## take care : too many bugs…

This course is an introduction to the  theory of manifolds. Manifold models arise in various area of mathematics, image processing, data mining or computer science. Surfaces of arbitrary dimension can be used to model non-linear datasets that one encounters in modern data processing. Numerical methods allow to exploit this geometric non-linear prior in order to extract relevant information from the data. These methods include in particular local differential computations (related to the Laplacian operator and its variants) and global distance methods (related to geodesic computations). In this course, you will learn how to perform differential and geodesic computations on images, volumes, surfaces and high dimensional graphs.

The course includes a set of Matlab experiments. These experiments give an overview of various tasks in computer vision, image processing, learning theory and mesh processing. This includes computation of shortest paths, voronoi segmentations, geodesic delaunay triangulations, surface flattening, dimensionality reduction and mesh processing.

Ref.

# Lecture 0 – Basic Matlab Instructions

Abstract : Learn the basic features of Matlab, and learn how to load and visualize signals and images.

## Basic Matlab commands.

• Create a variable and an array
• % this is a comment
a = 1; a = 2+1i; % real and complex numbers
b = [1 2 3 4]; % row vector
c = [1; 2; 3; 4]; % column vector
d = 1:2:7; % here one has d=[1 3 5 7]
A = eye(4); B = ones(4); C = rand(4); % identity, 1 and random matrices
c = b'; % transpose

• Modification of vectors and matrices
• A(2,2) = B(1,1) + b(1); % to access an entry in a vector or matrix
b(1:3) = 0; % to access a set of indices in a matrix
b(end-2:end) = 1: % to access the last entries
b = b(end:-1:1); % reverse a vector
b = sort(b); % sort values
b = b .* (b>2); % set to zeros (threshold) the values below 2
b(3) = []; % suppress the 3rd entry of a vector
B = [b; b]; % create a matrix of size 2×4
c = B(:,2); % to access 2nd column

• a = cos(b); a = sqrt(b); % usual function
help perform_wavelet_transform; % print the help
a = abs(b); a = real(b); a = imag(b); a = angle(b); % norm, real part and imaginary part of a complex
disp(‘Hello’); % display a text
disp( sprintf(‘Value of x=%.2f’, x) ); % print a values with 2 digits
A(A==Inf) = 3; % replace Inf values by 3
A(:); % flatten a matrix into a column vector
max(A(:)); % max of a matrix
M = M .* (abs(M)>T); % threshold to 0 values below T.

• Display
• plot( 1:10, (1:10).^2 ); % display a 1D function
title(‘My title’); % title
xlabel(‘variable x’); ylabel(‘variable y’); % axis
subplot(2, 2, 1); % divide the screen in 2×2 and select 1st quadrant

• Programmation
• for i=1:4 % repeat the loop for i=1, i=2, i=3 et i=4
disp(i); % make here something
end
i = 4;
while i>0 % while syntax
disp(i); % do smth
i = i-1;
end

## Load and visualize signals and images

• Load and plot a signal: (function load_signal.m should be in the toolbox of each course)
• f = load_signal(‘Piece-Regular’, n); % signal of size n
plot(f);

• Load and display an image (download function load_image.m should be in the toolboxes)
• I = load_image(‘lena’);
imagesc(I); axis image; colormap gray(256);

# Lecture 1 – Active Contours and Level Sets

Abstract : The goals of this lecture is to use the level set framework in order to do curve evolution. The mean curvature motion is the basic tool, and it can be extended into edge-based (geodesic active contours) and region-based (Chan-Vese) snakes.

## Setting up Matlab.

• First download the Matlab toolbox toolbox_fast_marching.zip. Unzip it into your working directory. You should have a directory toolbox_fast_marching/ in your path.
• The first thing to do is to install this toolbox in your path.
• path(path, ‘toolbox_fast_marching/’);
path(path, ‘toolbox_fast_marching/data/’);
path(path, ‘toolbox_fast_marching/toolbox/’);

• Recompile the mex file for your machine (this can produce some warning). If it does not work, either use the already compiled mex file (they should be available in toolbox_fast_marching/ for MacOs and Unix) or try to set up matlab with a C compiler (e.g. gcc) using ‘mex -setup‘.
• cd toolbox_fast_marching
compile_mex;
cd ..

