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<title>Spectral Mesh Flattening</title>
<NOSCRIPT>
<DIV STYLE="color:#CC0000; text-align:center"><B>Warning: <A HREF="http://www.math.union.edu/locate/jsMath">jsMath</A>
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<meta name="generator" content="MATLAB 8.2">
<meta name="date" content="2014-10-21">
<meta name="m-file" content="index">
<LINK REL="stylesheet" HREF="../style.css" TYPE="text/css">
</head>
<body>
<div class="content">
<h1>Spectral Mesh Flattening</h1>
<introduction>
<p>This tour explores 2D flattening of 3D surfaces using spectral methods.</p>
</introduction>
<h2>Contents</h2>
<div>
<ul>
<li><a href="#1">Installing toolboxes and setting up the path.</a></li>
<li><a href="#8">Spectral Mesh Flattening</a></li>
<li><a href="#21">Geodesic Embedding (Isomap)</a></li>
</ul>
</div>
<h2>Installing toolboxes and setting up the path.<a name="1"></a></h2>
<p>You need to download the following files: <a href="../toolbox_signal.zip">signal toolbox</a>, <a href="../toolbox_general.zip">general toolbox</a> and <a href="../toolbox_graph.zip">graph toolbox</a>.
</p>
<p>You need to unzip these toolboxes in your working directory, so that you have <tt>toolbox_signal</tt>, <tt>toolbox_general</tt> and <tt>toolbox_graph</tt> in your directory.
</p>
<p><b>For Scilab user:</b> you must replace the Matlab comment '%' by its Scilab counterpart '//'.
</p>
<p><b>Recommandation:</b> You should create a text file named for instance <tt>numericaltour.sce</tt> (in Scilab) or <tt>numericaltour.m</tt> (in Matlab) to write all the Scilab/Matlab command you want to execute. Then, simply run <tt>exec('numericaltour.sce');</tt> (in Scilab) or <tt>numericaltour;</tt> (in Matlab) to run the commands.
</p>
<p>Execute this line only if you are using Matlab.</p><pre class="codeinput">getd = @(p)path(p,path); <span class="comment">% scilab users must *not* execute this</span>
</pre><p>Then you can add the toolboxes to the path.</p><pre class="codeinput">getd(<span class="string">'toolbox_signal/'</span>);
getd(<span class="string">'toolbox_general/'</span>);
getd(<span class="string">'toolbox_graph/'</span>);
</pre><h2>Spectral Mesh Flattening<a name="8"></a></h2>
<p>Mesh flattening finds 2D locations that minimize a variational energy with a non-degenracy constraint (for instance maximal
variance).
</p>
<p>For the Dirichlet energy (Sobolev norm), the resulting location are described by the first eigenvectors of the Laplacian.</p>
<p>This method is refered to as "Laplacian eigenmaps" in manifold learning, see:</p>
<p>The advantage over fixed boundary harmonic parameterization is that the boundary of the flattened domain is not fixed, but
the drawback is that the parameterization is not guaranteed to be valid (bijective).
</p>
<p>First load a mesh.</p><pre class="codeinput">name = <span class="string">'nefertiti'</span>;
options.name = name;
[vertex,faces] = read_mesh(name);
n = size(vertex,2);
</pre><p>Display it.</p><pre class="codeinput">clf;
plot_mesh(vertex,faces, options);
shading <span class="string">faceted</span>;
</pre><img vspace="5" hspace="5" src="index_01.png"> <p>Compute the mesh Laplacian matrix.</p><pre class="codeinput">options.symmetrize = 1;
options.normalize = 0;
L = compute_mesh_laplacian(vertex,faces,<span class="string">'conformal'</span>,options);
</pre><p>Compute the eigenvalues and eigenvectors</p><pre class="codeinput">[U,S] = eig(full(L)); S = diag(S);
[S,I] = sort(S,<span class="string">'ascend'</span>); U = U(:,I);
</pre><p>The vertex positions are the eigenvectors 2 and 3.</p><pre class="codeinput">vertexF = U(:,2:3)';
</pre><p>Use translation / rotation to align the parameterization.</p><pre class="codeinput">icenter = 88;
irotate = 154;
vertexF = vertexF - repmat(vertexF(:,icenter), [1 n]);
theta = -pi/2+atan2(vertexF(2,irotate),vertexF(1,irotate));
vertexF = [vertexF(1,:)*cos(theta)+vertexF(2,:)*sin(theta); <span class="keyword">...</span>
-vertexF(1,:)*sin(theta)+vertexF(2,:)*cos(theta)];
</pre><p>Display the flattened mesh.</p><pre class="codeinput">clf;
plot_mesh(vertexF,faces);
</pre><img vspace="5" hspace="5" src="index_02.png"> <p><i>Exercice 1:</i> (<a href="../missing-exo/">check the solution</a>) Perform the same flattening, but with the combinatorial Laplacian.
