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<title>libigl Tutorial</title>
<meta name="author" content="Daniele Panozzo and Alec Jacobson"/>
<meta name="date" content="20 June 2014"/>
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<blockquote>
<p>Warning: This tutorial has been prepared for the static material accompanying
SGP Grad School 2014. Please find our <a href="http://htmlpreview.github.io/?https://github.com/libigl/libigl/blob/master/tutorial/tutorial.html">up-to-date tutorial notes</a>.</p>
</blockquote>
<h1 id="libigltutorialnotes">libigl tutorial notes</h1>
<h4 id="danielepanozzoandalecjacobsonsgpgraduateschool2014">Daniele Panozzo and Alec Jacobson, SGP Graduate School 2014</h4>
<p>Libigl is an open source C++ library for geometry processing research and
development. Dropping the heavy data structures of tradition geometry
libraries, libigl is a simple header-only library of encapsulated functions.
This combines the rapid prototyping familiar to Matlab or Python programmers
with the performance and versatility of C++. The tutorial is a self-contained,
hands-on introduction to libigl. Via interactive, step-by-step examples, we
demonstrate how to accomplish common geometry processing tasks such as
computation of differential quantities and operators, real-time deformation,
parametrization, numerical optimization and remeshing. Each section of the
lecture notes links to a cross-platform example application.</p>
<h1 id="tableofcontents">Table of contents</h1>
<ul>
<li><a href="#100">Chapter 1: Introduction to libigl</a>
<ul>
<li><a href="#100b">Libigl design principles</a></li>
<li><a href="#101">101 Mesh representation</a></li>
<li><a href="#102">102 Visualizing surfaces</a></li>
<li><a href="#103">103 Interaction with keyboard and mouse</a></li>
<li><a href="#104">104 Scalar field visualization</a></li>
<li><a href="#105">105 Overlays</a></li>
</ul></li>
<li><a href="#chapter2:discretegeometricquantitiesandoperators">Chapter 2: Discrete Geometric Quantities and
Operators</a>
<ul>
<li><a href="#normals">201 Normals</a>
<ul>
<li><a href="#per-face">Per-face</a></li>
<li><a href="#per-vertex">Per-vertex</a></li>
<li><a href="#per-corner">Per-corner</a></li>
</ul></li>
<li><a href="#gaussiancurvature">202 Gaussian Curvature</a></li>
<li><a href="#curvaturedirections">203 Curvature Directions</a></li>
<li><a href="#gradient">204 Gradient</a></li>
<li><a href="#laplacian">204 Laplacian</a>
<ul>
<li><a href="#massmatrix">Mass matrix</a></li>
<li><a href="#alternativeconstructionoflaplacian">Alternative construction of
Laplacian</a></li>
</ul></li>
</ul></li>
<li><a href="#chapter3:matricesandlinearalgebra">Chapter 3: Matrices and Linear Algebra</a>
<ul>
<li><a href="#slice">301 Slice</a></li>
<li><a href="#sort">302 Sort</a>
<ul>
<li><a href="#othermatlab-stylefunctions">Other Matlab-style functions</a></li>
</ul></li>
<li><a href="#laplaceequation">303 Laplace Equation</a>
<ul>
<li><a href="#quadraticenergyminimization">Quadratic energy minimization</a></li>
</ul></li>
<li><a href="#linearequalityconstraints">304 Linear Equality Constraints</a></li>
<li><a href="#quadraticprogramming">305 Quadratic Programming</a></li>
</ul></li>
<li><a href="#chapter4:shapedeformation">Chapter 4: Shape Deformation</a>
<ul>
<li><a href="#biharmonicdeformation">401 Biharmonic Deformation</a></li>
<li><a href="#polyharmonicdeformation">402 Polyharmonic Deformation</a></li>
<li><a href="#boundedbiharmonicweights">403 Bounded Biharmonic Weights</a></li>
<li><a href="#dualquaternionskinning">404 Dual Quaternion Skinning</a></li>
<li><a href="#as-rigid-as-possible">405 As-rigid-as-possible</a></li>
<li><a href="#fastautomaticskinningtransformations">406 Fast automatic skinning
transformations</a>
<ul>
<li><a href="#arapwithgroupededge-sets">ARAP with grouped edge-sets</a></li>
</ul></li>
</ul></li>
<li><a href="#500">Chapter 5: Parametrization</a>
<ul>
<li><a href="#501">501 Harmonic parametrization</a></li>
<li><a href="#502">502 Least-Square Conformal Maps</a></li>
<li><a href="#503">503 As-Rigid-As-Possible</a></li>
<li><a href="#504">504 N-Rotationally symmetric tangent fields</a></li>
<li><a href="#505">505 Global, seamless integer-grid parametrization</a></li>
<li><a href="#506">506 Anisotropic remeshing using frame fields</a></li>
<li><a href="#507">507 N-PolyVector fields</a></li>
<li><a href="#508">508 Conjugate vector fields</a></li>
<li><a href="#509">509 Planarization</a></li>
</ul></li>
<li><a href="#600">Chapter 6: External libraries</a>
<ul>
<li><a href="#601">601 State serialization</a></li>
<li><a href="#602">602 Mixing Matlab code</a>
<ul>
<li><a href="#savingamatlabworkspace">Saving a Matlab workspace</a></li>
<li><a href="#dumpingeigenmatricestocopyandpasteintomatlab">Dumping Eigen matrices to copy and paste into
Matlab</a></li>
</ul></li>
<li><a href="#603">603 Calling libigl functions from Matlab</a></li>
<li><a href="#604">604 Triangulation of closed polygons</a></li>
<li><a href="#605">605 Tetrahedralization of closed surfaces</a></li>
<li><a href="#606">606 Baking ambient occlusion</a></li>
<li><a href="#607">607 Picking vertices and faces</a></li>
<li><a href="#608">608 Locally Injective Maps</a></li>
</ul></li>
<li><p><a href="#future">Chapter 7: Outlook for continuing development</a></p></li>
</ul>
<h1 id="100">Chapter 1</h1>
<p>We introduce libigl with a series of self-contained examples. The purpose of
each example is to showcase a feature of libigl while applying to a practical
problem in geometry processing. In this chapter, we will present the basic
concepts of libigl and introduce a simple mesh viewer that allows to
visualize a surface mesh and its attributes. All the tutorial examples are
cross-platform and can be compiled on MacOSX, Linux and Windows.</p>
<h2 id="100b">libigl design principles</h2>
<p>Before getting into the examples, we summarize the main design principles in
libigl:</p>
<ol>
<li><p><strong>No complex data types.</strong> We mostly use matrices and vectors. This greatly
favors code reusability and forces the function authors to expose all the
parameters used by the algorithm. </p></li>
<li><p><strong>Minimal dependencies.</strong> We use external libraries only when necessary and
we wrap them in a small set of functions.</p></li>
<li><p><strong>Header-only.</strong> It is straight forward to use our library since it is only
one additional include directory in your project. (if you are worried about
compilation speed, it is also possible to build the library as a <a href="../build/">static
library</a>)</p></li>
<li><p><strong>Function encapsulation.</strong> Every function (including its full
implementation) is contained in a pair of .h/.cpp files with the same name of
the function.</p></li>
</ol>
<h3 id="downloadinglibigl">Downloading libigl</h3>
<p>libigl can be downloaded from our <a href="https://github.com/libigl/libigl">github
repository</a> or cloned with git:</p>
<pre><code class="bash">git clone https://github.com/libigl/libigl.git
</code></pre>
<p>The core libigl functionality only depends on the C++ Standard Library and
Eigen.</p>
<p>The examples in this tutorial depend on <a href="http://www.glfw.org">glfw</a>,
<a href="http://glew.sourceforge.net">glew</a> and <a href="http://anttweakbar.sourceforge.net/doc/">AntTweakBar</a>.
The source code of each library is bundled with libigl
and they can be compiled all at once using:</p>
<pre><code class="bash">sh compile_dependencies_macosx.sh (MACOSX)
sh compile_dependencies_linux.sh (LINUX)
</code></pre>
<p>For windows, precompiled binaries are provided (Visual Studio 2014 64bit).</p>
<p>To build all the examples in the tutorial, you can use the CMakeLists.txt in
the tutorial folder:</p>
<pre><code class="bash">cd tutorial
mkdir build
cd build
cmake -DCMAKE_BUILD_TYPE=Release ../
make
</code></pre>
<p>The examples can also be built independently using the CMakeLists.txt
inside each example folder.</p>
<p>A few examples in Chapter 5 requires the <a href="http://www.graphics.rwth-aachen.de/software/comiso">CoMiSo
solver</a> which has to be
downloaded and compiled separately.</p>
<h2 id="101">Mesh representation</h2>
<p>libigl uses the <a href="http://eigen.tuxfamily.org/">Eigen</a> library to encode vector
and matrices. We suggest that you keep the
<a href="http://eigen.tuxfamily.org/dox/group__QuickRefPage.html">dense</a> and
<a href="http://eigen.tuxfamily.org/dox/group__SparseQuickRefPage.html">sparse</a> quick
reference guides at hand while you read the examples in this tutorial.</p>
<p>A triangular mesh is encoded as a pair of matrices:</p>
<pre><code class="cpp">Eigen::MatrixXd V;
Eigen::MatrixXi F;
</code></pre>
<p><code>V</code> is a #N by 3 matrix which stores the coordinates of the vertices. Each
row stores the coordinate of a vertex, with its x,y and z coordinates in the first,
second and third column, respectively. The matrix <code>F</code> stores the triangle
connectivity: each line of <code>F</code> denotes a triangle whose 3 vertices are
represented as indices pointing to rows of <code>V</code>.</p>
<figure>
<img src="images/VF.png" alt="A simple mesh made of 2 triangles and 4 vertices." />
<figcaption>A simple mesh made of 2 triangles and 4 vertices.</figcaption></figure>
<p>Note that the order of the vertex indices in <code>F</code> determines the orientation of
the triangles and it should thus be consistent for the entire surface.
This simple representation has many advantages:</p>
<ol>
<li>it is memory efficient and cache friendly</li>
<li>the use of indices instead of pointers greatly simplifies debugging</li>
<li>the data can be trivially copied and serialized</li>
</ol>
<p>libigl provides input [output] functions to read [write] many common mesh formats.
The IO functions are contained in the files read*.h and write*.h. As a general
rule each libigl function is contained in a pair of .h/.cpp files with the same name.
By default, the .h files include the corresponding cpp files, making the library header-only.</p>
<p>Reading a mesh from a file requires a single libigl function call:</p>
<pre><code class="cpp">igl::readOFF(&quot;../shared/cube.off&quot;, V, F);
</code></pre>
<p>The function reads the mesh cube.off and it fills the provided <code>V</code> and <code>F</code> matrices.
Similarly, a mesh can be written in an OBJ file using:</p>
<pre><code class="cpp">igl::writeOBJ(&quot;cube.obj&quot;,V,F);
</code></pre>
<p><a href="101_FileIO/main.cpp">Example 101</a> contains a simple mesh
converter from OFF to OBJ format.</p>
<h2 id="102">Visualizing surfaces</h2>
<p>Libigl provides an glfw-based OpenGL 3.2 viewer to visualize surfaces, their
properties and additional debugging informations.</p>
<p>The following code (<a href="102_DrawMesh/main.cpp">Example 102</a>) is a basic skeleton
for all the examples that will be used in the tutorial.
It is a standalone application that loads a mesh and uses the viewer to
render it.</p>
<pre><code class="cpp">#include &lt;igl/readOFF.h&gt;
#include &lt;igl/viewer/Viewer.h&gt;
Eigen::MatrixXd V;
Eigen::MatrixXi F;
int main(int argc, char *argv[])
{
// Load a mesh in OFF format
igl::readOFF(&quot;../shared/bunny.off&quot;, V, F);
// Plot the mesh
igl::Viewer viewer;
viewer.data.set_mesh(V, F);
viewer.launch();
}
</code></pre>
<p>The function <code>set_mesh</code> copies the mesh into the viewer.
<code>Viewer.launch()</code> creates a window, an OpenGL context and it starts the draw loop.
Additional properties can be plotted on the mesh (as we will see later),
and it is possible to extend the viewer with standard OpenGL code.
Please see the documentation in
<a href="../include/igl/Viewer/Viewer.h">Viewer.h</a> for more details.</p>
<figure>
<img src="images/102_DrawMesh.png" alt="(Example 102) loads and draws a
mesh." />
<figcaption>(<a href="102_DrawMesh/main.cpp">Example 102</a>) loads and draws a
mesh.</figcaption></figure>
<h2 id="103">Interaction with keyboard and mouse</h2>
<p>Keyboard and mouse events triggers callbacks that can be registered in the
viewer. The viewer supports the following callbacks:</p>
<pre><code class="cpp">bool (*callback_pre_draw)(Viewer&amp; viewer);
bool (*callback_post_draw)(Viewer&amp; viewer);
bool (*callback_mouse_down)(Viewer&amp; viewer, int button, int modifier);
bool (*callback_mouse_up)(Viewer&amp; viewer, int button, int modifier);
bool (*callback_mouse_move)(Viewer&amp; viewer, int mouse_x, int mouse_y);
bool (*callback_mouse_scroll)(Viewer&amp; viewer, float delta_y);
bool (*callback_key_down)(Viewer&amp; viewer, unsigned char key, int modifiers);
bool (*callback_key_up)(Viewer&amp; viewer, unsigned char key, int modifiers);
</code></pre>
<p>A keyboard callback can be used to visualize multiple meshes or different
stages of an algorithm, as demonstrated in <a href="103_Events/main.cpp">Example 103</a>, where
the keyboard callback changes the visualized mesh depending on the key pressed:</p>
<pre><code class="cpp">bool key_down(igl::Viewer&amp; viewer, unsigned char key, int modifier)
{
if (key == '1')
{
viewer.data.clear();
viewer.data.set_mesh(V1, F1);
}
else if (key == '2')
{
viewer.data.clear();
viewer.data.set_mesh(V2, F2);
}
return false;
}
</code></pre>
<p>The callback is registered in the viewer as follows:</p>
<pre><code class="cpp">viewer.callback_key_down = &amp;key_down;
</code></pre>
<p>Note that the mesh is cleared before using set_mesh. This has to be called
every time the number of vertices or faces of the plotted mesh changes. Every
callback returns a boolean value that tells the viewer if the event has been
handled by the plugin, or if the viewer should process it normally. This is
useful, for example, to disable the default mouse event handling if you want to
control the camera directly in your code.</p>
<p>The viewer can be extended using plugins, which are classes that implements all
the viewer&#8217;s callbacks. See the
<a href="../include/igl/viewer/ViewerPlugin.h">Viewer_plugin</a> for more details.</p>
<h2 id="104">Scalar field visualization</h2>
<p>Colors and normals can be associated to faces or vertices using the
set_colors function:</p>
<pre><code class="cpp">viewer.data.set_colors(C);
</code></pre>
<p><code>C</code> is a #C by 3 matrix with one RGB color per row. <code>C</code> must have as many
rows as the number of faces <strong>or</strong> the number of vertices of the mesh.
Depending on the size of <code>C</code>, the viewer applies the color to the faces or to
the vertices.</p>
<p>Colors can be used to visualize a scalar function defined on a surface. The
scalar function is converted to colors using a color transfer function, which
maps a scalar value between 0 and 1 to a color. A simple example of a scalar
field defined on a surface is the z coordinate of each point, which can be
extract from our mesh representation by taking the last column of <code>V</code>
(<a href="104_Colors/main.cpp">Example 104</a>). The function <code>igl::jet</code> can be used to
convert it to colors:</p>
<pre><code class="cpp">Eigen::VectorXd Z = V.col(2);
igl::jet(Z,true,C);
</code></pre>
<p>The first row extracts the third column from <code>V</code> (the z coordinate of each
vertex) and the second calls a libigl functions that converts a scalar field to colors. The second parameter of jet normalizes the scalar field to lie between 0 and 1 before applying the transfer function.</p>
<figure>
<img src="images/104_Colors.png" alt="(Example 104) igl::jet converts a scalar field to a
color field." />
<figcaption>(<a href="104_Colors/main.cpp">Example 104</a>) igl::jet converts a scalar field to a
color field.</figcaption></figure>
<p><code>igl::jet</code> is an example of a standard function in libigl: it takes simple
types and can be easily reused for many different tasks. Not committing to
heavy data structures types favors simplicity, ease of use and reusability.</p>
<h2 id="105">Overlays</h2>
<p>In addition to plotting the surface, the viewer supports the visualization of points, lines and text labels: these overlays can be very helful while developing geometric processing algorithms to plot debug informations.</p>
<pre><code class="cpp">viewer.data.add_points(P,Eigen::RowVector3d(r,g,b));
</code></pre>
<p>Draws a point of color r,g,b for each row of P. The point is placed at the coordinates specified in each row of P, which is a #P by 3 matrix.</p>
<pre><code class="cpp">viewer.data.add_edges(P1,P2,Eigen::RowVector3d(r,g,b);
</code></pre>
<p>Draws a line of color r,g,b for each row of P1 and P2, which connects the 3D point in to the point in P2. Both P1 and P2 are of size #P by 3.</p>
<pre><code class="cpp">viewer.data.add_label(p,str);
</code></pre>
<p>Draws a label containing the string str at the position p, which is a vector of length 3.</p>
<p>These functions are demonstrate in <a href="105_Overlays/main.cpp">Example 105</a> where
the bounding box of a mesh is plotted using lines and points.
