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Interactive HTML Slides<br>for Geometry Processing: using the PMP library
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<section>
<div class="white-on-blue">
<div class="title"> Interactive HTML Slides<br>for Geometry Processing </div>
<div class="subtitle"> using the PMP library </div>
</div>
<div class="author"> Mario Botsch </div>
<div class="affiliation"> Graphics & Geometry Group<br>Bielefeld University </div>
</section>
<!-- Table of Contents -->
<!-- all the slides from markdown document: DO NOT INDENT THE body LINE!!! -->
<section id="how-to-use-these-html-slides" class="slide level1">
<h1>How to use these HTML slides</h1>
<ul>
<li>Use the cursor keys <strong>left</strong>/<strong>right</strong> to navigate through the slides</li>
<li>Click <strong>page number</strong> (bottom right) to open navigation menu</li>
<li>Press <strong>f</strong>/<strong>ESC</strong> to enter/leave fullscreen mode</li>
<li>Press <strong>o</strong> or <strong>ESC</strong> to enter/leave overview mode</li>
<li><strong>Double-click</strong> an item (e.g. an image) to zoom in/out.</li>
</ul>
</section>
<section id="interactive-demo-applications" class="slide level1">
<h1>Interactive Demo Applications</h1>
<ul>
<li>The demos are written in C++ using the <a href="http://pmp-library.org">PMP library</a></li>
<li>They are cross-compiled to Javascript using <a href="http://emscripten.org">emscripten</a></li>
<li>They require support for WebGL 2
<ul>
<li>See <a href="https://caniuse.com/webgl2" class="uri">https://caniuse.com/webgl2</a></li>
<li>The demos work nicely on Chrome, Chromium, or Firefox</li>
<li>They do not run on Apple Safari in MacOS and iOS</li>
</ul></li>
</ul>
</section>
<section id="subdivision-surfaces" class="slide level1 section">
<h1>Subdivision Surfaces</h1>
</section>
<section id="loop-subdivision" class="slide level1">
<h1>Loop Subdivision</h1>
<ul>
<li>Subdivision scheme for triangle meshes<br />
<img class="" style="" src="images/subdivision-triangle.png" style=""></img></li>
<li>Generates <span class="math inline">\(C^2\)</span> continuous limit surfaces:
<ul>
<li><span class="math inline">\(C^1\)</span> for extraordinary vertices (valence ≠ 6)</li>
<li><span class="math inline">\(C^2\)</span> continuous everywhere else</li>
</ul></li>
</ul>
<p><a href="https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/thesis-10.pdf" class="footer" target="_blank">Loop, <em>Smooth Subdivision Surfaces Based on Triangles</em>, M.S. thesis, 1987</a></p>
</section>
<section id="loop-subdivision-rules" class="slide level1">
<h1>Loop Subdivision Rules</h1>
<p><figure class="" style=""><img src="images/loop-rule-1.png" style=""></img><figcaption>New edge vertices</figcaption></figure> <span style="display:inline-block; width:100px;"></span> <figure class="" style=""><img src="images/loop-rule-2.png" style=""></img><figcaption>Update old vertices (valence <span class="math inline">\(k\)</span>)</figcaption></figure></p>
<p><a href="https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/thesis-10.pdf" class="footer" target="_blank">Loop, <em>Smooth Subdivision Surfaces Based on Triangles</em>, M.S. thesis, 1987</a></p>
</section>
<section id="generalized-catmull-clark-subdivision" class="slide level1">
<h1>Generalized Catmull-Clark Subdivision</h1>
<ul>
<li>Subdivision scheme for <strong>arbitrary polygons</strong><br />
<img class="" style="" src="images/subdivision-polygon.png" style=""></img></li>
<li>Connect new face points to edge-vertex-edge triple
<ul>
<li>Turns all polygon faces into quads</li>
</ul></li>
<li>Generates <span class="math inline">\(C^2\)</span> continuous limit surfaces:
<ul>
<li><span class="math inline">\(C^1\)</span> for extraordinary vertices (valence ≠ 4)</li>
<li><span class="math inline">\(C^2\)</span> continuous everywhere else</li>
</ul></li>
</ul>
<p><a href="http://graphics.pixar.com/library/Geri/paper.pdf" class="footer" target="_blank">DeRose et al, <em>Subdivision Surfaces in Character Animation</em>, SIGGRAPH 1998</a></p>
</section>
<section id="catmull-clark-rules" class="slide level1">