</p><pre class="codeinput">exo1;
</pre><img vspace="5" hspace="5" src="index_03.png"> <h2>Geodesic Embedding (Isomap)<a name="21"></a></h2>
<p>Another (nonlinear) embedding can be computed by minimizing the geodesic distortion between points on the surface and points
over the parameterized domain.
</p>
<p>First we compute the geodesic distance on the mesh using the Fast Marching algorithm.</p><pre class="codeinput">D = zeros(n);
<span class="keyword">for</span> i=1:n
D(:,i) = perform_fast_marching_mesh(vertex,faces,i);
<span class="keyword">end</span>
</pre><p>Enforce symmetry.</p><pre class="codeinput">D = (D+D')/2;
</pre><p>Compute the centered matrix.</p><pre class="codeinput">J = eye(n) - ones(n)/n;
W = -J*(D.^2)*J;
</pre><p>Diagonalize the centered matrix.</p><pre class="codeinput">[U,S] = eig(W);
S = diag(S);
[S,I] = sort(S,<span class="string">'descend'</span>); U = U(:,I);
</pre><p>Display the decay of the eigenvalues. If the mesh was isometric to the plane, then only the two largest eigenvalues would
be non zero.
</p><pre class="codeinput">clf;
hh = plot(S(1:30), <span class="string">'.-'</span>); axis(<span class="string">'tight'</span>);
set(hh, <span class="string">'LineWidth'</span>, 2);
</pre><img vspace="5" hspace="5" src="index_04.png"> <p>Isomap embedding is defined from the two largest eigenvalues.</p><pre class="codeinput">vertexF = U(:,1:2)' .* repmat(sqrt(S(1:2)), [1 n]);
</pre><p>Align the parameters.</p><pre class="codeinput">vertexF = vertexF - repmat(vertexF(:,icenter), [1 n]);
theta = -pi/2+atan2(vertexF(2,irotate),vertexF(1,irotate));
vertexF = [vertexF(1,:)*cos(theta)+vertexF(2,:)*sin(theta); <span class="keyword">...</span>
-vertexF(1,:)*sin(theta)+vertexF(2,:)*cos(theta)];
</pre><p>Display and compare with Laplacian embeddedding.</p><pre class="codeinput">clf;
plot_mesh(vertexF,faces,options);
</pre><img vspace="5" hspace="5" src="index_05.png"> <p><i>Exercice 2:</i> (<a href="../missing-exo/">check the solution</a>) Compute the embedding using Stress minimization with SMACOF. See the numerical tours on bending invariants for more details.
</p><pre class="codeinput">exo2;
</pre><img vspace="5" hspace="5" src="index_06.png"> <p>Plot stress evolution during minimization.</p><pre class="codeinput">clf;
plot(stress(2:end), <span class="string">'.-'</span>);
axis(<span class="string">'tight'</span>);
</pre><img vspace="5" hspace="5" src="index_07.png"> <p><i>Exercice 3:</i> (<a href="../missing-exo/">check the solution</a>) Compute mesh parameterization using a circle as boundary.
</p><pre class="codeinput">exo3;
</pre><img vspace="5" hspace="5" src="index_08.png"> <p class="footer"><br>
Copyright (c) 2010 Gabriel Peyre<br></p>
</div>
<!--
##### SOURCE BEGIN #####
%% Spectral Mesh Flattening
% This tour explores 2D flattening of 3D surfaces using spectral methods.
%% Installing toolboxes and setting up the path.
%%
% You need to download the following files:
% <../toolbox_signal.zip signal toolbox>,
% <../toolbox_general.zip general toolbox> and
% <../toolbox_graph.zip graph toolbox>.
%%
% You need to unzip these toolboxes in your working directory, so
% that you have
% |toolbox_signal|,
% |toolbox_general| and
% |toolbox_graph|
% in your directory.
%%
% *For Scilab user:* you must replace the Matlab comment '%' by its Scilab
% counterpart '//'.
%%
% *Recommandation:* You should create a text file named for instance |numericaltour.sce| (in Scilab) or |numericaltour.m| (in Matlab) to write all the
% Scilab/Matlab command you want to execute. Then, simply run |exec('numericaltour.sce');| (in Scilab) or |numericaltour;| (in Matlab) to run the commands.