Using matrices to encode the mesh and its attributes allows to write short and
efficient code for many operations, avoiding to write for loops. For example,
the bounding box of a mesh can be found by taking the colwise maximum and minimum of <code>V</code>:</p>
<pre><code class="cpp">Eigen::Vector3d m = V.colwise().minCoeff();
Eigen::Vector3d M = V.colwise().maxCoeff();
</code></pre>
<figure>
<img src="images/105_Overlays.png" alt="(Example 105) The bounding box of a mesh is shown
using overlays." />
<figcaption>(<a href="105_Overlays/main.cpp">Example 105</a>) The bounding box of a mesh is shown
using overlays.</figcaption></figure>
<h1 id="chapter2:discretegeometricquantitiesandoperators">Chapter 2: Discrete Geometric Quantities and Operators</h1>
<p>This chapter illustrates a few discrete quantities that libigl can compute on a
mesh and the libigl functions that construct popular discrete differential
geometry operators. It also provides an introduction to basic drawing and coloring routines of our viewer.</p>
<h2 id="normals">Normals</h2>
<p>Surface normals are a basic quantity necessary for rendering a surface. There
are a variety of ways to compute and store normals on a triangle mesh. <a href="201_Normals/main.cpp">Example 201</a> demonstrates how to compute and visualize normals with libigl.</p>
<h3 id="per-face">Per-face</h3>
<p>Normals are well defined on each triangle of a mesh as the vector orthogonal to
triangle&#8217;s plane. These piecewise-constant normals produce piecewise-flat
renderings: the surface appears non-smooth and reveals its underlying
discretization.</p>
<h3 id="per-vertex">Per-vertex</h3>
<p>Normals can be computed and stored on vertices, and interpolated in the interior of the triangles to produce smooth renderings (<a href="http://en.wikipedia.org/wiki/Phong_shading">Phong shading</a>).
Most techniques for computing per-vertex normals take an average of incident face normals. The main difference between these techniques is their weighting scheme: Uniform
weighting is heavily biased by the discretization choice, whereas area-based
or angle-based weighting is more forgiving.</p>
<p>The typical half-edge style computation of area-based weights has this structure:</p>
<pre><code class="cpp">N.setZero(V.rows(),3);
for(int i : vertices)
{
for(face : incident_faces(i))
{
N.row(i) += face.area * face.normal;
}
}
N.rowwise().normalize();
</code></pre>
<p>At first glance, it might seem inefficient to loop over incident faces&#8212;and thus constructing the per-vertex normals&#8212; without using an half-edge data structure. However, per-vertex normals may be <em>throwing</em> each face normal to
running sums on its corner vertices:</p>
<pre><code class="cpp">N.setZero(V.rows(),3);
for(int f = 0; f &lt; F.rows();f++)
{
for(int c = 0; c &lt; 3;c++)
{
N.row(F(f,c)) += area(f) * face_normal.row(f);
}
}
N.rowwise().normalize();
</code></pre>
<h3 id="per-corner">Per-corner</h3>
<p>Storing normals per-corner is an efficient and convenient way of supporting both
smooth and sharp (e.g. creases and corners) rendering. This format is common to
OpenGL and the .obj mesh file format. Often such normals are tuned by the mesh
designer, but creases and corners can also be computed automatically. Libigl
implements a simple scheme which computes corner normals as averages of
normals of faces incident on the corresponding vertex which do not deviate by more than a specified dihedral angle (e.g. 20°).</p>
<figure>
<img src="images/fandisk-normals.jpg" alt="The Normals example computes per-face (left), per-vertex (middle) and
per-corner (right) normals" />
<figcaption>The <code>Normals</code> example computes per-face (left), per-vertex (middle) and
per-corner (right) normals</figcaption></figure>
<h2 id="gaussiancurvature">Gaussian curvature</h2>
<p>Gaussian curvature on a continuous surface is defined as the product of the
principal curvatures:</p>
<p><span class="math">\(k_G = k_1 k_2.\)</span></p>
<p>As an <em>intrinsic</em> measure, it depends on the metric and
not the surface&#8217;s embedding.</p>
<p>Intuitively, Gaussian curvature tells how locally spherical or <em>elliptic</em> the
surface is ( <span class="math">\(k_G>0\)</span> ), how locally saddle-shaped or <em>hyperbolic</em> the surface
is ( <span class="math">\(k_G<0\)</span> ), or how locally cylindrical or <em>parabolic</em> ( <span class="math">\(k_G=0\)</span> ) the
surface is.</p>
<p>In the discrete setting, one definition for a &#8220;discrete Gaussian curvature&#8221;
on a triangle mesh is via a vertex&#8217;s <em>angular deficit</em>:</p>
<p><span class="math">\(k_G(v_i) = 2π - \sum\limits_{j\in N(i)}θ_{ij},\)</span></p>
<p>where <span class="math">\(N(i)\)</span> are the triangles incident on vertex <span class="math">\(i\)</span> and <span class="math">\(θ_{ij}\)</span> is the angle
at vertex <span class="math">\(i\)</span> in triangle <span class="math">\(j\)</span> <a class="citation" href="#fn:1" title="Jump to citation">[1]<span class="citekey" style="display:none">meyer_2003</span></a>.</p>
<p>Just like the continuous analog, our discrete Gaussian curvature reveals
elliptic, hyperbolic and parabolic vertices on the domain, as demonstrated in <a href="202GaussianCurvature/main.cpp">Example 202</a>.</p>
<figure>
<img src="images/bumpy-gaussian-curvature.jpg" alt="The GaussianCurvature example computes discrete Gaussian curvature and
visualizes it in pseudocolor." />
<figcaption>The <code>GaussianCurvature</code> example computes discrete Gaussian curvature and
visualizes it in pseudocolor.</figcaption></figure>
<h2 id="curvaturedirections">Curvature directions</h2>
<p>The two principal curvatures <span class="math">\((k_1,k_2)\)</span> at a point on a surface measure how
much the surface bends in different directions. The directions of maximum and
minimum (signed) bending are called principal directions and are always
orthogonal.</p>
<p>Mean curvature is defined as the average of principal curvatures:</p>
<p><span class="math">\(H = \frac{1}{2}(k_1 + k_2).\)</span></p>
<p>One way to extract mean curvature is by examining the Laplace-Beltrami operator
applied to the surface positions. The result is a so-called mean-curvature
normal:</p>
<p><span class="math">\(-\Delta \mathbf{x} = H \mathbf{n}.\)</span></p>
<p>It is easy to compute this on a discrete triangle mesh in libigl using the
cotangent Laplace-Beltrami operator <a class="citation" href="#fn:1" title="Jump to citation">[1]<span class="citekey" style="display:none">meyer_2003</span></a>.</p>
<pre><code class="cpp">#include &lt;igl/cotmatrix.h&gt;
#include &lt;igl/massmatrix.h&gt;
#include &lt;igl/invert_diag.h&gt;
...
MatrixXd HN;
SparseMatrix&lt;double&gt; L,M,Minv;
igl::cotmatrix(V,F,L);
igl::massmatrix(V,F,igl::MASSMATRIX_TYPE_VORONOI,M);
igl::invert_diag(M,Minv);
HN = -Minv*(L*V);
H = (HN.rowwise().squaredNorm()).array().sqrt();
</code></pre>
<p>Combined with the angle defect definition of discrete Gaussian curvature, one
can define principal curvatures and use least squares fitting to find
directions <a class="citation" href="#fn:1" title="Jump to citation">[1]<span class="citekey" style="display:none">meyer_2003</span></a>.</p>
<p>Alternatively, a robust method for determining principal curvatures is via
quadric fitting <a class="citation" href="#fn:2" title="Jump to citation">[2]<span class="citekey" style="display:none">panozzo_2010</span></a>. In the neighborhood
around every vertex, a best-fit quadric is found and principal curvature values
and directions are analytically computed on this quadric (<a href="203_curvatureDirections/main.cpp">Example 203</a>).</p>
<figure>
<img src="images/fertility-principal-curvature.jpg" alt="The CurvatureDirections example computes principal curvatures via quadric
fitting and visualizes mean curvature in pseudocolor and principal directions
with a cross field." />
<figcaption>The <code>CurvatureDirections</code> example computes principal curvatures via quadric
fitting and visualizes mean curvature in pseudocolor and principal directions
with a cross field.</figcaption></figure>
<h2 id="gradient">Gradient</h2>
<p>Scalar functions on a surface can be discretized as a piecewise linear function
with values defined at each mesh vertex:</p>
<p><span class="math">\(f(\mathbf{x}) \approx \sum\limits_{i=1}^n \phi_i(\mathbf{x})\, f_i,\)</span></p>
<p>where <span class="math">\(\phi_i\)</span> is a piecewise linear hat function defined by the mesh so that
for each triangle <span class="math">\(\phi_i\)</span> is <em>the</em> linear function which is one only at
vertex <span class="math">\(i\)</span> and zero at the other corners.</p>
<figure>
<img src="images/hat-function.jpg" alt="Hat function $\phi_i$ is one at vertex $i$, zero at all other vertices, and
linear on incident triangles." />
<figcaption>Hat function <span class="math">\(\phi_i\)</span> is one at vertex <span class="math">\(i\)</span>, zero at all other vertices, and
linear on incident triangles.</figcaption></figure>
<p>Thus gradients of such piecewise linear functions are simply sums of gradients
of the hat functions:</p>
<p><span class="math">\(\nabla f(\mathbf{x}) \approx
\nabla \sum\limits_{i=1}^n \nabla \phi_i(\mathbf{x})\, f_i =
\sum\limits_{i=1}^n \nabla \phi_i(\mathbf{x})\, f_i.\)</span></p>
<p>This reveals that the gradient is a linear function of the vector of <span class="math">\(f_i\)</span>
values. Because the <span class="math">\(\phi_i\)</span> are linear in each triangle, their gradients are
<em>constant</em> in each triangle. Thus our discrete gradient operator can be written
as a matrix multiplication taking vertex values to triangle values:</p>
<p><span class="math">\(\nabla f \approx \mathbf{G}\,\mathbf{f},\)</span></p>
<p>where <span class="math">\(\mathbf{f}\)</span> is <span class="math">\(n\times 1\)</span> and <span class="math">\(\mathbf{G}\)</span> is an <span class="math">\(md\times n\)</span> sparse
matrix. This matrix <span class="math">\(\mathbf{G}\)</span> can be derived geometrically, e.g.
<a class="citation" href="#fn:3" title="Jump to citation">[<span class="locator">ch. 2</span>, 3]<span class="citekey" style="display:none">jacobson_thesis_2013</span></a>.
Libigl&#8217;s <code>grad</code> function computes <span class="math">\(\mathbf{G}\)</span> for
triangle and tetrahedral meshes (<a href="204_Gradient/main.cpp">Example 204</a>):</p>
<figure>
<img src="images/cheburashka-gradient.jpg" alt="The Gradient example computes gradients of an input function on a mesh and
visualizes the vector field." />
<figcaption>The <code>Gradient</code> example computes gradients of an input function on a mesh and
visualizes the vector field.</figcaption></figure>
<h2 id="laplacian">Laplacian</h2>
<p>The discrete Laplacian is an essential geometry processing tool. Many
interpretations and flavors of the Laplace and Laplace-Beltrami operator exist.</p>
<p>In open Euclidean space, the <em>Laplace</em> operator is the usual divergence of
gradient (or equivalently the Laplacian of a function is the trace of its
Hessian):</p>
<p><span class="math">\(\Delta f =
\frac{\partial^2 f}{\partial x^2} +
\frac{\partial^2 f}{\partial y^2} +
\frac{\partial^2 f}{\partial z^2}.\)</span></p>
<p>The <em>Laplace-Beltrami</em> operator generalizes this to surfaces.</p>
<p>When considering piecewise-linear functions on a triangle mesh, a discrete
Laplacian may be derived in a variety of ways. The most popular in geometry
processing is the so-called ``cotangent Laplacian&#8217;&#8217; <span class="math">\(\mathbf{L}\)</span>, arising
simultaneously from FEM, DEC and applying divergence theorem to vertex
one-rings. As a linear operator taking vertex values to vertex values, the
Laplacian <span class="math">\(\mathbf{L}\)</span> is a <span class="math">\(n\times n\)</span> matrix with elements:</p>
<p><span class="math">\(L_{ij} = \begin{cases}j \in N(i) &\cot \alpha_{ij} + \cot \beta_{ij},\\
j \notin N(i) & 0,\\
i = j & -\sum\limits_{k\neq i} L_{ik},
\end{cases}\)</span></p>
<p>where <span class="math">\(N(i)\)</span> are the vertices adjacent to (neighboring) vertex <span class="math">\(i\)</span>, and
<span class="math">\(\alpha_{ij},\beta_{ij}\)</span> are the angles opposite to edge <span class="math">\({ij}\)</span>.
This formula leads to a typical half-edge style implementation for
constructing <span class="math">\(\mathbf{L}\)</span>:</p>
<pre><code class="cpp">for(int i : vertices)
{
for(int j : one_ring(i))
{
for(int k : triangle_on_edge(i,j))
{
L(i,j) = cot(angle(i,j,k));
L(i,i) -= cot(angle(i,j,k));
}
}
}
</code></pre>
<p>Similarly as before, it may seem to loop over one-rings without having an half-edge data structure. However, this is not the case, since the Laplacian may be built by summing together contributions for each triangle, much in spirit with its FEM discretization
of the Dirichlet energy (sum of squared gradients):</p>
<pre><code class="cpp">for(triangle t : triangles)
{
for(edge i,j : t)
{
L(i,j) += cot(angle(i,j,k));
L(j,i) += cot(angle(i,j,k));
L(i,i) -= cot(angle(i,j,k));
L(j,j) -= cot(angle(i,j,k));
}
}
</code></pre>
<p>Libigl implements discrete &#8220;cotangent&#8221; Laplacians for triangles meshes and
tetrahedral meshes, building both with fast geometric rules rather than &#8220;by the
book&#8221; FEM construction which involves many (small) matrix inversions, cf.
