U, A, V);
-			break;
-	}
-
-	unit3(V+(3*0+0));
-	unit3(V+(3*1+0));
-	unit3(V+(3*2+0));
-
-	unit3(U+(3*0+0));
-	unit3(U+(3*1+0));
-	unit3(U+(3*2+0));
-
-	S[0] = (T)sqrt(S[0]);
-	S[1] = (T)sqrt(S[1]);
-	S[2] = (T)sqrt(S[2]);
-}
-
-template<typename T>
-void thomas(T *x, const T *a, const T *b, T *c, int n) {
-	T tmp;
-	int i;
-
-	c[0] /= b[0];
-	x[0] /= b[0];
-
-	for (i = 1; i < n; ++i) {
-		tmp = 1/(b[i] - c[i-1] * a[i]);
-		c[i] *= tmp;
-		x[i] = (x[i] - x[i-1] * a[i])*tmp;
-	}
-
-	for (i = n-2; i >= 0; --i) {
-		x[i] -= c[i]*x[i+1];
-	}
-}
-
-template<typename T>
-void spline(T *D, const T *y, int n) {
-	int i;
-	T *a, *b, *c;
-
-	a = new T[n];
-	b = new T[n];
-	c = new T[n];
-
-	if (n < 4) {
-		a[0] = 0;
-		b[0] = 2;
-		c[0] = 1;
-		D[0] = 3*(y[1]-y[0]);
-
-		a[n-1] = 1;
-		b[n-1] = 2;
-		c[n-1] = 0;
-		D[n-1] = 3*(y[n-1]-y[n-2]);
-	}
-	else {
-		a[0] = 0;
-		b[0] = 2;
-		c[0] = 4;
-		D[0] = -5*y[0] + 4*y[1] + y[2];
-
-		a[n-1] = 4;
-		b[n-1] = 2;
-		c[n-1] = 0;
-		D[n-1] = 5*y[n-1] - 4*y[n-2] - y[n-3];
-	}
-
-	for (i = 1; i < n-1; ++i) {
-		a[i] = 1;
-		b[i] = 4;
-		c[i] = 1;
-		D[i] = 3*(y[i+1]-y[i-1]);
-	}
-
-	thomas(D, a, b, c, n);
-
-	delete [] a;
-	delete [] b;
-	delete [] c;
-}
-
-template<typename T>
+  ldu3((T *)LDU, P);
+
+  y[0] = y[1] = y[2] = 0;
+  k = (fabs(LDU[P[1]][1]) < fabs(LDU[P[0]][0]) ||
+       fabs(LDU[P[2]][2]) < fabs(LDU[P[0]][0])) +
+      (fabs(LDU[P[2]][2]) < fabs(LDU[P[0]][0]) &&
+       fabs(LDU[P[2]][2]) < fabs(LDU[P[1]][1]));
+  y[k] = 1;
+
+  ldubsolve3(V + (3 * 0 + 0), y, (T *)LDU, P);
+
+  memcpy(LDU, AA, sizeof(LDU));
+  LDU[0][0] -= S[2];
+  LDU[1][1] -= S[2];
+  LDU[2][2] -= S[2];
+
+  ldu3((T *)LDU, P);
+
+  y[0] = y[1] = y[2] = 0;
+  k = (fabs(LDU[P[1]][1]) <= fabs(LDU[P[0]][0]) ||
+       fabs(LDU[P[2]][2]) <= fabs(LDU[P[0]][0])) +
+      (fabs(LDU[P[2]][2]) <= fabs(LDU[P[0]][0]) &&
+       fabs(LDU[P[2]][2]) <= fabs(LDU[P[1]][1]));
+  y[k] = 1;
+
+  ldubsolve3(V + (3 * 2 + 0), y, (T *)LDU, P);
+
+  cross(V + (3 * 1 + 0), V + (3 * 2 + 0), V + (3 * 0 + 0));
+
+  k = (S[0] > thr) + (S[1] > thr) + (S[2] > thr);
+
+  switch (k) {
+  case 0:
+    memcpy(U, V, 9 * sizeof(T));
+    break;
+  case 1:
+    matvec3(U + (3 * 0 + 0), A, V + (3 * 0 + 0));
+
+    y[0] = y[1] = y[2] = 0;
+    k = (fabs(U[3 * 0 + 1]) <= fabs(U[3 * 0 + 0]) ||
+         fabs(U[3 * 0 + 2]) <= fabs(U[3 * 0 + 0])) +
+        (fabs(U[3 * 0 + 2]) <= fabs(U[3 * 0 + 0]) &&
+         fabs(U[3 * 0 + 2]) <= fabs(U[3 * 0 + 1]));
+    y[k] = 1;
+
+    cross(U + (3 * 1 + 0), y, U + (3 * 0 + 0));
+    cross(U + (3 * 2 + 0), U + (3 * 0 + 0), U + (3 * 1 + 0));
+    break;
+  case 2:
+    matvec3(U + (3 * 0 + 0), A, V + (3 * 0 + 0));
+    matvec3(U + (3 * 1 + 0), A, V + (3 * 1 + 0));
+
+    cross(U + (3 * 2 + 0), U + (3 * 0 + 0), U + (3 * 1 + 0));
+    break;
+  case 3:
+    matmul3(U, A, V);
+    break;
+  }
+
+  unit3(V + (3 * 0 + 0));
+  unit3(V + (3 * 1 + 0));
+  unit3(V + (3 * 2 + 0));
+
+  unit3(U + (3 * 0 + 0));
+  unit3(U + (3 * 1 + 0));
+  unit3(U + (3 * 2 + 0));
+
+  S[0] = (T)sqrt(S[0]);
+  S[1] = (T)sqrt(S[1]);
+  S[2] = (T)sqrt(S[2]);
+}
+
+template <typename T> void thomas(T *x, const T *a, const T *b, T *c, int n) {
+  T tmp;
+  int i;
+
+  c[0] /= b[0];
+  x[0] /= b[0];
+
+  for (i = 1; i < n; ++i) {
+    tmp = 1 / (b[i] - c[i - 1] * a[i]);
+    c[i] *= tmp;
+    x[i] = (x[i] - x[i - 1] * a[i]) * tmp;
+  }
+
+  for (i = n - 2; i >= 0; --i) {
+    x[i] -= c[i] * x[i + 1];
+  }
+}
+
+template <typename T> void spline(T *D, const T *y, int n) {
+  int i;
+  T *a, *b, *c;
+
+  a = new T[n];
+  b = new T[n];
+  c = new T[n];
+
+  if (n < 4) {
+    a[0] = 0;
+    b[0] = 2;
+    c[0] = 1;
+    D[0] = 3 * (y[1] - y[0]);
+
+    a[n - 1] = 1;
+    b[n - 1] = 2;
+    c[n - 1] = 0;
+    D[n - 1] = 3 * (y[n - 1] - y[n - 2]);
+  } else {
+    a[0] = 0;
+    b[0] = 2;
+    c[0] = 4;
+    D[0] = -5 * y[0] + 4 * y[1] + y[2];
+
+    a[n - 1] = 4;
+    b[n - 1] = 2;
+    c[n - 1] = 0;
+    D[n - 1] = 5 * y[n - 1] - 4 * y[n - 2] - y[n - 3];
+  }
+
+  for (i = 1; i < n - 1; ++i) {
+    a[i] = 1;
+    b[i] = 4;
+    c[i] = 1;
+    D[i] = 3 * (y[i + 1] - y[i - 1]);
+  }
+
+  thomas(D, a, b, c, n);
+
+  delete[] a;
+  delete[] b;
+  delete[] c;
+}
+
+template <typename T>
 inline void lookupspline(T &t, int &k, T dist, T len, int n) {
-	t = (n-1)*dist/len;
-	k = (int)floor(t);
+  t = (n - 1) * dist / len;
+  k = (int)floor(t);
 
-	k = (k > 0)*k;
-	k += (k > n-2)*(n-2-k);
+  k = (k > 0) * k;
+  k += (k > n - 2) * (n - 2 - k);
 
-	t -= k;
+  t -= k;
 }
 
-template<typename T>
-inline T evalspline(T t, const T D[2], const T y[2]) {
-	const T c[4] = { y[0], D[0], 3*(y[1]-y[0])-2*D[0]-D[1], 2*(y[0]-y[1])+D[0]+D[1] };
+template <typename T> inline T evalspline(T t, const T D[2], const T y[2]) {
+  const T c[4] = {y[0], D[0], 3 * (y[1] - y[0]) - 2 * D[0] - D[1],
+                  2 * (y[0] - y[1]) + D[0] + D[1]};
 
-	return t*(t*(t*c[3] + c[2]) + c[1]) + c[0];
+  return t * (t * (t * c[3] + c[2]) + c[1]) + c[0];
 }
 
-template<typename T>
-void interp2(T *Dy, const T *Z, int nx, int ny) {
+template <typename T> void interp2(T *Dy, const T *Z, int nx, int ny) {
 
-	for (int i = 0; i < ny; ++i) {
-		spline(Dy+(nx*i+0), Z+(nx*i+0), nx);
-	}
+  for (int i = 0; i < ny; ++i) {
+    spline(Dy + (nx * i + 0), Z + (nx * i + 0), nx);
+  }
 }
 
-template<typename T>
+template <typename T>
 void evalinterp2(T u, T *Dx, T *zx, const T *Dy, const T *Z, int nx, int ny) {
 
-	for (int i = 0; i < ny; ++i) {
-		zx[i] = evalspline(u, Dy+(nx*i+0), Z+(nx*i+0));
-	}
+  for (int i = 0; i < ny; ++i) {
+    zx[i] = evalspline(u, Dy + (nx * i + 0), Z + (nx * i + 0));
+  }
 
-	spline(Dx, zx, ny);
+  spline(Dx, zx, ny);
 }
 
 #endif // !SURFACE_H
diff --git a/src/UnitSquareImage.cpp b/src/UnitSquareImage.cpp
index c4e18605..7426e11f 100644
--- a/src/UnitSquareImage.cpp
+++ b/src/UnitSquareImage.cpp
@@ -6,48 +6,6 @@
 #include "ImageRegister.h"
 
 using namespace std;
-using namespace Rcpp;
-
-RcppExport SEXP find_grad_2D(SEXP dfdui, SEXP dfdvi, SEXP fi, SEXP ni, SEXP ti, SEXP di){
-    int n = as<int>(ni);
-    int t = as<int>(ti);
-    int d = as<int>(di);
-    NumericVector f(fi);
-    NumericVector dfdu(dfdui);
-    NumericVector dfdv(dfdvi);
-
-    findgrad2D(dfdu.begin(), dfdv.begin(), f.begin(), n, t, d);
-    List ret;
-    ret["dfdu"] = dfdu;
-    ret["dfdv"] = dfdv;
-    return(ret);
-}
-
-RcppExport SEXP check_cross(SEXP fi, SEXP ni, SEXP ti, SEXP Di){
-    int temp;
-    int n = as<int>(ni);
-    int t = as<int>(ti);
-    int D = as<int>(Di);
-    NumericVector f(fi);
-
-    temp = check_crossing(f.begin(), n, t, D);
-
-    return wrap(temp);
-}
-
-RcppExport SEXP find_phistar(SEXP wi, SEXP qi, SEXP bi, SEXP ni, SEXP ti, SEXP di, SEXP Ki){
-    int n = as<int>(ni);
-    int t = as<int>(ti);
-    int d = as<int>(di);
-    int K = as<int>(Ki);
-    NumericVector w(wi);
-    NumericVector q(qi);
-    NumericVector b(bi);
-
-    findphistar(w.begin(), q.begin(), b.begin(), n, t, d, K);
-
-    return(w);
-}
 
 void findgrad(double *dfdu, double *dfdv, const double *f, int n, int t) {
     int i, j, k, N = n*t;
@@ -533,7 +491,6 @@ int ReparamSurf(double *Fnew, double *gamnew, double *H,
     // compute initial energy
     Hdiff = 100;
     Calculate_Distance(H+iter,qt,qm,n,t,d);
-    Rprintf("iter %d, energy %f \n", iter, *(H+iter));
 
     // main iteration
     for (iter = 1; iter < itermax && Hdiff > tol; ++iter) {
@@ -567,8 +524,6 @@ int ReparamSurf(double *Fnew, double *gamnew, double *H,
         Calculate_Distance(H+iter,qt,qm,n,t,d);
         Hdiff = (H[iter-1]-H[iter])/H[iter-1];
 
-        Rprintf("iter %d, energy %f \n", iter, *(H+iter));
-
         // update iteration or break out
         if (H[iter] <= H[iter-1]) {
             cpyArray(gamold,gamnew,n*t*D);
@@ -704,4 +659,4 @@ void jacob_image(double *A, const double *F, int n, int t) {
 
     return;
 
-}
+}
\ No newline at end of file
diff --git a/src/UnitSquareImage.h b/src/UnitSquareImage.h
index f4e150b8..c144549f 100644
--- a/src/UnitSquareImage.h
+++ b/src/UnitSquareImage.h
@@ -29,4 +29,4 @@ int ReparamSurf(double *Fnew, double *gamnew, double *H,
 void InvtGamma(double *gaminv, const double *gam, const double *gamid, int n);
 void jacob_image(double *A, const double *F, int n, int t);
 