<h1>Catmull-Clark Rules</h1>
<div style="clear:both; height:50px;">
</div>
<div class="col30">
<p><figure class="" style=""><img src="images/catmull-clark-rule-6.png" style=""></img><figcaption>New face vertices</figcaption></figure></p>
<div class="tiny">
<p><span class="math display">\[\vec{f} = \frac{1}{n} \sum_{i=1}^n \vec{v}_i\]</span></p>
</div>
</div>
<div class="col30">
<p><figure class="" style=""><img src="images/catmull-clark-rule-7.png" style=""></img><figcaption>New edge vertices</figcaption></figure></p>
<div class="tiny">
<p><span class="math display">\[\vec{e} = \frac{1}{4} \left(\vec{v}_1 + \vec{v}_2 + \vec{f}_1 + \vec{f}_2\right)\]</span></p>
</div>
</div>
<div class="col40">
<p><figure class="" style=""><img src="images/catmull-clark-rule-8.png" style=""></img><figcaption>Update old vertices (valence <span class="math inline">\(k\)</span>)</figcaption></figure></p>
<div class="tiny">
<p><span class="math display">\[
\vec{v} = \frac{k-2}{k} \vec{v}
+ \frac{1}{k^2} \sum_{i=1}^k \vec{v}_i
+ \frac{1}{k^2} \sum_{i=1}^k \vec{f}_i
\]</span></p>
</div>
</div>
<p><a href="http://graphics.pixar.com/library/Geri/paper.pdf" class="footer" target="_blank">DeRose et al, <em>Subdivision Surfaces in Character Animation</em>, SIGGRAPH 1998</a></p>
</section>
<section id="try-it-yourself" class="slide level1">
<h1>Try it yourself!</h1>
<p><figure class="" style="width:1000px;height:550px;"><iframe data-src="demos/subdiv/subdiv.html?model=../../meshes/cross.off"class="" style="width:1000px;height:550px;" ></iframe><figcaption>pure quad mesh (triangulate for Loop subdivision)</figcaption></figure></p>
</section>
<section id="try-it-yourself-1" class="slide level1">
<h1>Try it yourself!</h1>
<p><figure class="" style="width:1000px;height:550px;"><iframe data-src="demos/subdiv/subdiv.html"class="" style="width:1000px;height:550px;" ></iframe><figcaption>mixed tri-quad mesh</figcaption></figure></p>
<p><span class="footer">Model created using Blender, original from Willem-Paul van Overbruggen</span></p>
</section>
<section id="surface-smoothing" class="slide level1 section">
<h1>Surface Smoothing</h1>
</section>
<section id="diffusion-flow-on-meshes" class="slide level1">
<h1>Diffusion Flow on Meshes</h1>
<ul>
<li>Continuous PDE: <span class="math inline">\(\frac{\partial \vec{x}}{\partial t} \;=\; \lambda \Delta \vec{x}\)</span></li>
<li>Explicit integration per vertex: <span class="math inline">\(\vec{x}_i \leftarrow \vec{x}_i + \delta t \, \lambda \Delta \vec{x}_i\)</span></li>
</ul>
<p><figure class="" style="height:300px;"><img src="images/smooth-bunny-0.png" style="height:300px;"></img><figcaption>0 iterations</figcaption></figure> <figure class="" style="height:300px;"><img src="images/smooth-bunny-10.png" style="height:300px;"></img><figcaption>10 iterations</figcaption></figure> <figure class="" style="height:300px;"><img src="images/smooth-bunny-100.png" style="height:300px;"></img><figcaption>100 iterations</figcaption></figure></p>
</section>
<section id="uniform-laplace-discretization" class="slide level1">
<h1>Uniform Laplace Discretization</h1>
<p><span class="math display">\[
\laplace \vec{x}\of{v_i}
\;:=\;
\frac{1}{\abs{\set{N}_1\of{v_i}}}
\sum_{v_j \in \set{N}_1\of{v_i}} \left( \vec{x}\of{v_j} - \vec{x}\of{v_i} \right)
\]</span></p>
<div id="section">
<div style="width:70%;float:left">
<ul>
<li>Properties
<ul>
<li>simple and efficient</li>
<li>depends only on connectivity</li>
<li>does not take into account geometry at all</li>
</ul></li>
</ul>
</div>
<div style="width:30%;float:left">
<p><img class="" style="height:200px;" src="images/uniformLaplace.png" style="height:200px;"></img></p>
</div>
<div style="clear: both">
</div>
</div>
<p><a href="http://mesh.brown.edu/taubin/pdfs/taubin-sg95.pdf" class="footer" target="_blank">Taubin, <em>A Signal Processing Approach to Fair Surface Design</em>, SIGGRAPH 1995</a></p>
</section>
<section id="cotan-laplace-discretization" class="slide level1">
<h1>Cotan Laplace Discretization</h1>
<p><span class="math display">\[
\laplace \vec{x}\of{v_i} \;:=\;
\frac{1}{2A\of{v_i}}
\sum_{v_j \in \set{N}_1\of{v_i}}
\left( \cot \alpha_{ij} + \cot \beta_{ij} \right)