<a class="citation" href="#fn:4" title="Jump to citation">[4]<span class="citekey" style="display:none">sharf_2007</span></a>.</p>
<p>The operator applied to mesh vertex positions amounts to smoothing by <em>flowing</em>
the surface along the mean curvature normal direction (<a href="205_Laplacian/main.cpp">Example 205</a>). Note that this is equivalent to minimizing surface area.</p>
<figure>
<img src="images/cow-curvature-flow.jpg" alt="The Laplacian example computes conformalized mean curvature flow using the
cotangent Laplacian ." />
<figcaption>The <code>Laplacian</code> example computes conformalized mean curvature flow using the
cotangent Laplacian <a class="citation" href="#fn:5" title="Jump to citation">[5]<span class="citekey" style="display:none">kazhdan_2012</span></a>.</figcaption></figure>
<h3 id="massmatrix">Mass matrix</h3>
<p>The mass matrix <span class="math">\(\mathbf{M}\)</span> is another <span class="math">\(n \times n\)</span> matrix which takes vertex
values to vertex values. From an FEM point of view, it is a discretization of
the inner-product: it accounts for the area around each vertex. Consequently,
<span class="math">\(\mathbf{M}\)</span> is often a diagonal matrix, such that <span class="math">\(M_{ii}\)</span> is the barycentric
or voronoi area around vertex <span class="math">\(i\)</span> in the mesh <a class="citation" href="#fn:1" title="Jump to citation">[1]<span class="citekey" style="display:none">meyer_2003</span></a>. The inverse of
this matrix is also very useful as it transforms integrated quantities into
point-wise quantities, e.g.:</p>
<p><span class="math">\(\nabla f \approx \mathbf{M}^{-1} \mathbf{L} \mathbf{f}.\)</span></p>
<p>In general, when encountering squared quantities integrated over the surface,
the mass matrix will be used as the discretization of the inner product when
sampling function values at vertices:</p>
<p><span class="math">\(\int_S x\, y\ dA \approx \mathbf{x}^T\mathbf{M}\,\mathbf{y}.\)</span></p>
<p>An alternative mass matrix <span class="math">\(\mathbf{T}\)</span> is a <span class="math">\(md \times md\)</span> matrix which takes
triangle vector values to triangle vector values. This matrix represents an
inner-product accounting for the area associated with each triangle (i.e. the
triangles true area).</p>
<h3 id="alternativeconstructionoflaplacian">Alternative construction of Laplacian</h3>
<p>An alternative construction of the discrete cotangent Laplacian is by
&#8220;squaring&#8221; the discrete gradient operator. This may be derived by applying
Green&#8217;s identity (ignoring boundary conditions for the moment):</p>
<p><span class="math">\(\int_S \|\nabla f\|^2 dA = \int_S f \Delta f dA\)</span></p>
<p>Or in matrix form which is immediately translatable to code:</p>
<p><span class="math">\(\mathbf{f}^T \mathbf{G}^T \mathbf{T} \mathbf{G} \mathbf{f} =
\mathbf{f}^T \mathbf{M} \mathbf{M}^{-1} \mathbf{L} \mathbf{f} =
\mathbf{f}^T \mathbf{L} \mathbf{f}.\)</span></p>
<p>So we have that <span class="math">\(\mathbf{L} = \mathbf{G}^T \mathbf{T} \mathbf{G}\)</span>. This also
hints that we may consider <span class="math">\(\mathbf{G}^T\)</span> as a discrete <em>divergence</em> operator,
since the Laplacian is the divergence of the gradient. Naturally, <span class="math">\(\mathbf{G}^T\)</span> is
a <span class="math">\(n \times md\)</span> sparse matrix which takes vector values stored at triangle faces
to scalar divergence values at vertices.</p>
<h1 id="chapter3:matricesandlinearalgebra">Chapter 3: Matrices and linear algebra</h1>
<p>Libigl relies heavily on the Eigen library for dense and sparse linear algebra
routines. Besides geometry processing routines, libigl has linear algebra
routines which bootstrap Eigen and make it feel even more similar to a high-level
algebra library such as Matlab.</p>
<h2 id="slice">Slice</h2>
<p>A very familiar and powerful routine in Matlab is array slicing. This allows
reading from or writing to a possibly non-contiguous sub-matrix. Let&#8217;s consider
the Matlab code:</p>
<pre><code class="matlab">B = A(R,C);
</code></pre>
<p>If <code>A</code> is a <span class="math">\(m \times n\)</span> matrix and <code>R</code> is a <span class="math">\(j\)</span>-long list of row-indices
(between 1 and <span class="math">\(m\)</span>) and <code>C</code> is a <span class="math">\(k\)</span>-long list of column-indices, then as a
result <code>B</code> will be a <span class="math">\(j \times k\)</span> matrix drawing elements from <code>A</code> according to
<code>R</code> and <code>C</code>. In libigl, the same functionality is provided by the <code>slice</code>
function (<a href="301_Slice/main.cpp">Example 301</a>):</p>
<pre><code class="cpp">VectorXi R,C;
MatrixXd A,B;
...
igl::slice(A,R,C,B);
</code></pre>
<p>Note that <code>A</code> and <code>B</code> could also be sparse matrices.</p>
<p>Similarly, consider the Matlab code:</p>
<pre><code class="matlab">A(R,C) = B;
</code></pre>
<p>Now, the selection is on the left-hand side so the <span class="math">\(j \times k\)</span> matrix <code>B</code> is
being <em>written into</em> the submatrix of <code>A</code> determined by <code>R</code> and <code>C</code>. This
functionality is provided in libigl using <code>slice_into</code>:</p>
<pre><code class="cpp">igl::slice_into(B,R,C,A);
</code></pre>
<figure>
<img src="images/decimated-knight-slice-color.jpg" alt="The example Slice shows how to use igl::slice to change the colors for
triangles on a mesh." />
<figcaption>The example <code>Slice</code> shows how to use <code>igl::slice</code> to change the colors for
triangles on a mesh.</figcaption></figure>
<h2 id="sort">Sort</h2>
<p>Matlab and other higher-level languages make it very easy to extract indices of
sorting and comparison routines. For example in Matlab, one can write:</p>
<pre><code class="matlab">[Y,I] = sort(X,1,'ascend');
</code></pre>
<p>so if <code>X</code> is a <span class="math">\(m \times n\)</span> matrix then <code>Y</code> will also be an <span class="math">\(m \times n\)</span> matrix
with entries sorted along dimension <code>1</code> in <code>'ascend'</code>ing order. The second
output <code>I</code> is a <span class="math">\(m \times n\)</span> matrix of indices such that <code>Y(i,j) =
X(I(i,j),j);</code>. That is, <code>I</code> reveals how <code>X</code> is sorted into <code>Y</code>.</p>
<p>This same functionality is supported in libigl:</p>
<pre><code class="cpp">igl::sort(X,1,true,Y,I);
</code></pre>
<p>Similarly, sorting entire rows can be accomplished in Matlab using:</p>
<pre><code class="matlab">[Y,I] = sortrows(X,'ascend');
</code></pre>
<p>where now <code>I</code> is a <span class="math">\(m\)</span> vector of indices such that <code>Y = X(I,:)</code>.</p>
<p>In libigl, this is supported with</p>
<pre><code class="cpp">igl::sortrows(X,true,Y,I);
</code></pre>
<p>where again <code>I</code> reveals the index of sort so that it can be reproduced with
<code>igl::slice(X,I,1,Y)</code>.</p>
<p>Analogous functions are available in libigl for: <code>max</code>, <code>min</code>, and <code>unique</code>.</p>
<figure>
<img src="images/decimated-knight-sort-color.jpg" alt="The example Sort shows how to use igl::sortrows to
pseudocolor triangles according to their barycenters sorted
order (Example 302)." />
<figcaption>The example <code>Sort</code> shows how to use <code>igl::sortrows</code> to
pseudocolor triangles according to their barycenters&#8217; sorted
order (<a href="302_Sort/main.cpp">Example 302</a>).</figcaption></figure>
<h3 id="othermatlab-stylefunctions">Other Matlab-style functions</h3>
<p>Libigl implements a variety of other routines with the same api and
functionality as common Matlab functions.</p>
<table>
<colgroup>
<col style="text-align:left;"/>
<col style="text-align:left;"/>
</colgroup>
<thead>
<tr>
<th style="text-align:left;">Name</th>
<th style="text-align:left;">Description</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;"><code>igl::any_of</code></td>
<td style="text-align:left;">Whether any elements are non-zero (true)</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::cat</code></td>
<td style="text-align:left;">Concatenate two matrices (especially useful for dealing with Eigen sparse matrices)</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::ceil</code></td>
<td style="text-align:left;">Round entries up to nearest integer</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::cumsum</code></td>
<td style="text-align:left;">Cumulative sum of matrix elements</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::colon</code></td>
<td style="text-align:left;">Act like Matlab&#8217;s <code>:</code>, similar to Eigen&#8217;s <code>LinSpaced</code></td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::cross</code></td>
<td style="text-align:left;">Cross product per-row</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::dot</code></td>
<td style="text-align:left;">dot product per-row</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::find</code></td>
<td style="text-align:left;">Find subscripts of non-zero entries</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::floot</code></td>
<td style="text-align:left;">Round entries down to nearest integer</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::histc</code></td>
<td style="text-align:left;">Counting occurrences for building a histogram</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::hsv_to_rgb</code></td>
<td style="text-align:left;">Convert HSV colors to RGB (cf. Matlab&#8217;s <code>hsv2rgb</code>)</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::intersect</code></td>
<td style="text-align:left;">Set intersection of matrix elements.</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::jet</code></td>
<td style="text-align:left;">Quantized colors along the rainbow.</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::kronecker_product</code></td>
<td style="text-align:left;">Compare to Matlab&#8217;s <code>kronprod</code></td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::median</code></td>
<td style="text-align:left;">Compute the median per column</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::mode</code></td>
<td style="text-align:left;">Compute the mode per column</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::orth</code></td>
<td style="text-align:left;">Orthogonalization of a basis</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::setdiff</code></td>
<td style="text-align:left;">Set difference of matrix elements</td>
</tr>
<tr>
<td style="text-align:left;"><code>igl::speye</code></td>
<td style="text-align:left;">Identity as sparse matrix</td>
</tr>
</tbody>
</table>
<h2 id="laplaceequation">Laplace equation</h2>
<p>A common linear system in geometry processing is the Laplace equation:</p>
<p><span class="math">\(∆z = 0\)</span></p>
<p>subject to some boundary conditions, for example Dirichlet boundary conditions
(fixed value):</p>
<p><span class="math">\(\left.z\right|_{\partial{S}} = z_{bc}\)</span></p>
<p>In the discrete setting, the linear system can be written as:</p>
<p><span class="math">\(\mathbf{L} \mathbf{z} = \mathbf{0}\)</span></p>
<p>where <span class="math">\(\mathbf{L}\)</span> is the <span class="math">\(n \times n\)</span> discrete Laplacian and <span class="math">\(\mathbf{z}\)</span> is a
vector of per-vertex values. Most of <span class="math">\(\mathbf{z}\)</span> correspond to interior
vertices and are unknown, but some of <span class="math">\(\mathbf{z}\)</span> represent values at boundary
vertices. Their values are known so we may move their corresponding terms to
the right-hand side.</p>
<p>Conceptually, this is very easy if we have sorted <span class="math">\(\mathbf{z}\)</span> so that interior
vertices come first and then boundary vertices:</p>
<p><span class="math">\[\left(\begin{array}{cc}
\mathbf{L}_{in,in} & \mathbf{L}_{in,b}\\
\mathbf{L}_{b,in} & \mathbf{L}_{b,b}\end{array}\right)
\left(\begin{array}{c}
\mathbf{z}_{in}\\
\mathbf{L}_{b}\end{array}\right) =
\left(\begin{array}{c}
\mathbf{0}_{in}\\
\mathbf{*}_{b}\end{array}\right)\]</span></p>
<p>The bottom block of equations is no longer meaningful so we&#8217;ll only consider
the top block:</p>
<p><span class="math">\[\left(\begin{array}{cc}
\mathbf{L}_{in,in} & \mathbf{L}_{in,b}\end{array}\right)
\left(\begin{array}{c}
\mathbf{z}_{in}\\
\mathbf{z}_{b}\end{array}\right) =
\mathbf{0}_{in}\]</span></p>
<p>We can move the known values to the right-hand side:</p>
<p><span class="math">\[\mathbf{L}_{in,in}
\mathbf{z}_{in} = -
\mathbf{L}_{in,b}
\mathbf{z}_{b}\]</span></p>
<p>Finally we can solve this equation for the unknown values at interior vertices
<span class="math">\(\mathbf{z}_{in}\)</span>.</p>
<p>However, our vertices will often not be sorted in this way. One option would be to sort <code>V</code>,
then proceed as above and then <em>unsort</em> the solution <code>Z</code> to match <code>V</code>. However,
this solution is not very general.</p>
<p>With array slicing no explicit sort is needed. Instead we can <em>slice-out</em>
submatrix blocks (<span class="math">\(\mathbf{L}_{in,in}\)</span>, <span class="math">\(\mathbf{L}_{in,b}\)</span>, etc.) and follow
the linear algebra above directly. Then we can slice the solution <em>into</em> the
rows of <code>Z</code> corresponding to the interior vertices (<a href="303_LaplaceEquation/main.cpp">Example 303</a>).</p>
<figure>
<img src="images/camelhead-laplace-equation.jpg" alt="The LaplaceEquation example solves a Laplace equation with Dirichlet
boundary conditions." />
<figcaption>The <code>LaplaceEquation</code> example solves a Laplace equation with Dirichlet
boundary conditions.</figcaption></figure>
<h3 id="quadraticenergyminimization">Quadratic energy minimization</h3>
<p>The same Laplace equation may be equivalently derived by minimizing Dirichlet
energy subject to the same boundary conditions:</p>
<p><span class="math">\(\mathop{\text{minimize }}_z \frac{1}{2}\int\limits_S \|\nabla z\|^2 dA\)</span></p>
<p>On our discrete mesh, recall that this becomes</p>
<p><span class="math">\(\mathop{\text{minimize }}_\mathbf{z} \frac{1}{2}\mathbf{z}^T \mathbf{G}^T \mathbf{D}
\mathbf{G} \mathbf{z} \rightarrow \mathop{\text{minimize }}_\mathbf{z} \mathbf{z}^T \mathbf{L} \mathbf{z}\)</span></p>
<p>The general problem of minimizing some energy over a mesh subject to fixed
value boundary conditions is so wide spread that libigl has a dedicated api for
solving such systems.</p>
<p>Let us consider a general quadratic minimization problem subject to different
common constraints:</p>
<p><span class="math">\[\mathop{\text{minimize }}_\mathbf{z} \frac{1}{2}\mathbf{z}^T \mathbf{Q} \mathbf{z} +
\mathbf{z}^T \mathbf{B} + \text{constant},\]</span></p>
<p>subject to</p>
<p><span class="math">\[\mathbf{z}_b = \mathbf{z}_{bc} \text{ and } \mathbf{A}_{eq} \mathbf{z} =
\mathbf{B}_{eq},\]</span></p>
<p>where</p>
<ul>
<li><span class="math">\(\mathbf{Q}\)</span> is a (usually sparse) <span class="math">\(n \times n\)</span> positive semi-definite
matrix of quadratic coefficients (Hessian),</li>
<li><span class="math">\(\mathbf{B}\)</span> is a <span class="math">\(n \times 1\)</span> vector of linear coefficients,</li>
<li><span class="math">\(\mathbf{z}_b\)</span> is a <span class="math">\(|b| \times 1\)</span> portion of
<span class="math">\(\mathbf{z}\)</span> corresponding to boundary or <em>fixed</em> vertices,</li>
<li><span class="math">\(\mathbf{z}_{bc}\)</span> is a <span class="math">\(|b| \times 1\)</span> vector of known values corresponding to
<span class="math">\(\mathbf{z}_b\)</span>,</li>