-#endif
+#endif
\ No newline at end of file
diff --git a/src/cUnitSquareImage.pxd b/src/cUnitSquareImage.pxd
new file mode 100644
index 00000000..481a3bb1
--- /dev/null
+++ b/src/cUnitSquareImage.pxd
@@ -0,0 +1,6 @@
+# cython: language_level=2
+from libcpp cimport bool
+
+cdef extern from "UnitSquareImage.h":
+    void findgrad2D(double *dfdu, double *dfdv, double *f, int n, int t, int d);
+    int check_crossing(double *f, int n, int t, int D);
diff --git a/src/imagecpp.cpp b/src/imagecpp.cpp
new file mode 100644
index 00000000..c7d14752
--- /dev/null
+++ b/src/imagecpp.cpp
@@ -0,0 +1,7761 @@
+/* Generated by Cython 0.29.33 */
+
+#ifndef PY_SSIZE_T_CLEAN
+#define PY_SSIZE_T_CLEAN
+#endif /* PY_SSIZE_T_CLEAN */
+#include "Python.h"
+#ifndef Py_PYTHON_H
+    #error Python headers needed to compile C extensions, please install development version of Python.
+#elif PY_VERSION_HEX < 0x02060000 || (0x03000000 <= PY_VERSION_HEX && PY_VERSION_HEX < 0x03030000)
+    #error Cython requires Python 2.6+ or Python 3.3+.
+#else
+#define CYTHON_ABI "0_29_33"
+#define CYTHON_HEX_VERSION 0x001D21F0
+#define CYTHON_FUTURE_DIVISION 0
+#include <stddef.h>
+#ifndef offsetof
+  #define offsetof(type, member) ( (size_t) & ((type*)0) -> member )
+#endif
+#if !defined(WIN32) && !defined(MS_WINDOWS)
+  #ifndef __stdcall
+    #define __stdcall
+  #endif
+  #ifndef __cdecl
+    #define __cdecl
+  #endif
+  #ifndef __fastcall
+    #define __fastcall
+  #endif
+#endif
+#ifndef DL_IMPORT
+  #define DL_IMPORT(t) t
+#endif
+#ifndef DL_EXPORT
+  #define DL_EXPORT(t) t
+#endif
+#define __PYX_COMMA ,
+#ifndef HAVE_LONG_LONG
+  #if PY_VERSION_HEX >= 0x02070000
+    #define HAVE_LONG_LONG
+  #endif
+#endif
+#ifndef PY_LONG_LONG
+  #define PY_LONG_LONG LONG_LONG
+#endif
+#ifndef Py_HUGE_VAL
+  #define Py_HUGE_VAL HUGE_VAL
+#endif
+#ifdef PYPY_VERSION
+  #define CYTHON_COMPILING_IN_PYPY 1
+  #define CYTHON_COMPILING_IN_PYSTON 0
+  #define CYTHON_COMPILING_IN_CPYTHON 0
+  #define CYTHON_COMPILING_IN_NOGIL 0
+  #undef CYTHON_USE_TYPE_SLOTS
+  #define CYTHON_USE_TYPE_SLOTS 0
+  #undef CYTHON_USE_PYTYPE_LOOKUP
+  #define CYTHON_USE_PYTYPE_LOOKUP 0
+  #if PY_VERSION_HEX < 0x03050000
+    #undef CYTHON_USE_ASYNC_SLOTS
+    #define CYTHON_USE_ASYNC_SLOTS 0
+  #elif !defined(CYTHON_USE_ASYNC_SLOTS)
+    #define CYTHON_USE_ASYNC_SLOTS 1
+  #endif
+  #undef CYTHON_USE_PYLIST_INTERNALS
+  #define CYTHON_USE_PYLIST_INTERNALS 0
+  #undef CYTHON_USE_UNICODE_INTERNALS
+  #define CYTHON_USE_UNICODE_INTERNALS 0
+  #undef CYTHON_USE_UNICODE_WRITER
+  #define CYTHON_USE_UNICODE_WRITER 0
+  #undef CYTHON_USE_PYLONG_INTERNALS
+  #define CYTHON_USE_PYLONG_INTERNALS 0
+  #undef CYTHON_AVOID_BORROWED_REFS
+  #define CYTHON_AVOID_BORROWED_REFS 1
+  #undef CYTHON_ASSUME_SAFE_MACROS
+  #define CYTHON_ASSUME_SAFE_MACROS 0
+  #undef CYTHON_UNPACK_METHODS
+  #define CYTHON_UNPACK_METHODS 0
+  #undef CYTHON_FAST_THREAD_STATE
+  #define CYTHON_FAST_THREAD_STATE 0
+  #undef CYTHON_FAST_PYCALL
+  #define CYTHON_FAST_PYCALL 0
+  #undef CYTHON_PEP489_MULTI_PHASE_INIT
+  #define CYTHON_PEP489_MULTI_PHASE_INIT 0
+  #undef CYTHON_USE_TP_FINALIZE
+  #define CYTHON_USE_TP_FINALIZE 0
+  #undef CYTHON_USE_DICT_VERSIONS
+  #define CYTHON_USE_DICT_VERSIONS 0
+  #undef CYTHON_USE_EXC_INFO_STACK
+  #define CYTHON_USE_EXC_INFO_STACK 0
+  #ifndef CYTHON_UPDATE_DESCRIPTOR_DOC
+    #define CYTHON_UPDATE_DESCRIPTOR_DOC 0
+  #endif
+#elif defined(PYSTON_VERSION)
+  #define CYTHON_COMPILING_IN_PYPY 0
+  #define CYTHON_COMPILING_IN_PYSTON 1
+  #define CYTHON_COMPILING_IN_CPYTHON 0
+  #define CYTHON_COMPILING_IN_NOGIL 0
+  #ifndef CYTHON_USE_TYPE_SLOTS
+    #define CYTHON_USE_TYPE_SLOTS 1
+  #endif
+  #undef CYTHON_USE_PYTYPE_LOOKUP
+  #define CYTHON_USE_PYTYPE_LOOKUP 0
+  #undef CYTHON_USE_ASYNC_SLOTS
+  #define CYTHON_USE_ASYNC_SLOTS 0
+  #undef CYTHON_USE_PYLIST_INTERNALS
+  #define CYTHON_USE_PYLIST_INTERNALS 0
+  #ifndef CYTHON_USE_UNICODE_INTERNALS
+    #define CYTHON_USE_UNICODE_INTERNALS 1
+  #endif
+  #undef CYTHON_USE_UNICODE_WRITER
+  #define CYTHON_USE_UNICODE_WRITER 0
+  #undef CYTHON_USE_PYLONG_INTERNALS
+  #define CYTHON_USE_PYLONG_INTERNALS 0
+  #ifndef CYTHON_AVOID_BORROWED_REFS
+    #define CYTHON_AVOID_BORROWED_REFS 0
+  #endif
+  #ifndef CYTHON_ASSUME_SAFE_MACROS
+    #define CYTHON_ASSUME_SAFE_MACROS 1
+  #endif
+  #ifndef CYTHON_UNPACK_METHODS
+    #define CYTHON_UNPACK_METHODS 1
+  #endif
+  #undef CYTHON_FAST_THREAD_STATE
+  #define CYTHON_FAST_THREAD_STATE 0
+  #undef CYTHON_FAST_PYCALL
+  #define CYTHON_FAST_PYCALL 0
+  #undef CYTHON_PEP489_MULTI_PHASE_INIT
+  #define CYTHON_PEP489_MULTI_PHASE_INIT 0
+  #undef CYTHON_USE_TP_FINALIZE
+  #define CYTHON_USE_TP_FINALIZE 0
+  #undef CYTHON_USE_DICT_VERSIONS
+  #define CYTHON_USE_DICT_VERSIONS 0
+  #undef CYTHON_USE_EXC_INFO_STACK
+  #define CYTHON_USE_EXC_INFO_STACK 0
+  #ifndef CYTHON_UPDATE_DESCRIPTOR_DOC
+    #define CYTHON_UPDATE_DESCRIPTOR_DOC 0
+  #endif
+#elif defined(PY_NOGIL)
+  #define CYTHON_COMPILING_IN_PYPY 0
+  #define CYTHON_COMPILING_IN_PYSTON 0
+  #define CYTHON_COMPILING_IN_CPYTHON 0
+  #define CYTHON_COMPILING_IN_NOGIL 1
+  #ifndef CYTHON_USE_TYPE_SLOTS
+    #define CYTHON_USE_TYPE_SLOTS 1
+  #endif
+  #undef CYTHON_USE_PYTYPE_LOOKUP
+  #define CYTHON_USE_PYTYPE_LOOKUP 0
+  #ifndef CYTHON_USE_ASYNC_SLOTS
+    #define CYTHON_USE_ASYNC_SLOTS 1
+  #endif
+  #undef CYTHON_USE_PYLIST_INTERNALS
+  #define CYTHON_USE_PYLIST_INTERNALS 0
+  #ifndef CYTHON_USE_UNICODE_INTERNALS
+    #define CYTHON_USE_UNICODE_INTERNALS 1
+  #endif
+  #undef CYTHON_USE_UNICODE_WRITER
+  #define CYTHON_USE_UNICODE_WRITER 0
+  #undef CYTHON_USE_PYLONG_INTERNALS
+  #define CYTHON_USE_PYLONG_INTERNALS 0
+  #ifndef CYTHON_AVOID_BORROWED_REFS
+    #define CYTHON_AVOID_BORROWED_REFS 0
+  #endif
+  #ifndef CYTHON_ASSUME_SAFE_MACROS
+    #define CYTHON_ASSUME_SAFE_MACROS 1
+  #endif
+  #ifndef CYTHON_UNPACK_METHODS
+    #define CYTHON_UNPACK_METHODS 1
+  #endif
+  #undef CYTHON_FAST_THREAD_STATE
+  #define CYTHON_FAST_THREAD_STATE 0
+  #undef CYTHON_FAST_PYCALL
+  #define CYTHON_FAST_PYCALL 0
+  #ifndef CYTHON_PEP489_MULTI_PHASE_INIT
+    #define CYTHON_PEP489_MULTI_PHASE_INIT 1
+  #endif
+  #ifndef CYTHON_USE_TP_FINALIZE
+    #define CYTHON_USE_TP_FINALIZE 1
+  #endif
+  #undef CYTHON_USE_DICT_VERSIONS
+  #define CYTHON_USE_DICT_VERSIONS 0
+  #undef CYTHON_USE_EXC_INFO_STACK
+  #define CYTHON_USE_EXC_INFO_STACK 0
+#else
+  #define CYTHON_COMPILING_IN_PYPY 0
+  #define CYTHON_COMPILING_IN_PYSTON 0
+  #define CYTHON_COMPILING_IN_CPYTHON 1
+  #define CYTHON_COMPILING_IN_NOGIL 0
+  #ifndef CYTHON_USE_TYPE_SLOTS
+    #define CYTHON_USE_TYPE_SLOTS 1
+  #endif
+  #if PY_VERSION_HEX < 0x02070000
+    #undef CYTHON_USE_PYTYPE_LOOKUP
+    #define CYTHON_USE_PYTYPE_LOOKUP 0
+  #elif !defined(CYTHON_USE_PYTYPE_LOOKUP)
+    #define CYTHON_USE_PYTYPE_LOOKUP 1
+  #endif
+  #if PY_MAJOR_VERSION < 3
+    #undef CYTHON_USE_ASYNC_SLOTS
+    #define CYTHON_USE_ASYNC_SLOTS 0
+  #elif !defined(CYTHON_USE_ASYNC_SLOTS)
+    #define CYTHON_USE_ASYNC_SLOTS 1
+  #endif
+  #if PY_VERSION_HEX < 0x02070000
+    #undef CYTHON_USE_PYLONG_INTERNALS
+    #define CYTHON_USE_PYLONG_INTERNALS 0
+  #elif !defined(CYTHON_USE_PYLONG_INTERNALS)
+    #define CYTHON_USE_PYLONG_INTERNALS 1
+  #endif
+  #ifndef CYTHON_USE_PYLIST_INTERNALS
+    #define CYTHON_USE_PYLIST_INTERNALS 1
+  #endif
+  #ifndef CYTHON_USE_UNICODE_INTERNALS
+    #define CYTHON_USE_UNICODE_INTERNALS 1
+  #endif
+  #if PY_VERSION_HEX < 0x030300F0 || PY_VERSION_HEX >= 0x030B00A2
+    #undef CYTHON_USE_UNICODE_WRITER
+    #define CYTHON_USE_UNICODE_WRITER 0
+  #elif !defined(CYTHON_USE_UNICODE_WRITER)
+    #define CYTHON_USE_UNICODE_WRITER 1
+  #endif
+  #ifndef CYTHON_AVOID_BORROWED_REFS
+    #define CYTHON_AVOID_BORROWED_REFS 0
+  #endif
+  #ifndef CYTHON_ASSUME_SAFE_MACROS
+    #define CYTHON_ASSUME_SAFE_MACROS 1
+  #endif
+  #ifndef CYTHON_UNPACK_METHODS
+    #define CYTHON_UNPACK_METHODS 1
+  #endif
+  #if PY_VERSION_HEX >= 0x030B00A4
+    #undef CYTHON_FAST_THREAD_STATE
+    #define CYTHON_FAST_THREAD_STATE 0
+  #elif !defined(CYTHON_FAST_THREAD_STATE)
+    #define CYTHON_FAST_THREAD_STATE 1
+  #endif
+  #ifndef CYTHON_FAST_PYCALL
+    #define CYTHON_FAST_PYCALL (PY_VERSION_HEX < 0x030A0000)
+  #endif
+  #ifndef CYTHON_PEP489_MULTI_PHASE_INIT
+    #define CYTHON_PEP489_MULTI_PHASE_INIT (PY_VERSION_HEX >= 0x03050000)
+  #endif
+  #ifndef CYTHON_USE_TP_FINALIZE
+    #define CYTHON_USE_TP_FINALIZE (PY_VERSION_HEX >= 0x030400a1)
+  #endif
+  #ifndef CYTHON_USE_DICT_VERSIONS
+    #define CYTHON_USE_DICT_VERSIONS (PY_VERSION_HEX >= 0x030600B1)
+  #endif
+  #if PY_VERSION_HEX >= 0x030B00A4
+    #undef CYTHON_USE_EXC_INFO_STACK
+    #define CYTHON_USE_EXC_INFO_STACK 0
+  #elif !defined(CYTHON_USE_EXC_INFO_STACK)
+    #define CYTHON_USE_EXC_INFO_STACK (PY_VERSION_HEX >= 0x030700A3)
+  #endif
+  #ifndef CYTHON_UPDATE_DESCRIPTOR_DOC
+    #define CYTHON_UPDATE_DESCRIPTOR_DOC 1
+  #endif
+#endif
+#if !defined(CYTHON_FAST_PYCCALL)
+#define CYTHON_FAST_PYCCALL  (CYTHON_FAST_PYCALL && PY_VERSION_HEX >= 0x030600B1)
+#endif
+#if CYTHON_USE_PYLONG_INTERNALS
+  #if PY_MAJOR_VERSION < 3
+    #include "longintrepr.h"
+  #endif
+  #undef SHIFT
+  #undef BASE
+  #undef MASK
+  #ifdef SIZEOF_VOID_P
+    enum { __pyx_check_sizeof_voidp = 1 / (int)(SIZEOF_VOID_P == sizeof(void*)) };
+  #endif
+#endif
+#ifndef __has_attribute
+  #define __has_attribute(x) 0
+#endif
+#ifndef __has_cpp_attribute
+  #define __has_cpp_attribute(x) 0
+#endif
+#ifndef CYTHON_RESTRICT
+  #if defined(__GNUC__)
+    #define CYTHON_RESTRICT __restrict__
+  #elif defined(_MSC_VER) && _MSC_VER >= 1400
+    #define CYTHON_RESTRICT __restrict
+  #elif defined (__STDC_VERSION__) && __STDC_VERSION__ >= 199901L
+    #define CYTHON_RESTRICT restrict
+  #else
+    #define CYTHON_RESTRICT
+  #endif
+#endif
+#ifndef CYTHON_UNUSED
+# if defined(__GNUC__)
+#   if !(defined(__cplusplus)) || (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 4))
+#     define CYTHON_UNUSED __attribute__ ((__unused__))
+#   else
+#     define CYTHON_UNUSED
+#   endif
+# elif defined(__ICC) || (defined(__INTEL_COMPILER) && !defined(_MSC_VER))
+#   define CYTHON_UNUSED __attribute__ ((__unused__))
+# else
+#   define CYTHON_UNUSED
+# endif
+#endif
+#ifndef CYTHON_MAYBE_UNUSED_VAR
+#  if defined(__cplusplus)
+     template<class T> void CYTHON_MAYBE_UNUSED_VAR( const T& ) { }
+#  else
+#    define CYTHON_MAYBE_UNUSED_VAR(x) (void)(x)
+#  endif
+#endif
+#ifndef CYTHON_NCP_UNUSED
+# if CYTHON_COMPILING_IN_CPYTHON
+#  define CYTHON_NCP_UNUSED
+# else
+#  define CYTHON_NCP_UNUSED CYTHON_UNUSED
+# endif
+#endif
+#define __Pyx_void_to_None(void_result) ((void)(void_result), Py_INCREF(Py_None), Py_None)
+#ifdef _MSC_VER
+    #ifndef _MSC_STDINT_H_
+        #if _MSC_VER < 1300
+           typedef unsigned char     uint8_t;
+           typedef unsigned int      uint32_t;
+        #else
+           typedef unsigned __int8   uint8_t;
+           typedef unsigned __int32  uint32_t;
+        #endif
+    #endif
+#else
+   #include <stdint.h>
+#endif
+#ifndef CYTHON_FALLTHROUGH
+  #if defined(__cplusplus) && __cplusplus >= 201103L
+    #if __has_cpp_attribute(fallthrough)
+      #define CYTHON_FALLTHROUGH [[fallthrough]]
+    #elif __has_cpp_attribute(clang::fallthrough)
+      #define CYTHON_FALLTHROUGH [[clang::fallthrough]]
+    #elif __has_cpp_attribute(gnu::fallthrough)
+      #define CYTHON_FALLTHROUGH [[gnu::fallthrough]]
+    #endif
+  #endif
+  #ifndef CYTHON_FALLTHROUGH
+    #if __has_attribute(fallthrough)
+      #define CYTHON_FALLTHROUGH __attribute__((fallthrough))
+    #else
+      #define CYTHON_FALLTHROUGH
+    #endif
+  #endif
+  #if defined(__clang__ ) && defined(__apple_build_version__)
+    #if __apple_build_version__ < 7000000
+      #undef  CYTHON_FALLTHROUGH
+      #define CYTHON_FALLTHROUGH
+    #endif
+  #endif
+#endif
+
+#ifndef __cplusplus
+  #error "Cython files generated with the C++ option must be compiled with a C++ compiler."
+#endif
+#ifndef CYTHON_INLINE
+  #if defined(__clang__)
+    #define CYTHON_INLINE __inline__ __attribute__ ((__unused__))
+  #else
+    #define CYTHON_INLINE inline
+  #endif
+#endif
+template<typename T>
+void __Pyx_call_destructor(T& x) {
+    x.~T();
+}
+template<typename T>
+class __Pyx_FakeReference {
+  public:
+    __Pyx_FakeReference() : ptr(NULL) { }
+    __Pyx_FakeReference(const T& ref) : ptr(const_cast<T*>(&ref)) { }
+    T *operator->() { return ptr; }
+    T *operator&() { return ptr; }
+    operator T&() { return *ptr; }
+    template<typename U> bool operator ==(U other) { return *ptr == other; }
+    template<typename U> bool operator !=(U other) { return *ptr != other; }
+  private:
+    T *ptr;
+};
+
+#if CYTHON_COMPILING_IN_PYPY && PY_VERSION_HEX < 0x02070600 && !defined(Py_OptimizeFlag)
+  #define Py_OptimizeFlag 0
+#endif
+#define __PYX_BUILD_PY_SSIZE_T "n"
+#define CYTHON_FORMAT_SSIZE_T "z"
+#if PY_MAJOR_VERSION < 3
+  #define __Pyx_BUILTIN_MODULE_NAME "__builtin__"
+  #define __Pyx_PyCode_New(a, k, l, s, f, code, c, n, v, fv, cell, fn, name, fline, lnos)\
+          PyCode_New(a+k, l, s, f, code, c, n, v, fv, cell, fn, name, fline, lnos)
+  #define __Pyx_DefaultClassType PyClass_Type
+#else
+  #define __Pyx_BUILTIN_MODULE_NAME "builtins"
+  #define __Pyx_DefaultClassType PyType_Type
+#if PY_VERSION_HEX >= 0x030B00A1
+    static CYTHON_INLINE PyCodeObject* __Pyx_PyCode_New(int a, int k, int l, int s, int f,
+                                                    PyObject *code, PyObject *c, PyObject* n, PyObject *v,
+                                                    PyObject *fv, PyObject *cell, PyObject* fn,
+                                                    PyObject *name, int fline, PyObject *lnos) {
+        PyObject *kwds=NULL, *argcount=NULL, *posonlyargcount=NULL, *kwonlyargcount=NULL;
+        PyObject *nlocals=NULL, *stacksize=NULL, *flags=NULL, *replace=NULL, *call_result=NULL, *empty=NULL;
+        const char *fn_cstr=NULL;
+        const char *name_cstr=NULL;
+        PyCodeObject* co=NULL;
+        PyObject *type, *value, *traceback;
+        PyErr_Fetch(&type, &value, &traceback);
+        if (!(kwds=PyDict_New())) goto end;
+        if (!(argcount=PyLong_FromLong(a))) goto end;
+        if (PyDict_SetItemString(kwds, "co_argcount", argcount) != 0) goto end;
+        if (!(posonlyargcount=PyLong_FromLong(0))) goto end;
+        if (PyDict_SetItemString(kwds, "co_posonlyargcount", posonlyargcount) != 0) goto end;
+        if (!(kwonlyargcount=PyLong_FromLong(k))) goto end;
+        if (PyDict_SetItemString(kwds, "co_kwonlyargcount", kwonlyargcount) != 0) goto end;
+        if (!(nlocals=PyLong_FromLong(l))) goto end;
+        if (PyDict_SetItemString(kwds, "co_nlocals", nlocals) != 0) goto end;
+        if (!(stacksize=PyLong_FromLong(s))) goto end;
+        if (PyDict_SetItemString(kwds, "co_stacksize", stacksize) != 0) goto end;
+        if (!(flags=PyLong_FromLong(f))) goto end;
+        if (PyDict_SetItemString(kwds, "co_flags", flags) != 0) goto end;
+        if (PyDict_SetItemString(kwds, "co_code", code) != 0) goto end;
+        if (PyDict_SetItemString(kwds, "co_consts", c) != 0) goto end;
+        if (PyDict_SetItemString(kwds, "co_names", n) != 0) goto end;
+        if (PyDict_SetItemString(kwds, "co_varnames", v) != 0) goto end;
+        if (PyDict_SetItemString(kwds, "co_freevars", fv) != 0) goto end;
+        if (PyDict_SetItemString(kwds, "co_cellvars", cell) != 0) goto end;
+        if (PyDict_SetItemString(kwds, "co_linetable", lnos) != 0) goto end;
+        if (!(fn_cstr=PyUnicode_AsUTF8AndSize(fn, NULL))) goto end;
+        if (!(name_cstr=PyUnicode_AsUTF8AndSize(name, NULL))) goto end;
+        if (!(co = PyCode_NewEmpty(fn_cstr, name_cstr, fline))) goto end;
+        if (!(replace = PyObject_GetAttrString((PyObject*)co, "replace"))) goto cleanup_code_too;
+        if (!(empty = PyTuple_New(0))) goto cleanup_code_too; // unfortunately __pyx_empty_tuple isn't available here
+        if (!(call_result = PyObject_Call(replace, empty, kwds))) goto cleanup_code_too;
+        Py_XDECREF((PyObject*)co);
+        co = (PyCodeObject*)call_result;
+        call_result = NULL;
+        if (0) {
+            cleanup_code_too:
+            Py_XDECREF((PyObject*)co);
+            co = NULL;
+        }
+        end:
+        Py_XDECREF(kwds);
+        Py_XDECREF(argcount);
+        Py_XDECREF(posonlyargcount);
+        Py_XDECREF(kwonlyargcount);
+        Py_XDECREF(nlocals);
+        Py_XDECREF(stacksize);
+        Py_XDECREF(replace);
+        Py_XDECREF(call_result);
+        Py_XDECREF(empty);
+        if (type) {
+            PyErr_Restore(type, value, traceback);
+        }
+        return co;
+    }
+#else
+  #define __Pyx_PyCode_New(a, k, l, s, f, code, c, n, v, fv, cell, fn, name, fline, lnos)\
+          PyCode_New(a, k, l, s, f, code, c, n, v, fv, cell, fn, name, fline, lnos)
+#endif
+  #define __Pyx_DefaultClassType PyType_Type
+#endif
+#ifndef Py_TPFLAGS_CHECKTYPES
+  #define Py_TPFLAGS_CHECKTYPES 0
+#endif
+#ifndef Py_TPFLAGS_HAVE_INDEX
+  #define Py_TPFLAGS_HAVE_INDEX 0
+#endif
+#ifndef Py_TPFLAGS_HAVE_NEWBUFFER
+  #define Py_TPFLAGS_HAVE_NEWBUFFER 0
+#endif
+#ifndef Py_TPFLAGS_HAVE_FINALIZE
+  #define Py_TPFLAGS_HAVE_FINALIZE 0
+#endif
+#ifndef METH_STACKLESS
+  #define METH_STACKLESS 0
+#endif
+#if PY_VERSION_HEX <= 0x030700A3 || !defined(METH_FASTCALL)
+  #ifndef METH_FASTCALL
+     #define METH_FASTCALL 0x80
+  #endif
+  typedef PyObject *(*__Pyx_PyCFunctionFast) (PyObject *self, PyObject *const *args, Py_ssize_t nargs);
+  typedef PyObject *(*__Pyx_PyCFunctionFastWithKeywords) (PyObject *self, PyObject *const *args,
+                                                          Py_ssize_t nargs, PyObject *kwnames);
+#else
+  #define __Pyx_PyCFunctionFast _PyCFunctionFast
+  #define __Pyx_PyCFunctionFastWithKeywords _PyCFunctionFastWithKeywords
+#endif
+#if CYTHON_FAST_PYCCALL
+#define __Pyx_PyFastCFunction_Check(func)\
+    ((PyCFunction_Check(func) && (METH_FASTCALL == (PyCFunction_GET_FLAGS(func) & ~(METH_CLASS | METH_STATIC | METH_COEXIST | METH_KEYWORDS | METH_STACKLESS)))))
+#else
+#define __Pyx_PyFastCFunction_Check(func) 0
+#endif
+#if CYTHON_COMPILING_IN_PYPY && !defined(PyObject_Malloc)
+  #define PyObject_Malloc(s)   PyMem_Malloc(s)
+  #define PyObject_Free(p)     PyMem_Free(p)
+  #define PyObject_Realloc(p)  PyMem_Realloc(p)
+#endif
+#if CYTHON_COMPILING_IN_CPYTHON && PY_VERSION_HEX < 0x030400A1
+  #define PyMem_RawMalloc(n)           PyMem_Malloc(n)
+  #define PyMem_RawRealloc(p, n)       PyMem_Realloc(p, n)
+  #define PyMem_RawFree(p)             PyMem_Free(p)
+#endif
+#if CYTHON_COMPILING_IN_PYSTON
+  #define __Pyx_PyCode_HasFreeVars(co)  PyCode_HasFreeVars(co)
+  #define __Pyx_PyFrame_SetLineNumber(frame, lineno) PyFrame_SetLineNumber(frame, lineno)
+#else
+  #define __Pyx_PyCode_HasFreeVars(co)  (PyCode_GetNumFree(co) > 0)
+  #define __Pyx_PyFrame_SetLineNumber(frame, lineno)  (frame)->f_lineno = (lineno)
+#endif
+#if !CYTHON_FAST_THREAD_STATE || PY_VERSION_HEX < 0x02070000
+  #define __Pyx_PyThreadState_Current PyThreadState_GET()
+#elif PY_VERSION_HEX >= 0x03060000
+  #define __Pyx_PyThreadState_Current _PyThreadState_UncheckedGet()
+#elif PY_VERSION_HEX >= 0x03000000
+  #define __Pyx_PyThreadState_Current PyThreadState_GET()
+#else
+  #define __Pyx_PyThreadState_Current _PyThreadState_Current
+#endif
+#if PY_VERSION_HEX < 0x030700A2 && !defined(PyThread_tss_create) && !defined(Py_tss_NEEDS_INIT)
+#include "pythread.h"
+#define Py_tss_NEEDS_INIT 0
+typedef int Py_tss_t;
+static CYTHON_INLINE int PyThread_tss_create(Py_tss_t *key) {
+  *key = PyThread_create_key();
+  return 0;
+}
+static CYTHON_INLINE Py_tss_t * PyThread_tss_alloc(void) {
+  Py_tss_t *key = (Py_tss_t *)PyObject_Malloc(sizeof(Py_tss_t));
+  *key = Py_tss_NEEDS_INIT;
+  return key;
+}
+static CYTHON_INLINE void PyThread_tss_free(Py_tss_t *key) {
+  PyObject_Free(key);
+}
+static CYTHON_INLINE int PyThread_tss_is_created(Py_tss_t *key) {
+  return *key != Py_tss_NEEDS_INIT;
+}
+static CYTHON_INLINE void PyThread_tss_delete(Py_tss_t *key) {
+  PyThread_delete_key(*key);
+  *key = Py_tss_NEEDS_INIT;
+}
+static CYTHON_INLINE int PyThread_tss_set(Py_tss_t *key, void *value) {
+  return PyThread_set_key_value(*key, value);
+}
+static CYTHON_INLINE void * PyThread_tss_get(Py_tss_t *key) {
+  return PyThread_get_key_value(*key);
+}
+#endif
+#if CYTHON_COMPILING_IN_CPYTHON || defined(_PyDict_NewPresized)
+#define __Pyx_PyDict_NewPresized(n)  ((n <= 8) ? PyDict_New() : _PyDict_NewPresized(n))
+#else
+#define __Pyx_PyDict_NewPresized(n)  PyDict_New()
+#endif
+#if PY_MAJOR_VERSION >= 3 || CYTHON_FUTURE_DIVISION
+  #define __Pyx_PyNumber_Divide(x,y)         PyNumber_TrueDivide(x,y)
+  #define __Pyx_PyNumber_InPlaceDivide(x,y)  PyNumber_InPlaceTrueDivide(x,y)
+#else
+  #define __Pyx_PyNumber_Divide(x,y)         PyNumber_Divide(x,y)
+  #define __Pyx_PyNumber_InPlaceDivide(x,y)  PyNumber_InPlaceDivide(x,y)
+#endif
+#if CYTHON_COMPILING_IN_CPYTHON && PY_VERSION_HEX >= 0x030500A1 && CYTHON_USE_UNICODE_INTERNALS
+#define __Pyx_PyDict_GetItemStr(dict, name)  _PyDict_GetItem_KnownHash(dict, name, ((PyASCIIObject *) name)->hash)
+#else
+#define __Pyx_PyDict_GetItemStr(dict, name)  PyDict_GetItem(dict, name)
+#endif
+#if PY_VERSION_HEX > 0x03030000 && defined(PyUnicode_KIND)
+  #define CYTHON_PEP393_ENABLED 1
+  #if PY_VERSION_HEX >= 0x030C0000
+    #define __Pyx_PyUnicode_READY(op)       (0)
+  #else
+    #define __Pyx_PyUnicode_READY(op)       (likely(PyUnicode_IS_READY(op)) ?\
+                                                0 : _PyUnicode_Ready((PyObject *)(op)))
+  #endif
+  #define __Pyx_PyUnicode_GET_LENGTH(u)   PyUnicode_GET_LENGTH(u)
+  #define __Pyx_PyUnicode_READ_CHAR(u, i) PyUnicode_READ_CHAR(u, i)
+  #define __Pyx_PyUnicode_MAX_CHAR_VALUE(u)   PyUnicode_MAX_CHAR_VALUE(u)
+  #define __Pyx_PyUnicode_KIND(u)         PyUnicode_KIND(u)
+  #define __Pyx_PyUnicode_DATA(u)         PyUnicode_DATA(u)
+  #define __Pyx_PyUnicode_READ(k, d, i)   PyUnicode_READ(k, d, i)
+  #define __Pyx_PyUnicode_WRITE(k, d, i, ch)  PyUnicode_WRITE(k, d, i, ch)
+  #if PY_VERSION_HEX >= 0x030C0000
+    #define __Pyx_PyUnicode_IS_TRUE(u)      (0 != PyUnicode_GET_LENGTH(u))
+  #else
+    #if CYTHON_COMPILING_IN_CPYTHON && PY_VERSION_HEX >= 0x03090000
+    #define __Pyx_PyUnicode_IS_TRUE(u)      (0 != (likely(PyUnicode_IS_READY(u)) ? PyUnicode_GET_LENGTH(u) : ((PyCompactUnicodeObject *)(u))->wstr_length))
+    #else
+    #define __Pyx_PyUnicode_IS_TRUE(u)      (0 != (likely(PyUnicode_IS_READY(u)) ? PyUnicode_GET_LENGTH(u) : PyUnicode_GET_SIZE(u)))
+    #endif
+  #endif
+#else
+  #define CYTHON_PEP393_ENABLED 0
+  #define PyUnicode_1BYTE_KIND  1
+  #define PyUnicode_2BYTE_KIND  2
+  #define PyUnicode_4BYTE_KIND  4
+  #define __Pyx_PyUnicode_READY(op)       (0)
+  #define __Pyx_PyUnicode_GET_LENGTH(u)   PyUnicode_GET_SIZE(u)
+  #define __Pyx_PyUnicode_READ_CHAR(u, i) ((Py_UCS4)(PyUnicode_AS_UNICODE(u)[i]))
+  #define __Pyx_PyUnicode_MAX_CHAR_VALUE(u)   ((sizeof(Py_UNICODE) == 2) ? 65535 : 1114111)
+  #define __Pyx_PyUnicode_KIND(u)         (sizeof(Py_UNICODE))
+  #define __Pyx_PyUnicode_DATA(u)         ((void*)PyUnicode_AS_UNICODE(u))
+  #define __Pyx_PyUnicode_READ(k, d, i)   ((void)(k), (Py_UCS4)(((Py_UNICODE*)d)[i]))
+  #define __Pyx_PyUnicode_WRITE(k, d, i, ch)  (((void)(k)), ((Py_UNICODE*)d)[i] = ch)
+  #define __Pyx_PyUnicode_IS_TRUE(u)      (0 != PyUnicode_GET_SIZE(u))
+#endif
+#if CYTHON_COMPILING_IN_PYPY
+  #define __Pyx_PyUnicode_Concat(a, b)      PyNumber_Add(a, b)
+  #define __Pyx_PyUnicode_ConcatSafe(a, b)  PyNumber_Add(a, b)
+#else
+  #define __Pyx_PyUnicode_Concat(a, b)      PyUnicode_Concat(a, b)
+  #define __Pyx_PyUnicode_ConcatSafe(a, b)  ((unlikely((a) == Py_None) || unlikely((b) == Py_None)) ?\
+      PyNumber_Add(a, b) : __Pyx_PyUnicode_Concat(a, b))
+#endif
+#if CYTHON_COMPILING_IN_PYPY && !defined(PyUnicode_Contains)
+  #define PyUnicode_Contains(u, s)  PySequence_Contains(u, s)
+#endif
+#if CYTHON_COMPILING_IN_PYPY && !defined(PyByteArray_Check)
+  #define PyByteArray_Check(obj)  PyObject_TypeCheck(obj, &PyByteArray_Type)
+#endif
+#if CYTHON_COMPILING_IN_PYPY && !defined(PyObject_Format)
+  #define PyObject_Format(obj, fmt)  PyObject_CallMethod(obj, "__format__", "O", fmt)
+#endif
+#define __Pyx_PyString_FormatSafe(a, b)   ((unlikely((a) == Py_None || (PyString_Check(b) && !PyString_CheckExact(b)))) ? PyNumber_Remainder(a, b) : __Pyx_PyString_Format(a, b))
+#define __Pyx_PyUnicode_FormatSafe(a, b)  ((unlikely((a) == Py_None || (PyUnicode_Check(b) && !PyUnicode_CheckExact(b)))) ? PyNumber_Remainder(a, b) : PyUnicode_Format(a, b))
+#if PY_MAJOR_VERSION >= 3
+  #define __Pyx_PyString_Format(a, b)  PyUnicode_Format(a, b)
+#else
+  #define __Pyx_PyString_Format(a, b)  PyString_Format(a, b)
+#endif
+#if PY_MAJOR_VERSION < 3 && !defined(PyObject_ASCII)
+  #define PyObject_ASCII(o)            PyObject_Repr(o)
+#endif
+#if PY_MAJOR_VERSION >= 3
+  #define PyBaseString_Type            PyUnicode_Type
+  #define PyStringObject               PyUnicodeObject
+  #define PyString_Type                PyUnicode_Type
+  #define PyString_Check               PyUnicode_Check
+  #define PyString_CheckExact          PyUnicode_CheckExact
+#ifndef PyObject_Unicode
+  #define PyObject_Unicode             PyObject_Str
+#endif
+#endif
+#if PY_MAJOR_VERSION >= 3
+  #define __Pyx_PyBaseString_Check(obj) PyUnicode_Check(obj)
+  #define __Pyx_PyBaseString_CheckExact(obj) PyUnicode_CheckExact(obj)
+#else
+  #define __Pyx_PyBaseString_Check(obj) (PyString_Check(obj) || PyUnicode_Check(obj))
+  #define __Pyx_PyBaseString_CheckExact(obj) (PyString_CheckExact(obj) || PyUnicode_CheckExact(obj))
+#endif
+#ifndef PySet_CheckExact
+  #define PySet_CheckExact(obj)        (Py_TYPE(obj) == &PySet_Type)
+#endif
+#if PY_VERSION_HEX >= 0x030900A4
+  #define __Pyx_SET_REFCNT(obj, refcnt) Py_SET_REFCNT(obj, refcnt)
+  #define __Pyx_SET_SIZE(obj, size) Py_SET_SIZE(obj, size)
+#else
+  #define __Pyx_SET_REFCNT(obj, refcnt) Py_REFCNT(obj) = (refcnt)
+  #define __Pyx_SET_SIZE(obj, size) Py_SIZE(obj) = (size)
+#endif
+#if CYTHON_ASSUME_SAFE_MACROS
+  #define __Pyx_PySequence_SIZE(seq)  Py_SIZE(seq)
+#else
+  #define __Pyx_PySequence_SIZE(seq)  PySequence_Size(seq)
+#endif
+#if PY_MAJOR_VERSION >= 3
+  #define PyIntObject                  PyLongObject
+  #define PyInt_Type                   PyLong_Type
+  #define PyInt_Check(op)              PyLong_Check(op)
+  #define PyInt_CheckExact(op)         PyLong_CheckExact(op)
+  #define PyInt_FromString             PyLong_FromString
+  #define PyInt_FromUnicode            PyLong_FromUnicode
+  #define PyInt_FromLong               PyLong_FromLong
+  #define PyInt_FromSize_t             PyLong_FromSize_t
+  #define PyInt_FromSsize_t            PyLong_FromSsize_t
+  #define PyInt_AsLong                 PyLong_AsLong
+  #define PyInt_AS_LONG                PyLong_AS_LONG
+  #define PyInt_AsSsize_t              PyLong_AsSsize_t
+  #define PyInt_AsUnsignedLongMask     PyLong_AsUnsignedLongMask
+  #define PyInt_AsUnsignedLongLongMask PyLong_AsUnsignedLongLongMask
+  #define PyNumber_Int                 PyNumber_Long
+#endif
+#if PY_MAJOR_VERSION >= 3
+  #define PyBoolObject                 PyLongObject
+#endif
+#if PY_MAJOR_VERSION >= 3 && CYTHON_COMPILING_IN_PYPY
+  #ifndef PyUnicode_InternFromString
+    #define PyUnicode_InternFromString(s) PyUnicode_FromString(s)
+  #endif
+#endif
+#if PY_VERSION_HEX < 0x030200A4
+  typedef long Py_hash_t;
+  #define __Pyx_PyInt_FromHash_t PyInt_FromLong
+  #define __Pyx_PyInt_AsHash_t   __Pyx_PyIndex_AsHash_t
+#else
+  #define __Pyx_PyInt_FromHash_t PyInt_FromSsize_t
+  #define __Pyx_PyInt_AsHash_t   __Pyx_PyIndex_AsSsize_t
+#endif
+#if PY_MAJOR_VERSION >= 3
+  #define __Pyx_PyMethod_New(func, self, klass) ((self) ? ((void)(klass), PyMethod_New(func, self)) : __Pyx_NewRef(func))
+#else
+  #define __Pyx_PyMethod_New(func, self, klass) PyMethod_New(func, self, klass)
+#endif
+#if CYTHON_USE_ASYNC_SLOTS
+  #if PY_VERSION_HEX >= 0x030500B1
+    #define __Pyx_PyAsyncMethodsStruct PyAsyncMethods
+    #define __Pyx_PyType_AsAsync(obj) (Py_TYPE(obj)->tp_as_async)
+  #else
+    #define __Pyx_PyType_AsAsync(obj) ((__Pyx_PyAsyncMethodsStruct*) (Py_TYPE(obj)->tp_reserved))
+  #endif
+#else
+  #define __Pyx_PyType_AsAsync(obj) NULL
+#endif
+#ifndef __Pyx_PyAsyncMethodsStruct
+    typedef struct {
+        unaryfunc am_await;
+        unaryfunc am_aiter;
+        unaryfunc am_anext;
+    } __Pyx_PyAsyncMethodsStruct;
+#endif
+
+#if defined(_WIN32) || defined(WIN32) || defined(MS_WINDOWS)
+  #if !defined(_USE_MATH_DEFINES)
+    #define _USE_MATH_DEFINES
+  #endif
+#endif
+#include <math.h>
+#ifdef NAN
+#define __PYX_NAN() ((float) NAN)
+#else
+static CYTHON_INLINE float __PYX_NAN() {
+  float value;
+  memset(&value, 0xFF, sizeof(value));
+  return value;
+}
+#endif
+#if defined(__CYGWIN__) && defined(_LDBL_EQ_DBL)
+#define __Pyx_truncl trunc
+#else
+#define __Pyx_truncl truncl
+#endif
+
+#define __PYX_MARK_ERR_POS(f_index, lineno) \
+    { __pyx_filename = __pyx_f[f_index]; (void)__pyx_filename; __pyx_lineno = lineno; (void)__pyx_lineno; __pyx_clineno = __LINE__; (void)__pyx_clineno; }
+#define __PYX_ERR(f_index, lineno, Ln_error) \
+    { __PYX_MARK_ERR_POS(f_index, lineno) goto Ln_error; }
+
+#ifndef __PYX_EXTERN_C
+  #ifdef __cplusplus
+    #define __PYX_EXTERN_C extern "C"
+  #else
+    #define __PYX_EXTERN_C extern
+  #endif
+#endif
+
+#define __PYX_HAVE__cimage
+#define __PYX_HAVE_API__cimage
+/* Early includes */
+#include "UnitSquareImage.h"
+#include <string.h>
+#include <stdio.h>
+#include "numpy/arrayobject.h"
+#include "numpy/ufuncobject.h"
+
+    /* NumPy API declarations from "numpy/__init__.pxd" */
+    
+#include "pythread.h"
+#ifdef _OPENMP
+#include <omp.h>
+#endif /* _OPENMP */
+
+#if defined(PYREX_WITHOUT_ASSERTIONS) && !defined(CYTHON_WITHOUT_ASSERTIONS)
+#define CYTHON_WITHOUT_ASSERTIONS
+#endif
+
+typedef struct {PyObject **p; const char *s; const Py_ssize_t n; const char* encoding;
+                const char is_unicode; const char is_str; const char intern; } __Pyx_StringTabEntry;
+
+#define __PYX_DEFAULT_STRING_ENCODING_IS_ASCII 0
+#define __PYX_DEFAULT_STRING_ENCODING_IS_UTF8 0
+#define __PYX_DEFAULT_STRING_ENCODING_IS_DEFAULT (PY_MAJOR_VERSION >= 3 && __PYX_DEFAULT_STRING_ENCODING_IS_UTF8)
+#define __PYX_DEFAULT_STRING_ENCODING ""
+#define __Pyx_PyObject_FromString __Pyx_PyBytes_FromString
+#define __Pyx_PyObject_FromStringAndSize __Pyx_PyBytes_FromStringAndSize
+#define __Pyx_uchar_cast(c) ((unsigned char)c)
+#define __Pyx_long_cast(x) ((long)x)
+#define __Pyx_fits_Py_ssize_t(v, type, is_signed)  (\
+    (sizeof(type) < sizeof(Py_ssize_t))  ||\
+    (sizeof(type) > sizeof(Py_ssize_t) &&\
+          likely(v < (type)PY_SSIZE_T_MAX ||\
+                 v == (type)PY_SSIZE_T_MAX)  &&\
+          (!is_signed || likely(v > (type)PY_SSIZE_T_MIN ||\
+                                v == (type)PY_SSIZE_T_MIN)))  ||\
+    (sizeof(type) == sizeof(Py_ssize_t) &&\
+          (is_signed || likely(v < (type)PY_SSIZE_T_MAX ||\
+                               v == (type)PY_SSIZE_T_MAX)))  )
+static CYTHON_INLINE int __Pyx_is_valid_index(Py_ssize_t i, Py_ssize_t limit) {
+    return (size_t) i < (size_t) limit;
+}
+#if defined (__cplusplus) && __cplusplus >= 201103L
+    #include <cstdlib>
+    #define __Pyx_sst_abs(value) std::abs(value)
+#elif SIZEOF_INT >= SIZEOF_SIZE_T
+    #define __Pyx_sst_abs(value) abs(value)
+#elif SIZEOF_LONG >= SIZEOF_SIZE_T
+    #define __Pyx_sst_abs(value) labs(value)
+#elif defined (_MSC_VER)
+    #define __Pyx_sst_abs(value) ((Py_ssize_t)_abs64(value))
+#elif defined (__STDC_VERSION__) && __STDC_VERSION__ >= 199901L
+    #define __Pyx_sst_abs(value) llabs(value)
+#elif defined (__GNUC__)
+    #define __Pyx_sst_abs(value) __builtin_llabs(value)
+#else
+    #define __Pyx_sst_abs(value) ((value<0) ? -value : value)
+#endif
+static CYTHON_INLINE const char* __Pyx_PyObject_AsString(PyObject*);
+static CYTHON_INLINE const char* __Pyx_PyObject_AsStringAndSize(PyObject*, Py_ssize_t* length);
+#define __Pyx_PyByteArray_FromString(s) PyByteArray_FromStringAndSize((const char*)s, strlen((const char*)s))
+#define __Pyx_PyByteArray_FromStringAndSize(s, l) PyByteArray_FromStringAndSize((const char*)s, l)
+#define __Pyx_PyBytes_FromString        PyBytes_FromString
+#define __Pyx_PyBytes_FromStringAndSize PyBytes_FromStringAndSize
+static CYTHON_INLINE PyObject* __Pyx_PyUnicode_FromString(const char*);
+#if PY_MAJOR_VERSION < 3
+    #define __Pyx_PyStr_FromString        __Pyx_PyBytes_FromString
+    #define __Pyx_PyStr_FromStringAndSize __Pyx_PyBytes_FromStringAndSize
+#else
+    #define __Pyx_PyStr_FromString        __Pyx_PyUnicode_FromString
+    #define __Pyx_PyStr_FromStringAndSize __Pyx_PyUnicode_FromStringAndSize
+#endif
+#define __Pyx_PyBytes_AsWritableString(s)     ((char*) PyBytes_AS_STRING(s))
+#define __Pyx_PyBytes_AsWritableSString(s)    ((signed char*) PyBytes_AS_STRING(s))
+#define __Pyx_PyBytes_AsWritableUString(s)    ((unsigned char*) PyBytes_AS_STRING(s))
+#define __Pyx_PyBytes_AsString(s)     ((const char*) PyBytes_AS_STRING(s))
+#define __Pyx_PyBytes_AsSString(s)    ((const signed char*) PyBytes_AS_STRING(s))
+#define __Pyx_PyBytes_AsUString(s)    ((const unsigned char*) PyBytes_AS_STRING(s))
+#define __Pyx_PyObject_AsWritableString(s)    ((char*) __Pyx_PyObject_AsString(s))
+#define __Pyx_PyObject_AsWritableSString(s)    ((signed char*) __Pyx_PyObject_AsString(s))
+#define __Pyx_PyObject_AsWritableUString(s)    ((unsigned char*) __Pyx_PyObject_AsString(s))
+#define __Pyx_PyObject_AsSString(s)    ((const signed char*) __Pyx_PyObject_AsString(s))
+#define __Pyx_PyObject_AsUString(s)    ((const unsigned char*) __Pyx_PyObject_AsString(s))
+#define __Pyx_PyObject_FromCString(s)  __Pyx_PyObject_FromString((const char*)s)
+#define __Pyx_PyBytes_FromCString(s)   __Pyx_PyBytes_FromString((const char*)s)
+#define __Pyx_PyByteArray_FromCString(s)   __Pyx_PyByteArray_FromString((const char*)s)
+#define __Pyx_PyStr_FromCString(s)     __Pyx_PyStr_FromString((const char*)s)
+#define __Pyx_PyUnicode_FromCString(s) __Pyx_PyUnicode_FromString((const char*)s)
+static CYTHON_INLINE size_t __Pyx_Py_UNICODE_strlen(const Py_UNICODE *u) {
+    const Py_UNICODE *u_end = u;
+    while (*u_end++) ;
+    return (size_t)(u_end - u - 1);
+}
+#define __Pyx_PyUnicode_FromUnicode(u)       PyUnicode_FromUnicode(u, __Pyx_Py_UNICODE_strlen(u))
+#define __Pyx_PyUnicode_FromUnicodeAndLength PyUnicode_FromUnicode
+#define __Pyx_PyUnicode_AsUnicode            PyUnicode_AsUnicode
+#define __Pyx_NewRef(obj) (Py_INCREF(obj), obj)
+#define __Pyx_Owned_Py_None(b) __Pyx_NewRef(Py_None)
+static CYTHON_INLINE PyObject * __Pyx_PyBool_FromLong(long b);
+static CYTHON_INLINE int __Pyx_PyObject_IsTrue(PyObject*);
+static CYTHON_INLINE int __Pyx_PyObject_IsTrueAndDecref(PyObject*);
+static CYTHON_INLINE PyObject* __Pyx_PyNumber_IntOrLong(PyObject* x);
+#define __Pyx_PySequence_Tuple(obj)\
+    (likely(PyTuple_CheckExact(obj)) ? __Pyx_NewRef(obj) : PySequence_Tuple(obj))
+static CYTHON_INLINE Py_ssize_t __Pyx_PyIndex_AsSsize_t(PyObject*);
+static CYTHON_INLINE PyObject * __Pyx_PyInt_FromSize_t(size_t);
+static CYTHON_INLINE Py_hash_t __Pyx_PyIndex_AsHash_t(PyObject*);
+#if CYTHON_ASSUME_SAFE_MACROS
+#define __pyx_PyFloat_AsDouble(x) (PyFloat_CheckExact(x) ? PyFloat_AS_DOUBLE(x) : PyFloat_AsDouble(x))
+#else
+#define __pyx_PyFloat_AsDouble(x) PyFloat_AsDouble(x)
+#endif
+#define __pyx_PyFloat_AsFloat(x) ((float) __pyx_PyFloat_AsDouble(x))
+#if PY_MAJOR_VERSION >= 3
+#define __Pyx_PyNumber_Int(x) (PyLong_CheckExact(x) ? __Pyx_NewRef(x) : PyNumber_Long(x))
+#else
+#define __Pyx_PyNumber_Int(x) (PyInt_CheckExact(x) ? __Pyx_NewRef(x) : PyNumber_Int(x))
+#endif
+#define __Pyx_PyNumber_Float(x) (PyFloat_CheckExact(x) ? __Pyx_NewRef(x) : PyNumber_Float(x))
+#if PY_MAJOR_VERSION < 3 && __PYX_DEFAULT_STRING_ENCODING_IS_ASCII
+static int __Pyx_sys_getdefaultencoding_not_ascii;
+static int __Pyx_init_sys_getdefaultencoding_params(void) {
+    PyObject* sys;
+    PyObject* default_encoding = NULL;
+    PyObject* ascii_chars_u = NULL;
+    PyObject* ascii_chars_b = NULL;
+    const char* default_encoding_c;
+    sys = PyImport_ImportModule("sys");
+    if (!sys) goto bad;
+    default_encoding = PyObject_CallMethod(sys, (char*) "getdefaultencoding", NULL);
+    Py_DECREF(sys);
+    if (!default_encoding) goto bad;
+    default_encoding_c = PyBytes_AsString(default_encoding);
+    if (!default_encoding_c) goto bad;
+    if (strcmp(default_encoding_c, "ascii") == 0) {
+        __Pyx_sys_getdefaultencoding_not_ascii = 0;
+    } else {
+        char ascii_chars[128];
+        int c;
+        for (c = 0; c < 128; c++) {
+            ascii_chars[c] = c;
+        }
+        __Pyx_sys_getdefaultencoding_not_ascii = 1;
+        ascii_chars_u = PyUnicode_DecodeASCII(ascii_chars, 128, NULL);
+        if (!ascii_chars_u) goto bad;
+        ascii_chars_b = PyUnicode_AsEncodedString(ascii_chars_u, default_encoding_c, NULL);
+        if (!ascii_chars_b || !PyBytes_Check(ascii_chars_b) || memcmp(ascii_chars, PyBytes_AS_STRING(ascii_chars_b), 128) != 0) {
+            PyErr_Format(
+                PyExc_ValueError,
+                "This module compiled with c_string_encoding=ascii, but default encoding '%.200s' is not a superset of ascii.",
+                default_encoding_c);
+            goto bad;
+        }
+        Py_DECREF(ascii_chars_u);
+        Py_DECREF(ascii_chars_b);
+    }
+    Py_DECREF(default_encoding);
+    return 0;
+bad:
+    Py_XDECREF(default_encoding);
+    Py_XDECREF(ascii_chars_u);
+    Py_XDECREF(ascii_chars_b);
+    return -1;
+}
+#endif
+#if __PYX_DEFAULT_STRING_ENCODING_IS_DEFAULT && PY_MAJOR_VERSION >= 3
+#define __Pyx_PyUnicode_FromStringAndSize(c_str, size) PyUnicode_DecodeUTF8(c_str, size, NULL)
+#else
+#define __Pyx_PyUnicode_FromStringAndSize(c_str, size) PyUnicode_Decode(c_str, size, __PYX_DEFAULT_STRING_ENCODING, NULL)
+#if __PYX_DEFAULT_STRING_ENCODING_IS_DEFAULT
+static char* __PYX_DEFAULT_STRING_ENCODING;
+static int __Pyx_init_sys_getdefaultencoding_params(void) {
+    PyObject* sys;
+    PyObject* default_encoding = NULL;
+    char* default_encoding_c;
+    sys = PyImport_ImportModule("sys");
+    if (!sys) goto bad;
+    default_encoding = PyObject_CallMethod(sys, (char*) (const char*) "getdefaultencoding", NULL);
+    Py_DECREF(sys);
+    if (!default_encoding) goto bad;
+    default_encoding_c = PyBytes_AsString(default_encoding);
+    if (!default_encoding_c) goto bad;
+    __PYX_DEFAULT_STRING_ENCODING = (char*) malloc(strlen(default_encoding_c) + 1);
+    if (!__PYX_DEFAULT_STRING_ENCODING) goto bad;
+    strcpy(__PYX_DEFAULT_STRING_ENCODING, default_encoding_c);
+    Py_DECREF(default_encoding);
+    return 0;
+bad:
+    Py_XDECREF(default_encoding);
+    return -1;
+}
+#endif
+#endif
+
+
+/* Test for GCC > 2.95 */
+#if defined(__GNUC__)     && (__GNUC__ > 2 || (__GNUC__ == 2 && (__GNUC_MINOR__ > 95)))
+  #define likely(x)   __builtin_expect(!!(x), 1)
+  #define unlikely(x) __builtin_expect(!!(x), 0)
+#else /* !__GNUC__ or GCC < 2.95 */
+  #define likely(x)   (x)
+  #define unlikely(x) (x)
+#endif /* __GNUC__ */
+static CYTHON_INLINE void __Pyx_pretend_to_initialize(void* ptr) { (void)ptr; }
+
+static PyObject *__pyx_m = NULL;
+static PyObject *__pyx_d;
+static PyObject *__pyx_b;
+static PyObject *__pyx_cython_runtime = NULL;
+static PyObject *__pyx_empty_tuple;
+static PyObject *__pyx_empty_bytes;
+static PyObject *__pyx_empty_unicode;
+static int __pyx_lineno;
+static int __pyx_clineno = 0;
+static const char * __pyx_cfilenm= __FILE__;
+static const char *__pyx_filename;
+
+/* Header.proto */
+#if !defined(CYTHON_CCOMPLEX)
+  #if defined(__cplusplus)
+    #define CYTHON_CCOMPLEX 1
+  #elif defined(_Complex_I)
+    #define CYTHON_CCOMPLEX 1
+  #else
+    #define CYTHON_CCOMPLEX 0
+  #endif
+#endif
+#if CYTHON_CCOMPLEX
+  #ifdef __cplusplus
+    #include <complex>
+  #else
+    #include <complex.h>
+  #endif
+#endif
+#if CYTHON_CCOMPLEX && !defined(__cplusplus) && defined(__sun__) && defined(__GNUC__)
+  #undef _Complex_I
+  #define _Complex_I 1.0fj
+#endif
+
+
+static const char *__pyx_f[] = {
+  "src/imagecpp.pyx",
+  "__init__.pxd",
+  "array.pxd",
+  "type.pxd",
+  "bool.pxd",
+  "complex.pxd",
+};
+/* BufferFormatStructs.proto */
+#define IS_UNSIGNED(type) (((type) -1) > 0)
+struct __Pyx_StructField_;
+#define __PYX_BUF_FLAGS_PACKED_STRUCT (1 << 0)
+typedef struct {
+  const char* name;
+  struct __Pyx_StructField_* fields;
+  size_t size;
+  size_t arraysize[8];
+  int ndim;
+  char typegroup;
+  char is_unsigned;
+  int flags;
+} __Pyx_TypeInfo;
+typedef struct __Pyx_StructField_ {
+  __Pyx_TypeInfo* type;
+  const char* name;
+  size_t offset;
+} __Pyx_StructField;
+typedef struct {
+  __Pyx_StructField* field;
+  size_t parent_offset;
+} __Pyx_BufFmt_StackElem;
+typedef struct {
+  __Pyx_StructField root;
+  __Pyx_BufFmt_StackElem* head;
+  size_t fmt_offset;
+  size_t new_count, enc_count;
+  size_t struct_alignment;
+  int is_complex;
+  char enc_type;
+  char new_packmode;
+  char enc_packmode;
+  char is_valid_array;
+} __Pyx_BufFmt_Context;
+
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":689
+ * # in Cython to enable them only on the right systems.
+ * 
+ * ctypedef npy_int8       int8_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_int16      int16_t
+ * ctypedef npy_int32      int32_t
+ */
+typedef npy_int8 __pyx_t_5numpy_int8_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":690
+ * 
+ * ctypedef npy_int8       int8_t
+ * ctypedef npy_int16      int16_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_int32      int32_t
+ * ctypedef npy_int64      int64_t
+ */
+typedef npy_int16 __pyx_t_5numpy_int16_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":691
+ * ctypedef npy_int8       int8_t
+ * ctypedef npy_int16      int16_t
+ * ctypedef npy_int32      int32_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_int64      int64_t
+ * #ctypedef npy_int96      int96_t
+ */
+typedef npy_int32 __pyx_t_5numpy_int32_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":692
+ * ctypedef npy_int16      int16_t
+ * ctypedef npy_int32      int32_t
+ * ctypedef npy_int64      int64_t             # <<<<<<<<<<<<<<
+ * #ctypedef npy_int96      int96_t
+ * #ctypedef npy_int128     int128_t
+ */
+typedef npy_int64 __pyx_t_5numpy_int64_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":696
+ * #ctypedef npy_int128     int128_t
+ * 
+ * ctypedef npy_uint8      uint8_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_uint16     uint16_t
+ * ctypedef npy_uint32     uint32_t
+ */
+typedef npy_uint8 __pyx_t_5numpy_uint8_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":697
+ * 
+ * ctypedef npy_uint8      uint8_t
+ * ctypedef npy_uint16     uint16_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_uint32     uint32_t
+ * ctypedef npy_uint64     uint64_t
+ */
+typedef npy_uint16 __pyx_t_5numpy_uint16_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":698
+ * ctypedef npy_uint8      uint8_t
+ * ctypedef npy_uint16     uint16_t
+ * ctypedef npy_uint32     uint32_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_uint64     uint64_t
+ * #ctypedef npy_uint96     uint96_t
+ */
+typedef npy_uint32 __pyx_t_5numpy_uint32_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":699
+ * ctypedef npy_uint16     uint16_t
+ * ctypedef npy_uint32     uint32_t
+ * ctypedef npy_uint64     uint64_t             # <<<<<<<<<<<<<<
+ * #ctypedef npy_uint96     uint96_t
+ * #ctypedef npy_uint128    uint128_t
+ */
+typedef npy_uint64 __pyx_t_5numpy_uint64_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":703
+ * #ctypedef npy_uint128    uint128_t
+ * 
+ * ctypedef npy_float32    float32_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_float64    float64_t
+ * #ctypedef npy_float80    float80_t
+ */
+typedef npy_float32 __pyx_t_5numpy_float32_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":704
+ * 
+ * ctypedef npy_float32    float32_t
+ * ctypedef npy_float64    float64_t             # <<<<<<<<<<<<<<
+ * #ctypedef npy_float80    float80_t
+ * #ctypedef npy_float128   float128_t
+ */
+typedef npy_float64 __pyx_t_5numpy_float64_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":713
+ * # The int types are mapped a bit surprising --
+ * # numpy.int corresponds to 'l' and numpy.long to 'q'
+ * ctypedef npy_long       int_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_longlong   long_t