\left( \vec{x}\of{v_j} - \vec{x}\of{v_i} \right)
\]</span></p>
<div id="section-1">
<div style="width:70%;float:left">
<ul>
<li>Properties
<ul>
<li>takes geometry and connectivity into account</li>
<li>more accurate discretization</li>
<li>can be derived through FEM</li>
</ul></li>
</ul>
</div>
<div style="width:30%;float:left">
<p><img class="" style="height:200px;" src="images/cotanLaplace.png" style="height:200px;"></img></p>
</div>
<div style="clear: both">
</div>
</div>
<p><a href="http://multires.caltech.edu/pubs/diffGeoOps.pdf" class="footer" target="_blank">Meyer et al, <em>Discrete Differential-Geometry Operators for Triangulated 2-Manifolds</em>, VisMath III, 2003</a></p>
</section>
<section id="uniform-or-cotan-discretization" class="slide level1">
<h1>Uniform or Cotan Discretization?</h1>
<div id="section-2">
<div style="width:50%;float:left">
<ul>
<li>Uniform Laplacian is an inaccurate discretization</li>
<li>Might be non-zero even for planar meshes</li>
<li>Smoothes geometry <strong>and</strong> triangulation</li>
<li>Might be desired for mesh regularization</li>
</ul>
</div>
<div style="width:50%;float:left">
<p><img class="" style="width:400px;" src="images/umbrella_distortion.png" style="width:400px;"></img></p>
</div>
<div style="clear: both">
</div>
</div>
<p><a href="http://w.multires.caltech.edu/pubs/ImplicitFairing.pdf" class="footer" target="_blank">Desbrun et al, <em>Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow</em>, SIGGRAPH 1999</a></p>
</section>
<section id="numerical-integration" class="slide level1">
<h1>Numerical Integration</h1>
<ul>
<li class="fragment">Let’s write the position update in matrix notation</li>
<li class="fragment">Write all points <span class="math inline">\(\vec{x}_i^{(t)}\)</span> in a large vector/matrix: <span class="math display">\[\vec{X}^{(t)} = \trans{\left( \vec{x}_1^{(t)}, \ldots, \vec{x}_n^{(t)} \right)} \in \R^{n\times 3}\]</span></li>
<li class="fragment">Matrix version of explicit integration <span class="math display">\[\vec{X}^{(t+1)} = (\vec{I} + \delta t \, \lambda \vec{L}) \, \vec{X}^{(t)}\]</span></li>
<li class="fragment">Matrix version of implicit integration <span class="math display">\[(\vec{I} - \delta t \, \lambda \vec{L}) \, \vec{X}^{(t+1)} = \vec{X}^{(t)}\]</span></li>
</ul>
<div class="comment fragment" style="left: 750px; top: 400px;">
Easy to implement, but requires small <span class="math inline">\(\delta t \lambda\)</span> for stability
</div>
<div class="comment fragment" style="left: 750px; top: 530px;">
Works for any <span class="math inline">\(\delta t \lambda\)</span>, but has to solve linear system(s)
</div>
<p><a href="http://w.multires.caltech.edu/pubs/ImplicitFairing.pdf" class="footer" target="_blank">Desbrun et al, <em>Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow</em>, SIGGRAPH 1999</a></p>
</section>
<section id="try-it-yourself-2" class="slide level1">
<h1>Try it yourself!</h1>
<p><iframe data-src="demos/smoothing/smoothing.html?model=../../meshes/kiss.off" class="" style="width:1000px;height:600px;"></iframe></p>
</section>
<section id="isotropic-remeshing" class="slide level1 section">
<h1>Isotropic Remeshing</h1>
</section>
<section id="isotropic-triangle-remeshing" class="slide level1">
<h1>Isotropic Triangle Remeshing</h1>
<div id="section-3">
<div style="width:33%;float:left">
<p><figure class="" style="height:500px;"><img src="images/remesh_max_original.png" style="height:500px;"></img><figcaption>original mesh</figcaption></figure></p>
</div>
<div style="width:33%;float:left">
<p><figure class="" style="height:500px;"><img src="images/remesh_max_uniform.png" style="height:500px;"></img><figcaption>uniform remeshing</figcaption></figure></p>
</div>
<div style="width:33%;float:left">
<p><figure class="" style="height:500px;"><img src="images/remesh_max_adaptive.png" style="height:500px;"></img><figcaption>adaptive remeshing</figcaption></figure></p>
</div>
<div style="clear: both">
</div>
</div>
</section>
<section id="uniform-remeshing" class="slide level1">
<h1>Uniform Remeshing</h1>
<div style="clear:both; height:50px;">
</div>
<ul>