<li><span class="math">\(\mathbf{A}_{eq}\)</span> is a (usually sparse) <span class="math">\(m \times n\)</span> matrix of linear
equality constraint coefficients (one row per constraint), and</li>
<li><span class="math">\(\mathbf{B}_{eq}\)</span> is a <span class="math">\(m \times 1\)</span> vector of linear equality constraint
right-hand side values.</li>
</ul>
<p>This specification is overly general as we could write <span class="math">\(\mathbf{z}_b =
\mathbf{z}_{bc}\)</span> as rows of <span class="math">\(\mathbf{A}_{eq} \mathbf{z} =
\mathbf{B}_{eq}\)</span>, but these fixed value constraints appear so often that they
merit a dedicated place in the API.</p>
<p>In libigl, solving such quadratic optimization problems is split into two
routines: precomputation and solve. Precomputation only depends on the
quadratic coefficients, known value indices and linear constraint coefficients:</p>
<pre><code class="cpp">igl::min_quad_with_fixed_data mqwf;
igl::min_quad_with_fixed_precompute(Q,b,Aeq,true,mqwf);
</code></pre>
<p>The output is a struct <code>mqwf</code> which contains the system matrix factorization
and is used during solving with arbitrary linear terms, known values, and
constraint in the right-hand sides:</p>
<pre><code class="cpp">igl::min_quad_with_fixed_solve(mqwf,B,bc,Beq,Z);
</code></pre>
<p>The output <code>Z</code> is a <span class="math">\(n \times 1\)</span> vector of solutions with fixed values
correctly placed to match the mesh vertices <code>V</code>.</p>
<h2 id="linearequalityconstraints">Linear equality constraints</h2>
<p>We saw above that <code>min_quad_with_fixed_*</code> in libigl provides a compact way to
solve general quadratic programs. Let&#8217;s consider another example, this time
with active linear equality constraints. Specifically let&#8217;s solve the
<code>bi-Laplace equation</code> or equivalently minimize the Laplace energy:</p>
<p><span class="math">\[\Delta^2 z = 0 \leftrightarrow \mathop{\text{minimize }}\limits_z \frac{1}{2}
\int\limits_S (\Delta z)^2 dA\]</span></p>
<p>subject to fixed value constraints and a linear equality constraint:</p>
<p><span class="math">\(z_{a} = 1, z_{b} = -1\)</span> and <span class="math">\(z_{c} = z_{d}\)</span>.</p>
<p>Notice that we can rewrite the last constraint in the familiar form from above:</p>
<p><span class="math">\(z_{c} - z_{d} = 0.\)</span></p>
<p>Now we can assembly <code>Aeq</code> as a <span class="math">\(1 \times n\)</span> sparse matrix with a coefficient
<span class="math">\(1\)</span> in the column corresponding to vertex <span class="math">\(c\)</span> and a <span class="math">\(-1\)</span> at <span class="math">\(d\)</span>. The right-hand
side <code>Beq</code> is simply zero.</p>
<p>Internally, <code>min_quad_with_fixed_*</code> solves using the Lagrange Multiplier
method. This method adds additional variables for each linear constraint (in
general a <span class="math">\(m \times 1\)</span> vector of variables <span class="math">\(\lambda\)</span>) and then solves the
saddle problem:</p>
<p><span class="math">\[\mathop{\text{find saddle }}_{\mathbf{z},\lambda}\, \frac{1}{2}\mathbf{z}^T \mathbf{Q} \mathbf{z} +
\mathbf{z}^T \mathbf{B} + \text{constant} + \lambda^T\left(\mathbf{A}_{eq}
\mathbf{z} - \mathbf{B}_{eq}\right)\]</span></p>
<p>This can be rewritten in a more familiar form by stacking <span class="math">\(\mathbf{z}\)</span> and
<span class="math">\(\lambda\)</span> into one <span class="math">\((m+n) \times 1\)</span> vector of unknowns:</p>
<p><span class="math">\[\mathop{\text{find saddle }}_{\mathbf{z},\lambda}\,
\frac{1}{2}
\left(
\mathbf{z}^T
\lambda^T
\right)
\left(
\begin{array}{cc}
\mathbf{Q} & \mathbf{A}_{eq}^T\\
\mathbf{A}_{eq} & 0
\end{array}
\right)
\left(
\begin{array}{c}
\mathbf{z}\\
\lambda
\end{array}
\right) +
\left(
\mathbf{z}^T
\lambda^T
\right)
\left(
\begin{array}{c}
\mathbf{B}\\
-\mathbf{B}_{eq}
\end{array}
\right)
+ \text{constant}\]</span></p>
<p>Differentiating with respect to <span class="math">\(\left( \mathbf{z}^T \lambda^T \right)\)</span> reveals
a linear system and we can solve for <span class="math">\(\mathbf{z}\)</span> and <span class="math">\(\lambda\)</span>. The only
difference from the straight quadratic <em>minimization</em> system, is that this
saddle problem system will not be positive definite. Thus, we must use a
different factorization technique (LDLT rather than LLT): libigl&#8217;s
<code>min_quad_with_fixed_precompute</code> automatically chooses the correct solver in
the presence of linear equality constraints (<a href="304_LinearEqualityConstraints/main.cpp">Example 304</a>).</p>
<figure>
<img src="images/cheburashka-biharmonic-leq.jpg" alt="The example LinearEqualityConstraints first solves with just fixed value
constraints (left: 1 and -1 on the left hand and foot respectively), then
solves with an additional linear equality constraint (right: points on right
hand and foot constrained to be equal)." />
<figcaption>The example <code>LinearEqualityConstraints</code> first solves with just fixed value
constraints (left: 1 and &#8211;1 on the left hand and foot respectively), then
solves with an additional linear equality constraint (right: points on right
hand and foot constrained to be equal).</figcaption></figure>
<h2 id="quadraticprogramming">Quadratic programming</h2>
<p>We can generalize the quadratic optimization in the previous section even more
by allowing inequality constraints. Specifically box constraints (lower and
upper bounds):</p>
<p><span class="math">\(\mathbf{l} \le \mathbf{z} \le \mathbf{u},\)</span></p>
<p>where <span class="math">\(\mathbf{l},\mathbf{u}\)</span> are <span class="math">\(n \times 1\)</span> vectors of lower and upper
bounds
and general linear inequality constraints:</p>
<p><span class="math">\(\mathbf{A}_{ieq} \mathbf{z} \le \mathbf{B}_{ieq},\)</span></p>
<p>where <span class="math">\(\mathbf{A}_{ieq}\)</span> is a <span class="math">\(k \times n\)</span> matrix of linear coefficients and
<span class="math">\(\mathbf{B}_{ieq}\)</span> is a <span class="math">\(k \times 1\)</span> matrix of constraint right-hand sides.</p>
<p>Again, we are overly general as the box constraints could be written as
rows of the linear inequality constraints, but bounds appear frequently enough
to merit a dedicated api.</p>
<p>Libigl implements its own active set routine for solving <em>quadratric programs</em>
(QPs). This algorithm works by iteratively &#8220;activating&#8221; violated inequality
constraints by enforcing them as equalities and &#8220;deactivating&#8221; constraints
which are no longer needed.</p>
<p>After deciding which constraints are active at each iteration, the problem
reduces to a quadratic minimization subject to linear <em>equality</em> constraints,
and the method from the previous section is invoked. This is repeated until convergence.</p>
<p>Currently the implementation is efficient for box constraints and sparse
non-overlapping linear inequality constraints.</p>
<p>Unlike alternative interior-point methods, the active set method benefits from
a warm-start (initial guess for the solution vector <span class="math">\(\mathbf{z}\)</span>).</p>
<pre><code class="cpp">igl::active_set_params as;
// Z is optional initial guess and output
igl::active_set(Q,B,b,bc,Aeq,Beq,Aieq,Bieq,lx,ux,as,Z);
</code></pre>
<figure>
<img src="images/cheburashka-multiscale-biharmonic-kernels.jpg" alt=" Example 305 uses an active set solver to optimize
discrete biharmonic kernels at multiple scales
." />
<figcaption> <a href="305_QuadraticProgramming/main.cpp">Example 305</a> uses an active set solver to optimize
discrete biharmonic kernels <a class="citation" href="#fn:6" title="Jump to citation">[6]<span class="citekey" style="display:none">rustamov_2011</span></a> at multiple scales
.</figcaption></figure>
<h1 id="chapter4:shapedeformation">Chapter 4: Shape deformation</h1>
<p>Modern mesh-based shape deformation methods satisfy user deformation
constraints at handles (selected vertices or regions on the mesh) and propagate
these handle deformations to the rest of shape <em>smoothly</em> and <em>without removing
or distorting details</em>. Libigl provides implementations of a variety of
state-of-the-art deformation techniques, ranging from quadratic mesh-based
energy minimizers, to skinning methods, to non-linear elasticity-inspired
techniques.</p>
<h2 id="biharmonicdeformation">Biharmonic deformation</h2>
<p>The period of research between 2000 and 2010 produced a collection of
techniques that cast the problem of handle-based shape deformation as a
quadratic energy minimization problem or equivalently the solution to a linear
partial differential equation.</p>
<p>There are many flavors of these techniques, but a prototypical subset are those
that consider solutions to the bi-Laplace equation, that is a biharmonic
function <a class="citation" href="#fn:7" title="Jump to citation">[7]<span class="citekey" style="display:none">botsch_2004</span></a>. This fourth-order PDE provides sufficient
flexibility in boundary conditions to ensure <span class="math">\(C^1\)</span> continuity at handle
constraints (in the limit under refinement) <a class="citation" href="#fn:8" title="Jump to citation">[8]<span class="citekey" style="display:none">jacobson_mixed_2010</span></a>.</p>
<h3 id="biharmonicsurfaces">Biharmonic surfaces</h3>
<p>Let us first begin our discussion of biharmonic <em>deformation</em>, by considering
biharmonic <em>surfaces</em>. We will casually define biharmonic surfaces as surface
whose <em>position functions</em> are biharmonic with respect to some initial
parameterization:</p>
<p><span class="math">\(\Delta \mathbf{x}' = 0\)</span></p>
<p>and subject to some handle constraints, conceptualized as &#8220;boundary
conditions&#8221;:</p>
<p><span class="math">\(\mathbf{x}'_{b} = \mathbf{x}_{bc}.\)</span></p>
<p>where <span class="math">\(\mathbf{x}'\)</span> is the unknown 3D position of a point on the surface. So we
are asking that the bi-Laplacian of each of spatial coordinate function to be
zero.</p>
<p>In libigl, one can solve a biharmonic problem with <code>igl::harmonic</code>
and setting <span class="math">\(k=2\)</span> (<em>bi</em>-harmonic):</p>
<pre><code class="cpp">// U_bc contains deformation of boundary vertices b
igl::harmonic(V,F,b,U_bc,2,U);
</code></pre>
<p>This produces a smooth surface that interpolates the handle constraints, but all
original details on the surface will be <em>smoothed away</em>. Most obviously, if the
original surface is not already biharmonic, then giving all handles the
identity deformation (keeping them at their rest positions) will <strong>not</strong>
reproduce the original surface. Rather, the result will be the biharmonic
surface that does interpolate those handle positions.</p>
<p>Thus, we may conclude that this is not an intuitive technique for shape
deformation.</p>
<h3 id="biharmonicdeformationfields">Biharmonic deformation fields</h3>
<p>Now we know that one useful property for a deformation technique is &#8220;rest pose
reproduction&#8221;: applying no deformation to the handles should apply no
deformation to the shape.</p>
<p>To guarantee this by construction we can work with <em>deformation fields</em> (ie.
displacements)
<span class="math">\(\mathbf{d}\)</span> rather
than directly with positions <span class="math">\(\mathbf{x}\)</span>. Then the deformed positions can be
recovered as</p>
<p><span class="math">\(\mathbf{x}' = \mathbf{x}+\mathbf{d}.\)</span></p>
<p>A smooth deformation field <span class="math">\(\mathbf{d}\)</span> which interpolates the deformation
fields of the handle constraints will impose a smooth deformed shape
<span class="math">\(\mathbf{x}'\)</span>. Naturally, we consider <em>biharmonic deformation fields</em>:</p>
<p><span class="math">\(\Delta \mathbf{d} = 0\)</span></p>
<p>subject to the same handle constraints, but rewritten in terms of their implied
deformation field at the boundary (handles):</p>
<p><span class="math">\(\mathbf{d}_b = \mathbf{x}_{bc} - \mathbf{x}_b.\)</span></p>
<p>Again we can use <code>igl::harmonic</code> with <span class="math">\(k=2\)</span>, but this time solve for the
deformation field and then recover the deformed positions:</p>
<pre><code class="cpp">// U_bc contains deformation of boundary vertices b
D_bc = U_bc - igl::slice(V,b,1);
igl::harmonic(V,F,b,D_bc,2,D);
U = V+D;
</code></pre>
<figure>
<img src="images/max-biharmonic.jpg" alt="The BiharmonicDeformation example deforms a statues head as a biharmonic
surface (top) and using a biharmonic displacements
(bottom)." />
<figcaption>The <a href="401_BiharmonicDeformation/main.cpp">BiharmonicDeformation</a> example deforms a statue&#8217;s head as a <em>biharmonic
surface</em> (top) and using a <em>biharmonic displacements</em>
(bottom).</figcaption></figure>
<h4 id="relationshiptodifferentialcoordinatesandlaplaciansurfaceediting">Relationship to &#8220;differential coordinates&#8221; and Laplacian surface editing</h4>
<p>Biharmonic functions (whether positions or displacements) are solutions to the
bi-Laplace equation, but also minimizers of the &#8220;Laplacian energy&#8221;. For
example, for displacements <span class="math">\(\mathbf{d}\)</span>, the energy reads</p>
<p><span class="math">\(\int\limits_S \|\Delta \mathbf{d}\|^2 dA.\)</span></p>
<p>By linearity of the Laplace(-Beltrami) operator we can reexpress this energy in
terms of the original positions <span class="math">\(\mathbf{x}\)</span> and the unknown positions
<span class="math">\(\mathbf{x}' = \mathbf{x} - \mathbf{d}\)</span>:</p>
<p><span class="math">\(\int\limits_S \|\Delta (\mathbf{x}' - \mathbf{x})\|^2 dA = \int\limits_S
\|\Delta \mathbf{x}' - \Delta \mathbf{x})\|^2 dA.\)</span></p>
<p>In the early work of Sorkine et al., the quantities <span class="math">\(\Delta \mathbf{x}'\)</span> and
<span class="math">\(\Delta \mathbf{x}\)</span> were dubbed &#8220;differential coordinates&#8221; <a class="citation" href="#fn:9" title="Jump to citation">[9]<span class="citekey" style="display:none">sorkine_2004</span></a>.
Their deformations (without linearized rotations) is thus equivalent to
biharmonic deformation fields.</p>
<h2 id="polyharmonicdeformation">Polyharmonic deformation</h2>
<p>We can generalize biharmonic deformation by considering different powers of
the Laplacian, resulting in a series of PDEs of the form:</p>
<p><span class="math">\(\Delta^k \mathbf{d} = 0.\)</span></p>
<p>with <span class="math">\(k\in{1,2,3,\dots}\)</span>. The choice of <span class="math">\(k\)</span> determines the level of continuity
at the handles. In particular, <span class="math">\(k=1\)</span> implies <span class="math">\(C^0\)</span> at the boundary, <span class="math">\(k=2\)</span>
implies <span class="math">\(C^1\)</span>, <span class="math">\(k=3\)</span> implies <span class="math">\(C^2\)</span> and in general <span class="math">\(k\)</span> implies <span class="math">\(C^{k-1}\)</span>.</p>
<pre><code class="cpp">int k = 2;// or 1,3,4,...
igl::harmonic(V,F,b,bc,k,Z);
</code></pre>
<figure>
<img src="images/bump-k-harmonic.jpg" alt="The PolyharmonicDeformation example deforms a flat domain (left) into a bump as a
solution to various $k$-harmonic PDEs." />
<figcaption>The <a href="402_PolyharmonicDeformation/main.cpp">PolyharmonicDeformation</a> example deforms a flat domain (left) into a bump as a
solution to various <span class="math">\(k\)</span>-harmonic PDEs.</figcaption></figure>
<h2 id="boundedbiharmonicweights">Bounded biharmonic weights</h2>
<p>In computer animation, shape deformation is often referred to as &#8220;skinning&#8221;.