+ * ctypedef npy_longlong   longlong_t
+ */
+typedef npy_long __pyx_t_5numpy_int_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":714
+ * # numpy.int corresponds to 'l' and numpy.long to 'q'
+ * ctypedef npy_long       int_t
+ * ctypedef npy_longlong   long_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_longlong   longlong_t
+ * 
+ */
+typedef npy_longlong __pyx_t_5numpy_long_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":715
+ * ctypedef npy_long       int_t
+ * ctypedef npy_longlong   long_t
+ * ctypedef npy_longlong   longlong_t             # <<<<<<<<<<<<<<
+ * 
+ * ctypedef npy_ulong      uint_t
+ */
+typedef npy_longlong __pyx_t_5numpy_longlong_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":717
+ * ctypedef npy_longlong   longlong_t
+ * 
+ * ctypedef npy_ulong      uint_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_ulonglong  ulong_t
+ * ctypedef npy_ulonglong  ulonglong_t
+ */
+typedef npy_ulong __pyx_t_5numpy_uint_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":718
+ * 
+ * ctypedef npy_ulong      uint_t
+ * ctypedef npy_ulonglong  ulong_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_ulonglong  ulonglong_t
+ * 
+ */
+typedef npy_ulonglong __pyx_t_5numpy_ulong_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":719
+ * ctypedef npy_ulong      uint_t
+ * ctypedef npy_ulonglong  ulong_t
+ * ctypedef npy_ulonglong  ulonglong_t             # <<<<<<<<<<<<<<
+ * 
+ * ctypedef npy_intp       intp_t
+ */
+typedef npy_ulonglong __pyx_t_5numpy_ulonglong_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":721
+ * ctypedef npy_ulonglong  ulonglong_t
+ * 
+ * ctypedef npy_intp       intp_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_uintp      uintp_t
+ * 
+ */
+typedef npy_intp __pyx_t_5numpy_intp_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":722
+ * 
+ * ctypedef npy_intp       intp_t
+ * ctypedef npy_uintp      uintp_t             # <<<<<<<<<<<<<<
+ * 
+ * ctypedef npy_double     float_t
+ */
+typedef npy_uintp __pyx_t_5numpy_uintp_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":724
+ * ctypedef npy_uintp      uintp_t
+ * 
+ * ctypedef npy_double     float_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_double     double_t
+ * ctypedef npy_longdouble longdouble_t
+ */
+typedef npy_double __pyx_t_5numpy_float_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":725
+ * 
+ * ctypedef npy_double     float_t
+ * ctypedef npy_double     double_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_longdouble longdouble_t
+ * 
+ */
+typedef npy_double __pyx_t_5numpy_double_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":726
+ * ctypedef npy_double     float_t
+ * ctypedef npy_double     double_t
+ * ctypedef npy_longdouble longdouble_t             # <<<<<<<<<<<<<<
+ * 
+ * ctypedef npy_cfloat      cfloat_t
+ */
+typedef npy_longdouble __pyx_t_5numpy_longdouble_t;
+/* Declarations.proto */
+#if CYTHON_CCOMPLEX
+  #ifdef __cplusplus
+    typedef ::std::complex< float > __pyx_t_float_complex;
+  #else
+    typedef float _Complex __pyx_t_float_complex;
+  #endif
+#else
+    typedef struct { float real, imag; } __pyx_t_float_complex;
+#endif
+static CYTHON_INLINE __pyx_t_float_complex __pyx_t_float_complex_from_parts(float, float);
+
+/* Declarations.proto */
+#if CYTHON_CCOMPLEX
+  #ifdef __cplusplus
+    typedef ::std::complex< double > __pyx_t_double_complex;
+  #else
+    typedef double _Complex __pyx_t_double_complex;
+  #endif
+#else
+    typedef struct { double real, imag; } __pyx_t_double_complex;
+#endif
+static CYTHON_INLINE __pyx_t_double_complex __pyx_t_double_complex_from_parts(double, double);
+
+
+/*--- Type declarations ---*/
+#ifndef _ARRAYARRAY_H
+struct arrayobject;
+typedef struct arrayobject arrayobject;
+#endif
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":728
+ * ctypedef npy_longdouble longdouble_t
+ * 
+ * ctypedef npy_cfloat      cfloat_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_cdouble     cdouble_t
+ * ctypedef npy_clongdouble clongdouble_t
+ */
+typedef npy_cfloat __pyx_t_5numpy_cfloat_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":729
+ * 
+ * ctypedef npy_cfloat      cfloat_t
+ * ctypedef npy_cdouble     cdouble_t             # <<<<<<<<<<<<<<
+ * ctypedef npy_clongdouble clongdouble_t
+ * 
+ */
+typedef npy_cdouble __pyx_t_5numpy_cdouble_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":730
+ * ctypedef npy_cfloat      cfloat_t
+ * ctypedef npy_cdouble     cdouble_t
+ * ctypedef npy_clongdouble clongdouble_t             # <<<<<<<<<<<<<<
+ * 
+ * ctypedef npy_cdouble     complex_t
+ */
+typedef npy_clongdouble __pyx_t_5numpy_clongdouble_t;
+
+/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":732
+ * ctypedef npy_clongdouble clongdouble_t
+ * 
+ * ctypedef npy_cdouble     complex_t             # <<<<<<<<<<<<<<
+ * 
+ * cdef inline object PyArray_MultiIterNew1(a):
+ */
+typedef npy_cdouble __pyx_t_5numpy_complex_t;
+
+/* --- Runtime support code (head) --- */
+/* Refnanny.proto */
+#ifndef CYTHON_REFNANNY
+  #define CYTHON_REFNANNY 0
+#endif
+#if CYTHON_REFNANNY
+  typedef struct {
+    void (*INCREF)(void*, PyObject*, int);
+    void (*DECREF)(void*, PyObject*, int);
+    void (*GOTREF)(void*, PyObject*, int);
+    void (*GIVEREF)(void*, PyObject*, int);
+    void* (*SetupContext)(const char*, int, const char*);
+    void (*FinishContext)(void**);
+  } __Pyx_RefNannyAPIStruct;
+  static __Pyx_RefNannyAPIStruct *__Pyx_RefNanny = NULL;
+  static __Pyx_RefNannyAPIStruct *__Pyx_RefNannyImportAPI(const char *modname);
+  #define __Pyx_RefNannyDeclarations void *__pyx_refnanny = NULL;
+#ifdef WITH_THREAD
+  #define __Pyx_RefNannySetupContext(name, acquire_gil)\
+          if (acquire_gil) {\
+              PyGILState_STATE __pyx_gilstate_save = PyGILState_Ensure();\
+              __pyx_refnanny = __Pyx_RefNanny->SetupContext((name), __LINE__, __FILE__);\
+              PyGILState_Release(__pyx_gilstate_save);\
+          } else {\
+              __pyx_refnanny = __Pyx_RefNanny->SetupContext((name), __LINE__, __FILE__);\
+          }
+#else
+  #define __Pyx_RefNannySetupContext(name, acquire_gil)\
+          __pyx_refnanny = __Pyx_RefNanny->SetupContext((name), __LINE__, __FILE__)
+#endif
+  #define __Pyx_RefNannyFinishContext()\
+          __Pyx_RefNanny->FinishContext(&__pyx_refnanny)
+  #define __Pyx_INCREF(r)  __Pyx_RefNanny->INCREF(__pyx_refnanny, (PyObject *)(r), __LINE__)
+  #define __Pyx_DECREF(r)  __Pyx_RefNanny->DECREF(__pyx_refnanny, (PyObject *)(r), __LINE__)
+  #define __Pyx_GOTREF(r)  __Pyx_RefNanny->GOTREF(__pyx_refnanny, (PyObject *)(r), __LINE__)
+  #define __Pyx_GIVEREF(r) __Pyx_RefNanny->GIVEREF(__pyx_refnanny, (PyObject *)(r), __LINE__)
+  #define __Pyx_XINCREF(r)  do { if((r) != NULL) {__Pyx_INCREF(r); }} while(0)
+  #define __Pyx_XDECREF(r)  do { if((r) != NULL) {__Pyx_DECREF(r); }} while(0)
+  #define __Pyx_XGOTREF(r)  do { if((r) != NULL) {__Pyx_GOTREF(r); }} while(0)
+  #define __Pyx_XGIVEREF(r) do { if((r) != NULL) {__Pyx_GIVEREF(r);}} while(0)
+#else
+  #define __Pyx_RefNannyDeclarations
+  #define __Pyx_RefNannySetupContext(name, acquire_gil)
+  #define __Pyx_RefNannyFinishContext()
+  #define __Pyx_INCREF(r) Py_INCREF(r)
+  #define __Pyx_DECREF(r) Py_DECREF(r)
+  #define __Pyx_GOTREF(r)
+  #define __Pyx_GIVEREF(r)
+  #define __Pyx_XINCREF(r) Py_XINCREF(r)
+  #define __Pyx_XDECREF(r) Py_XDECREF(r)
+  #define __Pyx_XGOTREF(r)
+  #define __Pyx_XGIVEREF(r)
+#endif
+#define __Pyx_XDECREF_SET(r, v) do {\
+        PyObject *tmp = (PyObject *) r;\
+        r = v; __Pyx_XDECREF(tmp);\
+    } while (0)
+#define __Pyx_DECREF_SET(r, v) do {\
+        PyObject *tmp = (PyObject *) r;\
+        r = v; __Pyx_DECREF(tmp);\
+    } while (0)
+#define __Pyx_CLEAR(r)    do { PyObject* tmp = ((PyObject*)(r)); r = NULL; __Pyx_DECREF(tmp);} while(0)
+#define __Pyx_XCLEAR(r)   do { if((r) != NULL) {PyObject* tmp = ((PyObject*)(r)); r = NULL; __Pyx_DECREF(tmp);}} while(0)
+
+/* ArgTypeTest.proto */
+#define __Pyx_ArgTypeTest(obj, type, none_allowed, name, exact)\
+    ((likely((Py_TYPE(obj) == type) | (none_allowed && (obj == Py_None)))) ? 1 :\
+        __Pyx__ArgTypeTest(obj, type, name, exact))
+static int __Pyx__ArgTypeTest(PyObject *obj, PyTypeObject *type, const char *name, int exact);
+
+/* IsLittleEndian.proto */
+static CYTHON_INLINE int __Pyx_Is_Little_Endian(void);
+
+/* BufferFormatCheck.proto */
+static const char* __Pyx_BufFmt_CheckString(__Pyx_BufFmt_Context* ctx, const char* ts);
+static void __Pyx_BufFmt_Init(__Pyx_BufFmt_Context* ctx,
+                              __Pyx_BufFmt_StackElem* stack,
+                              __Pyx_TypeInfo* type);
+
+/* BufferGetAndValidate.proto */
+#define __Pyx_GetBufferAndValidate(buf, obj, dtype, flags, nd, cast, stack)\
+    ((obj == Py_None || obj == NULL) ?\
+    (__Pyx_ZeroBuffer(buf), 0) :\
+    __Pyx__GetBufferAndValidate(buf, obj, dtype, flags, nd, cast, stack))
+static int  __Pyx__GetBufferAndValidate(Py_buffer* buf, PyObject* obj,
+    __Pyx_TypeInfo* dtype, int flags, int nd, int cast, __Pyx_BufFmt_StackElem* stack);
+static void __Pyx_ZeroBuffer(Py_buffer* buf);
+static CYTHON_INLINE void __Pyx_SafeReleaseBuffer(Py_buffer* info);
+static Py_ssize_t __Pyx_minusones[] = { -1, -1, -1, -1, -1, -1, -1, -1 };
+static Py_ssize_t __Pyx_zeros[] = { 0, 0, 0, 0, 0, 0, 0, 0 };
+
+/* PyObjectGetAttrStr.proto */
+#if CYTHON_USE_TYPE_SLOTS
+static CYTHON_INLINE PyObject* __Pyx_PyObject_GetAttrStr(PyObject* obj, PyObject* attr_name);
+#else
+#define __Pyx_PyObject_GetAttrStr(o,n) PyObject_GetAttr(o,n)
+#endif
+
+/* GetBuiltinName.proto */
+static PyObject *__Pyx_GetBuiltinName(PyObject *name);
+
+/* PyDictVersioning.proto */
+#if CYTHON_USE_DICT_VERSIONS && CYTHON_USE_TYPE_SLOTS
+#define __PYX_DICT_VERSION_INIT  ((PY_UINT64_T) -1)
+#define __PYX_GET_DICT_VERSION(dict)  (((PyDictObject*)(dict))->ma_version_tag)
+#define __PYX_UPDATE_DICT_CACHE(dict, value, cache_var, version_var)\
+    (version_var) = __PYX_GET_DICT_VERSION(dict);\
+    (cache_var) = (value);
+#define __PYX_PY_DICT_LOOKUP_IF_MODIFIED(VAR, DICT, LOOKUP) {\
+    static PY_UINT64_T __pyx_dict_version = 0;\
+    static PyObject *__pyx_dict_cached_value = NULL;\
+    if (likely(__PYX_GET_DICT_VERSION(DICT) == __pyx_dict_version)) {\
+        (VAR) = __pyx_dict_cached_value;\
+    } else {\
+        (VAR) = __pyx_dict_cached_value = (LOOKUP);\
+        __pyx_dict_version = __PYX_GET_DICT_VERSION(DICT);\
+    }\
+}
+static CYTHON_INLINE PY_UINT64_T __Pyx_get_tp_dict_version(PyObject *obj);
+static CYTHON_INLINE PY_UINT64_T __Pyx_get_object_dict_version(PyObject *obj);
+static CYTHON_INLINE int __Pyx_object_dict_version_matches(PyObject* obj, PY_UINT64_T tp_dict_version, PY_UINT64_T obj_dict_version);
+#else
+#define __PYX_GET_DICT_VERSION(dict)  (0)
+#define __PYX_UPDATE_DICT_CACHE(dict, value, cache_var, version_var)
+#define __PYX_PY_DICT_LOOKUP_IF_MODIFIED(VAR, DICT, LOOKUP)  (VAR) = (LOOKUP);
+#endif
+
+/* GetModuleGlobalName.proto */
+#if CYTHON_USE_DICT_VERSIONS
+#define __Pyx_GetModuleGlobalName(var, name)  do {\
+    static PY_UINT64_T __pyx_dict_version = 0;\
+    static PyObject *__pyx_dict_cached_value = NULL;\
+    (var) = (likely(__pyx_dict_version == __PYX_GET_DICT_VERSION(__pyx_d))) ?\
+        (likely(__pyx_dict_cached_value) ? __Pyx_NewRef(__pyx_dict_cached_value) : __Pyx_GetBuiltinName(name)) :\
+        __Pyx__GetModuleGlobalName(name, &__pyx_dict_version, &__pyx_dict_cached_value);\
+} while(0)
+#define __Pyx_GetModuleGlobalNameUncached(var, name)  do {\
+    PY_UINT64_T __pyx_dict_version;\
+    PyObject *__pyx_dict_cached_value;\
+    (var) = __Pyx__GetModuleGlobalName(name, &__pyx_dict_version, &__pyx_dict_cached_value);\
+} while(0)
+static PyObject *__Pyx__GetModuleGlobalName(PyObject *name, PY_UINT64_T *dict_version, PyObject **dict_cached_value);
+#else
+#define __Pyx_GetModuleGlobalName(var, name)  (var) = __Pyx__GetModuleGlobalName(name)
+#define __Pyx_GetModuleGlobalNameUncached(var, name)  (var) = __Pyx__GetModuleGlobalName(name)
+static CYTHON_INLINE PyObject *__Pyx__GetModuleGlobalName(PyObject *name);
+#endif
+
+/* PyCFunctionFastCall.proto */
+#if CYTHON_FAST_PYCCALL
+static CYTHON_INLINE PyObject *__Pyx_PyCFunction_FastCall(PyObject *func, PyObject **args, Py_ssize_t nargs);
+#else
+#define __Pyx_PyCFunction_FastCall(func, args, nargs)  (assert(0), NULL)
+#endif
+
+/* PyFunctionFastCall.proto */
+#if CYTHON_FAST_PYCALL
+#define __Pyx_PyFunction_FastCall(func, args, nargs)\
+    __Pyx_PyFunction_FastCallDict((func), (args), (nargs), NULL)
+#if 1 || PY_VERSION_HEX < 0x030600B1
+static PyObject *__Pyx_PyFunction_FastCallDict(PyObject *func, PyObject **args, Py_ssize_t nargs, PyObject *kwargs);
+#else
+#define __Pyx_PyFunction_FastCallDict(func, args, nargs, kwargs) _PyFunction_FastCallDict(func, args, nargs, kwargs)
+#endif
+#define __Pyx_BUILD_ASSERT_EXPR(cond)\
+    (sizeof(char [1 - 2*!(cond)]) - 1)
+#ifndef Py_MEMBER_SIZE
+#define Py_MEMBER_SIZE(type, member) sizeof(((type *)0)->member)
+#endif
+#if CYTHON_FAST_PYCALL
+  static size_t __pyx_pyframe_localsplus_offset = 0;
+  #include "frameobject.h"
+#if PY_VERSION_HEX >= 0x030b00a6
+  #ifndef Py_BUILD_CORE
+    #define Py_BUILD_CORE 1
+  #endif
+  #include "internal/pycore_frame.h"
+#endif
+  #define __Pxy_PyFrame_Initialize_Offsets()\
+    ((void)__Pyx_BUILD_ASSERT_EXPR(sizeof(PyFrameObject) == offsetof(PyFrameObject, f_localsplus) + Py_MEMBER_SIZE(PyFrameObject, f_localsplus)),\
+     (void)(__pyx_pyframe_localsplus_offset = ((size_t)PyFrame_Type.tp_basicsize) - Py_MEMBER_SIZE(PyFrameObject, f_localsplus)))
+  #define __Pyx_PyFrame_GetLocalsplus(frame)\
+    (assert(__pyx_pyframe_localsplus_offset), (PyObject **)(((char *)(frame)) + __pyx_pyframe_localsplus_offset))
+#endif // CYTHON_FAST_PYCALL
+#endif
+
+/* PyObjectCall.proto */
+#if CYTHON_COMPILING_IN_CPYTHON
+static CYTHON_INLINE PyObject* __Pyx_PyObject_Call(PyObject *func, PyObject *arg, PyObject *kw);
+#else
+#define __Pyx_PyObject_Call(func, arg, kw) PyObject_Call(func, arg, kw)
+#endif
+
+/* PyObjectCall2Args.proto */
+static CYTHON_UNUSED PyObject* __Pyx_PyObject_Call2Args(PyObject* function, PyObject* arg1, PyObject* arg2);
+
+/* PyObjectCallMethO.proto */
+#if CYTHON_COMPILING_IN_CPYTHON
+static CYTHON_INLINE PyObject* __Pyx_PyObject_CallMethO(PyObject *func, PyObject *arg);
+#endif
+
+/* PyObjectCallOneArg.proto */
+static CYTHON_INLINE PyObject* __Pyx_PyObject_CallOneArg(PyObject *func, PyObject *arg);
+
+/* ExtTypeTest.proto */
+static CYTHON_INLINE int __Pyx_TypeTest(PyObject *obj, PyTypeObject *type);
+
+/* BufferFallbackError.proto */
+static void __Pyx_RaiseBufferFallbackError(void);
+
+/* BufferIndexError.proto */
+static void __Pyx_RaiseBufferIndexError(int axis);
+
+#define __Pyx_BufPtrCContig1d(type, buf, i0, s0) ((type)buf + i0)
+/* PyThreadStateGet.proto */
+#if CYTHON_FAST_THREAD_STATE
+#define __Pyx_PyThreadState_declare  PyThreadState *__pyx_tstate;
+#define __Pyx_PyThreadState_assign  __pyx_tstate = __Pyx_PyThreadState_Current;
+#define __Pyx_PyErr_Occurred()  __pyx_tstate->curexc_type
+#else
+#define __Pyx_PyThreadState_declare
+#define __Pyx_PyThreadState_assign
+#define __Pyx_PyErr_Occurred()  PyErr_Occurred()
+#endif
+
+/* PyErrFetchRestore.proto */
+#if CYTHON_FAST_THREAD_STATE
+#define __Pyx_PyErr_Clear() __Pyx_ErrRestore(NULL, NULL, NULL)
+#define __Pyx_ErrRestoreWithState(type, value, tb)  __Pyx_ErrRestoreInState(PyThreadState_GET(), type, value, tb)
+#define __Pyx_ErrFetchWithState(type, value, tb)    __Pyx_ErrFetchInState(PyThreadState_GET(), type, value, tb)
+#define __Pyx_ErrRestore(type, value, tb)  __Pyx_ErrRestoreInState(__pyx_tstate, type, value, tb)
+#define __Pyx_ErrFetch(type, value, tb)    __Pyx_ErrFetchInState(__pyx_tstate, type, value, tb)
+static CYTHON_INLINE void __Pyx_ErrRestoreInState(PyThreadState *tstate, PyObject *type, PyObject *value, PyObject *tb);
+static CYTHON_INLINE void __Pyx_ErrFetchInState(PyThreadState *tstate, PyObject **type, PyObject **value, PyObject **tb);
+#if CYTHON_COMPILING_IN_CPYTHON
+#define __Pyx_PyErr_SetNone(exc) (Py_INCREF(exc), __Pyx_ErrRestore((exc), NULL, NULL))
+#else
+#define __Pyx_PyErr_SetNone(exc) PyErr_SetNone(exc)
+#endif
+#else
+#define __Pyx_PyErr_Clear() PyErr_Clear()
+#define __Pyx_PyErr_SetNone(exc) PyErr_SetNone(exc)
+#define __Pyx_ErrRestoreWithState(type, value, tb)  PyErr_Restore(type, value, tb)
+#define __Pyx_ErrFetchWithState(type, value, tb)  PyErr_Fetch(type, value, tb)
+#define __Pyx_ErrRestoreInState(tstate, type, value, tb)  PyErr_Restore(type, value, tb)
+#define __Pyx_ErrFetchInState(tstate, type, value, tb)  PyErr_Fetch(type, value, tb)
+#define __Pyx_ErrRestore(type, value, tb)  PyErr_Restore(type, value, tb)
+#define __Pyx_ErrFetch(type, value, tb)  PyErr_Fetch(type, value, tb)
+#endif
+
+/* GetTopmostException.proto */
+#if CYTHON_USE_EXC_INFO_STACK
+static _PyErr_StackItem * __Pyx_PyErr_GetTopmostException(PyThreadState *tstate);
+#endif
+
+/* SaveResetException.proto */
+#if CYTHON_FAST_THREAD_STATE
+#define __Pyx_ExceptionSave(type, value, tb)  __Pyx__ExceptionSave(__pyx_tstate, type, value, tb)
+static CYTHON_INLINE void __Pyx__ExceptionSave(PyThreadState *tstate, PyObject **type, PyObject **value, PyObject **tb);
+#define __Pyx_ExceptionReset(type, value, tb)  __Pyx__ExceptionReset(__pyx_tstate, type, value, tb)
+static CYTHON_INLINE void __Pyx__ExceptionReset(PyThreadState *tstate, PyObject *type, PyObject *value, PyObject *tb);
+#else
+#define __Pyx_ExceptionSave(type, value, tb)   PyErr_GetExcInfo(type, value, tb)
+#define __Pyx_ExceptionReset(type, value, tb)  PyErr_SetExcInfo(type, value, tb)
+#endif
+
+/* PyErrExceptionMatches.proto */
+#if CYTHON_FAST_THREAD_STATE
+#define __Pyx_PyErr_ExceptionMatches(err) __Pyx_PyErr_ExceptionMatchesInState(__pyx_tstate, err)
+static CYTHON_INLINE int __Pyx_PyErr_ExceptionMatchesInState(PyThreadState* tstate, PyObject* err);
+#else
+#define __Pyx_PyErr_ExceptionMatches(err)  PyErr_ExceptionMatches(err)
+#endif
+
+/* GetException.proto */
+#if CYTHON_FAST_THREAD_STATE
+#define __Pyx_GetException(type, value, tb)  __Pyx__GetException(__pyx_tstate, type, value, tb)
+static int __Pyx__GetException(PyThreadState *tstate, PyObject **type, PyObject **value, PyObject **tb);
+#else
+static int __Pyx_GetException(PyObject **type, PyObject **value, PyObject **tb);
+#endif
+
+/* RaiseException.proto */
+static void __Pyx_Raise(PyObject *type, PyObject *value, PyObject *tb, PyObject *cause);
+
+/* TypeImport.proto */
+#ifndef __PYX_HAVE_RT_ImportType_proto
+#define __PYX_HAVE_RT_ImportType_proto
+enum __Pyx_ImportType_CheckSize {
+   __Pyx_ImportType_CheckSize_Error = 0,
+   __Pyx_ImportType_CheckSize_Warn = 1,
+   __Pyx_ImportType_CheckSize_Ignore = 2
+};
+static PyTypeObject *__Pyx_ImportType(PyObject* module, const char *module_name, const char *class_name, size_t size, enum __Pyx_ImportType_CheckSize check_size);
+#endif
+
+/* Import.proto */
+static PyObject *__Pyx_Import(PyObject *name, PyObject *from_list, int level);
+
+/* CLineInTraceback.proto */
+#ifdef CYTHON_CLINE_IN_TRACEBACK
+#define __Pyx_CLineForTraceback(tstate, c_line)  (((CYTHON_CLINE_IN_TRACEBACK)) ? c_line : 0)
+#else
+static int __Pyx_CLineForTraceback(PyThreadState *tstate, int c_line);
+#endif
+
+/* CodeObjectCache.proto */
+typedef struct {
+    PyCodeObject* code_object;
+    int code_line;
+} __Pyx_CodeObjectCacheEntry;
+struct __Pyx_CodeObjectCache {
+    int count;
+    int max_count;
+    __Pyx_CodeObjectCacheEntry* entries;
+};
+static struct __Pyx_CodeObjectCache __pyx_code_cache = {0,0,NULL};
+static int __pyx_bisect_code_objects(__Pyx_CodeObjectCacheEntry* entries, int count, int code_line);
+static PyCodeObject *__pyx_find_code_object(int code_line);
+static void __pyx_insert_code_object(int code_line, PyCodeObject* code_object);
+
+/* AddTraceback.proto */
+static void __Pyx_AddTraceback(const char *funcname, int c_line,
+                               int py_line, const char *filename);
+
+/* ArrayAPI.proto */
+#ifndef _ARRAYARRAY_H
+#define _ARRAYARRAY_H
+typedef struct arraydescr {
+    int typecode;
+    int itemsize;
+    PyObject * (*getitem)(struct arrayobject *, Py_ssize_t);
+    int (*setitem)(struct arrayobject *, Py_ssize_t, PyObject *);
+#if PY_MAJOR_VERSION >= 3
+    char *formats;
+#endif
+} arraydescr;
+struct arrayobject {
+    PyObject_HEAD
+    Py_ssize_t ob_size;
+    union {
+        char *ob_item;
+        float *as_floats;
+        double *as_doubles;
+        int *as_ints;
+        unsigned int *as_uints;
+        unsigned char *as_uchars;
+        signed char *as_schars;
+        char *as_chars;
+        unsigned long *as_ulongs;
+        long *as_longs;
+#if PY_MAJOR_VERSION >= 3
+        unsigned long long *as_ulonglongs;
+        long long *as_longlongs;
+#endif
+        short *as_shorts;
+        unsigned short *as_ushorts;
+        Py_UNICODE *as_pyunicodes;
+        void *as_voidptr;
+    } data;
+    Py_ssize_t allocated;
+    struct arraydescr *ob_descr;
+    PyObject *weakreflist;
+#if PY_MAJOR_VERSION >= 3
+        int ob_exports;
+#endif
+};
+#ifndef NO_NEWARRAY_INLINE
+static CYTHON_INLINE PyObject * newarrayobject(PyTypeObject *type, Py_ssize_t size,
+    struct arraydescr *descr) {
+    arrayobject *op;
+    size_t nbytes;
+    if (size < 0) {
+        PyErr_BadInternalCall();
+        return NULL;
+    }
+    nbytes = size * descr->itemsize;
+    if (nbytes / descr->itemsize != (size_t)size) {
+        return PyErr_NoMemory();
+    }
+    op = (arrayobject *) type->tp_alloc(type, 0);
+    if (op == NULL) {
+        return NULL;
+    }
+    op->ob_descr = descr;
+    op->allocated = size;
+    op->weakreflist = NULL;
+    __Pyx_SET_SIZE(op, size);
+    if (size <= 0) {
+        op->data.ob_item = NULL;
+    }
+    else {
+        op->data.ob_item = PyMem_NEW(char, nbytes);
+        if (op->data.ob_item == NULL) {
+            Py_DECREF(op);
+            return PyErr_NoMemory();
+        }
+    }
+    return (PyObject *) op;
+}
+#else
+PyObject* newarrayobject(PyTypeObject *type, Py_ssize_t size,
+    struct arraydescr *descr);
+#endif
+static CYTHON_INLINE int resize(arrayobject *self, Py_ssize_t n) {
+    void *items = (void*) self->data.ob_item;
+    PyMem_Resize(items, char, (size_t)(n * self->ob_descr->itemsize));
+    if (items == NULL) {
+        PyErr_NoMemory();
+        return -1;
+    }
+    self->data.ob_item = (char*) items;
+    __Pyx_SET_SIZE(self, n);
+    self->allocated = n;
+    return 0;
+}
+static CYTHON_INLINE int resize_smart(arrayobject *self, Py_ssize_t n) {
+    void *items = (void*) self->data.ob_item;
+    Py_ssize_t newsize;
+    if (n < self->allocated && n*4 > self->allocated) {
+        __Pyx_SET_SIZE(self, n);
+        return 0;
+    }
+    newsize = n + (n / 2) + 1;
+    if (newsize <= n) {
+        PyErr_NoMemory();
+        return -1;
+    }
+    PyMem_Resize(items, char, (size_t)(newsize * self->ob_descr->itemsize));
+    if (items == NULL) {
+        PyErr_NoMemory();
+        return -1;
+    }
+    self->data.ob_item = (char*) items;
+    __Pyx_SET_SIZE(self, n);
+    self->allocated = newsize;
+    return 0;
+}
+#endif
+
+/* BufferStructDeclare.proto */
+typedef struct {
+  Py_ssize_t shape, strides, suboffsets;
+} __Pyx_Buf_DimInfo;
+typedef struct {
+  size_t refcount;
+  Py_buffer pybuffer;
+} __Pyx_Buffer;
+typedef struct {
+  __Pyx_Buffer *rcbuffer;
+  char *data;
+  __Pyx_Buf_DimInfo diminfo[8];
+} __Pyx_LocalBuf_ND;
+
+#if PY_MAJOR_VERSION < 3
+    static int __Pyx_GetBuffer(PyObject *obj, Py_buffer *view, int flags);
+    static void __Pyx_ReleaseBuffer(Py_buffer *view);
+#else
+    #define __Pyx_GetBuffer PyObject_GetBuffer
+    #define __Pyx_ReleaseBuffer PyBuffer_Release
+#endif
+
+
+/* GCCDiagnostics.proto */
+#if defined(__GNUC__) && (__GNUC__ > 4 || (__GNUC__ == 4 && __GNUC_MINOR__ >= 6))
+#define __Pyx_HAS_GCC_DIAGNOSTIC
+#endif
+
+/* RealImag.proto */
+#if CYTHON_CCOMPLEX
+  #ifdef __cplusplus
+    #define __Pyx_CREAL(z) ((z).real())
+    #define __Pyx_CIMAG(z) ((z).imag())
+  #else
+    #define __Pyx_CREAL(z) (__real__(z))
+    #define __Pyx_CIMAG(z) (__imag__(z))
+  #endif
+#else
+    #define __Pyx_CREAL(z) ((z).real)
+    #define __Pyx_CIMAG(z) ((z).imag)
+#endif
+#if defined(__cplusplus) && CYTHON_CCOMPLEX\
+        && (defined(_WIN32) || defined(__clang__) || (defined(__GNUC__) && (__GNUC__ >= 5 || __GNUC__ == 4 && __GNUC_MINOR__ >= 4 )) || __cplusplus >= 201103)
+    #define __Pyx_SET_CREAL(z,x) ((z).real(x))
+    #define __Pyx_SET_CIMAG(z,y) ((z).imag(y))
+#else
+    #define __Pyx_SET_CREAL(z,x) __Pyx_CREAL(z) = (x)
+    #define __Pyx_SET_CIMAG(z,y) __Pyx_CIMAG(z) = (y)
+#endif
+
+/* Arithmetic.proto */
+#if CYTHON_CCOMPLEX
+    #define __Pyx_c_eq_float(a, b)   ((a)==(b))
+    #define __Pyx_c_sum_float(a, b)  ((a)+(b))
+    #define __Pyx_c_diff_float(a, b) ((a)-(b))
+    #define __Pyx_c_prod_float(a, b) ((a)*(b))
+    #define __Pyx_c_quot_float(a, b) ((a)/(b))
+    #define __Pyx_c_neg_float(a)     (-(a))
+  #ifdef __cplusplus
+    #define __Pyx_c_is_zero_float(z) ((z)==(float)0)
+    #define __Pyx_c_conj_float(z)    (::std::conj(z))
+    #if 1
+        #define __Pyx_c_abs_float(z)     (::std::abs(z))
+        #define __Pyx_c_pow_float(a, b)  (::std::pow(a, b))
+    #endif
+  #else
+    #define __Pyx_c_is_zero_float(z) ((z)==0)
+    #define __Pyx_c_conj_float(z)    (conjf(z))
+    #if 1
+        #define __Pyx_c_abs_float(z)     (cabsf(z))
+        #define __Pyx_c_pow_float(a, b)  (cpowf(a, b))
+    #endif
+ #endif
+#else
+    static CYTHON_INLINE int __Pyx_c_eq_float(__pyx_t_float_complex, __pyx_t_float_complex);
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_sum_float(__pyx_t_float_complex, __pyx_t_float_complex);
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_diff_float(__pyx_t_float_complex, __pyx_t_float_complex);
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_prod_float(__pyx_t_float_complex, __pyx_t_float_complex);
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_quot_float(__pyx_t_float_complex, __pyx_t_float_complex);
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_neg_float(__pyx_t_float_complex);
+    static CYTHON_INLINE int __Pyx_c_is_zero_float(__pyx_t_float_complex);
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_conj_float(__pyx_t_float_complex);
+    #if 1
+        static CYTHON_INLINE float __Pyx_c_abs_float(__pyx_t_float_complex);
+        static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_pow_float(__pyx_t_float_complex, __pyx_t_float_complex);
+    #endif
+#endif
+
+/* Arithmetic.proto */
+#if CYTHON_CCOMPLEX
+    #define __Pyx_c_eq_double(a, b)   ((a)==(b))
+    #define __Pyx_c_sum_double(a, b)  ((a)+(b))
+    #define __Pyx_c_diff_double(a, b) ((a)-(b))
+    #define __Pyx_c_prod_double(a, b) ((a)*(b))
+    #define __Pyx_c_quot_double(a, b) ((a)/(b))
+    #define __Pyx_c_neg_double(a)     (-(a))
+  #ifdef __cplusplus
+    #define __Pyx_c_is_zero_double(z) ((z)==(double)0)
+    #define __Pyx_c_conj_double(z)    (::std::conj(z))
+    #if 1
+        #define __Pyx_c_abs_double(z)     (::std::abs(z))
+        #define __Pyx_c_pow_double(a, b)  (::std::pow(a, b))
+    #endif
+  #else
+    #define __Pyx_c_is_zero_double(z) ((z)==0)
+    #define __Pyx_c_conj_double(z)    (conj(z))
+    #if 1
+        #define __Pyx_c_abs_double(z)     (cabs(z))
+        #define __Pyx_c_pow_double(a, b)  (cpow(a, b))
+    #endif
+ #endif
+#else
+    static CYTHON_INLINE int __Pyx_c_eq_double(__pyx_t_double_complex, __pyx_t_double_complex);
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_sum_double(__pyx_t_double_complex, __pyx_t_double_complex);
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_diff_double(__pyx_t_double_complex, __pyx_t_double_complex);
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_prod_double(__pyx_t_double_complex, __pyx_t_double_complex);
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_quot_double(__pyx_t_double_complex, __pyx_t_double_complex);
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_neg_double(__pyx_t_double_complex);
+    static CYTHON_INLINE int __Pyx_c_is_zero_double(__pyx_t_double_complex);
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_conj_double(__pyx_t_double_complex);
+    #if 1
+        static CYTHON_INLINE double __Pyx_c_abs_double(__pyx_t_double_complex);
+        static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_pow_double(__pyx_t_double_complex, __pyx_t_double_complex);
+    #endif
+#endif
+
+/* CIntToPy.proto */
+static CYTHON_INLINE PyObject* __Pyx_PyInt_From_int(int value);
+
+/* CIntToPy.proto */
+static CYTHON_INLINE PyObject* __Pyx_PyInt_From_long(long value);
+
+/* CIntFromPy.proto */
+static CYTHON_INLINE long __Pyx_PyInt_As_long(PyObject *);
+
+/* CIntFromPy.proto */
+static CYTHON_INLINE int __Pyx_PyInt_As_int(PyObject *);
+
+/* FastTypeChecks.proto */
+#if CYTHON_COMPILING_IN_CPYTHON
+#define __Pyx_TypeCheck(obj, type) __Pyx_IsSubtype(Py_TYPE(obj), (PyTypeObject *)type)
+static CYTHON_INLINE int __Pyx_IsSubtype(PyTypeObject *a, PyTypeObject *b);
+static CYTHON_INLINE int __Pyx_PyErr_GivenExceptionMatches(PyObject *err, PyObject *type);
+static CYTHON_INLINE int __Pyx_PyErr_GivenExceptionMatches2(PyObject *err, PyObject *type1, PyObject *type2);
+#else
+#define __Pyx_TypeCheck(obj, type) PyObject_TypeCheck(obj, (PyTypeObject *)type)
+#define __Pyx_PyErr_GivenExceptionMatches(err, type) PyErr_GivenExceptionMatches(err, type)
+#define __Pyx_PyErr_GivenExceptionMatches2(err, type1, type2) (PyErr_GivenExceptionMatches(err, type1) || PyErr_GivenExceptionMatches(err, type2))
+#endif
+#define __Pyx_PyException_Check(obj) __Pyx_TypeCheck(obj, PyExc_Exception)
+
+/* CheckBinaryVersion.proto */
+static int __Pyx_check_binary_version(void);
+
+/* InitStrings.proto */
+static int __Pyx_InitStrings(__Pyx_StringTabEntry *t);
+
+
+/* Module declarations from 'libcpp' */
+
+/* Module declarations from 'cUnitSquareImage' */
+
+/* Module declarations from 'cpython.buffer' */
+
+/* Module declarations from 'libc.string' */
+
+/* Module declarations from 'libc.stdio' */
+
+/* Module declarations from '__builtin__' */
+
+/* Module declarations from 'cpython.type' */
+static PyTypeObject *__pyx_ptype_7cpython_4type_type = 0;
+
+/* Module declarations from 'cpython.version' */
+
+/* Module declarations from 'cpython.exc' */
+
+/* Module declarations from 'cpython.module' */
+
+/* Module declarations from 'cpython.mem' */
+
+/* Module declarations from 'cpython.tuple' */
+
+/* Module declarations from 'cpython.list' */
+
+/* Module declarations from 'cpython.sequence' */
+
+/* Module declarations from 'cpython.mapping' */
+
+/* Module declarations from 'cpython.iterator' */
+
+/* Module declarations from 'cpython.number' */
+
+/* Module declarations from 'cpython.int' */
+
+/* Module declarations from '__builtin__' */
+
+/* Module declarations from 'cpython.bool' */
+static PyTypeObject *__pyx_ptype_7cpython_4bool_bool = 0;
+
+/* Module declarations from 'cpython.long' */
+
+/* Module declarations from 'cpython.float' */
+
+/* Module declarations from '__builtin__' */
+
+/* Module declarations from 'cpython.complex' */
+static PyTypeObject *__pyx_ptype_7cpython_7complex_complex = 0;
+
+/* Module declarations from 'cpython.string' */
+
+/* Module declarations from 'cpython.unicode' */
+
+/* Module declarations from 'cpython.dict' */
+
+/* Module declarations from 'cpython.instance' */
+
+/* Module declarations from 'cpython.function' */
+
+/* Module declarations from 'cpython.method' */
+
+/* Module declarations from 'cpython.weakref' */
+
+/* Module declarations from 'cpython.getargs' */
+
+/* Module declarations from 'cpython.pythread' */
+
+/* Module declarations from 'cpython.pystate' */
+
+/* Module declarations from 'cpython.cobject' */
+
+/* Module declarations from 'cpython.oldbuffer' */
+
+/* Module declarations from 'cpython.set' */
+
+/* Module declarations from 'cpython.bytes' */
+
+/* Module declarations from 'cpython.pycapsule' */
+
+/* Module declarations from 'cpython' */
+
+/* Module declarations from 'cpython.object' */
+
+/* Module declarations from 'cpython.ref' */
+
+/* Module declarations from 'numpy' */
+
+/* Module declarations from 'numpy' */
+static PyTypeObject *__pyx_ptype_5numpy_dtype = 0;
+static PyTypeObject *__pyx_ptype_5numpy_flatiter = 0;
+static PyTypeObject *__pyx_ptype_5numpy_broadcast = 0;
+static PyTypeObject *__pyx_ptype_5numpy_ndarray = 0;
+static PyTypeObject *__pyx_ptype_5numpy_ufunc = 0;
+static CYTHON_INLINE int __pyx_f_5numpy_import_array(void); /*proto*/
+
+/* Module declarations from 'array' */
+
+/* Module declarations from 'cpython.array' */
+static PyTypeObject *__pyx_ptype_7cpython_5array_array = 0;
+static CYTHON_INLINE int __pyx_f_7cpython_5array_extend_buffer(arrayobject *, char *, Py_ssize_t); /*proto*/
+
+/* Module declarations from 'cimage' */
+static __Pyx_TypeInfo __Pyx_TypeInfo_double = { "double", NULL, sizeof(double), { 0 }, 0, 'R', 0, 0 };
+#define __Pyx_MODULE_NAME "cimage"
+extern int __pyx_module_is_main_cimage;
+int __pyx_module_is_main_cimage = 0;
+
+/* Implementation of 'cimage' */
+static PyObject *__pyx_builtin_ImportError;
+static PyObject *__pyx_builtin_MemoryError;
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+static const char __pyx_k_F[] = "F";
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+static const char __pyx_k_n[] = "n";
+static const char __pyx_k_t[] = "t";
+static const char __pyx_k_Fi[] = "Fi";
+static const char __pyx_k_np[] = "np";
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+static const char __pyx_k_gami[] = "gami";
+static const char __pyx_k_main[] = "__main__";
+static const char __pyx_k_name[] = "__name__";
+static const char __pyx_k_test[] = "__test__";
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+static const char __pyx_k_dfdvi[] = "dfdvi";
+static const char __pyx_k_numpy[] = "numpy";
+static const char __pyx_k_order[] = "order";
+static const char __pyx_k_zeros[] = "zeros";
+static const char __pyx_k_cimage[] = "cimage";
+static const char __pyx_k_import[] = "__import__";
+static const char __pyx_k_reshape[] = "reshape";
+static const char __pyx_k_is_diffeo[] = "is_diffeo";
+static const char __pyx_k_compgrad2D[] = "compgrad2D";
+static const char __pyx_k_ImportError[] = "ImportError";
+static const char __pyx_k_MemoryError[] = "MemoryError";
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+static const char __pyx_k_cline_in_traceback[] = "cline_in_traceback";
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+static PyObject *__pyx_n_s_F;
+static PyObject *__pyx_n_s_Fi;
+static PyObject *__pyx_n_s_ImportError;
+static PyObject *__pyx_n_s_MemoryError;
+static PyObject *__pyx_n_s_check_crossing;
+static PyObject *__pyx_n_s_cimage;
+static PyObject *__pyx_n_s_cline_in_traceback;
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+static PyObject *__pyx_n_s_gam;
+static PyObject *__pyx_n_s_gami;
+static PyObject *__pyx_n_s_import;
+static PyObject *__pyx_n_s_is_diffeo;
+static PyObject *__pyx_n_s_lam;
+static PyObject *__pyx_n_s_main;
+static PyObject *__pyx_n_s_n;
+static PyObject *__pyx_n_s_name;
+static PyObject *__pyx_n_s_np;
+static PyObject *__pyx_n_s_numpy;
+static PyObject *__pyx_kp_s_numpy_core_multiarray_failed_to;
+static PyObject *__pyx_kp_s_numpy_core_umath_failed_to_impor;
+static PyObject *__pyx_n_s_order;
+static PyObject *__pyx_n_s_reshape;
+static PyObject *__pyx_kp_s_src_imagecpp_pyx;
+static PyObject *__pyx_n_s_t;
+static PyObject *__pyx_n_s_test;
+static PyObject *__pyx_n_s_zeros;
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+static PyObject *__pyx_tuple__3;
+static PyObject *__pyx_tuple__5;
+static PyObject *__pyx_codeobj__4;
+static PyObject *__pyx_codeobj__6;
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+ * 
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+                 parent->type->name, field->name);
+  }
+}
+static int __Pyx_BufFmt_ProcessTypeChunk(__Pyx_BufFmt_Context* ctx) {
+  char group;
+  size_t size, offset, arraysize = 1;
+  if (ctx->enc_type == 0) return 0;
+  if (ctx->head->field->type->arraysize[0]) {
+    int i, ndim = 0;
+    if (ctx->enc_type == 's' || ctx->enc_type == 'p') {
+        ctx->is_valid_array = ctx->head->field->type->ndim == 1;
+        ndim = 1;
+        if (ctx->enc_count != ctx->head->field->type->arraysize[0]) {
+            PyErr_Format(PyExc_ValueError,
+                         "Expected a dimension of size %zu, got %zu",
+                         ctx->head->field->type->arraysize[0], ctx->enc_count);
+            return -1;
+        }
+    }
+    if (!ctx->is_valid_array) {
+      PyErr_Format(PyExc_ValueError, "Expected %d dimensions, got %d",
+                   ctx->head->field->type->ndim, ndim);
+      return -1;
+    }
+    for (i = 0; i < ctx->head->field->type->ndim; i++) {
+      arraysize *= ctx->head->field->type->arraysize[i];
+    }
+    ctx->is_valid_array = 0;
+    ctx->enc_count = 1;
+  }
+  group = __Pyx_BufFmt_TypeCharToGroup(ctx->enc_type, ctx->is_complex);
+  do {
+    __Pyx_StructField* field = ctx->head->field;
+    __Pyx_TypeInfo* type = field->type;
+    if (ctx->enc_packmode == '@' || ctx->enc_packmode == '^') {
+      size = __Pyx_BufFmt_TypeCharToNativeSize(ctx->enc_type, ctx->is_complex);
+    } else {
+      size = __Pyx_BufFmt_TypeCharToStandardSize(ctx->enc_type, ctx->is_complex);
+    }
+    if (ctx->enc_packmode == '@') {
+      size_t align_at = __Pyx_BufFmt_TypeCharToAlignment(ctx->enc_type, ctx->is_complex);
+      size_t align_mod_offset;
+      if (align_at == 0) return -1;
+      align_mod_offset = ctx->fmt_offset % align_at;
+      if (align_mod_offset > 0) ctx->fmt_offset += align_at - align_mod_offset;
+      if (ctx->struct_alignment == 0)
+          ctx->struct_alignment = __Pyx_BufFmt_TypeCharToPadding(ctx->enc_type,
+                                                                 ctx->is_complex);
+    }
+    if (type->size != size || type->typegroup != group) {
+      if (type->typegroup == 'C' && type->fields != NULL) {
+        size_t parent_offset = ctx->head->parent_offset + field->offset;
+        ++ctx->head;
+        ctx->head->field = type->fields;
+        ctx->head->parent_offset = parent_offset;
+        continue;
+      }
+      if ((type->typegroup == 'H' || group == 'H') && type->size == size) {
+      } else {
+          __Pyx_BufFmt_RaiseExpected(ctx);
+          return -1;
+      }
+    }
+    offset = ctx->head->parent_offset + field->offset;
+    if (ctx->fmt_offset != offset) {
+      PyErr_Format(PyExc_ValueError,
+                   "Buffer dtype mismatch; next field is at offset %" CYTHON_FORMAT_SSIZE_T "d but %" CYTHON_FORMAT_SSIZE_T "d expected",
+                   (Py_ssize_t)ctx->fmt_offset, (Py_ssize_t)offset);
+      return -1;
+    }
+    ctx->fmt_offset += size;
+    if (arraysize)
+      ctx->fmt_offset += (arraysize - 1) * size;
+    --ctx->enc_count;
+    while (1) {
+      if (field == &ctx->root) {
+        ctx->head = NULL;
+        if (ctx->enc_count != 0) {
+          __Pyx_BufFmt_RaiseExpected(ctx);
+          return -1;
+        }
+        break;
+      }
+      ctx->head->field = ++field;
+      if (field->type == NULL) {
+        --ctx->head;
+        field = ctx->head->field;
+        continue;
+      } else if (field->type->typegroup == 'S') {
+        size_t parent_offset = ctx->head->parent_offset + field->offset;
+        if (field->type->fields->type == NULL) continue;
+        field = field->type->fields;
+        ++ctx->head;
+        ctx->head->field = field;
+        ctx->head->parent_offset = parent_offset;
+        break;
+      } else {
+        break;
+      }
+    }
+  } while (ctx->enc_count);
+  ctx->enc_type = 0;
+  ctx->is_complex = 0;
+  return 0;
+}
+static PyObject *
+__pyx_buffmt_parse_array(__Pyx_BufFmt_Context* ctx, const char** tsp)
+{
+    const char *ts = *tsp;
+    int i = 0, number, ndim;
+    ++ts;
+    if (ctx->new_count != 1) {
+        PyErr_SetString(PyExc_ValueError,
+                        "Cannot handle repeated arrays in format string");
+        return NULL;
+    }
+    if (__Pyx_BufFmt_ProcessTypeChunk(ctx) == -1) return NULL;
+    ndim = ctx->head->field->type->ndim;
+    while (*ts && *ts != ')') {
+        switch (*ts) {
+            case ' ': case '\f': case '\r': case '\n': case '\t': case '\v':  continue;
+            default:  break;
+        }
+        number = __Pyx_BufFmt_ExpectNumber(&ts);
+        if (number == -1) return NULL;
+        if (i < ndim && (size_t) number != ctx->head->field->type->arraysize[i])
+            return PyErr_Format(PyExc_ValueError,
+                        "Expected a dimension of size %zu, got %d",
+                        ctx->head->field->type->arraysize[i], number);
+        if (*ts != ',' && *ts != ')')
+            return PyErr_Format(PyExc_ValueError,
+                                "Expected a comma in format string, got '%c'", *ts);
+        if (*ts == ',') ts++;
+        i++;
+    }
+    if (i != ndim)
+        return PyErr_Format(PyExc_ValueError, "Expected %d dimension(s), got %d",
+                            ctx->head->field->type->ndim, i);
+    if (!*ts) {
+        PyErr_SetString(PyExc_ValueError,
+                        "Unexpected end of format string, expected ')'");
+        return NULL;
+    }
+    ctx->is_valid_array = 1;
+    ctx->new_count = 1;
+    *tsp = ++ts;
+    return Py_None;
+}
+static const char* __Pyx_BufFmt_CheckString(__Pyx_BufFmt_Context* ctx, const char* ts) {
+  int got_Z = 0;
+  while (1) {
+    switch(*ts) {
+      case 0:
+        if (ctx->enc_type != 0 && ctx->head == NULL) {
+          __Pyx_BufFmt_RaiseExpected(ctx);
+          return NULL;
+        }
+        if (__Pyx_BufFmt_ProcessTypeChunk(ctx) == -1) return NULL;
+        if (ctx->head != NULL) {
+          __Pyx_BufFmt_RaiseExpected(ctx);
+          return NULL;
+        }
+        return ts;
+      case ' ':
+      case '\r':
+      case '\n':
+        ++ts;
+        break;
+      case '<':
+        if (!__Pyx_Is_Little_Endian()) {
+          PyErr_SetString(PyExc_ValueError, "Little-endian buffer not supported on big-endian compiler");
+          return NULL;
+        }
+        ctx->new_packmode = '=';
+        ++ts;
+        break;
+      case '>':
+      case '!':
+        if (__Pyx_Is_Little_Endian()) {
+          PyErr_SetString(PyExc_ValueError, "Big-endian buffer not supported on little-endian compiler");
+          return NULL;
+        }
+        ctx->new_packmode = '=';
+        ++ts;
+        break;
+      case '=':
+      case '@':
+      case '^':
+        ctx->new_packmode = *ts++;
+        break;
+      case 'T':
+        {
+          const char* ts_after_sub;
+          size_t i, struct_count = ctx->new_count;
+          size_t struct_alignment = ctx->struct_alignment;
+          ctx->new_count = 1;
+          ++ts;
+          if (*ts != '{') {
+            PyErr_SetString(PyExc_ValueError, "Buffer acquisition: Expected '{' after 'T'");
+            return NULL;
+          }
+          if (__Pyx_BufFmt_ProcessTypeChunk(ctx) == -1) return NULL;
+          ctx->enc_type = 0;
+          ctx->enc_count = 0;
+          ctx->struct_alignment = 0;
+          ++ts;
+          ts_after_sub = ts;
+          for (i = 0; i != struct_count; ++i) {
+            ts_after_sub = __Pyx_BufFmt_CheckString(ctx, ts);
+            if (!ts_after_sub) return NULL;
+          }
+          ts = ts_after_sub;
+          if (struct_alignment) ctx->struct_alignment = struct_alignment;
+        }
+        break;
+      case '}':
+        {
+          size_t alignment = ctx->struct_alignment;
+          ++ts;
+          if (__Pyx_BufFmt_ProcessTypeChunk(ctx) == -1) return NULL;
+          ctx->enc_type = 0;
+          if (alignment && ctx->fmt_offset % alignment) {
+            ctx->fmt_offset += alignment - (ctx->fmt_offset % alignment);
+          }
+        }
+        return ts;
+      case 'x':
+        if (__Pyx_BufFmt_ProcessTypeChunk(ctx) == -1) return NULL;
+        ctx->fmt_offset += ctx->new_count;
+        ctx->new_count = 1;
+        ctx->enc_count = 0;
+        ctx->enc_type = 0;
+        ctx->enc_packmode = ctx->new_packmode;
+        ++ts;
+        break;
+      case 'Z':
+        got_Z = 1;
+        ++ts;
+        if (*ts != 'f' && *ts != 'd' && *ts != 'g') {
+          __Pyx_BufFmt_RaiseUnexpectedChar('Z');
+          return NULL;
+        }
+        CYTHON_FALLTHROUGH;
+      case '?': case 'c': case 'b': case 'B': case 'h': case 'H': case 'i': case 'I':