%%
% Execute this line only if you are using Matlab.
getd = @(p)path(p,path); % scilab users must *not* execute this
%%
% Then you can add the toolboxes to the path.
getd('toolbox_signal/');
getd('toolbox_general/');
getd('toolbox_graph/');
%% Spectral Mesh Flattening
% Mesh flattening finds 2D locations that minimize a variational energy
% with a non-degenracy constraint (for instance maximal variance).
%%
% For the Dirichlet energy (Sobolev norm), the
% resulting location are described by the first eigenvectors of the
% Laplacian.
%%
% This method is refered to as "Laplacian eigenmaps" in manifold learning,
% see:
%%
%
%%
% The advantage over fixed boundary harmonic parameterization is
% that the boundary of the flattened domain is not fixed, but the drawback
% is that the parameterization is not guaranteed to be valid (bijective).
%%
% First load a mesh.
name = 'nefertiti';
options.name = name;
[vertex,faces] = read_mesh(name);
n = size(vertex,2);
%%
% Display it.
clf;
plot_mesh(vertex,faces, options);
shading faceted;
%%
% Compute the mesh Laplacian matrix.
options.symmetrize = 1;
options.normalize = 0;
L = compute_mesh_laplacian(vertex,faces,'conformal',options);
%%
% Compute the eigenvalues and eigenvectors
[U,S] = eig(full(L)); S = diag(S);
[S,I] = sort(S,'ascend'); U = U(:,I);
%%
% The vertex positions are the eigenvectors 2 and 3.
vertexF = U(:,2:3)';
%%
% Use translation / rotation to align the parameterization.
icenter = 88;
irotate = 154;
vertexF = vertexF - repmat(vertexF(:,icenter), [1 n]);
theta = -pi/2+atan2(vertexF(2,irotate),vertexF(1,irotate));
vertexF = [vertexF(1,:)*cos(theta)+vertexF(2,:)*sin(theta); ...
-vertexF(1,:)*sin(theta)+vertexF(2,:)*cos(theta)];
%%
% Display the flattened mesh.
clf;
plot_mesh(vertexF,faces);
%%
% _Exercice 1:_ (<../missing-exo/ check the solution>)
% Perform the same flattening, but with the combinatorial Laplacian.
exo1;
%% Geodesic Embedding (Isomap)
% Another (nonlinear) embedding can be computed by minimizing the geodesic
% distortion between points on the surface and points over the
% parameterized domain.
%%
% First we compute the geodesic distance on the mesh using the Fast Marching algorithm.
D = zeros(n);
for i=1:n
D(:,i) = perform_fast_marching_mesh(vertex,faces,i);
end
%%
% Enforce symmetry.
D = (D+D')/2;
%%
% Compute the centered matrix.
J = eye(n) - ones(n)/n;
W = -J*(D.^2)*J;
%%
% Diagonalize the centered matrix.
[U,S] = eig(W);
S = diag(S);
[S,I] = sort(S,'descend'); U = U(:,I);
%%
% Display the decay of the eigenvalues. If the mesh was isometric to the
% plane, then only the two largest eigenvalues would be non zero.
clf;
hh = plot(S(1:30), '.-'); axis('tight');
set(hh, 'LineWidth', 2);
%%
% Isomap embedding is defined from the two largest eigenvalues.
vertexF = U(:,1:2)' .* repmat(sqrt(S(1:2)), [1 n]);
%%
% Align the parameters.
vertexF = vertexF - repmat(vertexF(:,icenter), [1 n]);
theta = -pi/2+atan2(vertexF(2,irotate),vertexF(1,irotate));
vertexF = [vertexF(1,:)*cos(theta)+vertexF(2,:)*sin(theta); ...
-vertexF(1,:)*sin(theta)+vertexF(2,:)*cos(theta)];
%%
% Display and compare with Laplacian embeddedding.
clf;
plot_mesh(vertexF,faces,options);
%%
% _Exercice 2:_ (<../missing-exo/ check the solution>)
% Compute the embedding using Stress minimization with SMACOF.
% See the numerical tours on bending invariants for more details.
exo2;
%%
% Plot stress evolution during minimization.
clf;
plot(stress(2:end), '.-');
axis('tight');
%%
% _Exercice 3:_ (<../missing-exo/ check the solution>)
% Compute mesh parameterization using a circle as boundary.
exo3;
##### SOURCE END #####
-->
</body>
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