Constraints are posed as relative rotations of internal rigid &#8220;bones&#8221; inside a
character. The deformation method, or skinning method, determines how the
surface of the character (i.e. its skin) should move as a function of the bone
rotations.</p>
<p>The most popular technique is linear blend skinning. Each point on the shape
computes its new location as a linear combination of bone transformations:</p>
<p><span class="math">\(\mathbf{x}' = \sum\limits_{i = 1}^m w_i(\mathbf{x}) \mathbf{T}_i
\left(\begin{array}{c}\mathbf{x}_i\\1\end{array}\right),\)</span></p>
<p>where <span class="math">\(w_i(\mathbf{x})\)</span> is the scalar <em>weight function</em> of the ith bone evaluated at
<span class="math">\(\mathbf{x}\)</span> and <span class="math">\(\mathbf{T}_i\)</span> is the bone transformation as a <span class="math">\(4 \times 3\)</span>
matrix.</p>
<p>This formula is embarassingly parallel (computation at one point does not
depend on shared data need by computation at another point). It is often
implemented as a vertex shader. The weights and rest positions for each vertex
are sent as vertex shader <em>attributes</em> and bone transformations are sent as
<em>uniforms</em>. Then vertices are transformed within the vertex shader, just in
time for rendering.</p>
<p>As the skinning formula is linear (hence its name), we can write it as matrix
multiplication:</p>
<p><span class="math">\(\mathbf{X}' = \mathbf{M} \mathbf{T},\)</span></p>
<p>where <span class="math">\(\mathbf{X}'\)</span> is <span class="math">\(n \times 3\)</span> stack of deformed positions as row
vectors, <span class="math">\(\mathbf{M}\)</span> is a <span class="math">\(n \times m\cdot dim\)</span> matrix containing weights and
rest positions and <span class="math">\(\mathbf{T}\)</span> is a <span class="math">\(m\cdot (dim+1) \times dim\)</span> stack of
transposed bone transformations.</p>
<p>Traditionally, the weight functions <span class="math">\(w_j\)</span> are painted manually by skilled
rigging professionals. Modern techniques now exist to compute weight functions
automatically given the shape and a description of the skeleton (or in general
any handle structure such as a cage, collection of points, selected regions,
etc.).</p>
<p>Bounded biharmonic weights are one such technique that casts weight computation
as a constrained optimization problem <a class="citation" href="#fn:10" title="Jump to citation">[10]<span class="citekey" style="display:none">jacobson_2011</span></a>. The weights enforce
smoothness by minimizing the familiar Laplacian energy:</p>
<p><span class="math">\(\sum\limits_{i = 1}^m \int_S (\Delta w_i)^2 dA\)</span></p>
<p>subject to constraints which enforce interpolation of handle constraints:</p>
<p><span class="math">\(w_i(\mathbf{x}) = \begin{cases} 1 & \text{ if } \mathbf{x} \in H_i\\ 0 &
\text{ otherwise } \end{cases},\)</span></p>
<p>where <span class="math">\(H_i\)</span> is the ith handle, and constraints which enforce non-negativity,
parition of unity and encourage sparsity:</p>
<p><span class="math">\(0\le w_i \le 1\)</span> and <span class="math">\(\sum\limits_{i=1}^m w_i = 1.\)</span></p>
<p>This is a quadratic programming problem and libigl solves it using its active
set solver or by calling out to <a href="http://www.mosek.com">Mosek</a>.</p>
<figure>
<img src="images/hand-bbw.jpg" alt="The example BoundedBiharmonicWeights computes weights for a tetrahedral
mesh given a skeleton (top) and then animates a linear blend skinning
deformation (bottom)." />
<figcaption>The example <a href="403_BoundedBiharmonicWeights/main.cpp">BoundedBiharmonicWeights</a> computes weights for a tetrahedral
mesh given a skeleton (top) and then animates a linear blend skinning
deformation (bottom).</figcaption></figure>
<h2 id="dualquaternionskinning">Dual quaternion skinning</h2>
<p>Even with high quality weights, linear blend skinning is limited. In
particular, it suffers from known artifacts stemming from blending rotations as
as matrices: a weight combination of rotation matrices is not necessarily a
rotation. Consider an equal blend between rotating by <span class="math">\(-\pi/2\)</span> and by <span class="math">\(\pi/2\)</span>
about the <span class="math">\(z\)</span>-axis. Intuitively one might expect to get the identity matrix,
but instead the blend is a degenerate matrix scaling the <span class="math">\(x\)</span> and <span class="math">\(y\)</span>
coordinates by zero:</p>
<p><span class="math">\(0.5\left(\begin{array}{ccc}0&-1&0\\1&0&0\\0&0&1\end{array}\right)+
0.5\left(\begin{array}{ccc}0&1&0\\-1&0&0\\0&0&1\end{array}\right)=
\left(\begin{array}{ccc}0&0&0\\0&0&0\\0&0&1\end{array}\right)\)</span></p>
<p>In practice, this means the shape shrinks and collapses in regions where bone
weights overlap: near joints.</p>
<p>Dual quaternion skinning presents a solution <a class="citation" href="#fn:11" title="Jump to citation">[11]<span class="citekey" style="display:none">kavan_2008</span></a>. This method
represents rigid transformations as a pair of unit quaternions,
<span class="math">\(\hat{\mathbf{q}}\)</span>. The linear blend skinning formula is replaced with a
linear blend of dual quaternions:</p>
<p><span class="math">\(\mathbf{x}' =
\cfrac{\sum\limits_{i=1}^m w_i(\mathbf{x})\hat{\mathbf{q}_i}}
{\left\|\sum\limits_{i=1}^m w_i(\mathbf{x})\hat{\mathbf{q}_i}\right\|}
\mathbf{x},\)</span></p>
<p>where <span class="math">\(\hat{\mathbf{q}_i}\)</span> is the dual quaternion representation of the rigid
transformation of bone <span class="math">\(i\)</span>. The normalization forces the result of the linear
blending to again be a unit dual quaternion and thus also a rigid
transformation.</p>
<p>Like linear blend skinning, dual quaternion skinning is best performed in the
vertex shader. The only difference being that bone transformations are sent as
dual quaternions rather than affine transformation matrices. Libigl supports
CPU-side dual quaternion skinning with the <code>igl::dqs</code> function, which takes a
more traditional representation of rigid transformations as input and
internally converts to the dual quaternion representation before blending:</p>
<pre><code class="cpp">// vQ is a list of rotations as quaternions
// vT is a list of translations
igl::dqs(V,W,vQ,vT,U);
</code></pre>
<figure>
<img src="images/arm-dqs.jpg" alt="The example DualQuaternionSkinning compares linear blend skinning (top) to dual
quaternion skinning (bottom), highlighting LBSs candy wrapper effect (middle)
and joint collapse (right)." />
<figcaption>The example <a href="404_DualQuaternionSkinning/main.cpp">DualQuaternionSkinning</a> compares linear blend skinning (top) to dual
quaternion skinning (bottom), highlighting LBS&#8217;s candy wrapper effect (middle)
and joint collapse (right).</figcaption></figure>
<h2 id="as-rigid-as-possible">As-rigid-as-possible</h2>
<p>Skinning and other linear methods for deformation are inherently limited.
Difficult arises especially when large rotations are imposed by the handle
constraints.</p>
<p>In the context of energy-minimization approaches, the problem stems from
comparing positions (our displacements) in the coordinate frame of the
undeformed shape. These quadratic energies are at best invariant to global
rotations of the entire shape, but not smoothly varying local rotations. Thus
linear techniques will not produce non-trivial bending and twisting.</p>
<p>Furthermore, when considering solid shapes (e.g. discretized with tetrahedral
meshes) linear methods struggle to maintain local volume, and they often suffer from
shrinking and bulging artifacts.</p>
<p>Non-linear deformation techniques present a solution to these problems.
They work by comparing the deformation of a mesh
vertex to its rest position <em>rotated</em> to a new coordinate frame which best
matches the deformation. The non-linearity stems from the mutual dependence of
the deformation and the best-fit rotation. These techniques are often labeled
&#8220;as-rigid-as-possible&#8221; as they penalize the sum of all local deformations&#8217;
deviations from rotations.</p>
<p>To arrive at such an energy, let&#8217;s consider a simple per-triangle energy:</p>
<p><span class="math">\(E_\text{linear}(\mathbf{X}') = \sum\limits_{t \in T} a_t \sum\limits_{\{i,j\}
\in t} w_{ij} \left\|
\left(\mathbf{x}'_i - \mathbf{x}'_j\right) -
\left(\mathbf{x}_i - \mathbf{x}_j\right)\right\|^2\)</span></p>
<p>where <span class="math">\(\mathbf{X}'\)</span> are the mesh&#8217;s unknown deformed vertex positions, <span class="math">\(t\)</span> is a
triangle in a list of triangles <span class="math">\(T\)</span>, <span class="math">\(a_t\)</span> is the area of triangle <span class="math">\(t\)</span> and
<span class="math">\(\{i,j\}\)</span> is an edge in triangle <span class="math">\(t\)</span>. Thus, this energy measures the norm of
change between an edge vector in the original mesh <span class="math">\(\left(\mathbf{x}_i -
\mathbf{x}_j\right)\)</span> and the unknown mesh <span class="math">\(\left(\mathbf{x}'_i -
\mathbf{x}'_j\right)\)</span>.</p>
<p>This energy is <strong>not</strong> rotation invariant. If we rotate the mesh by 90 degrees
the change in edge vectors not aligned with the axis of rotation will be large,
despite the overall deformation being perfectly rigid.</p>
<p>So, the &#8220;as-rigid-as-possible&#8221; solution is to append auxiliary variables
<span class="math">\(\mathbf{R}_t\)</span>
for each triangle <span class="math">\(t\)</span> which are constrained to be rotations. Then the energy is
rewritten, this time comparing deformed edge vectors to their rotated rest
counterparts:</p>
<p><span class="math">\(E_\text{arap}(\mathbf{X}',\{\mathbf{R}_1,\dots,\mathbf{R}_{|T|}\}) = \sum\limits_{t \in T} a_t \sum\limits_{\{i,j\}
\in t} w_{ij} \left\|
\left(\mathbf{x}'_i - \mathbf{x}'_j\right)-
\mathbf{R}_t\left(\mathbf{x}_i - \mathbf{x}_j\right)\right\|^2.\)</span></p>
<p>The separation into the primary vertex position variables <span class="math">\(\mathbf{X}'\)</span> and the
rotations <span class="math">\(\{\mathbf{R}_1,\dots,\mathbf{R}_{|T|}\}\)</span> lead to strategy for
optimization, too. If the rotations <span class="math">\(\{\mathbf{R}_1,\dots,\mathbf{R}_{|T|}\}\)</span>
are held fixed then the energy is quadratic in the remaining variables
<span class="math">\(\mathbf{X}'\)</span> and can be optimized by solving a (sparse) global linear system.
Alternatively, if <span class="math">\(\mathbf{X}'\)</span> are held fixed then each rotation is the
solution to a localized <em>Procrustes</em> problem (found via <span class="math">\(3 \times 3\)</span> SVD or
polar decompostion). These two steps&#8212;local and global&#8212;each weakly decrease
the energy, thus we may safely iterate them until convergence.</p>
<p>The different flavors of &#8220;as-rigid-as-possible&#8221; depend on the dimension and
codimension of the domain and the edge-sets <span class="math">\(T\)</span>. The proposed surface
manipulation technique by Sorkine and Alexa <a class="citation" href="#fn:12" title="Jump to citation">[12]<span class="citekey" style="display:none">sorkine_2007</span></a>, considers <span class="math">\(T\)</span> to
be the set of sets of edges emanating from each vertex (spokes). Later, Chao et
al. derived the relationship between &#8220;as-rigid-as-possible&#8221; mesh energies and
co-rotational elasticity considering 0-codimension elements as edge-sets:
triangles in 2D and tetrahedra in 3D <a class="citation" href="#fn:13" title="Jump to citation">[13]<span class="citekey" style="display:none">chao_2010</span></a>. They also showed how
Sorkine and Alexa&#8217;s edge-sets are not a discretization of a continuous energy,
proposing instead edge-sets for surfaces containing all edges of elements
incident on a vertex (spokes and rims). They show that this amounts to
measuring bending, albeit in a discretization-dependent way.</p>
<p>Libigl, supports these common flavors. Selecting one is a matter of setting the
energy type before the precompuation phase:</p>
<pre><code class="cpp">igl::ARAPData data;
arap_data.energy = igl::ARAP_ENERGY_TYPE_SPOKES;
//arap_data.energy = igl::ARAP_ENERGY_TYPE_SPOKES_AND_RIMS;
//arap_data.energy = igl::ARAP_ENERGY_TYPE_ELEMENTS; //triangles or tets
igl::arap_precomputation(V,F,dim,b,data);
</code></pre>
<p>Just like <code>igl::min_quad_with_fixed_*</code>, this precomputation phase only depends
on the mesh, fixed vertex indices <code>b</code> and the energy parameters. To solve with
certain constraints on the positions of vertices in <code>b</code>, we may call:</p>
<pre><code class="cpp">igl::arap_solve(bc,data,U);
</code></pre>
<p>which uses <code>U</code> as an initial guess and then computes the solution into it.</p>
<p>Libigl&#8217;s implementation of as-rigid-as-possible deformation takes advantage of
the highly optimized singular value decomposition code from McAdams et al.
<a class="citation" href="#fn:14" title="Jump to citation">[14]<span class="citekey" style="display:none">mcadams_2011</span></a> which leverages SSE intrinsics.</p>
<figure>
<img src="images/decimated-knight-arap.jpg" alt="The example AsRigidAsPossible deforms a surface as if it were made of an
elastic material" />
<figcaption>The example <a href="405_AsRigidAsPossible/main.cpp">AsRigidAsPossible</a> deforms a surface as if it were made of an
elastic material</figcaption></figure>
<p>The concept of local rigidity will be revisited shortly in the context of
surface parameterization.</p>
<h2 id="fastautomaticskinningtransformations">Fast automatic skinning transformations</h2>
<p>Non-linear optimization is, unsurprisingly, slower than its linear cousins. In
the case of the as-rigid-as-possible optimization, the bottleneck is typically
the large number of polar decompositions necessary to recover best fit
rotations for each edge-set (i.e. for each triangle, tetrahedron, or vertex
cell). Even if this code is optimized, the number of primary degrees of freedom
is tied to the discretization level, despite the deformations&#8217; low frequency
behavior.</p>
<p>This invites two routes toward fast non-linear optimization. First, is it
necessary (or even advantageous) to find so many best-fit rotations? Second,
can we reduce the degrees of freedom to better reflect the frequency of the
desired deformations.</p>
<p>Taken in turn, these optimizations culminate in a method which optimizes over
the space of linear blend skinning deformations spanned by high-quality weights
(i.e. manually painted ones or bounded biharmonic weights). This space is a
low-dimensional subspace of all possible mesh deformations, captured by writing
linear blend skinning in matrix form:</p>
<p><span class="math">\(\mathbf{X}' = \mathbf{M}\mathbf{T}\)</span></p>
<p>where the mesh vertex positions in the <span class="math">\(n \times 3\)</span> matrix <span class="math">\(\mathbf{X}'\)</span> are
replaced by a linear combination of a small number of degrees of freedom in the
<span class="math">\((3+1)m \times 3\)</span> stack of transposed &#8220;handle&#8221; transformations. Swapping in
<span class="math">\(\mathbf{M}\mathbf{T}\)</span> for <span class="math">\(\mathbf{X}'\)</span> in the ARAP energies above immediately
sees performance gains during the global solve step as <span class="math">\(m << n\)</span>.</p>
<p>The complexity of the local step&#8212;fitting rotations&#8212;is still bound
to the original mesh discretization. However, if the skinning is well behaved,
we can make the assumption that places on the shape with similar skinning
weights will deform similarly and thus imply similar best-fit rotations.
Therefore, we cluster edge-sets according to their representation in
<em>weight-space</em>: where a vertex <span class="math">\(\mathbf{x}\)</span> takes the coordinates
<span class="math">\([w_1(\mathbf{x}),w_2(\mathbf{x}),\dots,w_m(\mathbf{x})]\)</span>. The number of
clustered edge-sets show diminishing returns on the deformation quality so we
may choose a small number of clusters, proportional to the number of skinning
weight functions (rather than the number of discrete mesh vertices).</p>
<p>This proposed deformation model <a class="citation" href="#fn:15" title="Jump to citation">[15]<span class="citekey" style="display:none">jacobson_2012</span></a>, can simultaneously be seen as a
fast, subspace optimization for ARAP and as an automatic method for finding
<em>the best</em> skinning transformation degrees of freedom.</p>
<p>A variety of user interfaces are supported via linear equality constraints on
the skinning transformations associated with handles. To fix a transformation
entirely we simply add the constraint:</p>
<p><span class="math">\(\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\end{array}\right)
\mathbf{T}_i^T = \hat{\mathbf{T}}_i^T,\)</span></p>
<p>where <span class="math">\(\hat{\mathbf{T}}_i^T\)</span> is the <span class="math">\((3+1) \times 3\)</span> transposed fixed
transformation for handle <span class="math">\(i\)</span>.</p>
<p>To fix only the origin of a handle, we add a constraint requiring the
transformation to interpolate a point in space (typically the centroid of all
points with <span class="math">\(w_i = 1\)</span>:</p>
<p><span class="math">\(\mathbf{c}'^T\mathbf{T}_i^T = \mathbf{c}^T,\)</span></p>
<p>where <span class="math">\(\mathbf{c}^T\)</span> is the <span class="math">\(1 \times (3+1)\)</span> position of the point at rest in
transposed homogeneous coordinates, and <span class="math">\(\mathbf{c}'^T\)</span> the point given by the
user.</p>
<p>We can similarly fix just the linear part of the transformation at a handle,
freeing the translation component (producing a &#8220;chickenhead&#8221; effect):</p>
<p><span class="math">\(\left(\begin{array}{cccc}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\end{array}\right)
\mathbf{T}_i^T = \hat{\mathbf{L}}_i^T,\)</span></p>
<p>where <span class="math">\(\hat{\mathbf{L}}_i^T\)</span> is the fixed <span class="math">\(3 \times 3\)</span> linear part of the
transformation at handle <span class="math">\(i\)</span>.</p>
<p>And lastly we can allow the user to entirely <em>free</em> the transformation&#8217;s
degrees of freedom, delegating the optimization to find the best possible
values for all elements. To do this, we simply abstain from adding a
corresponding constraint.</p>
<h3 id="arapwithgroupededge-sets">ARAP with grouped edge-sets</h3>
<p>Being a subspace method, an immediate disadvantage is the reduced degrees of
freedom. This brings performance, but in some situations limits behavior too
much. In such cases one can use the skinning subspace to build an effective
clustering of rotation edge-sets for a traditional ARAP optimization: forgoing
the subspace substitution. This has an two-fold effect. The cost of the
rotation fitting, local step drastically reduces, and the deformations are
&#8220;regularized&#8221; according the clusters. From a high level point of view, if the clusters
are derived from skinning weights, then they will discourage bending,
especially along isolines of the weight functions.</p>
<p>In this light, we can think of the &#8220;spokes+rims&#8221; style surface ARAP as a (slight and
redundant) clustering of the per-triangle edge-sets.</p>
<figure>
<img src="images/armadillo-fast.jpg" alt="The example FastAutomaticSkinningTransformations compares a full (slow)
ARAP deformation on a detailed shape (left of middle), to ARAP with grouped
rotation edge sets (right of middle), to the very fast subpsace method
(right)." />
<figcaption>The example <a href="406_FastAutomaticSkinningTransformations/main.cpp">FastAutomaticSkinningTransformations</a> compares a full (slow)
ARAP deformation on a detailed shape (left of middle), to ARAP with grouped
rotation edge sets (right of middle), to the very fast subpsace method
(right).</figcaption></figure>
<h1 id="500">Chapter 5: Parametrization</h1>
<p>In computer graphics, we denote as surface parametrization a map from the
surface to <span class="math">\(\mathbf{R}^2\)</span>. It is usually encoded by a new set of 2D
coordinates for each vertex of the mesh (and possibly also by a new set of
faces in one to one correspondence with the faces of the original surface).