+      case 'l': case 'L': case 'q': case 'Q':
+      case 'f': case 'd': case 'g':
+      case 'O': case 'p':
+        if ((ctx->enc_type == *ts) && (got_Z == ctx->is_complex) &&
+            (ctx->enc_packmode == ctx->new_packmode) && (!ctx->is_valid_array)) {
+          ctx->enc_count += ctx->new_count;
+          ctx->new_count = 1;
+          got_Z = 0;
+          ++ts;
+          break;
+        }
+        CYTHON_FALLTHROUGH;
+      case 's':
+        if (__Pyx_BufFmt_ProcessTypeChunk(ctx) == -1) return NULL;
+        ctx->enc_count = ctx->new_count;
+        ctx->enc_packmode = ctx->new_packmode;
+        ctx->enc_type = *ts;
+        ctx->is_complex = got_Z;
+        ++ts;
+        ctx->new_count = 1;
+        got_Z = 0;
+        break;
+      case ':':
+        ++ts;
+        while(*ts != ':') ++ts;
+        ++ts;
+        break;
+      case '(':
+        if (!__pyx_buffmt_parse_array(ctx, &ts)) return NULL;
+        break;
+      default:
+        {
+          int number = __Pyx_BufFmt_ExpectNumber(&ts);
+          if (number == -1) return NULL;
+          ctx->new_count = (size_t)number;
+        }
+    }
+  }
+}
+
+/* BufferGetAndValidate */
+  static CYTHON_INLINE void __Pyx_SafeReleaseBuffer(Py_buffer* info) {
+  if (unlikely(info->buf == NULL)) return;
+  if (info->suboffsets == __Pyx_minusones) info->suboffsets = NULL;
+  __Pyx_ReleaseBuffer(info);
+}
+static void __Pyx_ZeroBuffer(Py_buffer* buf) {
+  buf->buf = NULL;
+  buf->obj = NULL;
+  buf->strides = __Pyx_zeros;
+  buf->shape = __Pyx_zeros;
+  buf->suboffsets = __Pyx_minusones;
+}
+static int __Pyx__GetBufferAndValidate(
+        Py_buffer* buf, PyObject* obj,  __Pyx_TypeInfo* dtype, int flags,
+        int nd, int cast, __Pyx_BufFmt_StackElem* stack)
+{
+  buf->buf = NULL;
+  if (unlikely(__Pyx_GetBuffer(obj, buf, flags) == -1)) {
+    __Pyx_ZeroBuffer(buf);
+    return -1;
+  }
+  if (unlikely(buf->ndim != nd)) {
+    PyErr_Format(PyExc_ValueError,
+                 "Buffer has wrong number of dimensions (expected %d, got %d)",
+                 nd, buf->ndim);
+    goto fail;
+  }
+  if (!cast) {
+    __Pyx_BufFmt_Context ctx;
+    __Pyx_BufFmt_Init(&ctx, stack, dtype);
+    if (!__Pyx_BufFmt_CheckString(&ctx, buf->format)) goto fail;
+  }
+  if (unlikely((size_t)buf->itemsize != dtype->size)) {
+    PyErr_Format(PyExc_ValueError,
+      "Item size of buffer (%" CYTHON_FORMAT_SSIZE_T "d byte%s) does not match size of '%s' (%" CYTHON_FORMAT_SSIZE_T "d byte%s)",
+      buf->itemsize, (buf->itemsize > 1) ? "s" : "",
+      dtype->name, (Py_ssize_t)dtype->size, (dtype->size > 1) ? "s" : "");
+    goto fail;
+  }
+  if (buf->suboffsets == NULL) buf->suboffsets = __Pyx_minusones;
+  return 0;
+fail:;
+  __Pyx_SafeReleaseBuffer(buf);
+  return -1;
+}
+
+/* PyObjectGetAttrStr */
+  #if CYTHON_USE_TYPE_SLOTS
+static CYTHON_INLINE PyObject* __Pyx_PyObject_GetAttrStr(PyObject* obj, PyObject* attr_name) {
+    PyTypeObject* tp = Py_TYPE(obj);
+    if (likely(tp->tp_getattro))
+        return tp->tp_getattro(obj, attr_name);
+#if PY_MAJOR_VERSION < 3
+    if (likely(tp->tp_getattr))
+        return tp->tp_getattr(obj, PyString_AS_STRING(attr_name));
+#endif
+    return PyObject_GetAttr(obj, attr_name);
+}
+#endif
+
+/* GetBuiltinName */
+  static PyObject *__Pyx_GetBuiltinName(PyObject *name) {
+    PyObject* result = __Pyx_PyObject_GetAttrStr(__pyx_b, name);
+    if (unlikely(!result)) {
+        PyErr_Format(PyExc_NameError,
+#if PY_MAJOR_VERSION >= 3
+            "name '%U' is not defined", name);
+#else
+            "name '%.200s' is not defined", PyString_AS_STRING(name));
+#endif
+    }
+    return result;
+}
+
+/* PyDictVersioning */
+  #if CYTHON_USE_DICT_VERSIONS && CYTHON_USE_TYPE_SLOTS
+static CYTHON_INLINE PY_UINT64_T __Pyx_get_tp_dict_version(PyObject *obj) {
+    PyObject *dict = Py_TYPE(obj)->tp_dict;
+    return likely(dict) ? __PYX_GET_DICT_VERSION(dict) : 0;
+}
+static CYTHON_INLINE PY_UINT64_T __Pyx_get_object_dict_version(PyObject *obj) {
+    PyObject **dictptr = NULL;
+    Py_ssize_t offset = Py_TYPE(obj)->tp_dictoffset;
+    if (offset) {
+#if CYTHON_COMPILING_IN_CPYTHON
+        dictptr = (likely(offset > 0)) ? (PyObject **) ((char *)obj + offset) : _PyObject_GetDictPtr(obj);
+#else
+        dictptr = _PyObject_GetDictPtr(obj);
+#endif
+    }
+    return (dictptr && *dictptr) ? __PYX_GET_DICT_VERSION(*dictptr) : 0;
+}
+static CYTHON_INLINE int __Pyx_object_dict_version_matches(PyObject* obj, PY_UINT64_T tp_dict_version, PY_UINT64_T obj_dict_version) {
+    PyObject *dict = Py_TYPE(obj)->tp_dict;
+    if (unlikely(!dict) || unlikely(tp_dict_version != __PYX_GET_DICT_VERSION(dict)))
+        return 0;
+    return obj_dict_version == __Pyx_get_object_dict_version(obj);
+}
+#endif
+
+/* GetModuleGlobalName */
+  #if CYTHON_USE_DICT_VERSIONS
+static PyObject *__Pyx__GetModuleGlobalName(PyObject *name, PY_UINT64_T *dict_version, PyObject **dict_cached_value)
+#else
+static CYTHON_INLINE PyObject *__Pyx__GetModuleGlobalName(PyObject *name)
+#endif
+{
+    PyObject *result;
+#if !CYTHON_AVOID_BORROWED_REFS
+#if CYTHON_COMPILING_IN_CPYTHON && PY_VERSION_HEX >= 0x030500A1
+    result = _PyDict_GetItem_KnownHash(__pyx_d, name, ((PyASCIIObject *) name)->hash);
+    __PYX_UPDATE_DICT_CACHE(__pyx_d, result, *dict_cached_value, *dict_version)
+    if (likely(result)) {
+        return __Pyx_NewRef(result);
+    } else if (unlikely(PyErr_Occurred())) {
+        return NULL;
+    }
+#else
+    result = PyDict_GetItem(__pyx_d, name);
+    __PYX_UPDATE_DICT_CACHE(__pyx_d, result, *dict_cached_value, *dict_version)
+    if (likely(result)) {
+        return __Pyx_NewRef(result);
+    }
+#endif
+#else
+    result = PyObject_GetItem(__pyx_d, name);
+    __PYX_UPDATE_DICT_CACHE(__pyx_d, result, *dict_cached_value, *dict_version)
+    if (likely(result)) {
+        return __Pyx_NewRef(result);
+    }
+    PyErr_Clear();
+#endif
+    return __Pyx_GetBuiltinName(name);
+}
+
+/* PyCFunctionFastCall */
+  #if CYTHON_FAST_PYCCALL
+static CYTHON_INLINE PyObject * __Pyx_PyCFunction_FastCall(PyObject *func_obj, PyObject **args, Py_ssize_t nargs) {
+    PyCFunctionObject *func = (PyCFunctionObject*)func_obj;
+    PyCFunction meth = PyCFunction_GET_FUNCTION(func);
+    PyObject *self = PyCFunction_GET_SELF(func);
+    int flags = PyCFunction_GET_FLAGS(func);
+    assert(PyCFunction_Check(func));
+    assert(METH_FASTCALL == (flags & ~(METH_CLASS | METH_STATIC | METH_COEXIST | METH_KEYWORDS | METH_STACKLESS)));
+    assert(nargs >= 0);
+    assert(nargs == 0 || args != NULL);
+    /* _PyCFunction_FastCallDict() must not be called with an exception set,
+       because it may clear it (directly or indirectly) and so the
+       caller loses its exception */
+    assert(!PyErr_Occurred());
+    if ((PY_VERSION_HEX < 0x030700A0) || unlikely(flags & METH_KEYWORDS)) {
+        return (*((__Pyx_PyCFunctionFastWithKeywords)(void*)meth)) (self, args, nargs, NULL);
+    } else {
+        return (*((__Pyx_PyCFunctionFast)(void*)meth)) (self, args, nargs);
+    }
+}
+#endif
+
+/* PyFunctionFastCall */
+  #if CYTHON_FAST_PYCALL
+static PyObject* __Pyx_PyFunction_FastCallNoKw(PyCodeObject *co, PyObject **args, Py_ssize_t na,
+                                               PyObject *globals) {
+    PyFrameObject *f;
+    PyThreadState *tstate = __Pyx_PyThreadState_Current;
+    PyObject **fastlocals;
+    Py_ssize_t i;
+    PyObject *result;
+    assert(globals != NULL);
+    /* XXX Perhaps we should create a specialized
+       PyFrame_New() that doesn't take locals, but does
+       take builtins without sanity checking them.
+       */
+    assert(tstate != NULL);
+    f = PyFrame_New(tstate, co, globals, NULL);
+    if (f == NULL) {
+        return NULL;
+    }
+    fastlocals = __Pyx_PyFrame_GetLocalsplus(f);
+    for (i = 0; i < na; i++) {
+        Py_INCREF(*args);
+        fastlocals[i] = *args++;
+    }
+    result = PyEval_EvalFrameEx(f,0);
+    ++tstate->recursion_depth;
+    Py_DECREF(f);
+    --tstate->recursion_depth;
+    return result;
+}
+#if 1 || PY_VERSION_HEX < 0x030600B1
+static PyObject *__Pyx_PyFunction_FastCallDict(PyObject *func, PyObject **args, Py_ssize_t nargs, PyObject *kwargs) {
+    PyCodeObject *co = (PyCodeObject *)PyFunction_GET_CODE(func);
+    PyObject *globals = PyFunction_GET_GLOBALS(func);
+    PyObject *argdefs = PyFunction_GET_DEFAULTS(func);
+    PyObject *closure;
+#if PY_MAJOR_VERSION >= 3
+    PyObject *kwdefs;
+#endif
+    PyObject *kwtuple, **k;
+    PyObject **d;
+    Py_ssize_t nd;
+    Py_ssize_t nk;
+    PyObject *result;
+    assert(kwargs == NULL || PyDict_Check(kwargs));
+    nk = kwargs ? PyDict_Size(kwargs) : 0;
+    if (Py_EnterRecursiveCall((char*)" while calling a Python object")) {
+        return NULL;
+    }
+    if (
+#if PY_MAJOR_VERSION >= 3
+            co->co_kwonlyargcount == 0 &&
+#endif
+            likely(kwargs == NULL || nk == 0) &&
+            co->co_flags == (CO_OPTIMIZED | CO_NEWLOCALS | CO_NOFREE)) {
+        if (argdefs == NULL && co->co_argcount == nargs) {
+            result = __Pyx_PyFunction_FastCallNoKw(co, args, nargs, globals);
+            goto done;
+        }
+        else if (nargs == 0 && argdefs != NULL
+                 && co->co_argcount == Py_SIZE(argdefs)) {
+            /* function called with no arguments, but all parameters have
+               a default value: use default values as arguments .*/
+            args = &PyTuple_GET_ITEM(argdefs, 0);
+            result =__Pyx_PyFunction_FastCallNoKw(co, args, Py_SIZE(argdefs), globals);
+            goto done;
+        }
+    }
+    if (kwargs != NULL) {
+        Py_ssize_t pos, i;
+        kwtuple = PyTuple_New(2 * nk);
+        if (kwtuple == NULL) {
+            result = NULL;
+            goto done;
+        }
+        k = &PyTuple_GET_ITEM(kwtuple, 0);
+        pos = i = 0;
+        while (PyDict_Next(kwargs, &pos, &k[i], &k[i+1])) {
+            Py_INCREF(k[i]);
+            Py_INCREF(k[i+1]);
+            i += 2;
+        }
+        nk = i / 2;
+    }
+    else {
+        kwtuple = NULL;
+        k = NULL;
+    }
+    closure = PyFunction_GET_CLOSURE(func);
+#if PY_MAJOR_VERSION >= 3
+    kwdefs = PyFunction_GET_KW_DEFAULTS(func);
+#endif
+    if (argdefs != NULL) {
+        d = &PyTuple_GET_ITEM(argdefs, 0);
+        nd = Py_SIZE(argdefs);
+    }
+    else {
+        d = NULL;
+        nd = 0;
+    }
+#if PY_MAJOR_VERSION >= 3
+    result = PyEval_EvalCodeEx((PyObject*)co, globals, (PyObject *)NULL,
+                               args, (int)nargs,
+                               k, (int)nk,
+                               d, (int)nd, kwdefs, closure);
+#else
+    result = PyEval_EvalCodeEx(co, globals, (PyObject *)NULL,
+                               args, (int)nargs,
+                               k, (int)nk,
+                               d, (int)nd, closure);
+#endif
+    Py_XDECREF(kwtuple);
+done:
+    Py_LeaveRecursiveCall();
+    return result;
+}
+#endif
+#endif
+
+/* PyObjectCall */
+  #if CYTHON_COMPILING_IN_CPYTHON
+static CYTHON_INLINE PyObject* __Pyx_PyObject_Call(PyObject *func, PyObject *arg, PyObject *kw) {
+    PyObject *result;
+    ternaryfunc call = Py_TYPE(func)->tp_call;
+    if (unlikely(!call))
+        return PyObject_Call(func, arg, kw);
+    if (unlikely(Py_EnterRecursiveCall((char*)" while calling a Python object")))
+        return NULL;
+    result = (*call)(func, arg, kw);
+    Py_LeaveRecursiveCall();
+    if (unlikely(!result) && unlikely(!PyErr_Occurred())) {
+        PyErr_SetString(
+            PyExc_SystemError,
+            "NULL result without error in PyObject_Call");
+    }
+    return result;
+}
+#endif
+
+/* PyObjectCall2Args */
+  static CYTHON_UNUSED PyObject* __Pyx_PyObject_Call2Args(PyObject* function, PyObject* arg1, PyObject* arg2) {
+    PyObject *args, *result = NULL;
+    #if CYTHON_FAST_PYCALL
+    if (PyFunction_Check(function)) {
+        PyObject *args[2] = {arg1, arg2};
+        return __Pyx_PyFunction_FastCall(function, args, 2);
+    }
+    #endif
+    #if CYTHON_FAST_PYCCALL
+    if (__Pyx_PyFastCFunction_Check(function)) {
+        PyObject *args[2] = {arg1, arg2};
+        return __Pyx_PyCFunction_FastCall(function, args, 2);
+    }
+    #endif
+    args = PyTuple_New(2);
+    if (unlikely(!args)) goto done;
+    Py_INCREF(arg1);
+    PyTuple_SET_ITEM(args, 0, arg1);
+    Py_INCREF(arg2);
+    PyTuple_SET_ITEM(args, 1, arg2);
+    Py_INCREF(function);
+    result = __Pyx_PyObject_Call(function, args, NULL);
+    Py_DECREF(args);
+    Py_DECREF(function);
+done:
+    return result;
+}
+
+/* PyObjectCallMethO */
+  #if CYTHON_COMPILING_IN_CPYTHON
+static CYTHON_INLINE PyObject* __Pyx_PyObject_CallMethO(PyObject *func, PyObject *arg) {
+    PyObject *self, *result;
+    PyCFunction cfunc;
+    cfunc = PyCFunction_GET_FUNCTION(func);
+    self = PyCFunction_GET_SELF(func);
+    if (unlikely(Py_EnterRecursiveCall((char*)" while calling a Python object")))
+        return NULL;
+    result = cfunc(self, arg);
+    Py_LeaveRecursiveCall();
+    if (unlikely(!result) && unlikely(!PyErr_Occurred())) {
+        PyErr_SetString(
+            PyExc_SystemError,
+            "NULL result without error in PyObject_Call");
+    }
+    return result;
+}
+#endif
+
+/* PyObjectCallOneArg */
+  #if CYTHON_COMPILING_IN_CPYTHON
+static PyObject* __Pyx__PyObject_CallOneArg(PyObject *func, PyObject *arg) {
+    PyObject *result;
+    PyObject *args = PyTuple_New(1);
+    if (unlikely(!args)) return NULL;
+    Py_INCREF(arg);
+    PyTuple_SET_ITEM(args, 0, arg);
+    result = __Pyx_PyObject_Call(func, args, NULL);
+    Py_DECREF(args);
+    return result;
+}
+static CYTHON_INLINE PyObject* __Pyx_PyObject_CallOneArg(PyObject *func, PyObject *arg) {
+#if CYTHON_FAST_PYCALL
+    if (PyFunction_Check(func)) {
+        return __Pyx_PyFunction_FastCall(func, &arg, 1);
+    }
+#endif
+    if (likely(PyCFunction_Check(func))) {
+        if (likely(PyCFunction_GET_FLAGS(func) & METH_O)) {
+            return __Pyx_PyObject_CallMethO(func, arg);
+#if CYTHON_FAST_PYCCALL
+        } else if (__Pyx_PyFastCFunction_Check(func)) {
+            return __Pyx_PyCFunction_FastCall(func, &arg, 1);
+#endif
+        }
+    }
+    return __Pyx__PyObject_CallOneArg(func, arg);
+}
+#else
+static CYTHON_INLINE PyObject* __Pyx_PyObject_CallOneArg(PyObject *func, PyObject *arg) {
+    PyObject *result;
+    PyObject *args = PyTuple_Pack(1, arg);
+    if (unlikely(!args)) return NULL;
+    result = __Pyx_PyObject_Call(func, args, NULL);
+    Py_DECREF(args);
+    return result;
+}
+#endif
+
+/* ExtTypeTest */
+  static CYTHON_INLINE int __Pyx_TypeTest(PyObject *obj, PyTypeObject *type) {
+    if (unlikely(!type)) {
+        PyErr_SetString(PyExc_SystemError, "Missing type object");
+        return 0;
+    }
+    if (likely(__Pyx_TypeCheck(obj, type)))
+        return 1;
+    PyErr_Format(PyExc_TypeError, "Cannot convert %.200s to %.200s",
+                 Py_TYPE(obj)->tp_name, type->tp_name);
+    return 0;
+}
+
+/* BufferFallbackError */
+  static void __Pyx_RaiseBufferFallbackError(void) {
+  PyErr_SetString(PyExc_ValueError,
+     "Buffer acquisition failed on assignment; and then reacquiring the old buffer failed too!");
+}
+
+/* BufferIndexError */
+  static void __Pyx_RaiseBufferIndexError(int axis) {
+  PyErr_Format(PyExc_IndexError,
+     "Out of bounds on buffer access (axis %d)", axis);
+}
+
+/* PyErrFetchRestore */
+  #if CYTHON_FAST_THREAD_STATE
+static CYTHON_INLINE void __Pyx_ErrRestoreInState(PyThreadState *tstate, PyObject *type, PyObject *value, PyObject *tb) {
+    PyObject *tmp_type, *tmp_value, *tmp_tb;
+    tmp_type = tstate->curexc_type;
+    tmp_value = tstate->curexc_value;
+    tmp_tb = tstate->curexc_traceback;
+    tstate->curexc_type = type;
+    tstate->curexc_value = value;
+    tstate->curexc_traceback = tb;
+    Py_XDECREF(tmp_type);
+    Py_XDECREF(tmp_value);
+    Py_XDECREF(tmp_tb);
+}
+static CYTHON_INLINE void __Pyx_ErrFetchInState(PyThreadState *tstate, PyObject **type, PyObject **value, PyObject **tb) {
+    *type = tstate->curexc_type;
+    *value = tstate->curexc_value;
+    *tb = tstate->curexc_traceback;
+    tstate->curexc_type = 0;
+    tstate->curexc_value = 0;
+    tstate->curexc_traceback = 0;
+}
+#endif
+
+/* GetTopmostException */
+  #if CYTHON_USE_EXC_INFO_STACK
+static _PyErr_StackItem *
+__Pyx_PyErr_GetTopmostException(PyThreadState *tstate)
+{
+    _PyErr_StackItem *exc_info = tstate->exc_info;
+    while ((exc_info->exc_type == NULL || exc_info->exc_type == Py_None) &&
+           exc_info->previous_item != NULL)
+    {
+        exc_info = exc_info->previous_item;
+    }
+    return exc_info;
+}
+#endif
+
+/* SaveResetException */
+  #if CYTHON_FAST_THREAD_STATE
+static CYTHON_INLINE void __Pyx__ExceptionSave(PyThreadState *tstate, PyObject **type, PyObject **value, PyObject **tb) {
+    #if CYTHON_USE_EXC_INFO_STACK
+    _PyErr_StackItem *exc_info = __Pyx_PyErr_GetTopmostException(tstate);
+    *type = exc_info->exc_type;
+    *value = exc_info->exc_value;
+    *tb = exc_info->exc_traceback;
+    #else
+    *type = tstate->exc_type;
+    *value = tstate->exc_value;
+    *tb = tstate->exc_traceback;
+    #endif
+    Py_XINCREF(*type);
+    Py_XINCREF(*value);
+    Py_XINCREF(*tb);
+}
+static CYTHON_INLINE void __Pyx__ExceptionReset(PyThreadState *tstate, PyObject *type, PyObject *value, PyObject *tb) {
+    PyObject *tmp_type, *tmp_value, *tmp_tb;
+    #if CYTHON_USE_EXC_INFO_STACK
+    _PyErr_StackItem *exc_info = tstate->exc_info;
+    tmp_type = exc_info->exc_type;
+    tmp_value = exc_info->exc_value;
+    tmp_tb = exc_info->exc_traceback;
+    exc_info->exc_type = type;
+    exc_info->exc_value = value;
+    exc_info->exc_traceback = tb;
+    #else
+    tmp_type = tstate->exc_type;
+    tmp_value = tstate->exc_value;
+    tmp_tb = tstate->exc_traceback;
+    tstate->exc_type = type;
+    tstate->exc_value = value;
+    tstate->exc_traceback = tb;
+    #endif
+    Py_XDECREF(tmp_type);
+    Py_XDECREF(tmp_value);
+    Py_XDECREF(tmp_tb);
+}
+#endif
+
+/* PyErrExceptionMatches */
+  #if CYTHON_FAST_THREAD_STATE
+static int __Pyx_PyErr_ExceptionMatchesTuple(PyObject *exc_type, PyObject *tuple) {
+    Py_ssize_t i, n;
+    n = PyTuple_GET_SIZE(tuple);
+#if PY_MAJOR_VERSION >= 3
+    for (i=0; i<n; i++) {
+        if (exc_type == PyTuple_GET_ITEM(tuple, i)) return 1;
+    }
+#endif
+    for (i=0; i<n; i++) {
+        if (__Pyx_PyErr_GivenExceptionMatches(exc_type, PyTuple_GET_ITEM(tuple, i))) return 1;
+    }
+    return 0;
+}
+static CYTHON_INLINE int __Pyx_PyErr_ExceptionMatchesInState(PyThreadState* tstate, PyObject* err) {
+    PyObject *exc_type = tstate->curexc_type;
+    if (exc_type == err) return 1;
+    if (unlikely(!exc_type)) return 0;
+    if (unlikely(PyTuple_Check(err)))
+        return __Pyx_PyErr_ExceptionMatchesTuple(exc_type, err);
+    return __Pyx_PyErr_GivenExceptionMatches(exc_type, err);
+}
+#endif
+
+/* GetException */
+  #if CYTHON_FAST_THREAD_STATE
+static int __Pyx__GetException(PyThreadState *tstate, PyObject **type, PyObject **value, PyObject **tb)
+#else
+static int __Pyx_GetException(PyObject **type, PyObject **value, PyObject **tb)
+#endif
+{
+    PyObject *local_type, *local_value, *local_tb;
+#if CYTHON_FAST_THREAD_STATE
+    PyObject *tmp_type, *tmp_value, *tmp_tb;
+    local_type = tstate->curexc_type;
+    local_value = tstate->curexc_value;
+    local_tb = tstate->curexc_traceback;
+    tstate->curexc_type = 0;
+    tstate->curexc_value = 0;
+    tstate->curexc_traceback = 0;
+#else
+    PyErr_Fetch(&local_type, &local_value, &local_tb);
+#endif
+    PyErr_NormalizeException(&local_type, &local_value, &local_tb);
+#if CYTHON_FAST_THREAD_STATE
+    if (unlikely(tstate->curexc_type))
+#else
+    if (unlikely(PyErr_Occurred()))
+#endif
+        goto bad;
+    #if PY_MAJOR_VERSION >= 3
+    if (local_tb) {
+        if (unlikely(PyException_SetTraceback(local_value, local_tb) < 0))
+            goto bad;
+    }
+    #endif
+    Py_XINCREF(local_tb);
+    Py_XINCREF(local_type);
+    Py_XINCREF(local_value);
+    *type = local_type;
+    *value = local_value;
+    *tb = local_tb;
+#if CYTHON_FAST_THREAD_STATE
+    #if CYTHON_USE_EXC_INFO_STACK
+    {
+        _PyErr_StackItem *exc_info = tstate->exc_info;
+        tmp_type = exc_info->exc_type;
+        tmp_value = exc_info->exc_value;
+        tmp_tb = exc_info->exc_traceback;
+        exc_info->exc_type = local_type;
+        exc_info->exc_value = local_value;
+        exc_info->exc_traceback = local_tb;
+    }
+    #else
+    tmp_type = tstate->exc_type;
+    tmp_value = tstate->exc_value;
+    tmp_tb = tstate->exc_traceback;
+    tstate->exc_type = local_type;
+    tstate->exc_value = local_value;
+    tstate->exc_traceback = local_tb;
+    #endif
+    Py_XDECREF(tmp_type);
+    Py_XDECREF(tmp_value);
+    Py_XDECREF(tmp_tb);
+#else
+    PyErr_SetExcInfo(local_type, local_value, local_tb);
+#endif
+    return 0;
+bad:
+    *type = 0;
+    *value = 0;
+    *tb = 0;
+    Py_XDECREF(local_type);
+    Py_XDECREF(local_value);
+    Py_XDECREF(local_tb);
+    return -1;
+}
+
+/* RaiseException */
+  #if PY_MAJOR_VERSION < 3
+static void __Pyx_Raise(PyObject *type, PyObject *value, PyObject *tb,
+                        CYTHON_UNUSED PyObject *cause) {
+    __Pyx_PyThreadState_declare
+    Py_XINCREF(type);
+    if (!value || value == Py_None)
+        value = NULL;
+    else
+        Py_INCREF(value);
+    if (!tb || tb == Py_None)
+        tb = NULL;
+    else {
+        Py_INCREF(tb);
+        if (!PyTraceBack_Check(tb)) {
+            PyErr_SetString(PyExc_TypeError,
+                "raise: arg 3 must be a traceback or None");
+            goto raise_error;
+        }
+    }
+    if (PyType_Check(type)) {
+#if CYTHON_COMPILING_IN_PYPY
+        if (!value) {
+            Py_INCREF(Py_None);
+            value = Py_None;
+        }
+#endif
+        PyErr_NormalizeException(&type, &value, &tb);
+    } else {
+        if (value) {
+            PyErr_SetString(PyExc_TypeError,
+                "instance exception may not have a separate value");
+            goto raise_error;
+        }
+        value = type;
+        type = (PyObject*) Py_TYPE(type);
+        Py_INCREF(type);
+        if (!PyType_IsSubtype((PyTypeObject *)type, (PyTypeObject *)PyExc_BaseException)) {
+            PyErr_SetString(PyExc_TypeError,
+                "raise: exception class must be a subclass of BaseException");
+            goto raise_error;
+        }
+    }
+    __Pyx_PyThreadState_assign
+    __Pyx_ErrRestore(type, value, tb);
+    return;
+raise_error:
+    Py_XDECREF(value);
+    Py_XDECREF(type);
+    Py_XDECREF(tb);
+    return;
+}
+#else
+static void __Pyx_Raise(PyObject *type, PyObject *value, PyObject *tb, PyObject *cause) {
+    PyObject* owned_instance = NULL;
+    if (tb == Py_None) {
+        tb = 0;
+    } else if (tb && !PyTraceBack_Check(tb)) {
+        PyErr_SetString(PyExc_TypeError,
+            "raise: arg 3 must be a traceback or None");
+        goto bad;
+    }
+    if (value == Py_None)
+        value = 0;
+    if (PyExceptionInstance_Check(type)) {
+        if (value) {
+            PyErr_SetString(PyExc_TypeError,
+                "instance exception may not have a separate value");
+            goto bad;
+        }
+        value = type;
+        type = (PyObject*) Py_TYPE(value);
+    } else if (PyExceptionClass_Check(type)) {
+        PyObject *instance_class = NULL;
+        if (value && PyExceptionInstance_Check(value)) {
+            instance_class = (PyObject*) Py_TYPE(value);
+            if (instance_class != type) {
+                int is_subclass = PyObject_IsSubclass(instance_class, type);
+                if (!is_subclass) {
+                    instance_class = NULL;
+                } else if (unlikely(is_subclass == -1)) {
+                    goto bad;
+                } else {
+                    type = instance_class;
+                }
+            }
+        }
+        if (!instance_class) {
+            PyObject *args;
+            if (!value)
+                args = PyTuple_New(0);
+            else if (PyTuple_Check(value)) {
+                Py_INCREF(value);
+                args = value;
+            } else
+                args = PyTuple_Pack(1, value);
+            if (!args)
+                goto bad;
+            owned_instance = PyObject_Call(type, args, NULL);
+            Py_DECREF(args);
+            if (!owned_instance)
+                goto bad;
+            value = owned_instance;
+            if (!PyExceptionInstance_Check(value)) {
+                PyErr_Format(PyExc_TypeError,
+                             "calling %R should have returned an instance of "
+                             "BaseException, not %R",
+                             type, Py_TYPE(value));
+                goto bad;
+            }
+        }
+    } else {
+        PyErr_SetString(PyExc_TypeError,
+            "raise: exception class must be a subclass of BaseException");
+        goto bad;
+    }
+    if (cause) {
+        PyObject *fixed_cause;
+        if (cause == Py_None) {
+            fixed_cause = NULL;
+        } else if (PyExceptionClass_Check(cause)) {
+            fixed_cause = PyObject_CallObject(cause, NULL);
+            if (fixed_cause == NULL)
+                goto bad;
+        } else if (PyExceptionInstance_Check(cause)) {
+            fixed_cause = cause;
+            Py_INCREF(fixed_cause);
+        } else {
+            PyErr_SetString(PyExc_TypeError,
+                            "exception causes must derive from "
+                            "BaseException");
+            goto bad;
+        }
+        PyException_SetCause(value, fixed_cause);
+    }
+    PyErr_SetObject(type, value);
+    if (tb) {
+#if CYTHON_COMPILING_IN_PYPY
+        PyObject *tmp_type, *tmp_value, *tmp_tb;
+        PyErr_Fetch(&tmp_type, &tmp_value, &tmp_tb);
+        Py_INCREF(tb);
+        PyErr_Restore(tmp_type, tmp_value, tb);
+        Py_XDECREF(tmp_tb);
+#else
+        PyThreadState *tstate = __Pyx_PyThreadState_Current;
+        PyObject* tmp_tb = tstate->curexc_traceback;
+        if (tb != tmp_tb) {
+            Py_INCREF(tb);
+            tstate->curexc_traceback = tb;
+            Py_XDECREF(tmp_tb);
+        }
+#endif
+    }
+bad:
+    Py_XDECREF(owned_instance);
+    return;
+}
+#endif
+
+/* TypeImport */
+  #ifndef __PYX_HAVE_RT_ImportType
+#define __PYX_HAVE_RT_ImportType
+static PyTypeObject *__Pyx_ImportType(PyObject *module, const char *module_name, const char *class_name,
+    size_t size, enum __Pyx_ImportType_CheckSize check_size)
+{
+    PyObject *result = 0;
+    char warning[200];
+    Py_ssize_t basicsize;
+#ifdef Py_LIMITED_API
+    PyObject *py_basicsize;
+#endif
+    result = PyObject_GetAttrString(module, class_name);
+    if (!result)
+        goto bad;
+    if (!PyType_Check(result)) {
+        PyErr_Format(PyExc_TypeError,
+            "%.200s.%.200s is not a type object",
+            module_name, class_name);
+        goto bad;
+    }
+#ifndef Py_LIMITED_API
+    basicsize = ((PyTypeObject *)result)->tp_basicsize;
+#else
+    py_basicsize = PyObject_GetAttrString(result, "__basicsize__");
+    if (!py_basicsize)
+        goto bad;
+    basicsize = PyLong_AsSsize_t(py_basicsize);
+    Py_DECREF(py_basicsize);
+    py_basicsize = 0;
+    if (basicsize == (Py_ssize_t)-1 && PyErr_Occurred())
+        goto bad;
+#endif
+    if ((size_t)basicsize < size) {
+        PyErr_Format(PyExc_ValueError,
+            "%.200s.%.200s size changed, may indicate binary incompatibility. "
+            "Expected %zd from C header, got %zd from PyObject",
+            module_name, class_name, size, basicsize);
+        goto bad;
+    }
+    if (check_size == __Pyx_ImportType_CheckSize_Error && (size_t)basicsize != size) {
+        PyErr_Format(PyExc_ValueError,
+            "%.200s.%.200s size changed, may indicate binary incompatibility. "
+            "Expected %zd from C header, got %zd from PyObject",
+            module_name, class_name, size, basicsize);
+        goto bad;
+    }
+    else if (check_size == __Pyx_ImportType_CheckSize_Warn && (size_t)basicsize > size) {
+        PyOS_snprintf(warning, sizeof(warning),
+            "%s.%s size changed, may indicate binary incompatibility. "
+            "Expected %zd from C header, got %zd from PyObject",
+            module_name, class_name, size, basicsize);
+        if (PyErr_WarnEx(NULL, warning, 0) < 0) goto bad;
+    }
+    return (PyTypeObject *)result;
+bad:
+    Py_XDECREF(result);
+    return NULL;
+}
+#endif
+
+/* Import */
+  static PyObject *__Pyx_Import(PyObject *name, PyObject *from_list, int level) {
+    PyObject *empty_list = 0;
+    PyObject *module = 0;
+    PyObject *global_dict = 0;
+    PyObject *empty_dict = 0;
+    PyObject *list;
+    #if PY_MAJOR_VERSION < 3
+    PyObject *py_import;
+    py_import = __Pyx_PyObject_GetAttrStr(__pyx_b, __pyx_n_s_import);
+    if (!py_import)
+        goto bad;
+    #endif
+    if (from_list)
+        list = from_list;
+    else {
+        empty_list = PyList_New(0);
+        if (!empty_list)
+            goto bad;
+        list = empty_list;
+    }
+    global_dict = PyModule_GetDict(__pyx_m);
+    if (!global_dict)
+        goto bad;
+    empty_dict = PyDict_New();
+    if (!empty_dict)
+        goto bad;
+    {
+        #if PY_MAJOR_VERSION >= 3
+        if (level == -1) {
+            if ((1) && (strchr(__Pyx_MODULE_NAME, '.'))) {
+                module = PyImport_ImportModuleLevelObject(
+                    name, global_dict, empty_dict, list, 1);
+                if (!module) {
+                    if (!PyErr_ExceptionMatches(PyExc_ImportError))
+                        goto bad;
+                    PyErr_Clear();
+                }
+            }
+            level = 0;
+        }
+        #endif
+        if (!module) {
+            #if PY_MAJOR_VERSION < 3
+            PyObject *py_level = PyInt_FromLong(level);
+            if (!py_level)
+                goto bad;
+            module = PyObject_CallFunctionObjArgs(py_import,
+                name, global_dict, empty_dict, list, py_level, (PyObject *)NULL);
+            Py_DECREF(py_level);
+            #else
+            module = PyImport_ImportModuleLevelObject(
+                name, global_dict, empty_dict, list, level);
+            #endif
+        }
+    }
+bad:
+    #if PY_MAJOR_VERSION < 3
+    Py_XDECREF(py_import);
+    #endif
+    Py_XDECREF(empty_list);
+    Py_XDECREF(empty_dict);
+    return module;
+}
+
+/* CLineInTraceback */
+  #ifndef CYTHON_CLINE_IN_TRACEBACK
+static int __Pyx_CLineForTraceback(CYTHON_UNUSED PyThreadState *tstate, int c_line) {
+    PyObject *use_cline;
+    PyObject *ptype, *pvalue, *ptraceback;
+#if CYTHON_COMPILING_IN_CPYTHON
+    PyObject **cython_runtime_dict;
+#endif
+    if (unlikely(!__pyx_cython_runtime)) {
+        return c_line;
+    }
+    __Pyx_ErrFetchInState(tstate, &ptype, &pvalue, &ptraceback);
+#if CYTHON_COMPILING_IN_CPYTHON
+    cython_runtime_dict = _PyObject_GetDictPtr(__pyx_cython_runtime);
+    if (likely(cython_runtime_dict)) {
+        __PYX_PY_DICT_LOOKUP_IF_MODIFIED(
+            use_cline, *cython_runtime_dict,
+            __Pyx_PyDict_GetItemStr(*cython_runtime_dict, __pyx_n_s_cline_in_traceback))
+    } else
+#endif
+    {
+      PyObject *use_cline_obj = __Pyx_PyObject_GetAttrStr(__pyx_cython_runtime, __pyx_n_s_cline_in_traceback);
+      if (use_cline_obj) {
+        use_cline = PyObject_Not(use_cline_obj) ? Py_False : Py_True;
+        Py_DECREF(use_cline_obj);
+      } else {
+        PyErr_Clear();
+        use_cline = NULL;
+      }
+    }
+    if (!use_cline) {
+        c_line = 0;
+        (void) PyObject_SetAttr(__pyx_cython_runtime, __pyx_n_s_cline_in_traceback, Py_False);
+    }
+    else if (use_cline == Py_False || (use_cline != Py_True && PyObject_Not(use_cline) != 0)) {
+        c_line = 0;
+    }
+    __Pyx_ErrRestoreInState(tstate, ptype, pvalue, ptraceback);
+    return c_line;
+}
+#endif
+
+/* CodeObjectCache */
+  static int __pyx_bisect_code_objects(__Pyx_CodeObjectCacheEntry* entries, int count, int code_line) {
+    int start = 0, mid = 0, end = count - 1;
+    if (end >= 0 && code_line > entries[end].code_line) {
+        return count;
+    }
+    while (start < end) {
+        mid = start + (end - start) / 2;
+        if (code_line < entries[mid].code_line) {
+            end = mid;
+        } else if (code_line > entries[mid].code_line) {
+             start = mid + 1;
+        } else {
+            return mid;
+        }
+    }
+    if (code_line <= entries[mid].code_line) {
+        return mid;
+    } else {
+        return mid + 1;
+    }
+}
+static PyCodeObject *__pyx_find_code_object(int code_line) {
+    PyCodeObject* code_object;
+    int pos;
+    if (unlikely(!code_line) || unlikely(!__pyx_code_cache.entries)) {
+        return NULL;
+    }
+    pos = __pyx_bisect_code_objects(__pyx_code_cache.entries, __pyx_code_cache.count, code_line);
+    if (unlikely(pos >= __pyx_code_cache.count) || unlikely(__pyx_code_cache.entries[pos].code_line != code_line)) {
+        return NULL;
+    }
+    code_object = __pyx_code_cache.entries[pos].code_object;
+    Py_INCREF(code_object);
+    return code_object;
+}
+static void __pyx_insert_code_object(int code_line, PyCodeObject* code_object) {
+    int pos, i;
+    __Pyx_CodeObjectCacheEntry* entries = __pyx_code_cache.entries;
+    if (unlikely(!code_line)) {
+        return;
+    }
+    if (unlikely(!entries)) {
+        entries = (__Pyx_CodeObjectCacheEntry*)PyMem_Malloc(64*sizeof(__Pyx_CodeObjectCacheEntry));
+        if (likely(entries)) {
+            __pyx_code_cache.entries = entries;
+            __pyx_code_cache.max_count = 64;
+            __pyx_code_cache.count = 1;
+            entries[0].code_line = code_line;
+            entries[0].code_object = code_object;
+            Py_INCREF(code_object);
+        }
+        return;
+    }
+    pos = __pyx_bisect_code_objects(__pyx_code_cache.entries, __pyx_code_cache.count, code_line);
+    if ((pos < __pyx_code_cache.count) && unlikely(__pyx_code_cache.entries[pos].code_line == code_line)) {
+        PyCodeObject* tmp = entries[pos].code_object;
+        entries[pos].code_object = code_object;
+        Py_DECREF(tmp);
+        return;
+    }
+    if (__pyx_code_cache.count == __pyx_code_cache.max_count) {
+        int new_max = __pyx_code_cache.max_count + 64;
+        entries = (__Pyx_CodeObjectCacheEntry*)PyMem_Realloc(
+            __pyx_code_cache.entries, ((size_t)new_max) * sizeof(__Pyx_CodeObjectCacheEntry));
+        if (unlikely(!entries)) {
+            return;
+        }
+        __pyx_code_cache.entries = entries;
+        __pyx_code_cache.max_count = new_max;
+    }
+    for (i=__pyx_code_cache.count; i>pos; i--) {
+        entries[i] = entries[i-1];
+    }
+    entries[pos].code_line = code_line;
+    entries[pos].code_object = code_object;
+    __pyx_code_cache.count++;
+    Py_INCREF(code_object);
+}
+
+/* AddTraceback */
+  #include "compile.h"
+#include "frameobject.h"
+#include "traceback.h"
+#if PY_VERSION_HEX >= 0x030b00a6
+  #ifndef Py_BUILD_CORE
+    #define Py_BUILD_CORE 1
+  #endif
+  #include "internal/pycore_frame.h"
+#endif
+static PyCodeObject* __Pyx_CreateCodeObjectForTraceback(
+            const char *funcname, int c_line,
+            int py_line, const char *filename) {
+    PyCodeObject *py_code = NULL;
+    PyObject *py_funcname = NULL;
+    #if PY_MAJOR_VERSION < 3
+    PyObject *py_srcfile = NULL;
+    py_srcfile = PyString_FromString(filename);
+    if (!py_srcfile) goto bad;
+    #endif
+    if (c_line) {
+        #if PY_MAJOR_VERSION < 3
+        py_funcname = PyString_FromFormat( "%s (%s:%d)", funcname, __pyx_cfilenm, c_line);
+        if (!py_funcname) goto bad;
+        #else
+        py_funcname = PyUnicode_FromFormat( "%s (%s:%d)", funcname, __pyx_cfilenm, c_line);
+        if (!py_funcname) goto bad;
+        funcname = PyUnicode_AsUTF8(py_funcname);
+        if (!funcname) goto bad;
+        #endif
+    }
+    else {
+        #if PY_MAJOR_VERSION < 3
+        py_funcname = PyString_FromString(funcname);
+        if (!py_funcname) goto bad;
+        #endif
+    }
+    #if PY_MAJOR_VERSION < 3
+    py_code = __Pyx_PyCode_New(
+        0,
+        0,
+        0,
+        0,
+        0,
+        __pyx_empty_bytes, /*PyObject *code,*/
+        __pyx_empty_tuple, /*PyObject *consts,*/
+        __pyx_empty_tuple, /*PyObject *names,*/
+        __pyx_empty_tuple, /*PyObject *varnames,*/
+        __pyx_empty_tuple, /*PyObject *freevars,*/
+        __pyx_empty_tuple, /*PyObject *cellvars,*/
+        py_srcfile,   /*PyObject *filename,*/
+        py_funcname,  /*PyObject *name,*/
+        py_line,
+        __pyx_empty_bytes  /*PyObject *lnotab*/
+    );
+    Py_DECREF(py_srcfile);
+    #else
+    py_code = PyCode_NewEmpty(filename, funcname, py_line);
+    #endif
+    Py_XDECREF(py_funcname);  // XDECREF since it's only set on Py3 if cline
+    return py_code;
+bad:
+    Py_XDECREF(py_funcname);
+    #if PY_MAJOR_VERSION < 3
+    Py_XDECREF(py_srcfile);
+    #endif
+    return NULL;
+}
+static void __Pyx_AddTraceback(const char *funcname, int c_line,
+                               int py_line, const char *filename) {
+    PyCodeObject *py_code = 0;
+    PyFrameObject *py_frame = 0;
+    PyThreadState *tstate = __Pyx_PyThreadState_Current;
+    PyObject *ptype, *pvalue, *ptraceback;
+    if (c_line) {
+        c_line = __Pyx_CLineForTraceback(tstate, c_line);
+    }
+    py_code = __pyx_find_code_object(c_line ? -c_line : py_line);
+    if (!py_code) {
+        __Pyx_ErrFetchInState(tstate, &ptype, &pvalue, &ptraceback);
+        py_code = __Pyx_CreateCodeObjectForTraceback(
+            funcname, c_line, py_line, filename);
+        if (!py_code) {
+            /* If the code object creation fails, then we should clear the
+               fetched exception references and propagate the new exception */
+            Py_XDECREF(ptype);
+            Py_XDECREF(pvalue);
+            Py_XDECREF(ptraceback);
+            goto bad;
+        }
+        __Pyx_ErrRestoreInState(tstate, ptype, pvalue, ptraceback);
+        __pyx_insert_code_object(c_line ? -c_line : py_line, py_code);
+    }
+    py_frame = PyFrame_New(
+        tstate,            /*PyThreadState *tstate,*/
+        py_code,           /*PyCodeObject *code,*/
+        __pyx_d,    /*PyObject *globals,*/
+        0                  /*PyObject *locals*/
+    );
+    if (!py_frame) goto bad;
+    __Pyx_PyFrame_SetLineNumber(py_frame, py_line);
+    PyTraceBack_Here(py_frame);
+bad:
+    Py_XDECREF(py_code);
+    Py_XDECREF(py_frame);
+}
+
+#if PY_MAJOR_VERSION < 3
+static int __Pyx_GetBuffer(PyObject *obj, Py_buffer *view, int flags) {
+    if (PyObject_CheckBuffer(obj)) return PyObject_GetBuffer(obj, view, flags);
+        if (__Pyx_TypeCheck(obj, __pyx_ptype_7cpython_5array_array)) return __pyx_pw_7cpython_5array_5array_1__getbuffer__(obj, view, flags);
+    PyErr_Format(PyExc_TypeError, "'%.200s' does not have the buffer interface", Py_TYPE(obj)->tp_name);
+    return -1;
+}
+static void __Pyx_ReleaseBuffer(Py_buffer *view) {
+    PyObject *obj = view->obj;
+    if (!obj) return;
+    if (PyObject_CheckBuffer(obj)) {
+        PyBuffer_Release(view);
+        return;
+    }
+    if ((0)) {}
+        else if (__Pyx_TypeCheck(obj, __pyx_ptype_7cpython_5array_array)) __pyx_pw_7cpython_5array_5array_3__releasebuffer__(obj, view);
+    view->obj = NULL;
+    Py_DECREF(obj);
+}
+#endif
+
+
+  /* Declarations */
+  #if CYTHON_CCOMPLEX
+  #ifdef __cplusplus
+    static CYTHON_INLINE __pyx_t_float_complex __pyx_t_float_complex_from_parts(float x, float y) {
+      return ::std::complex< float >(x, y);
+    }
+  #else
+    static CYTHON_INLINE __pyx_t_float_complex __pyx_t_float_complex_from_parts(float x, float y) {
+      return x + y*(__pyx_t_float_complex)_Complex_I;
+    }
+  #endif
+#else
+    static CYTHON_INLINE __pyx_t_float_complex __pyx_t_float_complex_from_parts(float x, float y) {
+      __pyx_t_float_complex z;
+      z.real = x;
+      z.imag = y;
+      return z;
+    }
+#endif
+
+/* Arithmetic */
+  #if CYTHON_CCOMPLEX
+#else
+    static CYTHON_INLINE int __Pyx_c_eq_float(__pyx_t_float_complex a, __pyx_t_float_complex b) {
+       return (a.real == b.real) && (a.imag == b.imag);
+    }
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_sum_float(__pyx_t_float_complex a, __pyx_t_float_complex b) {
+        __pyx_t_float_complex z;
+        z.real = a.real + b.real;
+        z.imag = a.imag + b.imag;
+        return z;
+    }
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_diff_float(__pyx_t_float_complex a, __pyx_t_float_complex b) {
+        __pyx_t_float_complex z;
+        z.real = a.real - b.real;
+        z.imag = a.imag - b.imag;
+        return z;
+    }
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_prod_float(__pyx_t_float_complex a, __pyx_t_float_complex b) {
+        __pyx_t_float_complex z;
+        z.real = a.real * b.real - a.imag * b.imag;
+        z.imag = a.real * b.imag + a.imag * b.real;
+        return z;
+    }
+    #if 1
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_quot_float(__pyx_t_float_complex a, __pyx_t_float_complex b) {
+        if (b.imag == 0) {
+            return __pyx_t_float_complex_from_parts(a.real / b.real, a.imag / b.real);
+        } else if (fabsf(b.real) >= fabsf(b.imag)) {
+            if (b.real == 0 && b.imag == 0) {
+                return __pyx_t_float_complex_from_parts(a.real / b.real, a.imag / b.imag);
+            } else {
+                float r = b.imag / b.real;
+                float s = (float)(1.0) / (b.real + b.imag * r);
+                return __pyx_t_float_complex_from_parts(
+                    (a.real + a.imag * r) * s, (a.imag - a.real * r) * s);
+            }
+        } else {
+            float r = b.real / b.imag;
+            float s = (float)(1.0) / (b.imag + b.real * r);
+            return __pyx_t_float_complex_from_parts(
+                (a.real * r + a.imag) * s, (a.imag * r - a.real) * s);
+        }
+    }
+    #else
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_quot_float(__pyx_t_float_complex a, __pyx_t_float_complex b) {
+        if (b.imag == 0) {
+            return __pyx_t_float_complex_from_parts(a.real / b.real, a.imag / b.real);
+        } else {
+            float denom = b.real * b.real + b.imag * b.imag;
+            return __pyx_t_float_complex_from_parts(
+                (a.real * b.real + a.imag * b.imag) / denom,
+                (a.imag * b.real - a.real * b.imag) / denom);
+        }
+    }
+    #endif
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_neg_float(__pyx_t_float_complex a) {
+        __pyx_t_float_complex z;
+        z.real = -a.real;
+        z.imag = -a.imag;
+        return z;
+    }
+    static CYTHON_INLINE int __Pyx_c_is_zero_float(__pyx_t_float_complex a) {
+       return (a.real == 0) && (a.imag == 0);
+    }
+    static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_conj_float(__pyx_t_float_complex a) {
+        __pyx_t_float_complex z;
+        z.real =  a.real;
+        z.imag = -a.imag;
+        return z;
+    }
+    #if 1
+        static CYTHON_INLINE float __Pyx_c_abs_float(__pyx_t_float_complex z) {
+          #if !defined(HAVE_HYPOT) || defined(_MSC_VER)
+            return sqrtf(z.real*z.real + z.imag*z.imag);
+          #else
+            return hypotf(z.real, z.imag);
+          #endif
+        }
+        static CYTHON_INLINE __pyx_t_float_complex __Pyx_c_pow_float(__pyx_t_float_complex a, __pyx_t_float_complex b) {
+            __pyx_t_float_complex z;
+            float r, lnr, theta, z_r, z_theta;
+            if (b.imag == 0 && b.real == (int)b.real) {
+                if (b.real < 0) {
+                    float denom = a.real * a.real + a.imag * a.imag;
+                    a.real = a.real / denom;
+                    a.imag = -a.imag / denom;
+                    b.real = -b.real;
+                }
+                switch ((int)b.real) {
+                    case 0:
+                        z.real = 1;
+                        z.imag = 0;
+                        return z;
+                    case 1:
+                        return a;
+                    case 2:
+                        return __Pyx_c_prod_float(a, a);
+                    case 3:
+                        z = __Pyx_c_prod_float(a, a);
+                        return __Pyx_c_prod_float(z, a);
+                    case 4:
+                        z = __Pyx_c_prod_float(a, a);
+                        return __Pyx_c_prod_float(z, z);
+                }
+            }
+            if (a.imag == 0) {
+                if (a.real == 0) {
+                    return a;
+                } else if ((b.imag == 0) && (a.real >= 0)) {
+                    z.real = powf(a.real, b.real);
+                    z.imag = 0;
+                    return z;
+                } else if (a.real > 0) {
+                    r = a.real;
+                    theta = 0;
+                } else {
+                    r = -a.real;
+                    theta = atan2f(0.0, -1.0);
+                }
+            } else {
+                r = __Pyx_c_abs_float(a);
+                theta = atan2f(a.imag, a.real);
+            }
+            lnr = logf(r);
+            z_r = expf(lnr * b.real - theta * b.imag);
+            z_theta = theta * b.real + lnr * b.imag;
+            z.real = z_r * cosf(z_theta);
+            z.imag = z_r * sinf(z_theta);
+            return z;
+        }
+    #endif
+#endif
+
+/* Declarations */
+  #if CYTHON_CCOMPLEX
+  #ifdef __cplusplus
+    static CYTHON_INLINE __pyx_t_double_complex __pyx_t_double_complex_from_parts(double x, double y) {
+      return ::std::complex< double >(x, y);
+    }
+  #else
+    static CYTHON_INLINE __pyx_t_double_complex __pyx_t_double_complex_from_parts(double x, double y) {
+      return x + y*(__pyx_t_double_complex)_Complex_I;
+    }
+  #endif
+#else
+    static CYTHON_INLINE __pyx_t_double_complex __pyx_t_double_complex_from_parts(double x, double y) {
+      __pyx_t_double_complex z;
+      z.real = x;
+      z.imag = y;
+      return z;
+    }
+#endif
+
+/* Arithmetic */
+  #if CYTHON_CCOMPLEX
+#else
+    static CYTHON_INLINE int __Pyx_c_eq_double(__pyx_t_double_complex a, __pyx_t_double_complex b) {
+       return (a.real == b.real) && (a.imag == b.imag);
+    }
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_sum_double(__pyx_t_double_complex a, __pyx_t_double_complex b) {
+        __pyx_t_double_complex z;
+        z.real = a.real + b.real;
+        z.imag = a.imag + b.imag;
+        return z;
+    }
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_diff_double(__pyx_t_double_complex a, __pyx_t_double_complex b) {
+        __pyx_t_double_complex z;
+        z.real = a.real - b.real;
+        z.imag = a.imag - b.imag;
+        return z;
+    }
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_prod_double(__pyx_t_double_complex a, __pyx_t_double_complex b) {
+        __pyx_t_double_complex z;
+        z.real = a.real * b.real - a.imag * b.imag;
+        z.imag = a.real * b.imag + a.imag * b.real;
+        return z;
+    }
+    #if 1
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_quot_double(__pyx_t_double_complex a, __pyx_t_double_complex b) {
+        if (b.imag == 0) {
+            return __pyx_t_double_complex_from_parts(a.real / b.real, a.imag / b.real);
+        } else if (fabs(b.real) >= fabs(b.imag)) {
+            if (b.real == 0 && b.imag == 0) {
+                return __pyx_t_double_complex_from_parts(a.real / b.real, a.imag / b.imag);
+            } else {
+                double r = b.imag / b.real;
+                double s = (double)(1.0) / (b.real + b.imag * r);
+                return __pyx_t_double_complex_from_parts(
+                    (a.real + a.imag * r) * s, (a.imag - a.real * r) * s);
+            }
+        } else {
+            double r = b.real / b.imag;
+            double s = (double)(1.0) / (b.imag + b.real * r);
+            return __pyx_t_double_complex_from_parts(
+                (a.real * r + a.imag) * s, (a.imag * r - a.real) * s);
+        }
+    }
+    #else
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_quot_double(__pyx_t_double_complex a, __pyx_t_double_complex b) {
+        if (b.imag == 0) {
+            return __pyx_t_double_complex_from_parts(a.real / b.real, a.imag / b.real);
+        } else {
+            double denom = b.real * b.real + b.imag * b.imag;
+            return __pyx_t_double_complex_from_parts(
+                (a.real * b.real + a.imag * b.imag) / denom,
+                (a.imag * b.real - a.real * b.imag) / denom);
+        }
+    }
+    #endif
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_neg_double(__pyx_t_double_complex a) {
+        __pyx_t_double_complex z;
+        z.real = -a.real;
+        z.imag = -a.imag;
+        return z;
+    }
+    static CYTHON_INLINE int __Pyx_c_is_zero_double(__pyx_t_double_complex a) {
+       return (a.real == 0) && (a.imag == 0);
+    }
+    static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_conj_double(__pyx_t_double_complex a) {
+        __pyx_t_double_complex z;
+        z.real =  a.real;
+        z.imag = -a.imag;
+        return z;
+    }
+    #if 1
+        static CYTHON_INLINE double __Pyx_c_abs_double(__pyx_t_double_complex z) {
+          #if !defined(HAVE_HYPOT) || defined(_MSC_VER)
+            return sqrt(z.real*z.real + z.imag*z.imag);
+          #else
+            return hypot(z.real, z.imag);
+          #endif
+        }
+        static CYTHON_INLINE __pyx_t_double_complex __Pyx_c_pow_double(__pyx_t_double_complex a, __pyx_t_double_complex b) {
+            __pyx_t_double_complex z;
+            double r, lnr, theta, z_r, z_theta;
+            if (b.imag == 0 && b.real == (int)b.real) {
+                if (b.real < 0) {
+                    double denom = a.real * a.real + a.imag * a.imag;
+                    a.real = a.real / denom;
+                    a.imag = -a.imag / denom;
+                    b.real = -b.real;
+                }
+                switch ((int)b.real) {
+                    case 0:
+                        z.real = 1;
+                        z.imag = 0;
+                        return z;
+                    case 1:
+                        return a;
+                    case 2:
+                        return __Pyx_c_prod_double(a, a);
+                    case 3:
+                        z = __Pyx_c_prod_double(a, a);
+                        return __Pyx_c_prod_double(z, a);
+                    case 4:
+                        z = __Pyx_c_prod_double(a, a);
+                        return __Pyx_c_prod_double(z, z);
+                }
+            }
+            if (a.imag == 0) {
+                if (a.real == 0) {
+                    return a;
+                } else if ((b.imag == 0) && (a.real >= 0)) {
+                    z.real = pow(a.real, b.real);
+                    z.imag = 0;
+                    return z;
+                } else if (a.real > 0) {
+                    r = a.real;
+                    theta = 0;
+                } else {
+                    r = -a.real;
+                    theta = atan2(0.0, -1.0);
+                }
+            } else {
+                r = __Pyx_c_abs_double(a);
+                theta = atan2(a.imag, a.real);
+            }
+            lnr = log(r);
+            z_r = exp(lnr * b.real - theta * b.imag);
+            z_theta = theta * b.real + lnr * b.imag;
+            z.real = z_r * cos(z_theta);
+            z.imag = z_r * sin(z_theta);
+            return z;
+        }
+    #endif
+#endif
+
+/* CIntToPy */
+  static CYTHON_INLINE PyObject* __Pyx_PyInt_From_int(int value) {
+#ifdef __Pyx_HAS_GCC_DIAGNOSTIC
+#pragma GCC diagnostic push
+#pragma GCC diagnostic ignored "-Wconversion"
+#endif
+    const int neg_one = (int) -1, const_zero = (int) 0;
+#ifdef __Pyx_HAS_GCC_DIAGNOSTIC
+#pragma GCC diagnostic pop
+#endif
+    const int is_unsigned = neg_one > const_zero;
+    if (is_unsigned) {
+        if (sizeof(int) < sizeof(long)) {
+            return PyInt_FromLong((long) value);
+        } else if (sizeof(int) <= sizeof(unsigned long)) {
+            return PyLong_FromUnsignedLong((unsigned long) value);
+#ifdef HAVE_LONG_LONG
+        } else if (sizeof(int) <= sizeof(unsigned PY_LONG_LONG)) {
+            return PyLong_FromUnsignedLongLong((unsigned PY_LONG_LONG) value);
+#endif
+        }
+    } else {
+        if (sizeof(int) <= sizeof(long)) {
+            return PyInt_FromLong((long) value);
+#ifdef HAVE_LONG_LONG
+        } else if (sizeof(int) <= sizeof(PY_LONG_LONG)) {
+            return PyLong_FromLongLong((PY_LONG_LONG) value);
+#endif
+        }
+    }
+    {
+        int one = 1; int little = (int)*(unsigned char *)&one;
+        unsigned char *bytes = (unsigned char *)&value;
+        return _PyLong_FromByteArray(bytes, sizeof(int),
+                                     little, !is_unsigned);
+    }
+}
+
+/* CIntToPy */
+  static CYTHON_INLINE PyObject* __Pyx_PyInt_From_long(long value) {
+#ifdef __Pyx_HAS_GCC_DIAGNOSTIC
+#pragma GCC diagnostic push
+#pragma GCC diagnostic ignored "-Wconversion"
+#endif
+    const long neg_one = (long) -1, const_zero = (long) 0;
+#ifdef __Pyx_HAS_GCC_DIAGNOSTIC
+#pragma GCC diagnostic pop
+#endif
+    const int is_unsigned = neg_one > const_zero;
+    if (is_unsigned) {
+        if (sizeof(long) < sizeof(long)) {
+            return PyInt_FromLong((long) value);
+        } else if (sizeof(long) <= sizeof(unsigned long)) {
+            return PyLong_FromUnsignedLong((unsigned long) value);
+#ifdef HAVE_LONG_LONG
+        } else if (sizeof(long) <= sizeof(unsigned PY_LONG_LONG)) {
+            return PyLong_FromUnsignedLongLong((unsigned PY_LONG_LONG) value);
+#endif
+        }
+    } else {
+        if (sizeof(long) <= sizeof(long)) {
+            return PyInt_FromLong((long) value);
+#ifdef HAVE_LONG_LONG
+        } else if (sizeof(long) <= sizeof(PY_LONG_LONG)) {
+            return PyLong_FromLongLong((PY_LONG_LONG) value);
+#endif
+        }
+    }
+    {
+        int one = 1; int little = (int)*(unsigned char *)&one;
+        unsigned char *bytes = (unsigned char *)&value;
+        return _PyLong_FromByteArray(bytes, sizeof(long),
+                                     little, !is_unsigned);
+    }
+}
+
+/* CIntFromPyVerify */
+  #define __PYX_VERIFY_RETURN_INT(target_type, func_type, func_value)\
+    __PYX__VERIFY_RETURN_INT(target_type, func_type, func_value, 0)
+#define __PYX_VERIFY_RETURN_INT_EXC(target_type, func_type, func_value)\
+    __PYX__VERIFY_RETURN_INT(target_type, func_type, func_value, 1)
+#define __PYX__VERIFY_RETURN_INT(target_type, func_type, func_value, exc)\
+    {\
+        func_type value = func_value;\
+        if (sizeof(target_type) < sizeof(func_type)) {\
+            if (unlikely(value != (func_type) (target_type) value)) {\
+                func_type zero = 0;\
+                if (exc && unlikely(value == (func_type)-1 && PyErr_Occurred()))\
+                    return (target_type) -1;\
+                if (is_unsigned && unlikely(value < zero))\
+                    goto raise_neg_overflow;\
+                else\
+                    goto raise_overflow;\
+            }\
+        }\
+        return (target_type) value;\
+    }
+
+/* CIntFromPy */
+  static CYTHON_INLINE long __Pyx_PyInt_As_long(PyObject *x) {
+#ifdef __Pyx_HAS_GCC_DIAGNOSTIC
+#pragma GCC diagnostic push
+#pragma GCC diagnostic ignored "-Wconversion"
+#endif
+    const long neg_one = (long) -1, const_zero = (long) 0;
+#ifdef __Pyx_HAS_GCC_DIAGNOSTIC
+#pragma GCC diagnostic pop
+#endif
+    const int is_unsigned = neg_one > const_zero;
+#if PY_MAJOR_VERSION < 3
+    if (likely(PyInt_Check(x))) {
+        if (sizeof(long) < sizeof(long)) {
+            __PYX_VERIFY_RETURN_INT(long, long, PyInt_AS_LONG(x))
+        } else {
+            long val = PyInt_AS_LONG(x);
+            if (is_unsigned && unlikely(val < 0)) {
+                goto raise_neg_overflow;
+            }
+            return (long) val;
+        }
+    } else
+#endif
+    if (likely(PyLong_Check(x))) {
+        if (is_unsigned) {
+#if CYTHON_USE_PYLONG_INTERNALS
+            const digit* digits = ((PyLongObject*)x)->ob_digit;
+            switch (Py_SIZE(x)) {
+                case  0: return (long) 0;
+                case  1: __PYX_VERIFY_RETURN_INT(long, digit, digits[0])
+                case 2:
+                    if (8 * sizeof(long) > 1 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 2 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(long, unsigned long, (((((unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(long) >= 2 * PyLong_SHIFT) {
+                            return (long) (((((long)digits[1]) << PyLong_SHIFT) | (long)digits[0]));
+                        }
+                    }
+                    break;
+                case 3:
+                    if (8 * sizeof(long) > 2 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 3 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(long, unsigned long, (((((((unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(long) >= 3 * PyLong_SHIFT) {
+                            return (long) (((((((long)digits[2]) << PyLong_SHIFT) | (long)digits[1]) << PyLong_SHIFT) | (long)digits[0]));
+                        }
+                    }
+                    break;
+                case 4:
+                    if (8 * sizeof(long) > 3 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 4 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(long, unsigned long, (((((((((unsigned long)digits[3]) << PyLong_SHIFT) | (unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(long) >= 4 * PyLong_SHIFT) {
+                            return (long) (((((((((long)digits[3]) << PyLong_SHIFT) | (long)digits[2]) << PyLong_SHIFT) | (long)digits[1]) << PyLong_SHIFT) | (long)digits[0]));
+                        }
+                    }
+                    break;
+            }
+#endif
+#if CYTHON_COMPILING_IN_CPYTHON
+            if (unlikely(Py_SIZE(x) < 0)) {
+                goto raise_neg_overflow;
+            }
+#else
+            {
+                int result = PyObject_RichCompareBool(x, Py_False, Py_LT);
+                if (unlikely(result < 0))
+                    return (long) -1;
+                if (unlikely(result == 1))
+                    goto raise_neg_overflow;
+            }
+#endif
+            if (sizeof(long) <= sizeof(unsigned long)) {
+                __PYX_VERIFY_RETURN_INT_EXC(long, unsigned long, PyLong_AsUnsignedLong(x))
+#ifdef HAVE_LONG_LONG
+            } else if (sizeof(long) <= sizeof(unsigned PY_LONG_LONG)) {
+                __PYX_VERIFY_RETURN_INT_EXC(long, unsigned PY_LONG_LONG, PyLong_AsUnsignedLongLong(x))
+#endif
+            }
+        } else {
+#if CYTHON_USE_PYLONG_INTERNALS
+            const digit* digits = ((PyLongObject*)x)->ob_digit;
+            switch (Py_SIZE(x)) {
+                case  0: return (long) 0;
+                case -1: __PYX_VERIFY_RETURN_INT(long, sdigit, (sdigit) (-(sdigit)digits[0]))
+                case  1: __PYX_VERIFY_RETURN_INT(long,  digit, +digits[0])
+                case -2:
+                    if (8 * sizeof(long) - 1 > 1 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 2 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(long, long, -(long) (((((unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(long) - 1 > 2 * PyLong_SHIFT) {
+                            return (long) (((long)-1)*(((((long)digits[1]) << PyLong_SHIFT) | (long)digits[0])));
+                        }
+                    }
+                    break;
+                case 2:
+                    if (8 * sizeof(long) > 1 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 2 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(long, unsigned long, (((((unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(long) - 1 > 2 * PyLong_SHIFT) {
+                            return (long) ((((((long)digits[1]) << PyLong_SHIFT) | (long)digits[0])));
+                        }
+                    }
+                    break;
+                case -3:
+                    if (8 * sizeof(long) - 1 > 2 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 3 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(long, long, -(long) (((((((unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(long) - 1 > 3 * PyLong_SHIFT) {
+                            return (long) (((long)-1)*(((((((long)digits[2]) << PyLong_SHIFT) | (long)digits[1]) << PyLong_SHIFT) | (long)digits[0])));
+                        }
+                    }
+                    break;
+                case 3:
+                    if (8 * sizeof(long) > 2 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 3 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(long, unsigned long, (((((((unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(long) - 1 > 3 * PyLong_SHIFT) {
+                            return (long) ((((((((long)digits[2]) << PyLong_SHIFT) | (long)digits[1]) << PyLong_SHIFT) | (long)digits[0])));
+                        }
+                    }
+                    break;
+                case -4:
+                    if (8 * sizeof(long) - 1 > 3 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 4 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(long, long, -(long) (((((((((unsigned long)digits[3]) << PyLong_SHIFT) | (unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(long) - 1 > 4 * PyLong_SHIFT) {
+                            return (long) (((long)-1)*(((((((((long)digits[3]) << PyLong_SHIFT) | (long)digits[2]) << PyLong_SHIFT) | (long)digits[1]) << PyLong_SHIFT) | (long)digits[0])));
+                        }
+                    }
+                    break;
+                case 4:
+                    if (8 * sizeof(long) > 3 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 4 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(long, unsigned long, (((((((((unsigned long)digits[3]) << PyLong_SHIFT) | (unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(long) - 1 > 4 * PyLong_SHIFT) {
+                            return (long) ((((((((((long)digits[3]) << PyLong_SHIFT) | (long)digits[2]) << PyLong_SHIFT) | (long)digits[1]) << PyLong_SHIFT) | (long)digits[0])));
+                        }
+                    }
+                    break;
+            }
+#endif
+            if (sizeof(long) <= sizeof(long)) {
+                __PYX_VERIFY_RETURN_INT_EXC(long, long, PyLong_AsLong(x))
+#ifdef HAVE_LONG_LONG
+            } else if (sizeof(long) <= sizeof(PY_LONG_LONG)) {
+                __PYX_VERIFY_RETURN_INT_EXC(long, PY_LONG_LONG, PyLong_AsLongLong(x))
+#endif
+            }
+        }
+        {
+#if CYTHON_COMPILING_IN_PYPY && !defined(_PyLong_AsByteArray)
+            PyErr_SetString(PyExc_RuntimeError,
+                            "_PyLong_AsByteArray() not available in PyPy, cannot convert large numbers");
+#else
+            long val;
+            PyObject *v = __Pyx_PyNumber_IntOrLong(x);
+ #if PY_MAJOR_VERSION < 3
+            if (likely(v) && !PyLong_Check(v)) {
+                PyObject *tmp = v;
+                v = PyNumber_Long(tmp);
+                Py_DECREF(tmp);
+            }
+ #endif
+            if (likely(v)) {
+                int one = 1; int is_little = (int)*(unsigned char *)&one;
+                unsigned char *bytes = (unsigned char *)&val;
+                int ret = _PyLong_AsByteArray((PyLongObject *)v,
+                                              bytes, sizeof(val),
+                                              is_little, !is_unsigned);
+                Py_DECREF(v);
+                if (likely(!ret))
+                    return val;
+            }
+#endif
+            return (long) -1;
+        }
+    } else {
+        long val;
+        PyObject *tmp = __Pyx_PyNumber_IntOrLong(x);
+        if (!tmp) return (long) -1;
+        val = __Pyx_PyInt_As_long(tmp);
+        Py_DECREF(tmp);
+        return val;
+    }
+raise_overflow:
+    PyErr_SetString(PyExc_OverflowError,
+        "value too large to convert to long");
+    return (long) -1;
+raise_neg_overflow:
+    PyErr_SetString(PyExc_OverflowError,
+        "can't convert negative value to long");
+    return (long) -1;
+}
+
+/* CIntFromPy */
+  static CYTHON_INLINE int __Pyx_PyInt_As_int(PyObject *x) {
+#ifdef __Pyx_HAS_GCC_DIAGNOSTIC
+#pragma GCC diagnostic push
+#pragma GCC diagnostic ignored "-Wconversion"
+#endif
+    const int neg_one = (int) -1, const_zero = (int) 0;
+#ifdef __Pyx_HAS_GCC_DIAGNOSTIC
+#pragma GCC diagnostic pop
+#endif
+    const int is_unsigned = neg_one > const_zero;
+#if PY_MAJOR_VERSION < 3
+    if (likely(PyInt_Check(x))) {
+        if (sizeof(int) < sizeof(long)) {
+            __PYX_VERIFY_RETURN_INT(int, long, PyInt_AS_LONG(x))
+        } else {
+            long val = PyInt_AS_LONG(x);
+            if (is_unsigned && unlikely(val < 0)) {
+                goto raise_neg_overflow;
+            }
+            return (int) val;
+        }
+    } else
+#endif
+    if (likely(PyLong_Check(x))) {
+        if (is_unsigned) {
+#if CYTHON_USE_PYLONG_INTERNALS
+            const digit* digits = ((PyLongObject*)x)->ob_digit;
+            switch (Py_SIZE(x)) {
+                case  0: return (int) 0;
+                case  1: __PYX_VERIFY_RETURN_INT(int, digit, digits[0])
+                case 2:
+                    if (8 * sizeof(int) > 1 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 2 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(int, unsigned long, (((((unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(int) >= 2 * PyLong_SHIFT) {
+                            return (int) (((((int)digits[1]) << PyLong_SHIFT) | (int)digits[0]));
+                        }
+                    }
+                    break;
+                case 3:
+                    if (8 * sizeof(int) > 2 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 3 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(int, unsigned long, (((((((unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(int) >= 3 * PyLong_SHIFT) {
+                            return (int) (((((((int)digits[2]) << PyLong_SHIFT) | (int)digits[1]) << PyLong_SHIFT) | (int)digits[0]));
+                        }
+                    }
+                    break;
+                case 4:
+                    if (8 * sizeof(int) > 3 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 4 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(int, unsigned long, (((((((((unsigned long)digits[3]) << PyLong_SHIFT) | (unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(int) >= 4 * PyLong_SHIFT) {
+                            return (int) (((((((((int)digits[3]) << PyLong_SHIFT) | (int)digits[2]) << PyLong_SHIFT) | (int)digits[1]) << PyLong_SHIFT) | (int)digits[0]));
+                        }
+                    }
+                    break;
+            }
+#endif
+#if CYTHON_COMPILING_IN_CPYTHON
+            if (unlikely(Py_SIZE(x) < 0)) {
+                goto raise_neg_overflow;
+            }
+#else
+            {
+                int result = PyObject_RichCompareBool(x, Py_False, Py_LT);
+                if (unlikely(result < 0))
+                    return (int) -1;
+                if (unlikely(result == 1))
+                    goto raise_neg_overflow;
+            }
+#endif
+            if (sizeof(int) <= sizeof(unsigned long)) {
+                __PYX_VERIFY_RETURN_INT_EXC(int, unsigned long, PyLong_AsUnsignedLong(x))
+#ifdef HAVE_LONG_LONG
+            } else if (sizeof(int) <= sizeof(unsigned PY_LONG_LONG)) {
+                __PYX_VERIFY_RETURN_INT_EXC(int, unsigned PY_LONG_LONG, PyLong_AsUnsignedLongLong(x))
+#endif
+            }
+        } else {
+#if CYTHON_USE_PYLONG_INTERNALS
+            const digit* digits = ((PyLongObject*)x)->ob_digit;
+            switch (Py_SIZE(x)) {
+                case  0: return (int) 0;
+                case -1: __PYX_VERIFY_RETURN_INT(int, sdigit, (sdigit) (-(sdigit)digits[0]))
+                case  1: __PYX_VERIFY_RETURN_INT(int,  digit, +digits[0])
+                case -2:
+                    if (8 * sizeof(int) - 1 > 1 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 2 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(int, long, -(long) (((((unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(int) - 1 > 2 * PyLong_SHIFT) {
+                            return (int) (((int)-1)*(((((int)digits[1]) << PyLong_SHIFT) | (int)digits[0])));
+                        }
+                    }
+                    break;
+                case 2:
+                    if (8 * sizeof(int) > 1 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 2 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(int, unsigned long, (((((unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(int) - 1 > 2 * PyLong_SHIFT) {
+                            return (int) ((((((int)digits[1]) << PyLong_SHIFT) | (int)digits[0])));
+                        }
+                    }
+                    break;
+                case -3:
+                    if (8 * sizeof(int) - 1 > 2 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 3 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(int, long, -(long) (((((((unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(int) - 1 > 3 * PyLong_SHIFT) {
+                            return (int) (((int)-1)*(((((((int)digits[2]) << PyLong_SHIFT) | (int)digits[1]) << PyLong_SHIFT) | (int)digits[0])));
+                        }
+                    }
+                    break;
+                case 3:
+                    if (8 * sizeof(int) > 2 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 3 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(int, unsigned long, (((((((unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(int) - 1 > 3 * PyLong_SHIFT) {
+                            return (int) ((((((((int)digits[2]) << PyLong_SHIFT) | (int)digits[1]) << PyLong_SHIFT) | (int)digits[0])));
+                        }
+                    }
+                    break;
+                case -4:
+                    if (8 * sizeof(int) - 1 > 3 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 4 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(int, long, -(long) (((((((((unsigned long)digits[3]) << PyLong_SHIFT) | (unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(int) - 1 > 4 * PyLong_SHIFT) {
+                            return (int) (((int)-1)*(((((((((int)digits[3]) << PyLong_SHIFT) | (int)digits[2]) << PyLong_SHIFT) | (int)digits[1]) << PyLong_SHIFT) | (int)digits[0])));
+                        }
+                    }
+                    break;
+                case 4:
+                    if (8 * sizeof(int) > 3 * PyLong_SHIFT) {
+                        if (8 * sizeof(unsigned long) > 4 * PyLong_SHIFT) {
+                            __PYX_VERIFY_RETURN_INT(int, unsigned long, (((((((((unsigned long)digits[3]) << PyLong_SHIFT) | (unsigned long)digits[2]) << PyLong_SHIFT) | (unsigned long)digits[1]) << PyLong_SHIFT) | (unsigned long)digits[0])))
+                        } else if (8 * sizeof(int) - 1 > 4 * PyLong_SHIFT) {
+                            return (int) ((((((((((int)digits[3]) << PyLong_SHIFT) | (int)digits[2]) << PyLong_SHIFT) | (int)digits[1]) << PyLong_SHIFT) | (int)digits[0])));
+                        }
+                    }
+                    break;
+            }
+#endif
+            if (sizeof(int) <= sizeof(long)) {
+                __PYX_VERIFY_RETURN_INT_EXC(int, long, PyLong_AsLong(x))
+#ifdef HAVE_LONG_LONG
+            } else if (sizeof(int) <= sizeof(PY_LONG_LONG)) {
+                __PYX_VERIFY_RETURN_INT_EXC(int, PY_LONG_LONG, PyLong_AsLongLong(x))
+#endif
+            }
+        }
+        {
+#if CYTHON_COMPILING_IN_PYPY && !defined(_PyLong_AsByteArray)
+            PyErr_SetString(PyExc_RuntimeError,
+                            "_PyLong_AsByteArray() not available in PyPy, cannot convert large numbers");
+#else
+            int val;
+            PyObject *v = __Pyx_PyNumber_IntOrLong(x);
+ #if PY_MAJOR_VERSION < 3
+            if (likely(v) && !PyLong_Check(v)) {
+                PyObject *tmp = v;
+                v = PyNumber_Long(tmp);
+                Py_DECREF(tmp);
+            }
+ #endif
+            if (likely(v)) {
+                int one = 1; int is_little = (int)*(unsigned char *)&one;
+                unsigned char *bytes = (unsigned char *)&val;
+                int ret = _PyLong_AsByteArray((PyLongObject *)v,
+                                              bytes, sizeof(val),
+                                              is_little, !is_unsigned);
+                Py_DECREF(v);
+                if (likely(!ret))
+                    return val;
+            }
+#endif
+            return (int) -1;
+        }
+    } else {
+        int val;
+        PyObject *tmp = __Pyx_PyNumber_IntOrLong(x);
+        if (!tmp) return (int) -1;
+        val = __Pyx_PyInt_As_int(tmp);
+        Py_DECREF(tmp);
+        return val;
+    }
+raise_overflow:
+    PyErr_SetString(PyExc_OverflowError,
+        "value too large to convert to int");
+    return (int) -1;
+raise_neg_overflow:
+    PyErr_SetString(PyExc_OverflowError,
+        "can't convert negative value to int");
+    return (int) -1;
+}
+
+/* FastTypeChecks */
+  #if CYTHON_COMPILING_IN_CPYTHON
+static int __Pyx_InBases(PyTypeObject *a, PyTypeObject *b) {
+    while (a) {
+        a = a->tp_base;
+        if (a == b)
+            return 1;
+    }
+    return b == &PyBaseObject_Type;
+}
+static CYTHON_INLINE int __Pyx_IsSubtype(PyTypeObject *a, PyTypeObject *b) {
+    PyObject *mro;
+    if (a == b) return 1;
+    mro = a->tp_mro;
+    if (likely(mro)) {
+        Py_ssize_t i, n;
+        n = PyTuple_GET_SIZE(mro);
+        for (i = 0; i < n; i++) {
+            if (PyTuple_GET_ITEM(mro, i) == (PyObject *)b)
+                return 1;
+        }
+        return 0;
+    }
+    return __Pyx_InBases(a, b);
+}
+#if PY_MAJOR_VERSION == 2
+static int __Pyx_inner_PyErr_GivenExceptionMatches2(PyObject *err, PyObject* exc_type1, PyObject* exc_type2) {
+    PyObject *exception, *value, *tb;
+    int res;
+    __Pyx_PyThreadState_declare
+    __Pyx_PyThreadState_assign
+    __Pyx_ErrFetch(&exception, &value, &tb);
+    res = exc_type1 ? PyObject_IsSubclass(err, exc_type1) : 0;
+    if (unlikely(res == -1)) {
+        PyErr_WriteUnraisable(err);
+        res = 0;
+    }
+    if (!res) {
+        res = PyObject_IsSubclass(err, exc_type2);
+        if (unlikely(res == -1)) {
+            PyErr_WriteUnraisable(err);
+            res = 0;
+        }
+    }
+    __Pyx_ErrRestore(exception, value, tb);
+    return res;
+}
+#else
+static CYTHON_INLINE int __Pyx_inner_PyErr_GivenExceptionMatches2(PyObject *err, PyObject* exc_type1, PyObject *exc_type2) {
+    int res = exc_type1 ? __Pyx_IsSubtype((PyTypeObject*)err, (PyTypeObject*)exc_type1) : 0;
+    if (!res) {
+        res = __Pyx_IsSubtype((PyTypeObject*)err, (PyTypeObject*)exc_type2);
+    }
+    return res;
+}
+#endif
+static int __Pyx_PyErr_GivenExceptionMatchesTuple(PyObject *exc_type, PyObject *tuple) {
+    Py_ssize_t i, n;
+    assert(PyExceptionClass_Check(exc_type));
+    n = PyTuple_GET_SIZE(tuple);
+#if PY_MAJOR_VERSION >= 3
+    for (i=0; i<n; i++) {
+        if (exc_type == PyTuple_GET_ITEM(tuple, i)) return 1;
+    }
+#endif
+    for (i=0; i<n; i++) {
+        PyObject *t = PyTuple_GET_ITEM(tuple, i);
+        #if PY_MAJOR_VERSION < 3
+        if (likely(exc_type == t)) return 1;
+        #endif
+        if (likely(PyExceptionClass_Check(t))) {
+            if (__Pyx_inner_PyErr_GivenExceptionMatches2(exc_type, NULL, t)) return 1;
+        } else {
+        }
+    }
+    return 0;
+}
+static CYTHON_INLINE int __Pyx_PyErr_GivenExceptionMatches(PyObject *err, PyObject* exc_type) {
+    if (likely(err == exc_type)) return 1;
+    if (likely(PyExceptionClass_Check(err))) {
+        if (likely(PyExceptionClass_Check(exc_type))) {
+            return __Pyx_inner_PyErr_GivenExceptionMatches2(err, NULL, exc_type);
+        } else if (likely(PyTuple_Check(exc_type))) {
+            return __Pyx_PyErr_GivenExceptionMatchesTuple(err, exc_type);
+        } else {
+        }
+    }
+    return PyErr_GivenExceptionMatches(err, exc_type);
+}
+static CYTHON_INLINE int __Pyx_PyErr_GivenExceptionMatches2(PyObject *err, PyObject *exc_type1, PyObject *exc_type2) {
+    assert(PyExceptionClass_Check(exc_type1));
+    assert(PyExceptionClass_Check(exc_type2));
+    if (likely(err == exc_type1 || err == exc_type2)) return 1;
+    if (likely(PyExceptionClass_Check(err))) {
+        return __Pyx_inner_PyErr_GivenExceptionMatches2(err, exc_type1, exc_type2);
+    }
+    return (PyErr_GivenExceptionMatches(err, exc_type1) || PyErr_GivenExceptionMatches(err, exc_type2));
+}
+#endif
+
+/* CheckBinaryVersion */
+  static int __Pyx_check_binary_version(void) {
+    char ctversion[5];
+    int same=1, i, found_dot;
+    const char* rt_from_call = Py_GetVersion();
+    PyOS_snprintf(ctversion, 5, "%d.%d", PY_MAJOR_VERSION, PY_MINOR_VERSION);
+    found_dot = 0;
+    for (i = 0; i < 4; i++) {
+        if (!ctversion[i]) {
+            same = (rt_from_call[i] < '0' || rt_from_call[i] > '9');
+            break;
+        }
+        if (rt_from_call[i] != ctversion[i]) {
+            same = 0;
+            break;
+        }
+    }
+    if (!same) {
+        char rtversion[5] = {'\0'};
+        char message[200];
+        for (i=0; i<4; ++i) {
+            if (rt_from_call[i] == '.') {
+                if (found_dot) break;
+                found_dot = 1;
+            } else if (rt_from_call[i] < '0' || rt_from_call[i] > '9') {
+                break;
+            }
+            rtversion[i] = rt_from_call[i];
+        }
+        PyOS_snprintf(message, sizeof(message),
+                      "compiletime version %s of module '%.100s' "
+                      "does not match runtime version %s",
+                      ctversion, __Pyx_MODULE_NAME, rtversion);
+        return PyErr_WarnEx(NULL, message, 1);
+    }
+    return 0;
+}
+
+/* InitStrings */
+  static int __Pyx_InitStrings(__Pyx_StringTabEntry *t) {
+    while (t->p) {
+        #if PY_MAJOR_VERSION < 3
+        if (t->is_unicode) {
+            *t->p = PyUnicode_DecodeUTF8(t->s, t->n - 1, NULL);
+        } else if (t->intern) {
+            *t->p = PyString_InternFromString(t->s);
+        } else {
+            *t->p = PyString_FromStringAndSize(t->s, t->n - 1);
+        }
+        #else
+        if (t->is_unicode | t->is_str) {
+            if (t->intern) {
+                *t->p = PyUnicode_InternFromString(t->s);
+            } else if (t->encoding) {
+                *t->p = PyUnicode_Decode(t->s, t->n - 1, t->encoding, NULL);
+            } else {
+                *t->p = PyUnicode_FromStringAndSize(t->s, t->n - 1);
+            }
+        } else {
+            *t->p = PyBytes_FromStringAndSize(t->s, t->n - 1);
+        }
+        #endif
+        if (!*t->p)
+            return -1;
+        if (PyObject_Hash(*t->p) == -1)
+            return -1;
+        ++t;
+    }
+    return 0;
+}
+
+static CYTHON_INLINE PyObject* __Pyx_PyUnicode_FromString(const char* c_str) {
+    return __Pyx_PyUnicode_FromStringAndSize(c_str, (Py_ssize_t)strlen(c_str));
+}
+static CYTHON_INLINE const char* __Pyx_PyObject_AsString(PyObject* o) {
+    Py_ssize_t ignore;
+    return __Pyx_PyObject_AsStringAndSize(o, &ignore);
+}
+#if __PYX_DEFAULT_STRING_ENCODING_IS_ASCII || __PYX_DEFAULT_STRING_ENCODING_IS_DEFAULT
+#if !CYTHON_PEP393_ENABLED
+static const char* __Pyx_PyUnicode_AsStringAndSize(PyObject* o, Py_ssize_t *length) {
+    char* defenc_c;
+    PyObject* defenc = _PyUnicode_AsDefaultEncodedString(o, NULL);
+    if (!defenc) return NULL;
+    defenc_c = PyBytes_AS_STRING(defenc);
+#if __PYX_DEFAULT_STRING_ENCODING_IS_ASCII
+    {
+        char* end = defenc_c + PyBytes_GET_SIZE(defenc);
+        char* c;
+        for (c = defenc_c; c < end; c++) {
+            if ((unsigned char) (*c) >= 128) {
+                PyUnicode_AsASCIIString(o);
+                return NULL;
+            }
+        }
+    }
+#endif
+    *length = PyBytes_GET_SIZE(defenc);
+    return defenc_c;
+}
+#else
+static CYTHON_INLINE const char* __Pyx_PyUnicode_AsStringAndSize(PyObject* o, Py_ssize_t *length) {
+    if (unlikely(__Pyx_PyUnicode_READY(o) == -1)) return NULL;
+#if __PYX_DEFAULT_STRING_ENCODING_IS_ASCII
+    if (likely(PyUnicode_IS_ASCII(o))) {
+        *length = PyUnicode_GET_LENGTH(o);
+        return PyUnicode_AsUTF8(o);
+    } else {
+        PyUnicode_AsASCIIString(o);
+        return NULL;
+    }
+#else
+    return PyUnicode_AsUTF8AndSize(o, length);
+#endif
+}
+#endif
+#endif
+static CYTHON_INLINE const char* __Pyx_PyObject_AsStringAndSize(PyObject* o, Py_ssize_t *length) {
+#if __PYX_DEFAULT_STRING_ENCODING_IS_ASCII || __PYX_DEFAULT_STRING_ENCODING_IS_DEFAULT
+    if (
+#if PY_MAJOR_VERSION < 3 && __PYX_DEFAULT_STRING_ENCODING_IS_ASCII
+            __Pyx_sys_getdefaultencoding_not_ascii &&
+#endif
+            PyUnicode_Check(o)) {
+        return __Pyx_PyUnicode_AsStringAndSize(o, length);
+    } else
+#endif
+#if (!CYTHON_COMPILING_IN_PYPY) || (defined(PyByteArray_AS_STRING) && defined(PyByteArray_GET_SIZE))
+    if (PyByteArray_Check(o)) {
+        *length = PyByteArray_GET_SIZE(o);
+        return PyByteArray_AS_STRING(o);
+    } else
+#endif
+    {
+        char* result;
+        int r = PyBytes_AsStringAndSize(o, &result, length);
+        if (unlikely(r < 0)) {
+            return NULL;
+        } else {
+            return result;
+        }
+    }
+}
+static CYTHON_INLINE int __Pyx_PyObject_IsTrue(PyObject* x) {
+   int is_true = x == Py_True;
+   if (is_true | (x == Py_False) | (x == Py_None)) return is_true;
+   else return PyObject_IsTrue(x);
+}
+static CYTHON_INLINE int __Pyx_PyObject_IsTrueAndDecref(PyObject* x) {
+    int retval;
+    if (unlikely(!x)) return -1;
+    retval = __Pyx_PyObject_IsTrue(x);
+    Py_DECREF(x);
+    return retval;
+}
+static PyObject* __Pyx_PyNumber_IntOrLongWrongResultType(PyObject* result, const char* type_name) {
+#if PY_MAJOR_VERSION >= 3
+    if (PyLong_Check(result)) {
+        if (PyErr_WarnFormat(PyExc_DeprecationWarning, 1,
+                "__int__ returned non-int (type %.200s).  "
+                "The ability to return an instance of a strict subclass of int "
+                "is deprecated, and may be removed in a future version of Python.",
+                Py_TYPE(result)->tp_name)) {
+            Py_DECREF(result);
+            return NULL;
+        }
+        return result;
+    }
+#endif
+    PyErr_Format(PyExc_TypeError,
+                 "__%.4s__ returned non-%.4s (type %.200s)",
+                 type_name, type_name, Py_TYPE(result)->tp_name);
+    Py_DECREF(result);
+    return NULL;
+}
+static CYTHON_INLINE PyObject* __Pyx_PyNumber_IntOrLong(PyObject* x) {
+#if CYTHON_USE_TYPE_SLOTS
+  PyNumberMethods *m;
+#endif
+  const char *name = NULL;
+  PyObject *res = NULL;
+#if PY_MAJOR_VERSION < 3
+  if (likely(PyInt_Check(x) || PyLong_Check(x)))
+#else
+  if (likely(PyLong_Check(x)))
+#endif
+    return __Pyx_NewRef(x);
+#if CYTHON_USE_TYPE_SLOTS
+  m = Py_TYPE(x)->tp_as_number;
+  #if PY_MAJOR_VERSION < 3
+  if (m && m->nb_int) {
+    name = "int";
+    res = m->nb_int(x);
+  }
+  else if (m && m->nb_long) {
+    name = "long";
+    res = m->nb_long(x);
+  }
+  #else
+  if (likely(m && m->nb_int)) {
+    name = "int";
+    res = m->nb_int(x);
+  }
+  #endif
+#else
+  if (!PyBytes_CheckExact(x) && !PyUnicode_CheckExact(x)) {
+    res = PyNumber_Int(x);
+  }
+#endif
+  if (likely(res)) {
+#if PY_MAJOR_VERSION < 3
+    if (unlikely(!PyInt_Check(res) && !PyLong_Check(res))) {
+#else
+    if (unlikely(!PyLong_CheckExact(res))) {
+#endif
+        return __Pyx_PyNumber_IntOrLongWrongResultType(res, name);
+    }
+  }
+  else if (!PyErr_Occurred()) {
+    PyErr_SetString(PyExc_TypeError,
+                    "an integer is required");
+  }
+  return res;
+}
+static CYTHON_INLINE Py_ssize_t __Pyx_PyIndex_AsSsize_t(PyObject* b) {
+  Py_ssize_t ival;
+  PyObject *x;
+#if PY_MAJOR_VERSION < 3
+  if (likely(PyInt_CheckExact(b))) {
+    if (sizeof(Py_ssize_t) >= sizeof(long))
+        return PyInt_AS_LONG(b);
+    else
+        return PyInt_AsSsize_t(b);
+  }
+#endif
+  if (likely(PyLong_CheckExact(b))) {
+    #if CYTHON_USE_PYLONG_INTERNALS
+    const digit* digits = ((PyLongObject*)b)->ob_digit;
+    const Py_ssize_t size = Py_SIZE(b);
+    if (likely(__Pyx_sst_abs(size) <= 1)) {
+        ival = likely(size) ? digits[0] : 0;
+        if (size == -1) ival = -ival;
+        return ival;
+    } else {
+      switch (size) {
+         case 2:
+           if (8 * sizeof(Py_ssize_t) > 2 * PyLong_SHIFT) {
+             return (Py_ssize_t) (((((size_t)digits[1]) << PyLong_SHIFT) | (size_t)digits[0]));
+           }
+           break;
+         case -2:
+           if (8 * sizeof(Py_ssize_t) > 2 * PyLong_SHIFT) {
+             return -(Py_ssize_t) (((((size_t)digits[1]) << PyLong_SHIFT) | (size_t)digits[0]));
+           }
+           break;
+         case 3:
+           if (8 * sizeof(Py_ssize_t) > 3 * PyLong_SHIFT) {
+             return (Py_ssize_t) (((((((size_t)digits[2]) << PyLong_SHIFT) | (size_t)digits[1]) << PyLong_SHIFT) | (size_t)digits[0]));
+           }
+           break;
+         case -3:
+           if (8 * sizeof(Py_ssize_t) > 3 * PyLong_SHIFT) {
+             return -(Py_ssize_t) (((((((size_t)digits[2]) << PyLong_SHIFT) | (size_t)digits[1]) << PyLong_SHIFT) | (size_t)digits[0]));
+           }
+           break;
+         case 4:
+           if (8 * sizeof(Py_ssize_t) > 4 * PyLong_SHIFT) {
+             return (Py_ssize_t) (((((((((size_t)digits[3]) << PyLong_SHIFT) | (size_t)digits[2]) << PyLong_SHIFT) | (size_t)digits[1]) << PyLong_SHIFT) | (size_t)digits[0]));
+           }
+           break;
+         case -4:
+           if (8 * sizeof(Py_ssize_t) > 4 * PyLong_SHIFT) {
+             return -(Py_ssize_t) (((((((((size_t)digits[3]) << PyLong_SHIFT) | (size_t)digits[2]) << PyLong_SHIFT) | (size_t)digits[1]) << PyLong_SHIFT) | (size_t)digits[0]));
+           }
+           break;
+      }
+    }
+    #endif
+    return PyLong_AsSsize_t(b);
+  }
+  x = PyNumber_Index(b);
+  if (!x) return -1;
+  ival = PyInt_AsSsize_t(x);
+  Py_DECREF(x);
+  return ival;
+}
+static CYTHON_INLINE Py_hash_t __Pyx_PyIndex_AsHash_t(PyObject* o) {
+  if (sizeof(Py_hash_t) == sizeof(Py_ssize_t)) {
+    return (Py_hash_t) __Pyx_PyIndex_AsSsize_t(o);
+#if PY_MAJOR_VERSION < 3
+  } else if (likely(PyInt_CheckExact(o))) {
+    return PyInt_AS_LONG(o);
+#endif
+  } else {
+    Py_ssize_t ival;
+    PyObject *x;
+    x = PyNumber_Index(o);
+    if (!x) return -1;
+    ival = PyInt_AsLong(x);
+    Py_DECREF(x);
+    return ival;
+  }
+}
+static CYTHON_INLINE PyObject * __Pyx_PyBool_FromLong(long b) {
+  return b ? __Pyx_NewRef(Py_True) : __Pyx_NewRef(Py_False);
+}
+static CYTHON_INLINE PyObject * __Pyx_PyInt_FromSize_t(size_t ival) {
+    return PyInt_FromSize_t(ival);
+}
+
+
+#endif /* Py_PYTHON_H */
diff --git a/src/imagecpp.pyx b/src/imagecpp.pyx
new file mode 100644
index 00000000..5ca75309
--- /dev/null
+++ b/src/imagecpp.pyx
@@ -0,0 +1,59 @@
+# cython: language_level=2
+cimport cUnitSquareImage
+import numpy as np
+
+cimport numpy as np
+from cpython cimport array
+
+from libcpp cimport bool
+
+np.import_array()
+
+def compgrad2D(np.ndarray[double, ndim=3, mode="c"] F):
+    """
+    cython interface calculates the 2D gradient of q-map image
+
+    :param F: matrix of three dimension 
+
+    :rtype numpy ndarray
+    :return dfdu: derivative
+    :return dfdv: derivative
+
+    """
+    cdef int n, t, d
+    cdef double lam
+
+    n = F.shape[0]
+    t = F.shape[1]
+    d = F.shape[2]
+
+    cdef np.ndarray[double, ndim=1, mode="c"] Fi = np.zeros(n*t*d)
+    cdef np.ndarray[double, ndim=1, mode="c"] dfdui = np.zeros(n*t*d)
+    cdef np.ndarray[double, ndim=1, mode="c"] dfdvi = np.zeros(n*t*d)
+
+    Fi = F.reshape(n*t*d, order='F')
+
+    cUnitSquareImage.findgrad2D(&dfdui[0], &dfdvi[0], &Fi[0], n, t, d)
+
+    dfdu = dfdui.reshape((n,t,d), order='F')
+    dfdv = dfdvi.reshape((n,t,d), order='F')
+
+    return dfdu, dfdv
+
+
+def check_crossing(np.ndarray[double, ndim=3, mode="c"] gam):
+    cdef int n, t, D, is_diffeo
+
+    n = gam.shape[0]
+    t = gam.shape[1]
+    D = gam.shape[2]
+
+    is_diffeo = 1
+
+    cdef np.ndarray[double, ndim=1, mode="c"] gami = np.zeros(n*t*D)
+
+    gami = gam.reshape(n*t*D, order='F')
+
+    is_diffeo = cUnitSquareImage.check_crossing(&gami[0], n, t, D)
+
+    return is_diffeo
diff --git a/src/myVector.cpp b/src/myVector.cpp
index 51676ccb..e47c5c8a 100644
--- a/src/myVector.cpp
+++ b/src/myVector.cpp
@@ -1,11 +1,9 @@
 #include <iostream>
 #include <cstring>
 #include <cmath>
-#include <Rcpp.h>
 #include "myVector.h"
 