## Managing level set function.

• In order to perform curve evolution, we will deal with a distance function stored in a 2D image D. The curve will be embedded in the level set D=0. This curve will be evolved by modifying the image D. A curve evolution ODE can be replaced by an PDE on D. This allows to deal with topological changes when a curve split or two curves merge.
• n = 200; % size of the image
% load a distance function
D0 = compute_levelset_shape(‘circlerect2′, n);
% type ‘help compute_levelset_shape’ to see other
% basic curve you can load.

% display the curve
clf; hold on;
imagesc(D); axis image; axis off; axis([1 n 1 n]);
[c,h] = contour(D,[0 0], ‘r’);
set(h, ‘LineWidth’, 2);
hold off;
colormap gray(256);

% do the union of two curves
options.center = [0.15 0.15]*n;
D1 = compute_levelset_shape(‘circle’, n,options);
imagesc(min(D0,D1)<0);

• During the curve evolution, the image D might become far from being a distance function. In order to stabilize the algorithm, one needs to re-compute this distance function.
• % here we simulate a modification of the distance function
[Y,X] = meshgrid(1:n,1:n);
D = (D0.^3) .* (X+n/3);
D1 = perform_redistancing(D);
% display both the original and the new,
% redistanced, curve (should be very close)

## Mean Curvature Motion.

• In order to compute differential quantities (tangent, normal, curvature, etc) on the curve, you can compute derivatives of the image D.
• % the gradient
% display the gradient (as arrow field with ‘quiver’, …)

% the normalized gradient
d = max(eps, sqrt(sum(g0.^2,3)) );
g = g0 ./ repmat( d, [1 1 2] );
% display

% the curvature
K = d .* divgrad( g );
% display

• The mean curvature motion of the level sets of some image u is driven be the following equation.

Implement this evolution explicitly in time using finite differences.

Tmax = 1000; % maximum time of evolution
dt = 0.4; % time step (should be small)
niter = round(Tmax/dt); % number of iterations
D = D0; % initialization

for i=1:niter
% compute the right hand size of the PDE

% update the distance field
D = …;
% redistance the function from time to time
if mod(i,30)==0
D = perform_redistancing(D);
end
% display from time to time
if mod(i,30)=1
% display here

end
end

 Curve evolution under the mean curvature motion (the background is the distance function D).

## Edge-based Segmentation with Geodesic Active Contour (snakes + level set).

• Given a background image M to segment, one needs to compute an edge-stopping function E. It should be small in area of high gradient, and high in area of large gradient.
• % load an image
name = ‘brain';
M = rescale( sum( load_image(name, n), 3) );
% display it

% compute a smoothed gradient
sigma = 4; % blurring size
G = divgrad( perform_blurring(M,sigma) );
% compute the norm of the gradient
d = …
% compute the edge-stopping function
E = …
% rescale it so that it is in realistic ranges
E = rescale(E,0.3,1);

• The geodesic active contour evolution is a mean curvature motion modulated by the edge-stopping function:

Implement this evolution explicitly in time using finite differences.
•  Segmentation with geodesic active contours.

## Region-based Segmentation with Chan-Vese (Mumord-Shah + level sets).

• The geodesic active contour uses an edge-based energy. It has lots of local minima and is very sensitive to initialization. In order to circumvent these drawbacks, one can use a region based energy like the Mumford-Shah functional. Re-casted into the level set framework, it reads:

The corresponding gradient descent is the Chan-Vese active contour method:

Implement this evolution explicitly in time using finite differences, when c1 and c2 are known in advanced.
• % initialize with a complex distance function
D0 = compute_levelset_shape(‘small-disks’, n);
% set parameters
lambda = 0.8;
c1 = …; % black
c2 = …; % gray

 Segmentation with Chan-Vese active contour without edges.
• In the case that one does not know in advance the constants c0 and c1, how to update them automatically during the evolution ? Implement this method.

# Lecture 2 – Front propagation in 2D and 3D

Abstract : The goals of this lecture is to manipulate the fast marching algorithm in 2D and 3D. Application to shortest path extraction (e.g. road tracking and tubular structure extraction in medical images) and Voronoi cell segmentation is presented.