Note that
this definition is the <em>inverse</em> of the classical differential geometry
definition.</p>
<p>A parametrization has many applications, ranging from texture mapping to
surface remeshing. Many algorithms have been proposed, and they can be broadly
divided in four families:</p>
<ol>
<li><p><strong>Single patch, fixed boundary</strong>: these algorithm can parametrize a
disk-like part of the surface given fixed 2D positions for its boundary. These
algorithms are efficient and simple, but they usually produce high-distortion maps due to the fixed boundary.</p></li>
<li><p><strong>Single patch, free boundary:</strong> these algorithms let the boundary
deform freely, greatly reducing the map distortion. Care should be taken to
prevent the border to self-intersect.</p></li>
<li><p><strong>Global parametrization</strong>: these algorithms work on meshes with arbitrary
genus. They initially cut the mesh in multiple patches that can be separately parametrized. The generated maps are discontinuous on the cuts (often referred as <em>seams</em>).</p></li>
<li><p><strong>Global seamless parametrization</strong>: these are global parametrization algorithm that hides the seams, making the parametrization &#8220;continuous&#8221;, under specific assumptions that we will discuss later.</p></li>
</ol>
<h2 id="501">Harmonic parametrization</h2>
<p>Harmonic parametrization <a class="citation" href="#fn:16" title="Jump to citation">[16]<span class="citekey" style="display:none">eck_2005</span></a> is a single patch, fixed boundary parametrization
algorithm that computes the 2D coordinates of the flattened mesh as two
harmonic functions.</p>
<p>The algorithm is divided in 3 steps:</p>
<ol>
<li>Detect of the boundary vertices</li>
<li>Map the boundary vertices to a circle</li>
<li>Compute two harmonic functions (one for u and one for the v coordinate). The harmonic functions use the fixed vertices on the circle as boundary constraints.</li>
</ol>
<p>The algorithm can be coded using libigl as follows:</p>
<pre><code class="cpp">Eigen::VectorXi bnd;
igl::boundary_loop(V,F,bnd);
Eigen::MatrixXd bnd_uv;
igl::map_vertices_to_circle(V,bnd,bnd_uv);
igl::harmonic(V,F,bnd,bnd_uv,1,V_uv);
</code></pre>
<p>where <code>bnd</code> contains the indices of the boundary vertices, bnd_uv their position on the UV plane, and &#8220;1&#8221; denotes that we want to compute an harmonic function (2 will be for biharmonic, 3 for triharmonic, etc.). Note that each of the three
functions is designed to be reusable in other parametrization algorithms.</p>
<p>A UV parametrization can be visualized in the viewer with:</p>
<pre><code class="cpp">viewer.data.set_uv(V_uv);
</code></pre>
<p>The UV coordinates are then used to apply a procedural checkerboard texture to the
mesh (<a href="501_HarmonicParam/main.cpp">Example 501</a>).</p>
<figure>
<img src="images/501_HarmonicParam.png" alt="(Example 501) Harmonic parametrization. (left)
mesh with texture, (right) UV parametrization with
texture" />
<figcaption>(<a href="501_HarmonicParam/main.cpp">Example 501</a>) Harmonic parametrization. (left)
mesh with texture, (right) UV parametrization with
texture</figcaption></figure>
<h2 id="502">Least squares conformal maps</h2>
<p>Least squares conformal maps parametrization <a class="citation" href="#fn:17" title="Jump to citation">[17]<span class="citekey" style="display:none">levy_2002</span></a> minimizes the
conformal (angular) distortion of the parametrization. Differently from
harmonic parametrization, it does not need to have a fixed boundary.</p>
<p>LSCM minimizes the following energy:</p>
<p><span class="math">\[ E_{LSCM}(\mathbf{u},\mathbf{v}) = \int_X \frac{1}{2}| \nabla \mathbf{u}^{\perp} - \nabla \mathbf{v} |^2 dA \]</span></p>
<p>which can be rewritten in matrix form as <a class="citation" href="#fn:18" title="Jump to citation">[18]<span class="citekey" style="display:none">mullen_2008</span></a>:</p>
<p><span class="math">\[ E_{LSCM}(\mathbf{u},\mathbf{v}) = \frac{1}{2} [\mathbf{u},\mathbf{v}]^t (L_c - 2A) [\mathbf{u},\mathbf{v}] \]</span></p>
<p>where <span class="math">\(L_c\)</span> is the cotangent Laplacian matrix and A is a matrix such that \(
[\mathbf{u},\mathbf{v}]^t A [\mathbf{u},\mathbf{v}]$ is equal to the
<a href="http://en.wikipedia.org/wiki/Vector_area">vector area</a> of the mesh.</p>
<p>Using libigl, this matrix energy can be written in a few lines of codes. The
cotangent matrix can be computed using <code>igl::cotmatrix</code>:</p>
<pre><code class="cpp">SparseMatrix&lt;double&gt; L;
igl::cotmatrix(V,F,L);
</code></pre>
<p>Note that we want to apply the Laplacian matrix to the u and v coordinates at
the same time, thus we need to extend it taking the left
Kronecker product with a 2x2 identity matrix:</p>
<pre><code class="cpp">SparseMatrix&lt;double&gt; L_flat;
repdiag(L,2,L_flat);
</code></pre>
<p>The area matrix is computed with <code>igl::vector_area_matrix</code>:</p>
<pre><code class="cpp">SparseMatrix&lt;double&gt; A;
igl::vector_area_matrix(F,A);
</code></pre>
<p>The final energy matrix is the sum of these two matrices. Note that in this
case we do not need to fix the boundary. To remove the null space of the energy and make the minimum unique, it is sufficient to fix two arbitrary
vertices to two arbitrary positions. The full source code is provided in <a href="502_LSCMParam/main.cpp">Example 502</a>.</p>
<figure>
<img src="images/502_LSCMParam.png" alt="(Example 502) LSCM parametrization. (left) mesh
with texture, (right) UV parametrization" />
<figcaption>(<a href="502_LSCMParam/main.cpp">Example 502</a>) LSCM parametrization. (left) mesh
with texture, (right) UV parametrization</figcaption></figure>
<h2 id="503">As-rigid-as-possible parametrization</h2>
<p>As-rigid-as-possible parametrization <a class="citation" href="#fn:19" title="Jump to citation">[19]<span class="citekey" style="display:none">liu_2008</span></a> is a powerful single-patch,
non-linear algorithm to compute a parametrization that strives to preserve
distances (and thus angles). The idea is very similar to ARAP surface
deformation: each triangle is mapped to the plane trying to preserve its
original shape, up to a rigid rotation.</p>
<p>The algorithm can be implemented reusing the functions discussed in the
deformation chapter: <code>igl::arap_precomputation</code> and <code>igl::arap_solve</code>. The only
difference is that the optimization has to be done in 2D instead of 3D and that
we need to compute a starting point. While for 3D deformation the optimization
is bootstrapped with the original mesh, this is not the case for ARAP
parametrization since the starting point must be a 2D mesh. In <a href="503_ARAPParam/main.cpp">Example
503</a>, we initialize the optimization with harmonic
parametrization. Similarly to LSCM, the boundary is free to deform to minimize
the distortion.</p>
<figure>
<img src="images/503_ARAPParam.png" alt="(Example 503) As-Rigid-As-Possible parametrization.
(left) mesh with texture, (right) UV parametrization with
texture" />
<figcaption>(<a href="502_ARAPParam/main.cpp">Example 503</a>) As-Rigid-As-Possible parametrization.
(left) mesh with texture, (right) UV parametrization with
texture</figcaption></figure>
<h2 id="504">N-rotationally symmetric tangent fields</h2>
<p>The design of tangent fields is a basic tool used to design guidance fields for
uniform quadrilateral and hexahedral remeshing. Libigl contains an
implementation of all the state-of-the-art algorithms to design N-RoSy fields
and their generalizations.</p>
<p>In libigl, tangent unit-length vector fields are piece-wise constant on the
faces of a triangle mesh, and they are described by one or more vectors per-face. The function</p>
<pre><code class="cpp">igl::nrosy(V,F,b,bc,b_soft,b_soft_weight,bc_soft,N,0.5,
output_field,output_singularities);
</code></pre>
<p>creates a smooth unit-length vector field (N=1) starting from a sparse set of
constrained faces, whose indices are listed in b and their constrained value is
specified in bc. The functions supports soft_constraints (b_soft,
b_soft_weight, bc_soft), and returns the interpolated field for each face of
the triangle mesh (output_field), plus the singularities of the field
(output_singularities).</p>
<figure>
<img src="images/504_vector_field.png" alt="Design of a unit-length vector field" />
<figcaption>Design of a unit-length vector field</figcaption></figure>
<p>The singularities are vertices where the field vanishes (highlighted in red in
the figure above). <code>igl::nrosy</code> can also generate N-RoSy fields <a class="citation" href="#fn:20" title="Jump to citation">[20]<span class="citekey" style="display:none">levy_2008</span></a>,
which are a generalization of vector fields where in every face the vector is
defined up to a constant rotation of <span class="math">\(2\pi / N\)</span>. As can be observed in
the following figure, the singularities of the fields generated with different
N are of different types and they appear in different positions.</p>
<figure>
<img src="images/504_nrosy_field.png" alt="Design of a 2-,4- and 9-RoSy field" />
<figcaption>Design of a 2-,4- and 9-RoSy field</figcaption></figure>
<p>We demonstrate how to call and plot N-RoSy fields in <a href="504_NRosyDesign/main.cpp">Example
504</a>, where the degree of the field can be change
pressing the number keys. <code>igl::nrosy</code> implements the algorithm proposed in
<a class="citation" href="#fn:21" title="Jump to citation">[21]<span class="citekey" style="display:none">bommes_2009</span></a>. N-RoSy fields can also be interpolated with the algorithm
proposed in <a class="citation" href="#fn:22" title="Jump to citation">[22]<span class="citekey" style="display:none">knoppel_2013</span></a>, see Section <a href="#507">507</a> for more details
(<a href="../include/igl/n_polyvector.h">igl::n_polyvector</a>).</p>
<h3 id="505">Global, seamless integer-grid parametrization</h3>
<p>The previous parametrization methods were focusing on creating parametrizations
of surface patches aimed at texture mapping or baking of other surface
properties such as normals and high-frequency details. Global, seamless
parametrization aims at parametrizing complex shapes with a parametrization
that is aligned with a given set of directions for the purpose of surface
remeshing. In libigl, we provide a reference implementation of the pipeline
proposed in the mixed integer quadrangulation paper <a class="citation" href="#fn:21" title="Jump to citation">[21]<span class="citekey" style="display:none">bommes_2009</span></a>.</p>
<p>The first step involves the design of a 4-RoSy field (sometimes called <em>cross</em>
field) that describes the alignment of the edges of the desired quadrilateral
remeshing. The field constraints are usually manually specified or extracted
from the principal curvature directions. In [<a href="506_FrameField/main.cpp">Example
506</a>], we simply fix one face in a random direction.</p>
<figure>
<img src="images/505_MIQ_1.png" alt="Initial cross field prescribing the edge alignment." />
<figcaption>Initial cross field prescribing the edge alignment.</figcaption></figure>
<h3 id="combingandcutting">Combing and cutting</h3>
<p>Given the cross field, we now want to cut the surface so that it becomes
homeomorphic to a disk. While this could be done directly on the cross-field, we
opt to perform this operation on its bisector field (a copy of the field
rotated by 45 degrees) since it is more stable and generic. Working on the
bisectors allow us to take as input generalized, non-orthogonal and non-unit
length cross fields.</p>
<p>We thus rotate the field,</p>
<figure>
<img src="images/505_MIQ_2.png" alt="Bisector field." />
<figcaption>Bisector field.</figcaption></figure>
<p>and we remove the rotation ambiguity by assigning to each face a u and a v
direction. The assignment is done with a breadth-first search starting from a
random face.</p>
<figure>
<img src="images/505_MIQ_3.png" alt="Combed bisector field." />
<figcaption>Combed bisector field.</figcaption></figure>
<p>You can imagine this process as combing an hairy surface: you will be able to
comb part of it, but at some point you will not be able to consistently comb
the entire surface (<a href="http://en.wikipedia.org/wiki/Hairy_ball_theorem">Hairy ball
theorem</a>). The discontinuities
in the combing define the cut graph:</p>
<figure>
<img src="images/505_MIQ_4.png" alt="Cut graph." />
<figcaption>Cut graph.</figcaption></figure>
<p>Finally, we rotate the combed field by 45 degrees to undo the initial degrees
rotation:</p>
<figure>
<img src="images/505_MIQ_5.png" alt="Combed cross field." />
<figcaption>Combed cross field.</figcaption></figure>
<p>The combed cross field can be seen as the ideal Jacobian of the parametrization
that will be computed in the next section.</p>
<h3 id="poissonparametrization">Poisson parametrization</h3>
<p>The mesh is cut along the seams and a parametrization is computed trying to
find two scalar functions whose gradient matches the combed cross field
directions. This is a classical Poisson problem, that is solved minimizing the
following quadratic energy:</p>
<p><span class="math">\[ E(\mathbf{u},\mathbf{v}) = |\nabla \mathbf{u} - X_u|^2 + |\nabla \mathbf{v} - X_v|^2 \]</span></p>
<p>where <span class="math">\(X_u\)</span> and <span class="math">\(X_u\)</span> denotes the combed cross field. Solving this
problem generates a parametrization whose u and v isolines are aligned with the
input cross field.</p>
<figure>
<img src="images/505_MIQ_8.png" alt="Poisson parametrization." />
<figcaption>Poisson parametrization.</figcaption></figure>
<p>We hide the seams by adding integer constraints to the Poisson problem
that align the isolines on both sides of each seam <a class="citation" href="#fn:21" title="Jump to citation">[21]<span class="citekey" style="display:none">bommes_2009</span></a>.</p>
<figure>
<img src="images/505_MIQ_7.png" alt="Seamless Poisson parametrization." />
<figcaption>Seamless Poisson parametrization.</figcaption></figure>
<p>Note that this parametrization can only be used for remeshing purposes, since
it contains many overlaps.</p>
<figure>
<img src="images/505_MIQ_6.png" alt="Seamless Poisson parametrization (in 2D)." />
<figcaption>Seamless Poisson parametrization (in 2D).</figcaption></figure>
<p>A quad mesh can be extracted from this parametrization using
<a href="https://github.com/hcebke/libQEx">libQEx</a> (not included in libigl).