 using namespace std;
-using namespace Rcpp;
 
 double InProd(const double *u, const double *v, int d)
 {
diff --git a/src/myVector.h b/src/myVector.h
index 0148c09f..72c20617 100644
--- a/src/myVector.h
+++ b/src/myVector.h
@@ -1,6 +1,3 @@
-#ifndef MYVECTOR_H
-#define MYVECTOR_H
-
 #include <iostream>
 #include <cstring>
 #include <cmath>
@@ -16,5 +13,3 @@ void GramSchmitd(double *x, int &n, int d);
 
 double innerSquare(const double *u, const double *v, int n, int t, int d);
 void GramSchmitdSquare(double *x, int &n, int n1, int n2, int d);
-
-#endif

From cf1a55c6860bb51a7bcde25f87f4183d1512a85a Mon Sep 17 00:00:00 2001
From: "J. Derek Tucker" <jdtuck@sandia.gov>
Date: Mon, 13 Feb 2023 11:53:33 -0700
Subject: [PATCH 06/12] start example

---
 bin/ex_image_align.py |  38 ++++++++++++++++++++++++++++++++++++++
 bin/image.mat         | Bin 0 -> 63117 bytes
 2 files changed, 38 insertions(+)
 create mode 100644 bin/ex_image_align.py
 create mode 100644 bin/image.mat

diff --git a/bin/ex_image_align.py b/bin/ex_image_align.py
new file mode 100644
index 00000000..7b8bd551
--- /dev/null
+++ b/bin/ex_image_align.py
@@ -0,0 +1,38 @@
+#%%
+import fdasrsf as fs
+import numpy as np
+from scipy.io import loadmat
+import matplotlib.pyplot as plt
+
+dat = loadmat('image.mat')
+I1 = dat['I1']
+I2 = dat['I2']
+
+(m,n) = I1.shape
+
+I1 -= I1.min()
+I1 /= I1.max()
+
+I2 -= I2.min()
+I2 /= I2.max()
+
+F1 = np.zeros((m,n,2))
+F1[:,:,0],F1[:,:,1] = np.gradient(I1, 1/699,1/699)
+F2 = np.zeros((m,n,2))
+F2[:,:,0],F2[:,:,1] = np.gradient(I2, 1/699,1/699)
+
+F1 -= F1.min()
+F1 /= F1.max()
+
+F2 -= F2.min()
+F2 /= F2.max()
+
+plt.figure()
+plt.imshow(I1)
+plt.title('I1')
+
+plt.figure()
+plt.imshow(I2)
+plt.title('I2')
+# %%
+M = 10
diff --git a/bin/image.mat b/bin/image.mat
new file mode 100644
index 0000000000000000000000000000000000000000..93dfe85d0b8bea58e52cca0de9f078f5c895defc
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HcmV?d00001


From 6ff5917b2b878dd035c49bd69da6c774513832ea Mon Sep 17 00:00:00 2001
From: "J. Derek Tucker" <jdtuck@sandia.gov>
Date: Mon, 13 Feb 2023 12:15:09 -0700
Subject: [PATCH 07/12] bugfix

---
 fdasrsf/image.py           | 2 +-
 fdasrsf/image_functions.py | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/fdasrsf/image.py b/fdasrsf/image.py
index 11e1eb2e..7147da55 100644
--- a/fdasrsf/image.py
+++ b/fdasrsf/image.py
@@ -7,7 +7,7 @@
 
 import numpy as np
 import fdasrsf.image_functions as fif
-import imagecpp as im
+import cimage as im
 
 
 def reparam_image(It, Im, gam=None, b=None, stepsize=1e-4, itermax=20):
diff --git a/fdasrsf/image_functions.py b/fdasrsf/image_functions.py
index b2220160..dfaa4fce 100644
--- a/fdasrsf/image_functions.py
+++ b/fdasrsf/image_functions.py
@@ -7,7 +7,7 @@
 
 import numpy as np
 from scipy.interpolate import griddata
-import imagecpp as im
+import cimage as im
 
 
 def apply_gam_imag(F, gam):

From 36e5442bfb83835e2435ac46823b41076ccc1ce7 Mon Sep 17 00:00:00 2001
From: "J. Derek Tucker" <jdtuck@sandia.gov>
Date: Mon, 13 Feb 2023 12:49:00 -0700
Subject: [PATCH 08/12] expose functions

---
 fdasrsf/__init__.py | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/fdasrsf/__init__.py b/fdasrsf/__init__.py
index d811217e..0fde81b4 100644
--- a/fdasrsf/__init__.py
+++ b/fdasrsf/__init__.py
@@ -7,7 +7,7 @@
 """
 __all__ = ["time_warping", "utility_functions", "bayesian_functions", "curve_stats", "geodesic", "curve_functions", 
            "geometry", "pcr_regression", "tolerance", "boxplots", "curve_regression", "regression", "fPCA", 
-           "elastic_glm_regression", "curve_pcr_regression", "image"]
+           "elastic_glm_regression", "curve_pcr_regression", "image", "image_functions"]
 
 __version__ = "2.3.12"
 
@@ -36,6 +36,7 @@
 from .boxplots import ampbox, phbox
 from .tolerance import bootTB, pcaTB
 from .image import reparam_image
+from .image_functions import formbasisTid
 from .curve_functions import resamplecurve, calculatecentroid, curve_to_q, optimum_reparam_curve, find_best_rotation, elastic_distance_curve
 from .curve_functions import q_to_curve, find_rotation_and_seed_coord
 from .curve_stats import fdacurve

From 3f2834b161dbb2e6ad8c3896b353d803cf8553d4 Mon Sep 17 00:00:00 2001
From: Derek Tucker <jdtuck@sandia.gov>
Date: Mon, 13 Feb 2023 18:15:07 -0700
Subject: [PATCH 09/12] bug fix and doc update

---
 bin/ex_image_align.py                         |   4 +++
 doc/build/html/.doctrees/environment.pickle   | Bin 4507148 -> 4516557 bytes
 doc/build/html/.doctrees/geodesic.doctree     | Bin 77768 -> 80275 bytes
 doc/build/html/.doctrees/index.doctree        | Bin 19529 -> 20858 bytes
 doc/build/html/.doctrees/tolerance.doctree    | Bin 32676 -> 32681 bytes
 doc/build/html/_modules/boxplots.html         |   5 +--
 doc/build/html/_modules/geodesic.html         |  30 +++++++++++++++---
 doc/build/html/_modules/index.html            |   2 ++
 doc/build/html/_modules/time_warping.html     |   4 ++-
 doc/build/html/_modules/tolerance.html        |   3 +-
 .../html/_modules/utility_functions.html      |   7 ++--
 doc/build/html/_sources/index.rst.txt         |   3 ++
 doc/build/html/boxplots.html                  |   1 +
 doc/build/html/curve_functions.html           |   1 +
 doc/build/html/curve_statistics.html          |   5 +--
 doc/build/html/fPCA.html                      |   1 +
 doc/build/html/fPLS.html                      |   1 +
 doc/build/html/genindex.html                  |  18 +++++++++--
 doc/build/html/geodesic.html                  |  20 +++++++++---
 doc/build/html/index.html                     |   5 +++
 doc/build/html/objects.inv                    | Bin 3866 -> 3949 bytes
 doc/build/html/pcr_regression.html            |   1 +
 doc/build/html/py-modindex.html               |  10 ++++++
 doc/build/html/regression.html                |   1 +
 doc/build/html/search.html                    |   1 +
 doc/build/html/searchindex.js                 |   2 +-
 doc/build/html/time_warping.html              |   1 +
 doc/build/html/tolerance.html                 |   7 ++--
 doc/build/html/umap_metric.html               |   1 +
 doc/build/html/utility_functions.html         |   1 +
 doc/build/latex/.doctrees/environment.pickle  | Bin 4507170 -> 4516579 bytes
 doc/build/latex/.doctrees/geodesic.doctree    | Bin 77768 -> 80275 bytes
 doc/build/latex/.doctrees/index.doctree       | Bin 19529 -> 20858 bytes
 doc/build/latex/.doctrees/tolerance.doctree   | Bin 32676 -> 32681 bytes
 doc/build/latex/fdasrsf.pdf                   | Bin 775380 -> 778634 bytes
 doc/build/man/.doctrees/geodesic.doctree      | Bin 74298 -> 76705 bytes
 doc/build/man/.doctrees/index.doctree         | Bin 19195 -> 20498 bytes
 doc/build/man/.doctrees/tolerance.doctree     | Bin 31249 -> 31254 bytes
 fdasrsf/image.py                              |   4 +--
 fdasrsf/image_functions.py                    |  10 +++---
 40 files changed, 119 insertions(+), 30 deletions(-)

diff --git a/bin/ex_image_align.py b/bin/ex_image_align.py
index 7b8bd551..80f7d08d 100644
--- a/bin/ex_image_align.py
+++ b/bin/ex_image_align.py
@@ -36,3 +36,7 @@
 plt.title('I2')
 # %%
 M = 10
+b = fs.formbasisTid(M, m, n, 't')
+
+out = fs.reparam_image(F1,F2,None,b)
+# %%
diff --git a/doc/build/html/.doctrees/environment.pickle b/doc/build/html/.doctrees/environment.pickle
index 0b408b07c70fe55c7d21c8292c4bf41e9f990718..074986577d977482b6aeb4457a60309aace9aae7 100644
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K`RVE2+W!wce8Bnu

diff --git a/doc/build/html/.doctrees/geodesic.doctree b/doc/build/html/.doctrees/geodesic.doctree
index ee8511207470e1fdae3531a31d867bb6c2f0ec7e..e7020adeb0cdae095295014051fe7982a8996b1f 100644
GIT binary patch
delta 3219
zcmb7Hdu)@}73bVIPv_wz;1Jv2=P!1!W9JFUmPbngt%Y@2Qm3J$T46K<Q^#sz2Pa#y
z@<^tZiY}!MC+w<2P%@-Fq(K}Q4|SWiYLlR)tw0)WS*f;6+oTFAlCEe_nr7$TFY!a`
zG%bJl`aFKWbMCqKp5t8nvjP8Uh+dq+)zOSnU!=QjJdnO+JWxMseAM4Ru(MZ_y6476
z#&?gG1zf=*oIbChR#1eFQ+D&Rr~IMq{z!M17Vhj1Xx+V9`=guIW7{d~e=(ijb?So#
z*np4UC`(=48tV0*9=`D!SZlZW`#V=wXkuBVFFoDcbqC+agRm@HSQb?>#j?PA7`<MB
zDYx#!uN+<hyU;Y##Dc|m<!GTQm=`x<kV1S1FBId-ANhpm^lL7B^425lIDyXAvRBD)
z*aK672wuhJnL74c-Ipx7?=sv|U&Y?n{r{-@|3d~Q;o*i#_NfjaNdSgFS;ulLl04HQ
z$eGoG`7@1-q}K;M%tJ~_g(RIm>$LGxQwn>6tfm@l`PZ}TaSMh=pJHJmDXWDZ4BvT!
z{Xhs6I8%|%p0^-`mW$=34>pSBN>vR!i}{~<VFWwpofv+7C;MgGKPvqpiY52j841Gu
zM1uS6d*4i<(&3u9B7=j@7Po%h#6FDUuj%+}_|fO>3yv0f2mU&I#4{eRG5HDVcN3^T
zo(n-5PS1OBxF?G-t4y4h6JNCF9@^WuME=nQBb9*8$Dn(mItDA(VdYjntH1_EBKfj1
zE0*+n<c>=*)cXW<!~pFQ0PXxFz}Fc`!rv$T*$9<6<bgdD$%&?pbk<|Rv6UVcwBXds
zZSbNbS-UEWy<o+@I}Sda0xj@k(t7Z6O$Y0+;OvP)cGQab54zZ4D^5J*5w0-`H$D(e
z$Bzy=;mt*=j0@H!O9ExVZr+y)jqFcWY?*Yyb!q;KJ1sZZZ6Uk2IgbkQzRq}olCAY*
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zP{3!+P?wpf$Xk7eb{Tf1w#n0-DzIBCnmr=T7A?}*6MovSg+txFv@=pQ^L&6p91AQ}
z6AtA+OYIf`DrIwaDx$|Nb&6Q>s<i6ITR-|RPL*5#XRw@vhDuzbb7_j`GLXkRHSn@*
zAvtg+lFn$KbpLaX#mYj*gnjGHsiQ1BG>+4?s`=ZLhyEV6if{5gY*oa2;Fa0%Aq?{A
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zgWp{aUHtnc;KgvckzLgRv<Lt#1mJ5NP>RvYbRFX40~YWD<6}-r&D5U3zR*Fm2oNnI
z%Dtp0pIE^M<#CAHE)#RxIohCxkti^w_Jqvv{xzFnV;thsAwIi4(Mc4LmLy2{vCXh6
z4jI%TgLXa%^do>o0U3xvXsEN}svO4$RXL7dQsqQHrE}CM8JUyzc!6Em^tH8cK^SSW
zk8NVRR5{syZ<mw(cXmlNX^&I6_-`Cg3Af^WH+8-PDy~z^`~fGpV4nM(P|iRTGdMSR
zKpd&J*+^}3PG`JH8P|Gp5)72*^tZg?r#Lpg>12u~MsTJ@lesQ%Un>PStJFAKLFCoR
z<k>`CuE}&QS|YhH5V@F?blF6w=u3Rc)G3PCRXd+Bf@Z!-ThaSO2L;yqM2ALwLOJ&{
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g(ubD}MsBErK>kr`RzWd=f+0#K*sq+6RKa8LZ!fANQ~&?~

delta 2328
zcmZXVe@s(X6vw>>2CD@uh!onsmY;>zf&~n}6cL9QCq_19E}NS}MN1<;1efTh2Ha#B
zQ3oF8^~cO)F)?v5fNVjh+eEiagMY{$i!SOM#%T<~V9uy9;?8{+eC^ghdfvO|en0oz
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zs)~)}>8@B2GGwT#T+_XNXcG$!7K3!x<_UhM4Id3xV$g6lHjLy!Qw;VR6g0#~!S)zT
zKj?rq3I`PI%(bxX*!$O1tQ?Jluf$*}4TL{T#*v&rG&~H#u)lNRw5IQo`iq*rN7cWb
zV}>hI-}4{69$QwsU_@iwmy9uuabIQpk!yqylUl?52}ZZ!VN?LjG0_Ix_~zq}VU`JN
zcdmn_ChSkiWTp7wlZ#L-Zd^Fx3I?wUSxvME?bnux(4Xdpbta6fcEB1FhDXMu_h}t)
zO@UHDq2NY7e)&|7yUtApx)sV6U*(=>B>`n6CAj2SG`gRgu<v>ToYg>QR1iCZ|2$hd
zKF=6^tKo~f^6E9W8V)(ycC<CNZEZ{O*u6AW&`<f&@$py<+|zJ>%DO0=e7^K0646zE
zS~~O7%q~~~5oS6Z$F$GCnK4QId6K-NY|<oBgqhXDpm48X0Z<T&>dn4~QCqeG@+dKW
zd|6UK;oZUoKtU)nUnqKO98gY2JKq7NCTvZ2z+w}2wY&zk0+9(-X58{59xtcrv8Hhz
z>=2cu;AQu6$TQ*S@u|>g#_%FLY%>c6tK$7Hvy?!)SEVH5oX%iw31SIwT*c{*nQ0p-
z-VzfQDn@j6#KHGwT-;%Y9|d?u7-eXJbI}yq>uHHRCXmihpTuGlA5CQ$yd#*v*ChC=
z4>$_XbXsB1jMqEkVaOkBAn-j2zUl`S2`YNvEho>=u}r4p17S?T*a)L8$f6Y;#ZT#2
zGK=8@I_6?#IjLcO<V1bpgfUvw^9-_JT&BY}G0wj-nI*AojX%?mU#e$Dwt#!|%)%D_
zj~_w&`I0}=k1rC$FSMqLhDtBa<V_(g5s<du9Xk!_c)Uvj=x+lp<}FpP_&^9-&WES5
z8?0F@WW(CM7HG2I?!EPV3l+iki{V+JO`^bLPXIl+eHI=P&9<=)Sy{`l5x%zY-e~58
zqZU3C%?jBsTI#%(YK>t<?538wrloRBlzOD49=u3lO?Ey^QRvl>UZ+=7uTv;UuLBxc
zn3D?A#Ym~RYfZOe?EVm{1{+t;f+9^kSBhWP#5B^o$pP<4G2Quy)ys+y|Gi`T<`r<z
zg3mXb#X0q-kuNp!E(6PtIY?*pz9-3ZNIfL=C1&(UKC<A>50bgt#wuZtsKw3uY-}rh
zrl_WOR1saKJEEv>*gis1-??{X@Y!)}Eu0fs>4HBfYU5J~cNPN$(Z;70wee5VU}QZ1
zE}pH1K}~X3N+<{k1tGcZBT3*{iL3%5trM#-1Mg2{F<`K&5(+{RY8_vNw<fVV$k(Rl
z%IUdQzRb?tkSphBP0UxlS@_vnO|nW#R#{a8qaapT>8B6|wp&v)OGUF)H82W7@yQE?
zFtBeGwc(D42k3xyMK!RW1R_({Yt@d{j(M-lh2N~w!&={f-HPgAztVn-?EFpwOGLx+
zVCc1KAUogYVyQsQA!uWgc9`Cn6UM!%)W_l6p30`fRGaXxQ14;!B%O=&^ysH>atJ4k
zQK@7DK8bck_*WUskW80Nt&o_7N)$hk#*(4H#?Pk_wa}j$LezXob=iEWB0<|0%}*~D
zHGk+wzAcLzij8oWw%r{Yb(eM^ouHD|l}=D;!?VRoPo<mf0N7{i8OdO`1Cr}RzJ~tl
zcj~Ikt132buC1u2&z)B!KCSme(3g+${Am_@HSl*D!?=JA*cYhhKW8&faHBAqCQHwp
I99G8u1-kX;Pyhe`

diff --git a/doc/build/html/.doctrees/index.doctree b/doc/build/html/.doctrees/index.doctree
index d0ffbc3622d1d5af3acddb8b3cd5ab9ba89b0afd..64b6d7b0608b31ed72632c8accf1fe359cb3ebc6 100644
GIT binary patch
delta 2342
zcmdUwT}WG39Kii2Mq|)UL9%J99;?>G#l(oSet)M{vxcRmrBE!rnp4lcGdJl??z)Y2
zD(nMC1}*1K`cl~XFlZ^0E|z`Nl~KD@DT4}o*<MyU`ZQ?W2YXn@*f}RL#xZDLeL0-}
z@BGg_zjN;Yp1XP;{<;X38OsNe;8jbL)nbvu(j@g*4HKcsW$QG+r*y5eaR*r-dx)9T
z69;kn>*<@e3e#dy@`CNp!ls}b9hI>Q9daYTuEp`oF63j~DJd4istXOfc{oTkITpt%
zL$t|WI(-1gHF=_OT#Dfkib}F}T8?2PsUZ~Br{Z#S3L|cUk%lMPL`{mz5!H=Gd_EuY
z=~^U;B^3=u)RS1l>I6m+9)+WN9Bb%^9AnAI;Ts%0;$&Vp8pq*rthrEglefiYGe~}N
z{@H4PZ`sxw=t8sA;v{x*Gx_Mb0KoSd=n_ybw*%bIK#zdSJ5K<tXJE5{$12_d_$LEf
z1uU!j5TKCWb-!Tg5a17V8K7K%9sz#X&45FI0Re8*o&{(XV4nbUd)5Jl1Q-=y(NPO<
zkb{x{2l_ZU?3^{u7!G2g_ZqI#9~<t#2{9okCfs|Tsi^`S7T}{sH^3VL92LOknE`lD
zfMWuD)3gL|UV!5QEH$eD9}6G}fLksCED8|ffDW|w<}MAn5%sox?%>BfW+=;$le3{l
z@&sDRLWwVUa<Zyzo}WBFrAcB}?$Y)4nu6Obj|JZWE$ObJogLlQn<=u&hpp-1TP&!W
zRvxk;<q>bzc~YM8EARito8MXEx9Ox#NzLcy^+d0)wp~qkjv1R(khs2`{6(pFNy#>|
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zUxIr2xS*06o$aP(zNV{mh^^@{v6|n!pu_Z;u3A{3EnO}crczf;?=Vkb-y<nV<jLxk
ztYU*28{egp87)azC$TE4A^fVrcbD4SA->Qq`cs$F8syKhD*?8py>Oh~>1(8V_s-%M
zIZpi?B5Zhs@0g<eLJN<4JgUq(kby0>0roYn(4rn!<8)e5Ilwv@xePtHh0>Efm8N%v
znc1O?(aFBr4ZF95-8p9Gu8FWo32QhKN^Oc;WXVRE-fTveM`xZ!_I7V??H0#t8|GUa
zuhE9SU8XPp*YPvqc#At$3&*?|mb3|C{KxPVb4q3md9=Wz?D`s@dVhOyA6M9X1s-~%
ze<C-mk7HWhQJ(A__!~^6=4AKK19RXCxum#Qy5i)`L*D$t8a+{<N?{y-EsA5}&0m=l
zX>%fFPA6r&94?W5F^?>WNs1>K-M<R(Iz8q0nogU=U7Z+hD6Quk(apSE#D)6`>(X;0
zW%TFK_hDXS(~az6k8+j`C-PzuU7NrLaU^m@jry~mqO7ueoYhMDLHl+DcOyqcV=q>k
lLf*af!AM(0@czt&Z0O9Y;;UlKPcm!PJ;j>rr}o$%_!k0v1LFVy

delta 1315
zcmaKsO-NKx6vsWIGmhg(PRgK-<4}&{_%(i%m`XZh8ih7$S%Rhb8Mlj$k_Zuqx+oGY
z#4}l&z$B6+!J3VPhDMqqRJI9Pxv52zDMX8E(Y@z6eg$pLyXW`*=e_?s=ic{g6qfFR
zx<`E{zx1g(U5ifbZQna>H^3xrn3HfaC@5@#*;pV;V0M;RV#9u&S@lEj8`u5R`|6^s
z0I!5FQ*CEPHtU;?b_2YZz>q9KX9GKz*z4``+L;<xOkKEax&^CL;GlxRm^y$p0vQDC
zu{Qwz5Xd6%*)js4!;1JUbq;~KgW~{E1db7S6eoa<fQvwX{2;&)0>uQ*C#(Y$5;#Sm
zH&GZbCvcj;YSIuuErD_Z6Dh(+3xP@k9)|<q8i6VTAJTdNZW5>_FqAP3FifC^K;K~x
zz(WFc1TwQm0iF?PlmPc-7r+dbWl!66UKU*A<#P}a;BOey5L(!Q2A<<g&J^f;UycsT
zMvJa!FEJy3BQ)r8SoC~etOQtAKn$w$@<V48_Ij`A<)$cue-$hKO@=iQn1+9s>vb76
zL@3d9vUsjMMh>+Z>*LK~FE;mt9Q_1Z3d+S0+6$^+AG(UnI1&_L;ChvK+V<|@7~-_T
z0y>;2V8sj0I1#&@W)bf@V<BDC#o|Y&J=7t~j67Lf#8<`OlcFfd!}+3AY;&0+66IZG
zJGKawO;OpkGC8Cq7L&gb7Af9x;O=bT3m9COl6p~c+3_T?@OPYV(X?F@>k4C`N!YVM
zF4h-W{RZoWYRjNU7<AZ51sc${wl!V$G&NuK%D(OJ+bS&9Dn)j!OsbXS|9<C=7r1uW
z91>MzewR%@WrhJwM|1s^M&`ZlRSut#-~HFUS9!P6-UrgXmE8M__jvaRyMuo3G8x$P
z9<tSIY%9(QH%oKd<I&?(abpmdn`WiZ<jXzr8&n%AU-rpQfsx~EjE4w0*T}Fe!y=}a
zM%u>vi1!ie>txaL4>HDFBxDnOQA+hbUb+OTsz7w*J2XaK8Mrx(=`sTjyAK#_@(a!g
jD$kmj{8^lHKUQ`4_3hjG{m_bU%Z%YWDmWdNlnVD><=}=G

diff --git a/doc/build/html/.doctrees/tolerance.doctree b/doc/build/html/.doctrees/tolerance.doctree
index afc2ec97cbec653d322066ee3133b9f5c7be61dd..ebc06a11810981e368ec145d5f9792e612df4bd4 100644
GIT binary patch
delta 106
zcmZ4TpK;}XMwSNFsq;3nu(69-C*|jtgg9xW=_r(B=BDZ>B-$G28Elqk=V9VEMirba
t&#t;Tjx&{!jSZ}5a;2x{<d0k@1zCVPOs8b9$4<#G?*VZ(pX9z>3;@5FATs~}

delta 62
zcmV-E0KxyM{{f``0R(^rm6@>w1_uQ?VsCGBlTk(+vnvMy0s%6U{zn*-D+e00U<qXb
UlX*!llk^J7lU7O_v)T*KbF6L_Z~y=R

diff --git a/doc/build/html/_modules/boxplots.html b/doc/build/html/_modules/boxplots.html
index 78d66bf2..ada296f5 100644
--- a/doc/build/html/_modules/boxplots.html
+++ b/doc/build/html/_modules/boxplots.html
@@ -96,6 +96,7 @@
 <li class="toctree-l1"><a class="reference internal" href="../pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../utility_functions.html">Utility Functions</a></li>
@@ -315,7 +316,7 @@ <h1>Source code for boxplots</h1><div class="highlight"><pre>
         <span class="n">outlier_index</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([])</span>
         <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">N</span><span class="p">):</span>
             <span class="k">if</span> <span class="n">dy</span><span class="p">[</span><span class="n">dy_ordering</span><span class="p">[</span><span class="n">N</span><span class="o">-</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span> <span class="o">&gt;</span> <span class="n">whisker_dis</span><span class="p">:</span>
-                <span class="n">outlier_index</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">outlier_index</span><span class="p">,</span><span class="n">dy</span><span class="p">[</span><span class="n">dy_ordering</span><span class="p">[</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="o">-</span><span class="n">i</span><span class="p">]])</span>
+                <span class="n">outlier_index</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">outlier_index</span><span class="p">,</span><span class="n">dy</span><span class="p">[</span><span class="n">dy_ordering</span><span class="p">[</span><span class="n">N</span><span class="o">-</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]])</span>
         