<li>Specify target edge length <span class="math inline">\(L\)</span></li>
<li>Iterate a few times
<ol type="1">
<li><strong>Split</strong> edges longer than <span class="math inline">\(\frac{4}{3} L\)</span></li>
<li><strong>Collapse</strong> edges shorter than <span class="math inline">\(\frac{4}{5}L\)</span></li>
<li><strong>Flip</strong> edges to get closer to valence 6</li>
<li><strong>Shift</strong> vertices by tangential relaxation</li>
<li><strong>Project</strong> vertices onto input mesh</li>
</ol></li>
</ul>
<p><a href="https://graphics.uni-bielefeld.de/publications/sgp04.pdf" class="footer" target="_blank">Botsch & Kobbelt, <em>A Remeshing Approach to Multiresolution Modeling</em>, SGP 2004</a></p>
</section>
<section id="local-remeshing-operators" class="slide level1">
<h1>Local Remeshing Operators</h1>
<p><img class="" style="height:500px;" src="images/edge_collapse.png" style="height:500px;"></img> <img class="" style="height:500px;" src="images/edge_split.png" style="height:500px;"></img> <img class="" style="height:500px;" src="images/edge_flip.png" style="height:500px;"></img> <img class="" style="height:500px;" src="images/vertex_shift.png" style="height:500px;"></img></p>
<p><a href="https://graphics.uni-bielefeld.de/publications/sgp04.pdf" class="footer" target="_blank">Botsch & Kobbelt, <em>A Remeshing Approach to Multiresolution Modeling</em>, SGP 2004</a></p>
</section>
<section id="adaptive-remeshing" class="slide level1">
<h1>Adaptive Remeshing</h1>
<div id="section-4">
<div style="width:60%;float:left">
<ul>
<li>Adapt edge length to local curvature</li>
<li>Compute maximum principle curvature on reference mesh</li>
<li>Determine local target edge length from max-curvature</li>
<li>Adjust split & collapse criteria accordingly</li>
</ul>
</div>
<div style="width:40%;float:left">
<p><img class="" style="width:400px;" src="images/remesh-julius-2.png" style="width:400px;"></img></p>
</div>
<div style="clear: both">
</div>
</div>
<p><a href="https://graphics.uni-bielefeld.de/publications/eg13-remeshing.pdf" class="footer" target="_blank">Dunyach et al, <em>Adaptive Remeshing for Real-Time Mesh Deformation</em>, EG 2013</a></p>
</section>
<section id="real-time-remeshing" class="slide level1">
<h1>Real-Time Remeshing</h1>
<p><video data-autoplay controls src="videos/remesh-snake-1.mp4"class="" style="height:600px;"></video></p>
<p><a href="https://graphics.uni-bielefeld.de/publications/eg13-remeshing.pdf" class="footer" target="_blank">Dunyach et al, <em>Adaptive Remeshing for Real-Time Mesh Deformation</em>, EG 2013</a></p>
</section>
<section id="real-time-remeshing-1" class="slide level1">
<h1>Real-Time Remeshing</h1>
<p><video data-autoplay controls src="videos/remesh-snake-2.mp4"class="" style="height:600px;"></video></p>
<p><a href="https://graphics.uni-bielefeld.de/publications/eg13-remeshing.pdf" class="footer" target="_blank">Dunyach et al, <em>Adaptive Remeshing for Real-Time Mesh Deformation</em>, EG 2013</a></p>
</section>
<section id="lets-try" class="slide level1">
<h1>Let’s try!</h1>
<p><iframe data-src="demos/remeshing/remeshing.html?model=../../meshes/amo-highres.off" class="" style="width:1000px;height:600px;"></iframe></p>
</section>
<section id="feature-preservation" class="slide level1">
<h1>Feature Preservation</h1>
<div id="section-5">
<div style="width:60%;float:left">
<ul>
<li>Define feature edges / vertices
<ul>
<li>Large dihedral angles</li>
<li>Material boundaries</li>
</ul></li>
<li>Adjust local operators
<ul>
<li>Don’t flip feature edges</li>
<li>Collapse only along features</li>
<li>Univariate smoothing</li>
<li>Project to feature curves</li>
<li>Don’t touch feature vertices</li>
</ul></li>
</ul>
</div>
<div style="width:40%;float:left">
<p><img class="" style="width:400px;" src="images/remesh-fan-2.png" style="width:400px;"></img></p>
</div>
<div style="clear: both">
</div>
</div>
</section>
<section id="lets-try-1" class="slide level1">
<h1>Let’s try!</h1>
<p><iframe data-src="demos/remeshing/remeshing.html?model=../../meshes/fan-bad.off" class="" style="width:1000px;height:600px;"></iframe></p>
</section>
</div>
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