The full pipeline is implemented in <a href="505_MIQ/main.cpp">Example 505</a>.</p>
<h2 id="506">Anisotropic remeshing</h2>
<p>Anisotropic and non-uniform quad remeshing is important to concentrate the
elements in the regions with more details. It is possible to extend the MIQ
quad meshing framework to generate anisotropic quad meshes using a mesh
deformation approach <a class="citation" href="#fn:23" title="Jump to citation">[23]<span class="citekey" style="display:none">panozzo_2014</span></a>.</p>
<p>The input of the anisotropic remeshing algorithm is a sparse set of constraints
that define the shape and scale of the desired quads. This can be encoded as a
frame field, which is a pair of non-orthogonal and non-unit length vectors. The
frame field can be interpolated by decomposing it in a 4-RoSy field and a
unique affine transformation. The two parts can then be interpolated
separately, using <code>igl::nrosy</code> for the cross field, and an harmonic interpolant
for the affine part.</p>
<figure>
<img src="images/506_FrameField_1.png" alt="Interpolation of a frame field. Colors on the vectors denote the desired
scale. The red faces contains the frame field
constraints." />
<figcaption>Interpolation of a frame field. Colors on the vectors denote the desired
scale. The red faces contains the frame field
constraints.</figcaption></figure>
<p>After the interpolation, the surface is warped to transform each frame into an
orthogonal and unit length cross (i.e. removing the scaling and skewness from
the frame). This deformation defines a new embedding (and a new metric) for the
surface.</p>
<figure>
<img src="images/506_FrameField_2.png" alt="The surface is deformed to transform the frame field in a cross
field." />
<figcaption>The surface is deformed to transform the frame field in a cross
field.</figcaption></figure>
<p>The deformed surface can the be isotropically remeshed using the MIQ algorithm
that has been presented in the previous section.</p>
<figure>
<img src="images/506_FrameField_3.png" alt="The deformed surface is isotropically remeshed." />
<figcaption>The deformed surface is isotropically remeshed.</figcaption></figure>
<p>The UV coordinates of the deformed surface can then be used to transport the
parametrization to the original surface, where the isolines will trace a quad
mesh whose elements are similar to the shape prescribed in the input frame
field.</p>
<figure>
<img src="images/506_FrameField_4.png" alt="The global parametrization is lifted to the original surface to create the
anisotropic quad meshing." />
<figcaption>The global parametrization is lifted to the original surface to create the
anisotropic quad meshing.</figcaption></figure>
<p>Our implementation (<a href="506_FrameField/main.cpp">Example 506</a>) uses MIQ to
generate the UV parametrization, but other algorithms could be applied: the
only desiderata is that the generated quad mesh should be as isotropic as
possible.</p>
<h2 id="507">N-PolyVector fields</h2>
<p>N-RoSy vector fields can be further generalized to represent arbitrary
vector-sets, with arbitrary angles between them and with arbitrary lengths
<a class="citation" href="#fn:24" title="Jump to citation">[24]<span class="citekey" style="display:none">diamanti_2014</span></a>. This generalization is called N-PolyVector field, and
libigl provides the function <code>igl::n_polyvector</code> to design them starting from a
sparse set of constraints (<a href="507_PolyVectorField/main.cpp">Example 507</a>).</p>
<figure>
<img src="images/507_PolyVectorField.png" alt="Interpolation of a 6-PolyVector field (right) and a 12-PolyVector field from a sparse set of random constraints." />
<figcaption>Interpolation of a 6-PolyVector field (right) and a 12-PolyVector field from a sparse set of random constraints.</figcaption></figure>
<p>The core idea is to represent the vector set as the roots of a complex
polynomial: The polynomial coefficients are then harmonically interpolated
leading to polynomials whose roots smoothly vary over the surface.</p>
<p>Globally optimal direction fields <a class="citation" href="#fn:22" title="Jump to citation">[22]<span class="citekey" style="display:none">knoppel_2013</span></a> are a special case of
PolyVector fields. If the constraints are taken from an N-RoSy field,
<code>igl::n_polyvector</code> generates a field that is equivalent, after normalization,
to a globally optimal direction field.</p>
<h2 id="508">Conjugate vector fields</h2>
<p>Two tangent vectors lying on a face of a triangle mesh are conjugate if</p>
<p><span class="math">\[ k_1 (u^T d_1)(v^T d_1) + k_2(u^T d_2)(v^T d_2) = 0. \]</span></p>
<p>This condition is very important in architectural geometry: The faces of an
infinitely dense quad mesh whose edges are aligned with a conjugate field are
planar. Thus, a quad mesh whose edges follow a conjugate field are easier to
planarize <a class="citation" href="#fn:25" title="Jump to citation">[25]<span class="citekey" style="display:none">liu_2011</span></a>.</p>
<p>Finding a conjugate vector field that satisfies given directional constraints
is a standard problem in architectural geometry, which can be tackled by
deforming a Poly-Vector field to the closest conjugate field.</p>
<p>This algorithm <a class="citation" href="#fn:24" title="Jump to citation">[24]<span class="citekey" style="display:none">diamanti_2014</span></a> alternates a global step, which enforces
smoothness, with a local step, that projects the field on every face to the
closest conjugate field (<a href="508_ConjugateField/main.cpp">Example 508</a>).</p>
<figure>
<img src="images/508_ConjugateField.png" alt="A smooth 4-PolyVector field (left) is deformed to become a conjugate field
(right)." />
<figcaption>A smooth 4-PolyVector field (left) is deformed to become a conjugate field
(right).</figcaption></figure>
<h2 id="509">Planarization</h2>
<p>A quad mesh can be transformed in a planar quad mesh with Shape-Up
<a class="citation" href="#fn:26" title="Jump to citation">[26]<span class="citekey" style="display:none">bouaziz_2012</span></a>, a local/global approach that uses the global step to enforce
surface continuity and the local step to enforce planarity.</p>
<p><a href="509_Planarization/main.cpp">Example 509</a> planarizes a quad mesh until it
satisfies a user-given planarity threshold.</p>
<figure>
<img src="images/509_Planarization.png" alt="A non-planar quad mesh (left) is planarized using the libigl function
igl::palanarize (right). The colors represent the planarity of the
quads." />
<figcaption>A non-planar quad mesh (left) is planarized using the libigl function
igl::palanarize (right). The colors represent the planarity of the
quads.</figcaption></figure>
<h1 id="600">Chapter 6: External libraries</h1>
<p>An additional positive side effect of using matrices as basic types is that it
is easy to exchange data between libigl and other softwares and libraries.</p>
<h2 id="601">State serialization</h2>
<p>Geometry processing applications often require a considerable amount of
computational time and/or manual input. Serializing the state of the application is a simple strategy to greatly increase the development efficiency. It allows to quickly start debugging just
before the crash happens, avoiding to wait for the precomputation to take place
every time and it also makes your experiments reproducible, allowing to quickly test algorithms variants on the same input data.</p>
<p>Serialization is often not considered in geometry processing due
to the extreme difficulty in serializing pointer-based data structured, such as
an half-edge data structure (OpenMesh, CGAL), or a pointer based indexed structure (VCG).</p>
<p>In libigl, serialization is much simpler, since the majority of the functions use basic types, and pointers are used in very rare cases (usually to interface
with external libraries). Libigl bundles a simple and self-contained XML serialization framework, that drastically reduces the overhead required to add
serialization to your applications.</p>
<p>Assume that the state of your application is a mesh and a set of
integer ids:</p>
<pre><code class="cpp">class State : public igl::XMLSerialization
{
public:
State() : XMLSerialization(&quot;dummy&quot;) {}
Eigen::MatrixXd V;
Eigen::MatrixXi F;
std::vector&lt;int&gt; ids;
void InitSerialization()
{
xmlSerializer-&gt;Add(V , &quot;V&quot;);
xmlSerializer-&gt;Add(F , &quot;F&quot;);
xmlSerializer-&gt;Add(ids, &quot;ids&quot;);
}
};
</code></pre>
<p>Any class can be made serializable by inheriting from `<code>igl::XMLSerialization</code> and trivially implementing the <code>InitSerialization</code> method. The library can serialize all the basic <code>stl</code> types, all <code>Eigen</code> types and any class inheriting
from <code>igl::XMLSerialization</code>.</p>
<p>The state can be saved into an xml file with:</p>
<pre><code class="cpp">igl::XMLSerializer serializer_save(&quot;601_Serialization&quot;);
serializer_save.Add(state,&quot;State&quot;);
serializer_save.Save(&quot;temp.xml&quot;,true);
</code></pre>
<p>This code generates the following xml file (assuming <code>V</code> and <code>F</code> contains a simple mesh with two triangles, and <code>ids</code> contains the numbers 6 and 7):</p>
<pre><code class="xml">&lt;:::601_Serialization&gt;
&lt;State&gt;
&lt;V rows=&quot;4&quot; cols=&quot;3&quot; matrix=&quot;
0,0,0,
1,0,0,
1,1,1,
2,1,0&quot;/&gt;
&lt;F rows=&quot;2&quot; cols=&quot;3&quot; matrix=&quot;
0,1,2,
1,3,2&quot;/&gt;
&lt;ids size=&quot;2&quot; vector_int=&quot;
6,7&quot;/&gt;
&lt;/State&gt;
&lt;/:::601_Serialization&gt;
</code></pre>
<p>The xml file can be loaded in a similar way:</p>
<pre><code class="cpp">State loaded_state;
igl::XMLSerializer serializer_load(&quot;601_Serialization&quot;);
serializer_load.Add(loaded_state,&quot;State&quot;);
serializer_load.Load(&quot;temp.xml&quot;);
</code></pre>
<p>The serialization framework can also be used as a convenient interface to
provide parameters to command line applications, since the xml files can be
directly edited with a standard text editor.</p>
<p>The code snippets above are extracted from <a href="601_Serialization/main.cpp">Example
601</a>. We strongly suggest that you make the entire
state of your application always serializable since it will save you a lot of
troubles when you will be preparing figures for a scientific report. It is very
common to have to do small changes to figures, and being able to serialize the
entire state just before you take screenshots will save you many painful hours
before a submission deadline.</p>
<h2 id="602">Mixing Matlab code</h2>
<p>Libigl can be interfaced with Matlab to offload numerically heavy computation
to a Matlab script. The major advantage of this approach is that you will be
able to develop efficient and complex user-interfaces in C++, while exploting
the syntax and fast protototyping features of matlab. In particular, the use of
an external Matlab script in a libigl application allows to change the Matlab
code while the C++ application is running, greatly increasing coding
efficiency.</p>
<p>We demonstrate how to integrate Matlab in a libigl application in <a href="602_Matlab/main.cpp">Example
602</a>. The example uses Matlab to compute the
Eigenfunctions of the discrete Laplacian operator, relying on libigl for mesh
IO, visualization and for computing the Laplacian operator.</p>
<p>Libigl can connect to an existing instance of Matlab (or launching a new one on
Linux/MacOSX) using:</p>
<pre><code class="cpp">igl::mlinit(&amp;engine);
</code></pre>
<p>The cotangent Laplacian is computed using igl::cotmatrix and uploaded to the
Matlab workspace:</p>
<pre><code class="cpp">igl::cotmatrix(V,F,L);
igl::mlsetmatrix(&amp;engine,&quot;L&quot;,L);
</code></pre>
<p>It is now possible to use any Matlab function on the data. For example, we can
see the sparsity pattern of L using spy:</p>
<pre><code class="cpp">igl::mleval(&amp;engine,&quot;spy(L)&quot;);
</code></pre>
<figure>
<img src="images/602_Matlab_1.png" alt="The Matlab spy function is called from a libigl-based
application." />
<figcaption>The Matlab spy function is called from a libigl-based
application.</figcaption></figure>
<p>The results of Matlab computations can be returned back to the C++ application</p>
<pre><code class="cpp">igl::mleval(&amp;engine,&quot;[EV,~] = eigs(-L,10,'sm')&quot;);
igl::mlgetmatrix(&amp;engine,&quot;EV&quot;,EV);
</code></pre>
<p>and plotted using the libigl viewer.</p>
<figure>
<img src="images/602_Matlab_2.png" alt="4 Eigenfunctions of the Laplacian plotted in the libigl
viewer." />
<figcaption>4 Eigenfunctions of the Laplacian plotted in the libigl
viewer.</figcaption></figure>
<h3 id="savingamatlabworkspace">Saving a Matlab workspace</h3>
<p>To aid debugging, libigl also supplies functions to write Matlab <code>.mat</code>
&#8220;Workspaces&#8221;. This C++ snippet saves a mesh and it&#8217;s sparse Laplacian matrix to
a file:</p>
<pre><code class="cpp">igl::readOFF(&quot;../shared/fertility.off&quot;, V, F);
igl::cotmatrix(V,F,L);
igl::MatlabWorkspace mw;
mw.save(V,&quot;V&quot;);
mw.save_index(F,&quot;F&quot;);
mw.save(L,&quot;L&quot;);
mw.write(&quot;fertility.mat&quot;);
</code></pre>
<p>Then this workspace can be loaded into a Matlab IDE:</p>
<pre><code class="matlab">load fertility.mat
</code></pre>
<p>The <code>igl::MatlabWorkspace</code> depends on Matlab libraries to compile and run,
but&#8212;in contrast to the engine routines above&#8212;will avoid launching a Matlab
instance upon execution.</p>
<h3 id="dumpingeigenmatricestocopyandpasteintomatlab">Dumping Eigen matrices to copy and paste into Matlab</h3>
<p>Eigen supplies a sophisticated API for printing its matrix types to the screen.
Libigl has wrapped up a particularly useful formatting which makes it simple to
copy standard output from a C++ program into a Matlab IDE. The code:</p>
<pre><code class="cpp">igl::readOFF(&quot;../shared/2triangles.off&quot;, V, F);
igl::cotmatrix(V,F,L);
std::cout&lt;&lt;igl::matlab_format(V,&quot;V&quot;)&lt;&lt;std::endl;
std::cout&lt;&lt;igl::matlab_format((F.array()+1).eval(),&quot;F&quot;)&lt;&lt;std::endl;
std::cout&lt;&lt;igl::matlab_format(L,&quot;L&quot;)&lt;&lt;std::endl;
</code></pre>
<p>produces the output:</p>
<pre><code class="matlab">V = [
0 0 0
1 0 0
1 1 1
2 1 0
];
F = [
1 2 3
2 4 3
];
LIJV = [
1 1 -0.7071067811865476
2 1 0.7071067811865475
3 1 1.570092458683775e-16
1 2 0.7071067811865475
2 2 -1.638010440969447
3 2 0.6422285251880865
4 2 0.2886751345948129
1 3 1.570092458683775e-16
2 3 0.6422285251880865
3 3 -0.9309036597828995
4 3 0.2886751345948129
2 4 0.2886751345948129
3 4 0.2886751345948129
4 4 -0.5773502691896258
];
L = sparse(LIJV(:,1),LIJV(:,2),LIJV(:,3));
</code></pre>
<p>which is easily copied and pasted into Matlab for debugging, etc.</p>
<h2 id="603">Calling libigl functions from Matlab</h2>
<p>It is also possible to call libigl functions from matlab, compiling them as MEX
functions. This can be used to offload to C++ code the computationally
intensive parts of a Matlab application.</p>
<p>We provide a wrapper for <code>igl::readOBJ</code> in <a href="603_MEX/compileMEX.m">Example 603</a>.