         <span class="c1"># identify amplitude extremes</span>
         <span class="n">distance_to_upper</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">)</span>
@@ -548,7 +549,7 @@ <h1>Source code for boxplots</h1><div class="highlight"><pre>
         <span class="n">outlier_index</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([])</span>
         <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">N</span><span class="p">):</span>
             <span class="k">if</span> <span class="n">dx</span><span class="p">[</span><span class="n">dx_ordering</span><span class="p">[</span><span class="n">N</span><span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">i</span><span class="p">]]</span> <span class="o">&gt;</span> <span class="n">whisker_dis</span><span class="p">:</span>
-                <span class="n">outlier_index</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">outlier_index</span><span class="p">,</span><span class="n">dx_ordering</span><span class="p">[</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="o">-</span><span class="n">i</span><span class="p">])</span>
+                <span class="n">outlier_index</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">outlier_index</span><span class="p">,</span><span class="n">dx_ordering</span><span class="p">[</span><span class="n">N</span><span class="o">-</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
         
         <span class="c1"># identify phase extremes</span>
         <span class="n">distance_to_upper</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">)</span>
diff --git a/doc/build/html/_modules/geodesic.html b/doc/build/html/_modules/geodesic.html
index be141ffb..2b47e752 100644
--- a/doc/build/html/_modules/geodesic.html
+++ b/doc/build/html/_modules/geodesic.html
@@ -96,6 +96,7 @@
 <li class="toctree-l1"><a class="reference internal" href="../pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../utility_functions.html">Utility Functions</a></li>
@@ -173,12 +174,31 @@ <h1>Source code for geodesic</h1><div class="highlight"><pre>
 
 <span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">tile</span><span class="p">,</span> <span class="n">eye</span><span class="p">,</span> <span class="n">arccos</span><span class="p">,</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">arange</span><span class="p">,</span> <span class="n">linspace</span><span class="p">,</span> <span class="n">empty</span><span class="p">,</span> <span class="n">isnan</span>
 <span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">sqrt</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
 <span class="kn">from</span> <span class="nn">scipy.integrate</span> <span class="kn">import</span> <span class="n">trapz</span>
 <span class="kn">from</span> <span class="nn">scipy.linalg</span> <span class="kn">import</span> <span class="n">norm</span>
 <span class="kn">import</span> <span class="nn">fdasrsf.utility_functions</span> <span class="k">as</span> <span class="nn">uf</span>
 <span class="kn">import</span> <span class="nn">fdasrsf.curve_functions</span> <span class="k">as</span> <span class="nn">cf</span>
 
 
+<div class="viewcode-block" id="plot_geod"><a class="viewcode-back" href="../geodesic.html#geodesic.plot_geod">[docs]</a><span class="k">def</span> <span class="nf">plot_geod</span><span class="p">(</span><span class="n">path</span><span class="p">):</span>
+    <span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Plots the geodesic path as a sequence of curves</span>
+
+<span class="sd">    :param path: numpy ndarray of shape (2,M,K) of M sample points of K samples along path</span>
+
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
+    <span class="n">mv</span> <span class="o">=</span> <span class="mf">0.2</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">path</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">2</span><span class="p">]):</span>
+        <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">mv</span><span class="o">*</span><span class="n">i</span> <span class="o">+</span> <span class="n">path</span><span class="p">[</span><span class="mi">0</span><span class="p">,:,</span><span class="n">i</span><span class="p">],</span><span class="n">path</span><span class="p">[</span><span class="mi">1</span><span class="p">,:,</span><span class="n">i</span><span class="p">],</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s1">&#39;equal&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;off&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+
+    <span class="k">return</span></div>
+
+
 <div class="viewcode-block" id="geod_sphere"><a class="viewcode-back" href="../geodesic.html#geodesic.geod_sphere">[docs]</a><span class="k">def</span> <span class="nf">geod_sphere</span><span class="p">(</span><span class="n">beta1</span><span class="p">,</span> <span class="n">beta2</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
     <span class="sd">&quot;&quot;&quot;</span>
 <span class="sd">    This function calculates the geodesics between open curves beta1 and</span>
@@ -194,7 +214,7 @@ <h1>Source code for geodesic</h1><div class="highlight"><pre>
 <span class="sd">    :rtype: numpy ndarray</span>
 <span class="sd">    :return dist: geodesic distance</span>
 <span class="sd">    :return path: geodesic path</span>
-<span class="sd">    :return O: rotation matrix</span>
+<span class="sd">    :return PsiQ: geodesic path in SRVF</span>
 
 <span class="sd">    &quot;&quot;&quot;</span>
     <span class="n">lam</span> <span class="o">=</span> <span class="mf">0.0</span>
@@ -267,7 +287,7 @@ <h1>Source code for geodesic</h1><div class="highlight"><pre>
     <span class="k">return</span><span class="p">(</span><span class="n">dist</span><span class="p">,</span> <span class="n">path</span><span class="p">,</span> <span class="n">PsiQ</span><span class="p">)</span></div>
 
 
-<div class="viewcode-block" id="path_straightening"><a class="viewcode-back" href="../geodesic.html#geodesic.path_straightening">[docs]</a><span class="k">def</span> <span class="nf">path_straightening</span><span class="p">(</span><span class="n">beta1</span><span class="p">,</span> <span class="n">beta2</span><span class="p">,</span> <span class="n">betamid</span><span class="p">,</span> <span class="n">init</span><span class="o">=</span><span class="s2">&quot;rand&quot;</span><span class="p">,</span> <span class="n">T</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">5</span><span class="p">):</span>
+<div class="viewcode-block" id="path_straightening"><a class="viewcode-back" href="../geodesic.html#geodesic.path_straightening">[docs]</a><span class="k">def</span> <span class="nf">path_straightening</span><span class="p">(</span><span class="n">beta1</span><span class="p">,</span> <span class="n">beta2</span><span class="p">,</span> <span class="n">betamid</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">init</span><span class="o">=</span><span class="s2">&quot;rand&quot;</span><span class="p">,</span> <span class="n">T</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">5</span><span class="p">):</span>
     <span class="sd">&quot;&quot;&quot;</span>
 <span class="sd">    Perform path straightening to find geodesic between two shapes in either</span>
 <span class="sd">    the space of closed curves or the space of affine standardized curves.</span>
@@ -277,7 +297,7 @@ <h1>Source code for geodesic</h1><div class="highlight"><pre>
 <span class="sd">    :param beta1: numpy ndarray of shape (2,M) of M samples (first curve)</span>
 <span class="sd">    :param beta2: numpy ndarray of shape (2,M) of M samples (end curve)</span>
 <span class="sd">    :param betamid: numpy ndarray of shape (2,M) of M samples (mid curve</span>
-<span class="sd">     Default = NULL, only needed for init &quot;rand&quot;)</span>
+<span class="sd">     Default = None, only needed for init &quot;geod&quot;)</span>
 <span class="sd">    :param init: initialize path geodesic or random (Default = &quot;rand&quot;)</span>
 <span class="sd">    :param T: Number of samples of curve (Default = 100)</span>
 <span class="sd">    :param k: number of samples along path (Default = 5)</span>
@@ -399,7 +419,7 @@ <h1>Source code for geodesic</h1><div class="highlight"><pre>
     <span class="n">q_mid</span> <span class="o">=</span> <span class="n">cf</span><span class="o">.</span><span class="n">curve_to_q</span><span class="p">(</span><span class="n">beta_mid</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
 
     <span class="c1"># find optimal rotation of q2</span>
-    <span class="n">beta2</span><span class="p">,</span> <span class="n">O1</span><span class="p">,</span> <span class="n">tau1</span> <span class="o">=</span> <span class="n">cf</span><span class="o">.</span><span class="n">find_rotation_and_seed_coord</span><span class="p">(</span><span class="n">beta1</span><span class="p">,</span> <span class="n">beta2</span><span class="p">)</span>
+    <span class="n">beta2</span><span class="p">,</span> <span class="n">q2best</span><span class="p">,</span> <span class="n">O1</span><span class="p">,</span> <span class="n">gamIbest</span> <span class="o">=</span> <span class="n">cf</span><span class="o">.</span><span class="n">find_rotation_and_seed_coord</span><span class="p">(</span><span class="n">beta1</span><span class="p">,</span> <span class="n">beta2</span><span class="p">)</span>
     <span class="n">q2</span> <span class="o">=</span> <span class="n">cf</span><span class="o">.</span><span class="n">curve_to_q</span><span class="p">(</span><span class="n">beta2</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
 
     <span class="c1"># find the optimal coorespondence</span>
@@ -410,7 +430,7 @@ <h1>Source code for geodesic</h1><div class="highlight"><pre>
     <span class="n">beta2n</span> <span class="o">=</span> <span class="n">cf</span><span class="o">.</span><span class="n">group_action_by_gamma_coord</span><span class="p">(</span><span class="n">beta2</span><span class="p">,</span> <span class="n">gamI</span><span class="p">)</span>
 
     <span class="c1"># find optimal rotation of q2</span>
-    <span class="n">beta2n</span><span class="p">,</span> <span class="n">O2</span><span class="p">,</span> <span class="n">tau1</span> <span class="o">=</span> <span class="n">cf</span><span class="o">.</span><span class="n">find_rotation_and_seed_coord</span><span class="p">(</span><span class="n">beta1</span><span class="p">,</span> <span class="n">beta2n</span><span class="p">)</span>
+    <span class="n">beta2n</span><span class="p">,</span> <span class="n">q2n1</span><span class="p">,</span> <span class="n">O2</span><span class="p">,</span> <span class="n">gamIbest1</span> <span class="o">=</span> <span class="n">cf</span><span class="o">.</span><span class="n">find_rotation_and_seed_coord</span><span class="p">(</span><span class="n">beta1</span><span class="p">,</span> <span class="n">beta2n</span><span class="p">)</span>
     <span class="n">centroid2</span> <span class="o">=</span> <span class="n">cf</span><span class="o">.</span><span class="n">calculatecentroid</span><span class="p">(</span><span class="n">beta2n</span><span class="p">)</span>
     <span class="n">beta2n</span> <span class="o">=</span> <span class="n">beta2n</span> <span class="o">-</span> <span class="n">tile</span><span class="p">(</span><span class="n">centroid2</span><span class="p">,</span> <span class="p">[</span><span class="n">T</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">T</span>
     <span class="n">q2n</span> <span class="o">=</span> <span class="n">cf</span><span class="o">.</span><span class="n">curve_to_q</span><span class="p">(</span><span class="n">beta2n</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
diff --git a/doc/build/html/_modules/index.html b/doc/build/html/_modules/index.html
index a360f36c..6a58011d 100644
--- a/doc/build/html/_modules/index.html
+++ b/doc/build/html/_modules/index.html
@@ -96,6 +96,7 @@
 <li class="toctree-l1"><a class="reference internal" href="../pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../utility_functions.html">Utility Functions</a></li>
@@ -169,6 +170,7 @@ <h1>All modules for which code is available</h1>
 <li><a href="fPCA.html">fPCA</a></li>
 <li><a href="fPLS.html">fPLS</a></li>
 <li><a href="geodesic.html">geodesic</a></li>
+<li><a href="image.html">image</a></li>
 <li><a href="pcr_regression.html">pcr_regression</a></li>
 <li><a href="regression.html">regression</a></li>
 <li><a href="time_warping.html">time_warping</a></li>
diff --git a/doc/build/html/_modules/time_warping.html b/doc/build/html/_modules/time_warping.html
index 6e3c6af5..d93bdb0e 100644
--- a/doc/build/html/_modules/time_warping.html
+++ b/doc/build/html/_modules/time_warping.html
@@ -96,6 +96,7 @@
 <li class="toctree-l1"><a class="reference internal" href="../pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../utility_functions.html">Utility Functions</a></li>
@@ -298,7 +299,8 @@ <h1>Source code for time_warping</h1><div class="highlight"><pre>
         <span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">g2</span> <span class="o">=</span> <span class="n">uf</span><span class="o">.</span><span class="n">gradient_spline</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">time</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span><span class="p">,</span> <span class="n">smoothdata</span><span class="p">)</span>
         <span class="n">q</span> <span class="o">=</span> <span class="n">g</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="o">+</span> <span class="n">eps</span><span class="p">)</span>
 
-        <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Initializing...&quot;</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
+            <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Initializing...&quot;</span><span class="p">)</span>
         <span class="n">mnq</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
         <span class="n">a</span> <span class="o">=</span> <span class="n">mnq</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
         <span class="n">d1</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
diff --git a/doc/build/html/_modules/tolerance.html b/doc/build/html/_modules/tolerance.html
index 9f5cf0ca..7f7e5345 100644
--- a/doc/build/html/_modules/tolerance.html
+++ b/doc/build/html/_modules/tolerance.html
@@ -96,6 +96,7 @@
 <li class="toctree-l1"><a class="reference internal" href="../pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../utility_functions.html">Utility Functions</a></li>
@@ -180,7 +181,7 @@ <h1>Source code for tolerance</h1><div class="highlight"><pre>
 <span class="kn">from</span> <span class="nn">fdasrsf.boxplots</span> <span class="kn">import</span> <span class="n">ampbox</span><span class="p">,</span> <span class="n">phbox</span>
 
 
-<div class="viewcode-block" id="bootTB"><a class="viewcode-back" href="../tolerance.html#tolerance.bootTB">[docs]</a><span class="k">def</span> <span class="nf">bootTB</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">time</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mf">.99</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">no</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">parallel</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+<div class="viewcode-block" id="bootTB"><a class="viewcode-back" href="../tolerance.html#tolerance.bootTB">[docs]</a><span class="k">def</span> <span class="nf">bootTB</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">time</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mf">.99</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">no</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">parallel</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
     <span class="sd">&quot;&quot;&quot;</span>
 <span class="sd">    This function computes tolerance bounds for functional data containing</span>
 <span class="sd">    phase and amplitude variation using bootstrap sampling</span>
diff --git a/doc/build/html/_modules/utility_functions.html b/doc/build/html/_modules/utility_functions.html
index 28e6ba69..7ae41cf6 100644
--- a/doc/build/html/_modules/utility_functions.html
+++ b/doc/build/html/_modules/utility_functions.html
@@ -96,6 +96,7 @@
 <li class="toctree-l1"><a class="reference internal" href="../pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../utility_functions.html">Utility Functions</a></li>
@@ -180,7 +181,7 @@ <h1>Source code for utility_functions</h1><div class="highlight"><pre>
 <span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">arccos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">arange</span><span class="p">,</span> <span class="n">ascontiguousarray</span><span class="p">,</span> <span class="nb">round</span>
 <span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">ones</span><span class="p">,</span> <span class="n">real</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">cumsum</span><span class="p">,</span> <span class="n">fabs</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="n">diagflat</span><span class="p">,</span> <span class="n">inner</span>
 <span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">gradient</span><span class="p">,</span> <span class="n">column_stack</span><span class="p">,</span> <span class="n">append</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">hstack</span><span class="p">,</span> <span class="n">median</span>
-<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">insert</span><span class="p">,</span> <span class="n">vectorize</span><span class="p">,</span> <span class="n">ceil</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">array</span><span class="p">,</span> <span class="n">quantile</span><span class="p">,</span> <span class="n">dot</span>
+<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">insert</span><span class="p">,</span> <span class="n">vectorize</span><span class="p">,</span> <span class="n">ceil</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">array</span><span class="p">,</span> <span class="n">percentile</span><span class="p">,</span> <span class="n">dot</span>
 <span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">exp</span>
 <span class="kn">from</span> <span class="nn">joblib</span> <span class="kn">import</span> <span class="n">Parallel</span><span class="p">,</span> <span class="n">delayed</span>
 <span class="kn">import</span> <span class="nn">numpy.random</span> <span class="k">as</span> <span class="nn">rn</span>
@@ -1163,8 +1164,8 @@ <h1>Source code for utility_functions</h1><div class="highlight"><pre>
 
 
 <span class="k">def</span> <span class="nf">statsFun</span><span class="p">(</span><span class="n">vec</span><span class="p">):</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">quantile</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span><span class="mf">0.025</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">quantile</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span><span class="mf">0.975</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">a</span> <span class="o">=</span> <span class="n">percentile</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span><span class="mi">100</span><span class="o">*</span><span class="mf">0.025</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">b</span> <span class="o">=</span> <span class="n">percentile</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span><span class="mi">100</span><span class="o">*</span><span class="mf">0.975</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
     <span class="n">out</span> <span class="o">=</span> <span class="n">column_stack</span><span class="p">((</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
     <span class="k">return</span><span class="p">(</span><span class="n">out</span><span class="p">)</span>
 
diff --git a/doc/build/html/_sources/index.rst.txt b/doc/build/html/_sources/index.rst.txt
index b9db6b1e..3e155564 100644
--- a/doc/build/html/_sources/index.rst.txt
+++ b/doc/build/html/_sources/index.rst.txt
@@ -28,6 +28,7 @@ using functional component analysis and regression.
    pcr_regression.rst
    elastic_glm_regression.rst
    tolerance.rst
+   image.rst
    curve_statistics.rst
    geodesic.rst
    utility_functions.rst
@@ -101,6 +102,8 @@ References
 
    J. D. Tucker, L. Shand, and K. Chowdhary. “Multimodal Bayesian Registration of Noisy Functions using Hamiltonian Monte Carlo”, Computational Statistics and Data Analysis, accepted, 2021.
 
+   Q. Xie, S. Kurtek, E. Klassen, G. E. Christensen and A. Srivastava. Metric-based pairwise and multiple image registration. IEEE European Conference on Computer Vision (ECCV), September, 2014
+
 
 Indices and tables
 ==================
diff --git a/doc/build/html/boxplots.html b/doc/build/html/boxplots.html
index 1fc48aee..98b650ac 100644
--- a/doc/build/html/boxplots.html
+++ b/doc/build/html/boxplots.html
@@ -101,6 +101,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
diff --git a/doc/build/html/curve_functions.html b/doc/build/html/curve_functions.html
index e174ecc5..1402b48e 100644
--- a/doc/build/html/curve_functions.html
+++ b/doc/build/html/curve_functions.html
@@ -99,6 +99,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
diff --git a/doc/build/html/curve_statistics.html b/doc/build/html/curve_statistics.html
index b3039053..2e7a5e2b 100644
--- a/doc/build/html/curve_statistics.html
+++ b/doc/build/html/curve_statistics.html
@@ -43,7 +43,7 @@
     <link rel="index" title="Index" href="genindex.html" />
     <link rel="search" title="Search" href="search.html" />
     <link rel="next" title="SRVF Geodesic Computation" href="geodesic.html" />
-    <link rel="prev" title="Elastic Functional Tolerance Bounds" href="tolerance.html" /> 
+    <link rel="prev" title="Elastic Image Warping" href="image.html" /> 
 </head>
 
 <body class="wy-body-for-nav">
@@ -101,6 +101,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1 current"><a class="current reference internal" href="#">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
@@ -357,7 +358,7 @@
         <a href="geodesic.html" class="btn btn-neutral float-right" title="SRVF Geodesic Computation" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right"></span></a>
       
       
-        <a href="tolerance.html" class="btn btn-neutral float-left" title="Elastic Functional Tolerance Bounds" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left"></span> Previous</a>
+        <a href="image.html" class="btn btn-neutral float-left" title="Elastic Image Warping" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left"></span> Previous</a>
       
     </div>
   
diff --git a/doc/build/html/fPCA.html b/doc/build/html/fPCA.html
index e524dbc1..7a07d09a 100644
--- a/doc/build/html/fPCA.html
+++ b/doc/build/html/fPCA.html
@@ -101,6 +101,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
diff --git a/doc/build/html/fPLS.html b/doc/build/html/fPLS.html
index 585a9072..a7e89b32 100644
--- a/doc/build/html/fPLS.html
+++ b/doc/build/html/fPLS.html
@@ -99,6 +99,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
diff --git a/doc/build/html/genindex.html b/doc/build/html/genindex.html
index a5421018..37a2d1f2 100644
--- a/doc/build/html/genindex.html
+++ b/doc/build/html/genindex.html
@@ -96,6 +96,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
@@ -431,14 +432,21 @@ <h2 id="G">G</h2>
 <h2 id="I">I</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%; vertical-align: top;"><ul>
+      <li>
+    image
+
+      <ul>
+        <li><a href="image.html#module-image">module</a>
+</li>
+      </ul></li>
       <li><a href="geodesic.html#geodesic.init_path_geod">init_path_geod() (in module geodesic)</a>
 </li>
       <li><a href="geodesic.html#geodesic.init_path_rand">init_path_rand() (in module geodesic)</a>
-</li>
-      <li><a href="utility_functions.html#utility_functions.innerprod_q">innerprod_q() (in module utility_functions)</a>
 </li>
   </ul></td>
   <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="utility_functions.html#utility_functions.innerprod_q">innerprod_q() (in module utility_functions)</a>
+</li>
       <li><a href="curve_functions.html#curve_functions.innerprod_q2">innerprod_q2() (in module curve_functions)</a>
 </li>
       <li><a href="curve_functions.html#curve_functions.inverse_exp">inverse_exp() (in module curve_functions)</a>
@@ -512,6 +520,8 @@ <h2 id="M">M</h2>
         <li><a href="fPLS.html#module-fPLS">fPLS</a>
 </li>
         <li><a href="geodesic.html#module-geodesic">geodesic</a>
+</li>
+        <li><a href="image.html#module-image">image</a>
 </li>
         <li><a href="pcr_regression.html#module-pcr_regression">pcr_regression</a>
 </li>
@@ -603,6 +613,8 @@ <h2 id="P">P</h2>
       </ul></li>
   </ul></td>
   <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="geodesic.html#geodesic.plot_geod">plot_geod() (in module geodesic)</a>
+</li>
       <li><a href="fPLS.html#fPLS.pls_svd">pls_svd() (in module fPLS)</a>
 </li>
       <li><a href="curve_functions.html#curve_functions.pre_proc_curve">pre_proc_curve() (in module curve_functions)</a>
@@ -663,6 +675,8 @@ <h2 id="R">R</h2>
   </ul></td>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="regression.html#regression.regression_warp">regression_warp() (in module regression)</a>
+</li>
+      <li><a href="image.html#image.reparam_image">reparam_image() (in module image)</a>
 </li>
       <li><a href="curve_functions.html#curve_functions.resamplecurve">resamplecurve() (in module curve_functions)</a>
 </li>
diff --git a/doc/build/html/geodesic.html b/doc/build/html/geodesic.html
index fbba77fa..4ae2b079 100644
--- a/doc/build/html/geodesic.html
+++ b/doc/build/html/geodesic.html
@@ -101,6 +101,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1 current"><a class="current reference internal" href="#">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
@@ -379,8 +380,8 @@
 <dt class="field-even">Return path<span class="colon">:</span></dt>
 <dd class="field-even"><p>geodesic path</p>
 </dd>
-<dt class="field-odd">Return O<span class="colon">:</span></dt>
-<dd class="field-odd"><p>rotation matrix</p>
+<dt class="field-odd">Return PsiQ<span class="colon">:</span></dt>
+<dd class="field-odd"><p>geodesic path in SRVF</p>
 </dd>
 </dl>
 </dd></dl>
@@ -448,7 +449,7 @@
 
 <dl class="py function">
 <dt class="sig sig-object py" id="geodesic.path_straightening">
-<span class="sig-prename descclassname"><span class="pre">geodesic.</span></span><span class="sig-name descname"><span class="pre">path_straightening</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">beta1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">beta2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">betamid</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">init</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rand'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">T</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">100</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">5</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geodesic.html#path_straightening"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#geodesic.path_straightening" title="Permalink to this definition">¶</a></dt>
+<span class="sig-prename descclassname"><span class="pre">geodesic.</span></span><span class="sig-name descname"><span class="pre">path_straightening</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">beta1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">beta2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">betamid</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">init</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'rand'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">T</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">100</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">5</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geodesic.html#path_straightening"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#geodesic.path_straightening" title="Permalink to this definition">¶</a></dt>
 <dd><p>Perform path straightening to find geodesic between two shapes in either
 the space of closed curves or the space of affine standardized curves.
 This algorithm follows the steps outlined in section 4.6 of the
@@ -459,7 +460,7 @@
 <li><p><strong>beta1</strong> – numpy ndarray of shape (2,M) of M samples (first curve)</p></li>
 <li><p><strong>beta2</strong> – numpy ndarray of shape (2,M) of M samples (end curve)</p></li>
 <li><p><strong>betamid</strong> – numpy ndarray of shape (2,M) of M samples (mid curve
-Default = NULL, only needed for init “rand”)</p></li>
+Default = None, only needed for init “geod”)</p></li>
 <li><p><strong>init</strong> – initialize path geodesic or random (Default = “rand”)</p></li>
 <li><p><strong>T</strong> – Number of samples of curve (Default = 100)</p></li>
 <li><p><strong>k</strong> – number of samples along path (Default = 5)</p></li>
@@ -483,6 +484,17 @@
 </dl>
 </dd></dl>
 
+<dl class="py function">
+<dt class="sig sig-object py" id="geodesic.plot_geod">
+<span class="sig-prename descclassname"><span class="pre">geodesic.</span></span><span class="sig-name descname"><span class="pre">plot_geod</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">path</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geodesic.html#plot_geod"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#geodesic.plot_geod" title="Permalink to this definition">¶</a></dt>
+<dd><p>Plots the geodesic path as a sequence of curves</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>path</strong> – numpy ndarray of shape (2,M,K) of M sample points of K samples along path</p>
+</dd>
+</dl>
+</dd></dl>
+
 <dl class="py function">
 <dt class="sig sig-object py" id="geodesic.update_path">
 <span class="sig-prename descclassname"><span class="pre">geodesic.</span></span><span class="sig-name descname"><span class="pre">update_path</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">alpha</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">beta</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">gradE</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">delta</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">T</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">100</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">5</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geodesic.html#update_path"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#geodesic.update_path" title="Permalink to this definition">¶</a></dt>
diff --git a/doc/build/html/index.html b/doc/build/html/index.html
index 01fd483c..f689b653 100644
--- a/doc/build/html/index.html
+++ b/doc/build/html/index.html
@@ -98,6 +98,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
@@ -236,6 +237,7 @@ <h1>Welcome to fdasrsf’s documentation!<a class="headerlink" href="#welcome-to
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a><ul class="simple">
 </ul>
 </li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
@@ -299,6 +301,9 @@ <h1>References<a class="headerlink" href="#references" title="Permalink to this
 </ol>
 </li>
 </ol>
+<ol class="upperalpha simple" start="17">
+<li><p>Xie, S. Kurtek, E. Klassen, G. E. Christensen and A. Srivastava. Metric-based pairwise and multiple image registration. IEEE European Conference on Computer Vision (ECCV), September, 2014</p></li>
+</ol>
 </div></blockquote>
 </section>
 <section id="indices-and-tables">
diff --git a/doc/build/html/objects.inv b/doc/build/html/objects.inv
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diff --git a/doc/build/html/time_warping.html b/doc/build/html/time_warping.html
index ee012b4f..a47d4060 100644
--- a/doc/build/html/time_warping.html
+++ b/doc/build/html/time_warping.html
@@ -103,6 +103,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
diff --git a/doc/build/html/tolerance.html b/doc/build/html/tolerance.html
index 2f73cbff..a168d303 100644
--- a/doc/build/html/tolerance.html
+++ b/doc/build/html/tolerance.html
@@ -40,7 +40,7 @@
   <link rel="stylesheet" href="_static/css/theme.css" type="text/css" />
     <link rel="index" title="Index" href="genindex.html" />
     <link rel="search" title="Search" href="search.html" />
-    <link rel="next" title="Curve Registration" href="curve_statistics.html" />
+    <link rel="next" title="Elastic Image Warping" href="image.html" />
     <link rel="prev" title="Elastic GLM Regression" href="elastic_glm_regression.html" /> 
 </head>
 
@@ -101,6 +101,7 @@
 <li class="toctree-l1 current"><a class="current reference internal" href="#">Elastic Functional Tolerance Bounds</a><ul class="simple">
 </ul>
 </li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
@@ -201,7 +202,7 @@
 <p>moduleauthor:: J. Derek Tucker &lt;<a class="reference external" href="mailto:jdtuck&#37;&#52;&#48;sandia&#46;gov">jdtuck<span>&#64;</span>sandia<span>&#46;</span>gov</a>&gt;</p>
 <dl class="py function">
 <dt class="sig sig-object py" id="tolerance.bootTB">
-<span class="sig-prename descclassname"><span class="pre">tolerance.</span></span><span class="sig-name descname"><span class="pre">bootTB</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">f</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">time</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.99</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">B</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">500</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">no</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">parallel</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/tolerance.html#bootTB"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#tolerance.bootTB" title="Permalink to this definition">¶</a></dt>
+<span class="sig-prename descclassname"><span class="pre">tolerance.</span></span><span class="sig-name descname"><span class="pre">bootTB</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">f</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">time</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.05</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.99</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">B</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">500</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">no</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">parallel</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/tolerance.html#bootTB"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#tolerance.bootTB" title="Permalink to this definition">¶</a></dt>
 <dd><p>This function computes tolerance bounds for functional data containing
 phase and amplitude variation using bootstrap sampling</p>
 <dl class="field-list simple">
@@ -327,7 +328,7 @@
   
     <div class="rst-footer-buttons" role="navigation" aria-label="footer navigation">
       
-        <a href="curve_statistics.html" class="btn btn-neutral float-right" title="Curve Registration" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right"></span></a>
+        <a href="image.html" class="btn btn-neutral float-right" title="Elastic Image Warping" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right"></span></a>
       
       
         <a href="elastic_glm_regression.html" class="btn btn-neutral float-left" title="Elastic GLM Regression" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left"></span> Previous</a>
diff --git a/doc/build/html/umap_metric.html b/doc/build/html/umap_metric.html
index e23678a6..e7b31e2f 100644
--- a/doc/build/html/umap_metric.html
+++ b/doc/build/html/umap_metric.html
@@ -98,6 +98,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1"><a class="reference internal" href="utility_functions.html">Utility Functions</a></li>
diff --git a/doc/build/html/utility_functions.html b/doc/build/html/utility_functions.html
index b1eff30f..ecd44086 100644
--- a/doc/build/html/utility_functions.html
+++ b/doc/build/html/utility_functions.html
@@ -99,6 +99,7 @@
 <li class="toctree-l1"><a class="reference internal" href="pcr_regression.html">Elastic Principal Component Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="elastic_glm_regression.html">Elastic GLM Regression</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tolerance.html">Elastic Functional Tolerance Bounds</a></li>
+<li class="toctree-l1"><a class="reference internal" href="image.html">Elastic Image Warping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="curve_statistics.html">Curve Registration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="geodesic.html">SRVF Geodesic Computation</a></li>
 <li class="toctree-l1 current"><a class="current reference internal" href="#">Utility Functions</a></li>
diff --git a/doc/build/latex/.doctrees/environment.pickle b/doc/build/latex/.doctrees/environment.pickle
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diff --git a/doc/build/latex/.doctrees/geodesic.doctree b/doc/build/latex/.doctrees/geodesic.doctree
index ee8511207470e1fdae3531a31d867bb6c2f0ec7e..e7020adeb0cdae095295014051fe7982a8996b1f 100644
GIT binary patch
delta 3219
zcmb7Hdu)@}73bVIPv_wz;1Jv2=P!1!W9JFUmPbngt%Y@2Qm3J$T46K<Q^#sz2Pa#y
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g(ubD}MsBErK>kr`RzWd=f+0#K*sq+6RKa8LZ!fANQ~&?~

delta 2328
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I99G8u1-kX;Pyhe`

diff --git a/doc/build/latex/.doctrees/index.doctree b/doc/build/latex/.doctrees/index.doctree
index d0ffbc3622d1d5af3acddb8b3cd5ab9ba89b0afd..64b6d7b0608b31ed72632c8accf1fe359cb3ebc6 100644
GIT binary patch
delta 2342
zcmdUwT}WG39Kii2Mq|)UL9%J99;?>G#l(oSet)M{vxcRmrBE!rnp4lcGdJl??z)Y2
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lLf*af!AM(0@czt&Z0O9Y;;UlKPcm!PJ;j>rr}o$%_!k0v1LFVy

delta 1315
zcmaKsO-NKx6vsWIGmhg(PRgK-<4}&{_%(i%m`XZh8ih7$S%Rhb8Mlj$k_Zuqx+oGY
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diff --git a/doc/build/latex/.doctrees/tolerance.doctree b/doc/build/latex/.doctrees/tolerance.doctree
index afc2ec97cbec653d322066ee3133b9f5c7be61dd..ebc06a11810981e368ec145d5f9792e612df4bd4 100644
GIT binary patch
delta 106
zcmZ4TpK;}XMwSNFsq;3nu(69-C*|jtgg9xW=_r(B=BDZ>B-$G28Elqk=V9VEMirba
t&#t;Tjx&{!jSZ}5a;2x{<d0k@1zCVPOs8b9$4<#G?*VZ(pX9z>3;@5FATs~}

delta 62
zcmV-E0KxyM{{f``0R(^rm6@>w1_uQ?VsCGBlTk(+vnvMy0s%6U{zn*-D+e00U<qXb
UlX*!llk^J7lU7O_v)T*KbF6L_Z~y=R

diff --git a/doc/build/latex/fdasrsf.pdf b/doc/build/latex/fdasrsf.pdf
index 2a2142ad84837d4bdd490a25e8111662dc68b9e9..eeeaf52582f2d0d1d33d0ac39d741e21a41ff06e 100644
GIT binary patch
delta 211621
zcmZs>V{m0_)HRx9$F@7RZQC7oY}*}mvSX{G9ouHdR>!ujj_up$obUbaty}M(U8|n4
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diff --git a/doc/build/man/.doctrees/geodesic.doctree b/doc/build/man/.doctrees/geodesic.doctree
index 2709a9dd3c5b936d6d111dc4964701ea00822917..576db98976a793dae3efe3da756aaab4cbfcd1e8 100644
GIT binary patch
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zg7sMt%kORMM_Mnv%wu#N$k>k3H^tcCq9WHfPp>|24s5m^6AS0@%Q~=<z2)P&JqEUq
zj-*EmSB1U}TfPb7pCm%6?3d@wlV$h(H3NY6nPIkiOp$D%P4Cumn+4J#o-eY13$)y=
zCmHMBu$UHc$gag2x)re&dCW1|amhKkKob{@X7PM0q`)jid%8s)*RTD96&!4yh2ON2
zc+o53(Ij4=h)=gn78eB)7j{8c&u<qob(7yU_EqZH1G8)%4`@M(d2FKgig|3Zka?Io
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ebu@Z$aIqKLRB`-@6TA_%G=;B0>MVA_3iuByY!u!A

diff --git a/doc/build/man/.doctrees/index.doctree b/doc/build/man/.doctrees/index.doctree
index 65ffbca09fbe5fd69a7072b17083f56a570c48f1..e1ff8c4a712f4ba5907b809c4ac887528906e067 100644
GIT binary patch
delta 2392
zcmchYT})g>6vsKh0=t0*;+D^VQwrVP>{mgBl#c?1Ei9Bxfs{09TD`C%b8p$Z>)pMe
zjdk0^M$wI0=GK__qD?g+R!QA9rdFdOshDWugRj15d}>l#oA}aboSFRsjHM*Tm&u&p
zoSFHbnKNhZUx3ATz;woRq4@9>Q+0{SB!{FKT2*4?6Po;><OP6_7oDhLfK-xYvWFB9
z8?h6|h>aez)aPF<OP;d)Q5qlfprbN&p$QKf(X|*p=|Vo%ot2_dth&&khnxLGlcO=L
zGK3^Gi26KeLX+pDXiS=uJZKchG<n)RB}H)%MI>39m!nuXhV<E(9GS(4OT$RRGi)7A
zipgQsgZw_9uLt>bEgZp;iu%Lq(^$jmG)7@=g(7+kYiLrAGH+z}_4iLYm`)st;m{P;
zT&Q(#%i~G4su|$sqSJ12usBzeUsaC+{K##Wu)At^0{qNvudw5FPXgTGcB`;U>(2rF
z#qBm>f6?$ZKq+l$becK@con?|P%FTG0i?zy0J{JO1n6vf9iUBsg96-dz6CHKz_0*S
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zw42BHDwRqgUp(jUx~t#>^>htc4{@IQKf-L>&rDtbY-Vox(pE!V=BXdx?%6LVz+Uhg
zz1QV($2sR^XOwla;<9vn1Eq($>)|CPZ6_S3XS$m*;$F;&JI}<q?Mj%<OO%Ad!BkxQ
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I4~GMP1KFYLIsgCw

delta 1349
zcma)*OH5Ni6oxsswzgQ6My#L|)Rs~n<zWy7gi=zC%A-VyC=s+mb(laQViPq%U_k;&
zgTh>N;X+6>L=qCT8<jw+K_DS<;m(~HU8y`0SIWYfQ%Z}va5K5{%|CO$IWsdj48Mjz
z*`*v>IsHr-r)Cy)zyF2$4!{I+#qZZSh@OlSGf|KfVj-zzW=5h7TYhMxqL?$T)34s<
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z(X7jwuFOYp{=gG}I0QuqGL26G%n0lV)X8@NauFOyFmL_}P>SFbg3lIiT7{qj!O^q<
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zL4yEHpJ#(97SK<n$Yg7sOkYD(RBE@}AzO%&s1VIYAm1x?A@8Ms^~jJYD=&%aD$&z-
zz}lRN+r1>2zU5&~76eIRQbCa>5F~SdW!@mkq~CaeeizM>h^pwa_+AmsDsRf7#k#>&
zxbUJQLB|1^UcUz7oX1&AaTPxRSMfR6$wq9uSXM$zEWa>REAbV>cZRa%ViU}=SX%~|
znB8V9GKm1+IMMi2O`hf^k6UJPr~q_hROeG)b9<URjqaOrk1XoL7tiLuI;P1KE9hyu
zggc(J?}0*QC^@N16+C!VoB6m|j3=^3R;@~O4`MelqGU;%R4ct#=q&qGl3~5LZb;Mk
z$A;Mk&Dum+q1^zD+$&b11+3L>3hBARdLFK4O{V31qEFq@+8ErGFo{rZ;g0oTWzntu
zFXe$!TgfKV!I1wZ(-0*(pQ_#My52y-b$##;e>v<wBK(a(|Ipk&>^)rfUf5oAec07T
z7M$fe@?#7_FMLc|*5POX>R)ukOVX<3Upw(dQ8`9NsfOpMN;C_indgnyr}4O2+&6Bu
zx!du@jP+rx4<mhE5-*<7D6!)_GLDm|)}K{215%BGbz4meJ*{S~&TLj*u~%mna%U9c
oI*|ZB|FdkkGJ&y*Ua374ZsXw`XkjBxeN050&hdxBW}KD30o{_E`2YX_

diff --git a/doc/build/man/.doctrees/tolerance.doctree b/doc/build/man/.doctrees/tolerance.doctree
index dbab82137f0caaeb557e0f5e5b32a1bbdc5f7364..a539a612fbd88dde7543e0bc44f1171ac6da3e41 100644
GIT binary patch
delta 102
zcmbREg>l*!MwSNFsoWb`&a;VGC*|jtgg9xW=_r(B=BDZ>B-$G28BBh|p}Luu{URg3
oF{;GoHykC5Y-|uksa$siS%9idr)03lPRTIu0dY15a39PA0GRb3xc~qF

delta 65
zcmbRCg>m8+MwSNFscair&a-h?CFSRrgg9wTUhg6?`6q|eW<K_djEqJg?&hBynT(S~
U+;k^T;My|zwwu)E1nzBl02WRdQ~&?~

diff --git a/fdasrsf/image.py b/fdasrsf/image.py
index 7147da55..cd9c1092 100644
--- a/fdasrsf/image.py
+++ b/fdasrsf/image.py
@@ -54,7 +54,7 @@ def reparam_image(It, Im, gam=None, b=None, stepsize=1e-4, itermax=20):
     iter = 0
     H[iter] = fif.compEnergy(qt,qm)
 
