Report_03_Finding_the_Center_of_a_Circle.Rmd

---
title: "Finding the Center of a Circle"
author: "Paul Pearson"
date: "December 19, 2015"
output: pdf_document
---

# Finding the center of a circle

Here is the tiny $20$ pixel by $15$ pixel image that we will analyze:

![White disk on black background](find_the_center_example.jpg)

We now read this image into R as an array (with three indices: two for pixel location and one for color) and flatten it to a grayscale image by averaging the RGB color channels.

```{r}
# read the image into an array
library(jpeg)
img <- readJPEG("find_the_center_example.jpg",native=FALSE)
str(img) # what is the structure of img?
img.gray <- (img[,,1] + img[,,2] + img[,,3])/3 # flatten to grayscale
str(img.gray) # what is the structure of img.gray?
```

Let's plot the `img.gray` matrix in grayscale and large, so that we can see individual pixels.

```{r, fig.width=3, fig.height=4}
image( t(img.gray[20:1,]), 
       col = grey(seq(0, 1, length = 256)), # make it grayscale
       xaxt= "n", yaxt= "n" # do not plot axes
       )
```

# The image is now just a matrix

When we print the `img.gray` object, we notice that the values are all between 0 and 1, with black pixels taking the value 0 and white pixels taking the value 1.

```{r}
img.gray # print the values in the matrix
```

# Examining the properties of the matrix

When we print the $20$ row sums of the matrix, we notice that row $7$ has the largest row sum.

```{r}
rowSums(img.gray)
which.max( rowSums(img.gray) ) # return the index of the row that has the max row sum
```

Similarly, when we print the $15$ column sums for this matrix, we notice that column $9$ has the largest column sum.

```{r}
colSums(img.gray)
which.max( colSums(img.gray) ) # return the index of the column that has the max column sum
```

So, a good estimate for the center of the white region in the image is row $7$ and column $9$.  Why does this work?  Because the more white pixels there are in a row (or a column), the larger the sum will be.  When the row sum (or column sum) is largest, we have found a diameter for the white disk-shaped region.  We are exploiting the fact that the region outside of the white disk is uniformly black.

Thought question: Will this method always work well to find the center of the white region?  When will it work well? When will it fail?