We plan to provide wrappers for all our functions in the future, if you are
interested in this feature (or if you want to help implementing it) please let
us know.</p>
<h2 id="604">Triangulation of closed polygons</h2>
<p>The generation of high-quality triangle and tetrahedral meshes is a very common
task in geometry processing. We provide wrappers in libigl to
<a href="http://www.cs.cmu.edu/~quake/triangle.html">triangle</a> and
<a href="http://wias-berlin.de/software/tetgen/">Tetgen</a>.</p>
<p>A triangle mesh with a given boundary can be created with:</p>
<pre><code class="cpp">igl::triangulate(V,E,H,V2,F2,&quot;a0.005q&quot;);
</code></pre>
<p>where <code>E</code> is a set of boundary edges (#E by 2), <code>H</code> is a set of 2D positions of
points contained in holes of the triangulation (#H by 2) and (<code>V2</code>,<code>F2</code>) is the
generated triangulation. Additional parameters can be passed to <code>triangle</code>, to
control the quality: <code>&quot;a0.005q&quot;</code> enforces a bound on the maximal area of the
triangles and a minimal angle of 20 degrees. In <a href="604_Triangle/main.m">Example
604</a>, the interior of a square (excluded a smaller square
in its interior) is triangulated.</p>
<figure>
<img src="images/604_Triangle.png" alt="Triangulation of the interior of a polygon." />
<figcaption>Triangulation of the interior of a polygon.</figcaption></figure>
<h2 id="605">Tetrahedralization of closed surfaces</h2>
<p>Similarly, the interior of a closed manifold surface can be tetrahedralized
using the function <code>igl::tetrahedralize</code> which wraps the Tetgen library (<a href="605_Tetgen/main.c">Example
605</a>):</p>
<pre><code class="cpp">igl::tetrahedralize(V,F,&quot;pq1.414&quot;, TV,TT,TF);
</code></pre>
<figure>
<img src="images/605_Tetgen.png" alt="Tetrahedralization of the interior of a surface mesh." />
<figcaption>Tetrahedralization of the interior of a surface mesh.</figcaption></figure>
<h2 id="606">Baking ambient occlusion</h2>
<p><a href="http://en.wikipedia.org/wiki/Ambient_occlusion">Ambient occlusion</a> is a
rendering technique used to calculate the exposure of each point in a surface
to ambient lighting. It is usually encoded as a scalar (normalized between 0
and 1) associated with the vertice of a mesh.</p>
<p>Formally, ambient occlusion is defined as:</p>
<p><span class="math">\[ A_p = \frac{1}{\pi} \int_\omega V_{p,\omega}(n \cdot \omega) d\omega \]</span></p>
<p>where <span class="math">\(V_{p,\omega}\)</span> is the visibility function at p, defined to be zero
if p is occluded in the direction <span class="math">\(\omega\)</span> and one otherwise, and \(
d\omega$ is the infinitesimal solid angle step of the integration variable
<span class="math">\(\omega\)</span>.</p>
<p>The integral is usually approximated by casting rays in random directions
around each vertex. This approximation can be computed using the function:</p>
<pre><code class="cpp">igl::ambient_occlusion(V,F,V_samples,N_samples,500,AO);
</code></pre>
<p>that given a scene described in <code>V</code> and <code>F</code>, computes the ambient occlusion of
the points in <code>V_samples</code> whose associated normals are <code>N_samples</code>. The
number of casted rays can be controlled (usually at least 300&#8211;500 rays are
required to get a smooth result) and the result is returned in <code>AO</code>, as a
single scalar for each sample.</p>
<p>Ambient occlusion can be used to darken the surface colors, as shown in
<a href="606_AmbientOcclusion/main.c">Example 606</a></p>
<figure>
<img src="images/606_AmbientOcclusion.png" alt="A mesh rendered without (left) and with (right) ambient
occlusion." />
<figcaption>A mesh rendered without (left) and with (right) ambient
occlusion.</figcaption></figure>
<h2 id="607">Picking</h2>
<p>Picking vertices and faces using the mouse is very common in geometry
processing applications. While this might seem a simple operation, its
implementation is not straighforward. Libigl contains a function that solves this problem using the
<a href="https://software.intel.com/en-us/articles/embree-photo-realistic-ray-tracing-kernels">Embree</a>
raycaster. Its usage is demonstrated in <a href="607_Picking/main.cpp">Example 607</a>:</p>
<pre><code class="cpp">bool hit = igl::unproject_in_mesh(
Vector2f(x,y),
F,
viewer.core.view * viewer.core.model,
viewer.core.proj,
viewer.core.viewport,
*ei,
fid,
vid);
</code></pre>
<p>This function casts a ray from the view plane in the view direction. Variables
<code>x</code> and <code>y</code> are
the mouse screen coordinates; <code>view</code>, <code>model</code>, <code>proj</code> are the view, model and
projection matrix respectively; <code>viewport</code> is the viewport in OpenGL format;
<code>ei</code>
contains a <a href="http://en.wikipedia.org/wiki/Bounding_volume_hierarchy">Bounding Volume
Hierarchy</a> constructed
by Embree, and <code>fid</code> and <code>vid</code> are the picked face and vertex, respectively.</p>
<figure>
<img src="images/607_Picking.png" alt="(Example 607) Picking via ray casting. The selected
vertices are colored in red." />
<figcaption>(<a href="607_Picking/main.cpp">Example 607</a>) Picking via ray casting. The selected
vertices are colored in red.</figcaption></figure>
<h2 id="608">Locally Injective Maps</h2>
<p>Extreme deformations or parametrizations with high-distortion might flip
elements. This is undesirable in many applications, and it is possible to
avoid it by introducing a non-linear constraints that guarantees that the area
of every element remain positive.</p>
<p>Libigl can be used to compute Locally Injective Maps <a class="citation" href="#fn:27" title="Jump to citation">[27]<span class="citekey" style="display:none">schuller_2013</span></a> using a variety of
deformation energies. A simple deformation of a 2D grid is computed in <a href="608_LIM/main.cpp">Example
608</a>.</p>
<figure>
<img src="images/608_LIM.png" alt="A mesh (left) deformed using Laplacian editing (middle) and with Laplacian
editing plus the anti-flipping constraints (right)." />
<figcaption>A mesh (left) deformed using Laplacian editing (middle) and with Laplacian
editing plus the anti-flipping constraints (right).</figcaption></figure>
<h1 id="future">Outlook for continuing development</h1>
<p>Libigl is in active development, and we plan to focus on the following features
in the next months:</p>
<ul>
<li><p>A better and more consistent <strong>documentation</strong>, plus extending this tutorial
to cover more libigl features.</p></li>
<li><p>Implement a <strong>mixed-integer solver</strong> which only uses Eigen to remove the
dependency on CoMiSo.</p></li>
<li><p>Improve the robustness and performance of the active set QP solver. In
particular, handle linearly dependent constraints.</p></li>
<li><p>Implement more mesh analysis functions, including structural analysis for
masonry and <em>3D-printability</em> analysis.</p></li>
<li><p>Increase support for point clouds and general polygonal meshes.</p></li>
<li><p>What would you like to see in libigl? <a href="&#109;&#97;&#105;&#108;&#x74;&#111;&#x3a;&#x61;&#x6c;&#101;&#x63;&#x6a;&#x61;&#x63;&#111;&#98;&#115;&#111;&#110;&#64;&#103;&#109;&#x61;&#105;&#108;&#x2e;&#x63;&#x6f;&#x6d;">&#67;&#x6f;&#x6e;&#116;&#97;&#99;&#x74;
&#117;&#x73;&#x21;</a> or post a <a href="https://github.com/libigl/libigl/issues/new">feature
request</a>.</p></li>
</ul>
<p>We encourage you to contribute to the library and to report problems and bugs.
The best way to contribute new feature or bug fixes is to fork the libigl
repository and to open a <a href="https://help.github.com/articles/using-pull-requests">pull
request</a> on <a href="https://github.com/libigl/libigl">our github
repository</a>.</p>
<div class="footnotes">
<hr />
<ol>
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<a href="https://www.google.com/search?q=Discrete+Differential-Geometry+Operators+for+Triangulated+2-Manifolds">&#8220;Discrete Differential-Geometry Operators for Triangulated
2-Manifolds,&#8221;</a>
2003.</p>
</li>
<li id="fn:2" class="citation"><span class="citekey" style="display:none">panozzo_2010</span><p>Daniele Panozzo, Enrico Puppo, Luigi Rocca, <a href="https://www.google.com/search?q=Efficient+Multi-scale+Curvature+and+Crease+Estimation">&#8220;Efficient
Multi-scale Curvature and Crease
Estimation,&#8221;</a>
2010.</p>
</li>
<li id="fn:3" class="citation"><span class="citekey" style="display:none">jacobson_thesis_2013</span><p>Alec Jacobson,
<a href="https://www.google.com/search?q=Algorithms+and+Interfaces+for+Real-Time+Deformation+of+2D+and+3D+Shapes"><em>Algorithms and Interfaces for Real-Time Deformation of 2D and 3D
Shapes</em></a>,
2013.</p>
</li>
<li id="fn:4" class="citation"><span class="citekey" style="display:none">sharf_2007</span><p>Andrei Sharf, Thomas Lewiner, Gil Shklarski, Sivan Toledo, and
Daniel Cohen-Or. <a href="https://www.google.com/search?q=Interactive+topology-aware+surface+reconstruction">&#8220;Interactive topology-aware surface
reconstruction,&#8221;</a> 2007.</p>
</li>
<li id="fn:5" class="citation"><span class="citekey" style="display:none">kazhdan_2012</span><p>Michael Kazhdan, Jake Solomon, Mirela Ben-Chen,
<a href="https://www.google.com/search?q=Can+Mean-Curvature+Flow+Be+Made+Non-Singular">&#8220;Can Mean-Curvature Flow Be Made
Non-Singular,&#8221;</a> 2012.</p>
</li>
<li id="fn:6" class="citation"><span class="citekey" style="display:none">rustamov_2011</span><p>Raid M. Rustamov, <a href="https://www.google.com/search?q=Multiscale+Biharmonic+Kernels">&#8220;Multiscale Biharmonic
Kernels&#8221;</a>, 2011.</p>
</li>
<li id="fn:7" class="citation"><span class="citekey" style="display:none">botsch_2004</span><p>Matrio Botsch and Leif Kobbelt. <a href="https://www.google.com/search?q=An+Intuitive+Framework+for+Real-Time+Freeform+Modeling">&#8220;An Intuitive Framework for
Real-Time Freeform
Modeling,&#8221;</a>
2004.</p>
</li>
<li id="fn:8" class="citation"><span class="citekey" style="display:none">jacobson_mixed_2010</span><p>Alec Jacobson, Elif Tosun, Olga Sorkine, and Denis
Zorin. <a href="https://www.google.com/search?q=Mixed+Finite+Elements+for+Variational+Surface+Modeling">&#8220;Mixed Finite Elements for Variational Surface
Modeling,&#8221;</a> 2010.</p>
</li>
<li id="fn:9" class="citation"><span class="citekey" style="display:none">sorkine_2004</span><p>Olga Sorkine, Yaron Lipman, Daniel Cohen-Or, Marc Alexa,
Christian Rössl and Hans-Peter Seidel. <a href="https://www.google.com/search?q=Laplacian+Surface+Editing">&#8220;Laplacian Surface
Editing,&#8221;</a> 2004.</p>
</li>
<li id="fn:10" class="citation"><span class="citekey" style="display:none">jacobson_2011</span><p>Alec Jacobson, Ilya Baran, Jovan Popović, and Olga Sorkine.
<a href="https://www.google.com/search?q=Bounded+biharmonic+weights+for+real-time+deformation">&#8220;Bounded Biharmonic Weights for Real-Time Deformation,&#8221;</a> 2011.</p>
</li>
<li id="fn:11" class="citation"><span class="citekey" style="display:none">kavan_2008</span><p>Ladislav Kavan, Steven Collins, Jiri Zara, and Carol O&#8217;Sullivan.
<a href="https://www.google.com/search?q=Geometric+Skinning+with+Approximate+Dual+Quaternion+Blending">&#8220;Geometric Skinning with Approximate Dual Quaternion
Blending,&#8221;</a> 2008.</p>
</li>
<li id="fn:12" class="citation"><span class="citekey" style="display:none">sorkine_2007</span><p>Olga Sorkine and Marc Alexa, <a href="https://www.google.com/search?q=As-rigid-as-possible+Surface+Modeling">&#8220;As-rigid-as-possible Surface
Modeling.&#8221;</a> 2007.</p>
</li>
<li id="fn:13" class="citation"><span class="citekey" style="display:none">chao_2010</span><p>Isaac Chao, Ulrich Pinkall, Patrick Sanan, Peter Schröder.
<a href="https://www.google.com/search?q=A+Simple+Geometric+Model+for+Elastic+Deformations">&#8220;A Simple Geometric Model for Elastic
Deformations,&#8221;</a> 2010.</p>
</li>
<li id="fn:14" class="citation"><span class="citekey" style="display:none">mcadams_2011</span><p>Alexa McAdams, Andrew Selle, Rasmus Tamstorf, Joseph Teran,
Eftychios Sifakis. <a href="https://www.google.com/search?q=Computing+the+Singular+Value+Decomposition+of+3x3+matrices+with+minimal+branching+and+elementary+floating+point+operations">&#8220;Computing the Singular Value Decomposition of 3x3 matrices
with minimal branching and elementary floating point
operations,&#8221;</a>
2011.</p>
</li>
<li id="fn:15" class="citation"><span class="citekey" style="display:none">jacobson_2012</span><p>Alec Jacobson, Ilya Baran, Ladislav Kavan, Jovan Popović, and
Olga Sorkine. <a href="https://www.google.com/search?q=Fast+Automatic+Skinning+Transformations">&#8220;Fast Automatic Skinning
Transformations,&#8221;</a> 2012.</p>
</li>
<li id="fn:16" class="citation"><span class="citekey" style="display:none">eck_2005</span><p><a href="http://research.microsoft.com/en-us/um/people/hoppe/mra.pdf">Multiresolution Analysis of Arbitrary
Meshes</a>, Matthias
Eck, Tony DeRose, Tom Duchamp, Hugues Hoppe, Michael Lounsbery, Werner
Stuetzle, SIGGRAPH 2005</p>
</li>
<li id="fn:17" class="citation"><span class="citekey" style="display:none">levy_2002</span><p><a href="http://www.cs.jhu.edu/~misha/Fall09/Levy02.pdf">Least Squares Conformal Maps, for Automatic Texture Atlas
Generation,</a> Bruno Lévy,
Sylvain Petitjean, Nicolas Ray, Jérome Maillot, SIGGRAPH 2002</p>
</li>
<li id="fn:18" class="citation"><span class="citekey" style="display:none">mullen_2008</span><p><a href="http://www.geometry.caltech.edu/pubs/MTAD08.pdf">Spectral Conformal
Parameterization</a>, Patrick
Mullen, Yiying Tong, Pierre Alliez, Mathieu Desbrun, CGF 2008</p>
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<li id="fn:19" class="citation"><span class="citekey" style="display:none">liu_2008</span><p><a href="http://cs.harvard.edu/~sjg/papers/arap.pdf">A Local/Global Approach to Mesh
Parameterization</a> Ligang Liu, Lei
Zhang, Yin Xu, Craig Gotsman, Steven J. Gortler SGP 2008</p>
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<li id="fn:20" class="citation"><span class="citekey" style="display:none">levy_2008</span><p><a href="http://alice.loria.fr/publications/papers/2008/DGF/NSDFD-TOG.pdf">N-Symmetry Direction Field
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Nicolas Ray, Bruno Vallet, Wan Chiu Li, Bruno Lévy TOG 2008</p>
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<li id="fn:21" class="citation"><span class="citekey" style="display:none">bommes_2009</span><p><a href="http://www-sop.inria.fr/members/David.Bommes/publications/miq.pdf">Mixed-integer
quadrangulation</a>, David Bommes, Henrik Zimmer, Leif Kobbelt SIGGRAPH 2009</p>
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<li id="fn:22" class="citation"><span class="citekey" style="display:none">knoppel_2013</span><p><a href="http://www.cs.columbia.edu/~keenan/Projects/GloballyOptimalDirectionFields/paper.pdf">Globally Optimal Direction
Fields</a> Knöppel, Crane, Pinkall, Schröder SIGGRAPH 2013</p>
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<li id="fn:23" class="citation"><span class="citekey" style="display:none">panozzo_2014</span><p><a href="http://www.inf.ethz.ch/personal/dpanozzo/papers/frame-fields-2014.pdf">Frame Fields: Anisotropic and Non-Orthogonal Cross
Fields</a>,
Daniele Panozzo, Enrico Puppo, Marco Tarini, Olga Sorkine-Hornung, SIGGRAPH,
2014</p>
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<li id="fn:24" class="citation"><span class="citekey" style="display:none">diamanti_2014</span><p><a href="http://igl.ethz.ch/projects/complex-roots/">Designing N-PolyVector Fields with Complex
Polynomials</a> Olga Diamanti, Amir
Vaxman, Daniele Panozzo, Olga Sorkine-Hornung, SGP 2014</p>
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<li id="fn:25" class="citation"><span class="citekey" style="display:none">liu_2011</span><p><a href="http://research.microsoft.com/en-us/um/people/yangliu/publication/cdf.pdf">General Planar Quadrilateral Mesh Design Using Conjugate Direction
Field</a> Yang Liu, Weiwei Xu, Jun Wang, Lifeng Zhu, Baining Guo, Falai Chen, Guoping
Wang SIGGRAPH Asia 2011</p>
</li>
<li id="fn:26" class="citation"><span class="citekey" style="display:none">bouaziz_2012</span><p><a href="http://lgg.epfl.ch/publications/2012/shapeup.pdf">Shape-Up: Shaping Discrete Geometry with
Projections</a> Sofien Bouaziz,
Mario Deuss, Yuliy Schwartzburg, Thibaut Weise, Mark Pauly
SGP 2012</p>
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<li id="fn:27" class="citation"><span class="citekey" style="display:none">schuller_2013</span><p><a href="http://igl.ethz.ch/projects/LIM/">Locally Injective Mappings</a>
Christian Schüller, Ladislav Kavan, Daniele Panozzo, Olga Sorkine-Hornung,
SGP 2013</p>
</li>
</ol>
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