-    print('Iteration %d, energy %f\n'% (iter-1,H(iter)))
+    print('Iteration %d, energy %f\n'% (iter-1,H[iter]))
 
     gamupdate = fif.updateGam(qt,qm,b)
 
@@ -80,7 +80,7 @@ def reparam_image(It, Im, gam=None, b=None, stepsize=1e-4, itermax=20):
         Inew /= Inew.max()
         qm = fif.image_to_q(Inew)
         H[iter] = fif.compEnergy(qt,qm)
-        print('Iteration %d, energy %f\n'% (iter-1,H(iter)))
+        print('Iteration %d, energy %f\n'% (iter-1,H[iter]))
 
         if (iter > 4):
             hstop = 1
diff --git a/fdasrsf/image_functions.py b/fdasrsf/image_functions.py
index dfaa4fce..d2dc0c33 100644
--- a/fdasrsf/image_functions.py
+++ b/fdasrsf/image_functions.py
@@ -6,7 +6,7 @@
 """
 
 import numpy as np
-from scipy.interpolate import griddata
+from scipy.interpolate import RegularGridInterpolator
 import cimage as im
 
 
@@ -19,7 +19,8 @@ def apply_gam_imag(F, gam):
     V = np.linspace(0,1,n)
 
     for j in range(d):
-        Fnew[:,:,j] = griddata((U,V), F[:,:,j], (gam[:,:,0], gam[:,:,1]), method='cubic')
+        interp = RegularGridInterpolator((U, V), F[:,:,j], method="linear")
+        Fnew[:,:,j] = interp((gam[:,:,0], gam[:,:,1]))
     
     return Fnew
 
@@ -142,7 +143,8 @@ def apply_gam_gamid(gamid, gaminc):
 
     gamcum = np.zeros((m,n,d))
     for j in range(d):
-        gamcum[:,:,j] = griddata((U,V), gamid[:,:,j], (gaminc[:,:,0], gaminc[:,:,1]), method='cubic')
+        interp = RegularGridInterpolator((U, V), gamid[:,:,j], method="linear")
+        gamcum[:,:,j] = interp((gaminc[:,:,0], gaminc[:,:,1]))
     
     return gamcum
 
@@ -164,7 +166,7 @@ def formbasisTid(M, m, n, base_type):
 
     if base_type == 't':
         b = np.zeros((m,n,2,2*M))
-        for s in range(M):
+        for s in range(1,M+1):
             const = np.sqrt(2)*np.pi*s
             sPI2 = 2*np.pi*s
 

From a728cf0c6c19d1724c23185e7c3d2bdf9e465104 Mon Sep 17 00:00:00 2001
From: Derek Tucker <jdtuck@sandia.gov>
Date: Mon, 13 Feb 2023 19:15:17 -0700
Subject: [PATCH 10/12] c code debugging

---
 bin/ex_image_align.py      |    4 +-
 fdasrsf/__init__.py        |    2 +-
 fdasrsf/image.py           |    4 +-
 fdasrsf/image_functions.py |   60 +-
 src/imagecpp.cpp           | 1283 +++++++++++++++++++++++++++---------
 src/imagecpp.pyx           |   40 +-
 6 files changed, 1073 insertions(+), 320 deletions(-)

diff --git a/bin/ex_image_align.py b/bin/ex_image_align.py
index 80f7d08d..269c0816 100644
--- a/bin/ex_image_align.py
+++ b/bin/ex_image_align.py
@@ -38,5 +38,7 @@
 M = 10
 b = fs.formbasisTid(M, m, n, 't')
 
-out = fs.reparam_image(F1,F2,None,b)
+gamnew,Inew,H,stepsize = fs.reparam_image(F1,F2,None,b,stepsize=1e-2)
+
+I2_new = fs.apply_gam_imag(I2,gamnew)
 # %%
diff --git a/fdasrsf/__init__.py b/fdasrsf/__init__.py
index 0fde81b4..bc099a91 100644
--- a/fdasrsf/__init__.py
+++ b/fdasrsf/__init__.py
@@ -36,7 +36,7 @@
 from .boxplots import ampbox, phbox
 from .tolerance import bootTB, pcaTB
 from .image import reparam_image
-from .image_functions import formbasisTid
+from .image_functions import formbasisTid, apply_gam_imag
 from .curve_functions import resamplecurve, calculatecentroid, curve_to_q, optimum_reparam_curve, find_best_rotation, elastic_distance_curve
 from .curve_functions import q_to_curve, find_rotation_and_seed_coord
 from .curve_stats import fdacurve
diff --git a/fdasrsf/image.py b/fdasrsf/image.py
index cd9c1092..65e96e79 100644
--- a/fdasrsf/image.py
+++ b/fdasrsf/image.py
@@ -54,7 +54,7 @@ def reparam_image(It, Im, gam=None, b=None, stepsize=1e-4, itermax=20):
     iter = 0
     H[iter] = fif.compEnergy(qt,qm)
 
-    print('Iteration %d, energy %f\n'% (iter-1,H[iter]))
+    print('Iteration %d, energy %f\n'% (iter,H[iter]))
 
     gamupdate = fif.updateGam(qt,qm,b)
 
@@ -80,7 +80,7 @@ def reparam_image(It, Im, gam=None, b=None, stepsize=1e-4, itermax=20):
         Inew /= Inew.max()
         qm = fif.image_to_q(Inew)
         H[iter] = fif.compEnergy(qt,qm)
-        print('Iteration %d, energy %f\n'% (iter-1,H[iter]))
+        print('Iteration %d, energy %f\n'% (iter,H[iter]))
 
         if (iter > 4):
             hstop = 1
diff --git a/fdasrsf/image_functions.py b/fdasrsf/image_functions.py
index d2dc0c33..065b2619 100644
--- a/fdasrsf/image_functions.py
+++ b/fdasrsf/image_functions.py
@@ -11,16 +11,25 @@
 
 
 def apply_gam_imag(F, gam):
-    (m,n,d) = F.shape
+    ndim = F.ndim
+    if ndim == 3:
+        (m,n,d) = F.shape
+        Fnew = np.zeros((m,n,d))
 
-    Fnew = np.zeros((m,n,d))
+        U = np.linspace(0,1,m)
+        V = np.linspace(0,1,n)
 
-    U = np.linspace(0,1,m)
-    V = np.linspace(0,1,n)
+        for j in range(d):
+            interp = RegularGridInterpolator((U, V), F[:,:,j], method="linear")
+            Fnew[:,:,j] = interp((gam[:,:,0], gam[:,:,1]))
+    elif ndim == 2:
+        (m,n) = F.shape
 
-    for j in range(d):
-        interp = RegularGridInterpolator((U, V), F[:,:,j], method="linear")
-        Fnew[:,:,j] = interp((gam[:,:,0], gam[:,:,1]))
+        U = np.linspace(0,1,m)
+        V = np.linspace(0,1,n)
+
+        interp = RegularGridInterpolator((U, V), F, method="linear")
+        Fnew = interp((gam[:,:,0], gam[:,:,1]))
     
     return Fnew
 
@@ -58,7 +67,10 @@ def Jacob_imag(F):
     if d < 2:
         raise NameError('Data dimension is wrong!')
     
-    dfdu, dfdv = im.compgrad2D(F)
+    F = np.ascontiguousarray(F)
+    dfdu, dfdv = im.compgrad3D(F)
+    dfdu = dfdu.reshape((m,n,d), order='F')
+    dfdv = dfdv.reshape((m,n,d), order='F')
 
     multFactor = np.zeros((m,n))
 
@@ -101,12 +113,20 @@ def findphistar(q,b):
     expr2 = np.zeros((m,n,d,K))
 
     for k in range(K):
-        dbxdu[:,:,k], tmp = im.compgrad2D(b[:,:,0,k])
-        tmp, dbydv[:,:,k] = im.compgrad2D(b[:,:,1,k])
+        btmp = b[:,:,0,k]
+        btmp = np.ascontiguousarray(btmp)
+        dbxdui, tmp = im.compgrad2D(btmp)
+        dbxdu[:,:,k] = dbxdui.reshape((m,n), order='F')
+        btmp = b[:,:,1,k]
+        btmp = np.ascontiguousarray(btmp)
+        tmp, dbydvi = im.compgrad2D(btmp)
+        dbydv[:,:,k] = dbydvi.reshape((m,n), order='F')
     
     divb = dbxdu + dbydv
 
-    dqdu, dqdv = im.compgrad2D(q)
+    dqdu, dqdv = im.compgrad3D(q)
+    dqdu = dqdu.reshape(q.shape, order='F')
+    dqdv = dqdv.reshape(q.shape, order='F')
 
     for k in range(K):
         for j in range(d):
@@ -190,16 +210,28 @@ def GramSchmidt(b):
     G = np.zeros((m,n,D,N))
     G[:,:,:,cnt] = b[:,:,:,cnt]
 
-    dvx, dvy = im.compgrad2D(G[:,:,:,cnt])
+    Gtmp = G[:,:,:,cnt]
+    Gtmp = np.ascontiguousarray(Gtmp)
+    dvx, dvy = im.compgrad3D(Gtmp)
+    dvx = dvx.reshape(Gtmp.shape, order='F')
+    dvy = dvy.reshape(Gtmp.shape, order='F')
     l = np.dot(dvx.ravel(),dvx.ravel())*ds + np.dot(dvy.ravel(),dvy.ravel())*ds
     G[:,:,:,cnt] = G[:,:,:,cnt]/np.sqrt(l)
 
     for i in range(1,N):
         G[:,:,:,i] = b[:,:,:,i]
-        dv1x, dv1y = im.compgrad2D(G[:,:,:,i])
+        Gtmp = G[:,:,:,i]
+        Gtmp = np.ascontiguousarray(Gtmp)
+        dv1x, dv1y = im.compgrad3D(Gtmp)
+        dv1x = dv1x.reshape(Gtmp.shape, order='F')
+        dv1y = dv1y.reshape(Gtmp.shape, order='F')
 
         for j in range(0,i-1):
-            dv2x, dv2y = im.compgrad2D(G[:,:,:,j])
+            Gtmp = G[:,:,:,j]
+            Gtmp = np.ascontiguousarray(Gtmp)
+            dv2x, dv2y = im.compgrad3D(Gtmp)
+            dv2x = dv2x.reshape(Gtmp.shape, order='F')
+            dv2y = dv2y.reshape(Gtmp.shape, order='F')
             t = np.dot(dv1x.ravel(),dv2x.ravel())*ds + np.dot(dv1y.ravel(),dv2y.ravel())*ds
             G[:,:,:,i] = G[:,:,:,i]-t*G[:,:,:,j]
         
diff --git a/src/imagecpp.cpp b/src/imagecpp.cpp
index c7d14752..ea4aba5c 100644
--- a/src/imagecpp.cpp
+++ b/src/imagecpp.cpp
@@ -759,6 +759,9 @@ static CYTHON_INLINE float __PYX_NAN() {
 #include <string.h>
 #include <stdio.h>
 #include "numpy/arrayobject.h"
+#include "numpy/ndarrayobject.h"
+#include "numpy/ndarraytypes.h"
+#include "numpy/arrayscalars.h"
 #include "numpy/ufuncobject.h"
 
     /* NumPy API declarations from "numpy/__init__.pxd" */
@@ -1039,7 +1042,7 @@ typedef struct {
 } __Pyx_BufFmt_Context;
 
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":689
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":690
  * # in Cython to enable them only on the right systems.
  * 
  * ctypedef npy_int8       int8_t             # <<<<<<<<<<<<<<
@@ -1048,7 +1051,7 @@ typedef struct {
  */
 typedef npy_int8 __pyx_t_5numpy_int8_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":690
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":691
  * 
  * ctypedef npy_int8       int8_t
  * ctypedef npy_int16      int16_t             # <<<<<<<<<<<<<<
@@ -1057,7 +1060,7 @@ typedef npy_int8 __pyx_t_5numpy_int8_t;
  */
 typedef npy_int16 __pyx_t_5numpy_int16_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":691
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":692
  * ctypedef npy_int8       int8_t
  * ctypedef npy_int16      int16_t
  * ctypedef npy_int32      int32_t             # <<<<<<<<<<<<<<
@@ -1066,7 +1069,7 @@ typedef npy_int16 __pyx_t_5numpy_int16_t;
  */
 typedef npy_int32 __pyx_t_5numpy_int32_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":692
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":693
  * ctypedef npy_int16      int16_t
  * ctypedef npy_int32      int32_t
  * ctypedef npy_int64      int64_t             # <<<<<<<<<<<<<<
@@ -1075,7 +1078,7 @@ typedef npy_int32 __pyx_t_5numpy_int32_t;
  */
 typedef npy_int64 __pyx_t_5numpy_int64_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":696
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":697
  * #ctypedef npy_int128     int128_t
  * 
  * ctypedef npy_uint8      uint8_t             # <<<<<<<<<<<<<<
@@ -1084,7 +1087,7 @@ typedef npy_int64 __pyx_t_5numpy_int64_t;
  */
 typedef npy_uint8 __pyx_t_5numpy_uint8_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":697
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":698
  * 
  * ctypedef npy_uint8      uint8_t
  * ctypedef npy_uint16     uint16_t             # <<<<<<<<<<<<<<
@@ -1093,7 +1096,7 @@ typedef npy_uint8 __pyx_t_5numpy_uint8_t;
  */
 typedef npy_uint16 __pyx_t_5numpy_uint16_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":698
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":699
  * ctypedef npy_uint8      uint8_t
  * ctypedef npy_uint16     uint16_t
  * ctypedef npy_uint32     uint32_t             # <<<<<<<<<<<<<<
@@ -1102,7 +1105,7 @@ typedef npy_uint16 __pyx_t_5numpy_uint16_t;
  */
 typedef npy_uint32 __pyx_t_5numpy_uint32_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":699
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":700
  * ctypedef npy_uint16     uint16_t
  * ctypedef npy_uint32     uint32_t
  * ctypedef npy_uint64     uint64_t             # <<<<<<<<<<<<<<
@@ -1111,7 +1114,7 @@ typedef npy_uint32 __pyx_t_5numpy_uint32_t;
  */
 typedef npy_uint64 __pyx_t_5numpy_uint64_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":703
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":704
  * #ctypedef npy_uint128    uint128_t
  * 
  * ctypedef npy_float32    float32_t             # <<<<<<<<<<<<<<
@@ -1120,7 +1123,7 @@ typedef npy_uint64 __pyx_t_5numpy_uint64_t;
  */
 typedef npy_float32 __pyx_t_5numpy_float32_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":704
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":705
  * 
  * ctypedef npy_float32    float32_t
  * ctypedef npy_float64    float64_t             # <<<<<<<<<<<<<<
@@ -1129,7 +1132,7 @@ typedef npy_float32 __pyx_t_5numpy_float32_t;
  */
 typedef npy_float64 __pyx_t_5numpy_float64_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":713
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":714
  * # The int types are mapped a bit surprising --
  * # numpy.int corresponds to 'l' and numpy.long to 'q'
  * ctypedef npy_long       int_t             # <<<<<<<<<<<<<<
@@ -1138,7 +1141,7 @@ typedef npy_float64 __pyx_t_5numpy_float64_t;
  */
 typedef npy_long __pyx_t_5numpy_int_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":714
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":715
  * # numpy.int corresponds to 'l' and numpy.long to 'q'
  * ctypedef npy_long       int_t
  * ctypedef npy_longlong   long_t             # <<<<<<<<<<<<<<
@@ -1147,7 +1150,7 @@ typedef npy_long __pyx_t_5numpy_int_t;
  */
 typedef npy_longlong __pyx_t_5numpy_long_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":715
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":716
  * ctypedef npy_long       int_t
  * ctypedef npy_longlong   long_t
  * ctypedef npy_longlong   longlong_t             # <<<<<<<<<<<<<<
@@ -1156,7 +1159,7 @@ typedef npy_longlong __pyx_t_5numpy_long_t;
  */
 typedef npy_longlong __pyx_t_5numpy_longlong_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":717
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":718
  * ctypedef npy_longlong   longlong_t
  * 
  * ctypedef npy_ulong      uint_t             # <<<<<<<<<<<<<<
@@ -1165,7 +1168,7 @@ typedef npy_longlong __pyx_t_5numpy_longlong_t;
  */
 typedef npy_ulong __pyx_t_5numpy_uint_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":718
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":719
  * 
  * ctypedef npy_ulong      uint_t
  * ctypedef npy_ulonglong  ulong_t             # <<<<<<<<<<<<<<
@@ -1174,7 +1177,7 @@ typedef npy_ulong __pyx_t_5numpy_uint_t;
  */
 typedef npy_ulonglong __pyx_t_5numpy_ulong_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":719
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":720
  * ctypedef npy_ulong      uint_t
  * ctypedef npy_ulonglong  ulong_t
  * ctypedef npy_ulonglong  ulonglong_t             # <<<<<<<<<<<<<<
@@ -1183,7 +1186,7 @@ typedef npy_ulonglong __pyx_t_5numpy_ulong_t;
  */
 typedef npy_ulonglong __pyx_t_5numpy_ulonglong_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":721
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":722
  * ctypedef npy_ulonglong  ulonglong_t
  * 
  * ctypedef npy_intp       intp_t             # <<<<<<<<<<<<<<
@@ -1192,7 +1195,7 @@ typedef npy_ulonglong __pyx_t_5numpy_ulonglong_t;
  */
 typedef npy_intp __pyx_t_5numpy_intp_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":722
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":723
  * 
  * ctypedef npy_intp       intp_t
  * ctypedef npy_uintp      uintp_t             # <<<<<<<<<<<<<<
@@ -1201,7 +1204,7 @@ typedef npy_intp __pyx_t_5numpy_intp_t;
  */
 typedef npy_uintp __pyx_t_5numpy_uintp_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":724
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":725
  * ctypedef npy_uintp      uintp_t
  * 
  * ctypedef npy_double     float_t             # <<<<<<<<<<<<<<
@@ -1210,7 +1213,7 @@ typedef npy_uintp __pyx_t_5numpy_uintp_t;
  */
 typedef npy_double __pyx_t_5numpy_float_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":725
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":726
  * 
  * ctypedef npy_double     float_t
  * ctypedef npy_double     double_t             # <<<<<<<<<<<<<<
@@ -1219,7 +1222,7 @@ typedef npy_double __pyx_t_5numpy_float_t;
  */
 typedef npy_double __pyx_t_5numpy_double_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":726
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":727
  * ctypedef npy_double     float_t
  * ctypedef npy_double     double_t
  * ctypedef npy_longdouble longdouble_t             # <<<<<<<<<<<<<<
@@ -1258,7 +1261,7 @@ struct arrayobject;
 typedef struct arrayobject arrayobject;
 #endif
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":728
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":729
  * ctypedef npy_longdouble longdouble_t
  * 
  * ctypedef npy_cfloat      cfloat_t             # <<<<<<<<<<<<<<
@@ -1267,7 +1270,7 @@ typedef struct arrayobject arrayobject;
  */
 typedef npy_cfloat __pyx_t_5numpy_cfloat_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":729
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":730
  * 
  * ctypedef npy_cfloat      cfloat_t
  * ctypedef npy_cdouble     cdouble_t             # <<<<<<<<<<<<<<
@@ -1276,7 +1279,7 @@ typedef npy_cfloat __pyx_t_5numpy_cfloat_t;
  */
 typedef npy_cdouble __pyx_t_5numpy_cdouble_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":730
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":731
  * ctypedef npy_cfloat      cfloat_t
  * ctypedef npy_cdouble     cdouble_t
  * ctypedef npy_clongdouble clongdouble_t             # <<<<<<<<<<<<<<
@@ -1285,7 +1288,7 @@ typedef npy_cdouble __pyx_t_5numpy_cdouble_t;
  */
 typedef npy_clongdouble __pyx_t_5numpy_clongdouble_t;
 
-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":732
+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":733
  * ctypedef npy_clongdouble clongdouble_t
  * 
  * ctypedef npy_cdouble     complex_t             # <<<<<<<<<<<<<<
@@ -1986,6 +1989,16 @@ static PyTypeObject *__pyx_ptype_5numpy_dtype = 0;
 static PyTypeObject *__pyx_ptype_5numpy_flatiter = 0;
 static PyTypeObject *__pyx_ptype_5numpy_broadcast = 0;
 static PyTypeObject *__pyx_ptype_5numpy_ndarray = 0;
+static PyTypeObject *__pyx_ptype_5numpy_generic = 0;
+static PyTypeObject *__pyx_ptype_5numpy_number = 0;
+static PyTypeObject *__pyx_ptype_5numpy_integer = 0;
+static PyTypeObject *__pyx_ptype_5numpy_signedinteger = 0;
+static PyTypeObject *__pyx_ptype_5numpy_unsignedinteger = 0;
+static PyTypeObject *__pyx_ptype_5numpy_inexact = 0;
+static PyTypeObject *__pyx_ptype_5numpy_floating = 0;
+static PyTypeObject *__pyx_ptype_5numpy_complexfloating = 0;
+static PyTypeObject *__pyx_ptype_5numpy_flexible = 0;
+static PyTypeObject *__pyx_ptype_5numpy_character = 0;
 static PyTypeObject *__pyx_ptype_5numpy_ufunc = 0;
 static CYTHON_INLINE int __pyx_f_5numpy_import_array(void); /*proto*/
 
@@ -2013,8 +2026,6 @@ static const char __pyx_k_Fi[] = "Fi";
 static const char __pyx_k_np[] = "np";
 static const char __pyx_k_gam[] = "gam";
 static const char __pyx_k_lam[] = "lam";
-static const char __pyx_k_dfdu[] = "dfdu";
-static const char __pyx_k_dfdv[] = "dfdv";
 static const char __pyx_k_gami[] = "gami";
 static const char __pyx_k_main[] = "__main__";
 static const char __pyx_k_name[] = "__name__";
@@ -2029,10 +2040,12 @@ static const char __pyx_k_import[] = "__import__";
 static const char __pyx_k_reshape[] = "reshape";
 static const char __pyx_k_is_diffeo[] = "is_diffeo";
 static const char __pyx_k_compgrad2D[] = "compgrad2D";
+static const char __pyx_k_compgrad3D[] = "compgrad3D";
 static const char __pyx_k_ImportError[] = "ImportError";
 static const char __pyx_k_MemoryError[] = "MemoryError";
 static const char __pyx_k_check_crossing[] = "check_crossing";
 static const char __pyx_k_src_imagecpp_pyx[] = "src/imagecpp.pyx";
+static const char __pyx_k_ascontiguousarray[] = "ascontiguousarray";
 static const char __pyx_k_cline_in_traceback[] = "cline_in_traceback";
 static const char __pyx_k_numpy_core_multiarray_failed_to[] = "numpy.core.multiarray failed to import";
 static const char __pyx_k_numpy_core_umath_failed_to_impor[] = "numpy.core.umath failed to import";
@@ -2041,14 +2054,14 @@ static PyObject *__pyx_n_s_F;
 static PyObject *__pyx_n_s_Fi;
 static PyObject *__pyx_n_s_ImportError;
 static PyObject *__pyx_n_s_MemoryError;
+static PyObject *__pyx_n_s_ascontiguousarray;
 static PyObject *__pyx_n_s_check_crossing;
 static PyObject *__pyx_n_s_cimage;
 static PyObject *__pyx_n_s_cline_in_traceback;
 static PyObject *__pyx_n_s_compgrad2D;
+static PyObject *__pyx_n_s_compgrad3D;
 static PyObject *__pyx_n_s_d;
-static PyObject *__pyx_n_s_dfdu;
 static PyObject *__pyx_n_s_dfdui;
-static PyObject *__pyx_n_s_dfdv;
 static PyObject *__pyx_n_s_dfdvi;
 static PyObject *__pyx_n_s_gam;
 static PyObject *__pyx_n_s_gami;
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 static PyObject *__pyx_n_s_t;
 static PyObject *__pyx_n_s_test;
 static PyObject *__pyx_n_s_zeros;
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+static PyObject *__pyx_pf_6cimage_compgrad3D(CYTHON_UNUSED PyObject *__pyx_self, PyArrayObject *__pyx_v_F); /* proto */
+static PyObject *__pyx_pf_6cimage_2compgrad2D(CYTHON_UNUSED PyObject *__pyx_self, PyArrayObject *__pyx_v_F); /* proto */
+static PyObject *__pyx_pf_6cimage_4check_crossing(CYTHON_UNUSED PyObject *__pyx_self, PyArrayObject *__pyx_v_gam); /* proto */
 static int __pyx_pf_7cpython_5array_5array___getbuffer__(arrayobject *__pyx_v_self, Py_buffer *__pyx_v_info, CYTHON_UNUSED int __pyx_v_flags); /* proto */
 static void __pyx_pf_7cpython_5array_5array_2__releasebuffer__(CYTHON_UNUSED arrayobject *__pyx_v_self, Py_buffer *__pyx_v_info); /* proto */
 static PyObject *__pyx_tuple_;
 static PyObject *__pyx_tuple__2;
 static PyObject *__pyx_tuple__3;
 static PyObject *__pyx_tuple__5;
+static PyObject *__pyx_tuple__7;
 static PyObject *__pyx_codeobj__4;
 static PyObject *__pyx_codeobj__6;
+static PyObject *__pyx_codeobj__8;
 /* Late includes */
 
 /* "src/imagecpp.pyx":12
  * np.import_array()
  * 
- * def compgrad2D(np.ndarray[double, ndim=3, mode="c"] F):             # <<<<<<<<<<<<<<
+ * def compgrad3D(np.ndarray[double, ndim=3, mode="c"] F):             # <<<<<<<<<<<<<<
  *     """
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  */
 
 /* Python wrapper */
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+static char __pyx_doc_6cimage_compgrad3D[] = "\n    cython interface calculates the 2D gradient of q-map image\n\n    :param F: matrix of three dimension \n\n    :rtype numpy ndarray\n    :return dfdu: derivative\n    :return dfdv: derivative\n\n    ";
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   return __pyx_r;
 }
 
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-  /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":740
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-/* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":743
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-  /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":743
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-  /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":746
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-    /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":750
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+  /* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":929
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-  /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":874
+  /* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":935
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-    /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":875
+    /* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":936
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-  /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":876
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-  /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":872
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-      /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":882
+      /* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":943
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-      /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":881
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-    /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":883
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-    /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":881
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+/* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":947
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-      /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":888
+      /* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":949
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-      /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":887
+      /* "../../../../../../../private/var/folders/f1/72rqtvz96rx2jg5jc16s70q0002lrh/T/pip-build-env-mgzjg_sj/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":948
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-    /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":889
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-    /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":887
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-  /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":886
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-    /* "../../../../../../../tmp/pip-build-env-dwksnbrb/overlay/lib/python3.9/site-packages/numpy/__init__.pxd":895
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+  __pyx_t_1 = PyCFunction_NewEx(&__pyx_mdef_6cimage_5check_crossing, NULL, __pyx_n_s_cimage); if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 74, __pyx_L1_error)
   __Pyx_GOTREF(__pyx_t_1);
-  if (PyDict_SetItem(__pyx_d, __pyx_n_s_check_crossing, __pyx_t_1) < 0) __PYX_ERR(0, 44, __pyx_L1_error)
+  if (PyDict_SetItem(__pyx_d, __pyx_n_s_check_crossing, __pyx_t_1) < 0) __PYX_ERR(0, 74, __pyx_L1_error)
   __Pyx_DECREF(__pyx_t_1); __pyx_t_1 = 0;
 
   /* "src/imagecpp.pyx":1
diff --git a/src/imagecpp.pyx b/src/imagecpp.pyx
index 5ca75309..87c44367 100644
--- a/src/imagecpp.pyx
+++ b/src/imagecpp.pyx
@@ -9,7 +9,7 @@ from libcpp cimport bool
 
 np.import_array()
 
-def compgrad2D(np.ndarray[double, ndim=3, mode="c"] F):
+def compgrad3D(np.ndarray[double, ndim=3, mode="c"] F):
     """
     cython interface calculates the 2D gradient of q-map image
 
@@ -33,12 +33,42 @@ def compgrad2D(np.ndarray[double, ndim=3, mode="c"] F):
 
     Fi = F.reshape(n*t*d, order='F')
 
+    Fi = np.ascontiguousarray(Fi)
+
     cUnitSquareImage.findgrad2D(&dfdui[0], &dfdvi[0], &Fi[0], n, t, d)
 
-    dfdu = dfdui.reshape((n,t,d), order='F')
-    dfdv = dfdvi.reshape((n,t,d), order='F')
+    return dfdui, dfdvi
+
+
+def compgrad2D(np.ndarray[double, ndim=2, mode="c"] F):
+    """
+    cython interface calculates the 2D gradient of q-map image
+
+    :param F: matrix of three dimension 
+
+    :rtype numpy ndarray
+    :return dfdu: derivative
+    :return dfdv: derivative
+
+    """
+    cdef int n, t, d
+    cdef double lam
+
+    n = F.shape[0]
+    t = F.shape[1]
+    d = 1
+
+    cdef np.ndarray[double, ndim=1, mode="c"] Fi = np.zeros(n*t*d)
+    cdef np.ndarray[double, ndim=1, mode="c"] dfdui = np.zeros(n*t*d)
+    cdef np.ndarray[double, ndim=1, mode="c"] dfdvi = np.zeros(n*t*d)
+
+    Fi = F.reshape(n*t*d, order='F')
 
-    return dfdu, dfdv
+    Fi = np.ascontiguousarray(Fi)
+
+    cUnitSquareImage.findgrad2D(&dfdui[0], &dfdvi[0], &Fi[0], n, t, d)
+
+    return dfdui, dfdvi
 
 
 def check_crossing(np.ndarray[double, ndim=3, mode="c"] gam):
@@ -54,6 +84,8 @@ def check_crossing(np.ndarray[double, ndim=3, mode="c"] gam):
 
     gami = gam.reshape(n*t*D, order='F')
 
+    gami = np.ascontiguousarray(gami)
+
     is_diffeo = cUnitSquareImage.check_crossing(&gami[0], n, t, D)
 
     return is_diffeo

From aa0b7e29b19398fccf4413b27f14f45beb82b87d Mon Sep 17 00:00:00 2001
From: Derek Tucker <jdtuck@sandia.gov>
Date: Mon, 13 Feb 2023 20:43:50 -0700
Subject: [PATCH 11/12] interpolation fix

---
 bin/ex_image_align.py      | 29 +++++++++++++++++++----------
 fdasrsf/__init__.py        |  2 +-
 fdasrsf/image_functions.py | 21 ++++++++++++++++-----
 3 files changed, 36 insertions(+), 16 deletions(-)

diff --git a/bin/ex_image_align.py b/bin/ex_image_align.py
index 269c0816..621833a2 100644
--- a/bin/ex_image_align.py
+++ b/bin/ex_image_align.py
@@ -17,9 +17,9 @@
 I2 /= I2.max()
 
 F1 = np.zeros((m,n,2))
-F1[:,:,0],F1[:,:,1] = np.gradient(I1, 1/699,1/699)
+F1[:,:,1],F1[:,:,0] = np.gradient(I1, 1/699,1/699)
 F2 = np.zeros((m,n,2))
-F2[:,:,0],F2[:,:,1] = np.gradient(I2, 1/699,1/699)
+F2[:,:,1],F2[:,:,0] = np.gradient(I2, 1/699,1/699)
 
 F1 -= F1.min()
 F1 /= F1.max()
@@ -27,18 +27,27 @@
 F2 -= F2.min()
 F2 /= F2.max()
 
-plt.figure()
-plt.imshow(I1)
-plt.title('I1')
-
-plt.figure()
-plt.imshow(I2)
-plt.title('I2')
 # %%
 M = 10
 b = fs.formbasisTid(M, m, n, 't')
 
-gamnew,Inew,H,stepsize = fs.reparam_image(F1,F2,None,b,stepsize=1e-2)
+gamnew,Inew,H,stepsize = fs.reparam_image(F1,F2,None,b,stepsize=1e-2, itermax=1000)
 
 I2_new = fs.apply_gam_imag(I2,gamnew)
+
+plt.figure()
+ax1 = plt.subplot(131)
+plt.imshow(I1)
+plt.title('I1');
+ax1 = plt.subplot(132)
+plt.imshow(I2)
+plt.title('I2');
+ax1 = plt.subplot(133)
+plt.imshow(I2_new)
+plt.title('I2 Warped');
+
+plt.figure()
+gamid = fs.makediffeoid(m,n)
+plt.quiver(gamid[:,:,0],gamid[:,:,1],gamid[:,:,0]-gamnew[:,:,0],gamid[:,:,1]-gamnew[:,:,1])
+
 # %%
diff --git a/fdasrsf/__init__.py b/fdasrsf/__init__.py
index bc099a91..cfaa0e30 100644
--- a/fdasrsf/__init__.py
+++ b/fdasrsf/__init__.py
@@ -36,7 +36,7 @@
 from .boxplots import ampbox, phbox
 from .tolerance import bootTB, pcaTB
 from .image import reparam_image
-from .image_functions import formbasisTid, apply_gam_imag
+from .image_functions import formbasisTid, apply_gam_imag, makediffeoid
 from .curve_functions import resamplecurve, calculatecentroid, curve_to_q, optimum_reparam_curve, find_best_rotation, elastic_distance_curve
 from .curve_functions import q_to_curve, find_rotation_and_seed_coord
 from .curve_stats import fdacurve
diff --git a/fdasrsf/image_functions.py b/fdasrsf/image_functions.py
index 065b2619..9075f3b8 100644
--- a/fdasrsf/image_functions.py
+++ b/fdasrsf/image_functions.py
@@ -12,6 +12,11 @@
 
 def apply_gam_imag(F, gam):
     ndim = F.ndim
+
+    x2 = gam[:,:,1].ravel(order='F')
+    gamtmp = gam[:,:,0].T
+    y2 = np.sort(gamtmp.ravel(order='F'))
+
     if ndim == 3:
         (m,n,d) = F.shape
         Fnew = np.zeros((m,n,d))
@@ -21,7 +26,8 @@ def apply_gam_imag(F, gam):
 
         for j in range(d):
             interp = RegularGridInterpolator((U, V), F[:,:,j], method="linear")
-            Fnew[:,:,j] = interp((gam[:,:,0], gam[:,:,1]))
+            tmp = interp((x2,y2))
+            Fnew[:,:,j] = tmp.reshape((m,n),order='F')
     elif ndim == 2:
         (m,n) = F.shape
 
@@ -29,7 +35,8 @@ def apply_gam_imag(F, gam):
         V = np.linspace(0,1,n)
 
         interp = RegularGridInterpolator((U, V), F, method="linear")
-        Fnew = interp((gam[:,:,0], gam[:,:,1]))
+        tmp = interp((x2,y2))
+        Fnew = tmp.reshape((m,n),order='F')
     
     return Fnew
 
@@ -75,10 +82,10 @@ def Jacob_imag(F):
     multFactor = np.zeros((m,n))
 
     if d==2:
-        multFactor = dfdu[:,:,0]@dfdv[:,:,1] - dfdu[:,:,1]@dfdv[:,:,0]
+        multFactor = dfdu[:,:,0]*dfdv[:,:,1] - dfdu[:,:,1]*dfdv[:,:,0]
         multFactor = np.abs(multFactor)
     elif d==3:
-        multFactor = (dfdu[:,:,1]@dfdv[:,:,2] - dfdu[:,:,2]@dfdv[:,:,1])**2 + (dfdu[:,:,0]@dfdv[:,:,2] - dfdu[:,:,2]@dfdv[:,:,0])**2 + (dfdu[:,:,0]@dfdv[:,:,1] - dfdu[:,:,1]@dfdv[:,:,0])**2
+        multFactor = (dfdu[:,:,1]*dfdv[:,:,2] - dfdu[:,:,2]*dfdv[:,:,1])**2 + (dfdu[:,:,0]*dfdv[:,:,2] - dfdu[:,:,2]*dfdv[:,:,0])**2 + (dfdu[:,:,0]*dfdv[:,:,1] - dfdu[:,:,1]*dfdv[:,:,0])**2
         multFactor = np.sqrt(multFactor)
     
     return multFactor
@@ -162,9 +169,13 @@ def apply_gam_gamid(gamid, gaminc):
     V = np.linspace(0,1,n)
 
     gamcum = np.zeros((m,n,d))
+    x2 = gaminc[:,:,1].ravel(order='F')
+    gamtmp = gaminc[:,:,0].T
+    y2 = np.sort(gamtmp.ravel(order='F'))
     for j in range(d):
         interp = RegularGridInterpolator((U, V), gamid[:,:,j], method="linear")
-        gamcum[:,:,j] = interp((gaminc[:,:,0], gaminc[:,:,1]))
+        tmp = interp((x2,y2))
+        gamcum[:,:,j] = tmp.reshape((m,n),order='F')
     
     return gamcum
 

From edba0f53d298153c9b53fb791a381d1cf6800ced Mon Sep 17 00:00:00 2001
From: Derek Tucker <jdtuck@sandia.gov>
Date: Mon, 13 Feb 2023 20:47:16 -0700
Subject: [PATCH 12/12] start documenting

---
 fdasrsf/image_functions.py | 32 ++++++++++++++++++++++++++++++++
 1 file changed, 32 insertions(+)

diff --git a/fdasrsf/image_functions.py b/fdasrsf/image_functions.py
index 9075f3b8..a857be2c 100644
--- a/fdasrsf/image_functions.py
+++ b/fdasrsf/image_functions.py
@@ -11,6 +11,16 @@
 
 
 def apply_gam_imag(F, gam):
+    """
+    apply diffeomorphism to image
+
+    :param F: numpy array of image
+    :param gam: numpy array of diffeomorphism
+
+    :rtype: np.array
+    :return Fnew: warped image
+
+    """
     ndim = F.ndim
 
     x2 = gam[:,:,1].ravel(order='F')
@@ -42,6 +52,16 @@ def apply_gam_imag(F, gam):
 
 
 def makediffeoid(nrow,ncol):
+    """
+    computes a 2-D identity diffeomorphism
+
+    :param nrow: number of rows
+    :param ncol: number of columns
+
+    :rtype: np.array
+    :return gamid: amplitude depth
+
+    """
     D = 2
     gamid = np.zeros((nrow,ncol,D))
 
@@ -191,6 +211,18 @@ def gen_basis(m, n, M=10, N=10, baseType = 't', ortho=False):
 
 
 def formbasisTid(M, m, n, base_type):
+    """
+    Generate Basis on tangent space at identity
+
+    :param M: number of basis functions
+    :param m: number of rows
+    :param n: number of colunns
+    :param base_type: basis type ('t' only implemented)
+
+    :rtype: np.array
+    :return b: basis functions
+
+    """
     U, V = np.meshgrid(np.linspace(0,1,n), np.linspace(0,1,m), indexing='xy')
 
     idx = 0