## Setting up Matlab.

• First download the Matlab toolbox toolbox_fast_marching.zip. Unzip it into your working directory. You should have a directory toolbox_fast_marching/ in your path.
• The first thing to do is to install this toolbox in your path.
• path(path, ‘toolbox_fast_marching/’);
path(path, ‘toolbox_fast_marching/data/’);
path(path, ‘toolbox_fast_marching/toolbox/’);

• Recompile the mex file for your machine (this can produce some warning). If it does not work, either use the already compiled mex file (they should be available in toolbox_fast_marching/ for MacOs and Unix) or try to set up matlab with a C compiler (e.g. gcc) using ‘mex -setup‘.
• cd toolbox_fast_marching
compile_mex;
cd ..

## Front propagation in 2D.

• We can now load an image and build a speed propagation function W. Note that area with W values near 0 (black pixels) are those in which the front propagate slowly, so that W is the inverse of the potential that weight the metric.
• n = 128;
name = ‘cavern'; % other possibilities are ‘mountain’ and ‘road2′
W = load_image(name, n);
W = rescale(W, 0.01, 1); % set up a reasonable range for the potential
% display the weighting function

clf; imagesc(W); colormap gray(256);

• After asking to the user the starting and ending points, we proceed to the front propagation. Note that the propagation terminates when the front reaches the ending point.
• [start_points,end_points] = pick_start_end_point(W);
options.end_points = end_points;
[D,S] = perform_fast_marching_2d(W, start_points, options);
% display the distance function
clf; imagesc(D);
D(I==Inf) = 0; % remove Inf values that make contour crash
figure; contour(D,50);

• The shortest path is extracted by performing a gradient descent of D, starting from the ending point.
• p = extract_path_2d(D,end_points, options);
% display the path
clf; plot_fast_marching_2d(W,S,p,start_points,end_points);

• Now that you know the basic commands, you can try other speed functions loaded from other files, and you can try more complicated potential than just the value of the image. You can try a potential based on the gradient of the image computed using grad(W).
 Left : distance function for 3 different W maps. Center : the corresponding level set maps. Right : shortest path together with the explored area (in red).

## 3D Volumetric Shortest Paths.

• Firs you need to load a 3D array of data M. Such a volumetric dataset is more difficult to visualize than a standard 2D image. You can render slices like M(:,:,i) or use a volumetric renderer such a vol3d. In order to do so, you need to set up a correct alpha mapping to make transparent some parts of the volume. If you rotate the data, then you need to re-render the volume using vol3d(h). You can download the 3D data set here.
• % load the whole volume
% crop to retain only the central part
n = 100;
M = rescale( crop(M,n) );
% display some horizontal slices
imageplot(M(:,:,50));
% same thing in the other directions

 Cross sections of the data-set.
• Another, more efficient way, to render volumetric data is to display a semi-transparent volumetric image. This can be achieved using the function vol3d than render semi-transparent slices of the data orthogonal to the viewing direction. In order to do so, you need to set up a correct alpha mapping to make transparent some parts of the volume. If you rotate the data, then you need to re-render the volume using vol3d(h).
• clf;
h = vol3d(‘cdata’,M,’texture’,’2D’);
view(3); axis off;
% set up a colormap
colormap bone(256);
% set up an alpha map
options.center = …; % here a value in [0,1]
options.sigma = .08; % control the width of the non-transparent region
a = compute_alpha_map(‘gaussian’, options); % you can plot(a) to see the alphamap
colormap bone(256);
% refresh the rendering
vol3d(h);
% try with other alphamapping and colormapping

 Volumetric rendering with different alpha-mapping. Each time the options.center value is increased.
• Geodesic distances can be computed on a 3D volume using the Fast Marching. The important point here is to define the correct potential field W that should be large in the region where you want the front to move fast. It means that geodesic will follow these regions. In order to do so, we will ask the user to click on a starting point in a given horizontal slice W(:,:,delta). The potential is then computed in order to be large for value of M that are close to the value at this starting point.
• % ask to the user for some input point
delta = 5;
clf; imageplot(M(:,:,delta));
title(‘Pick starting point’);
start_point = round( ginput(1) );
start_point = [start_point(2); start_point(1); delta];
% compute a potential that is high only very close
% to the value of M at the selected point
W = …;
W = rescale(W,.001,1);
% perform the front propagation

options.nb_iter_max = Inf;
[D,S] = perform_fast_marching_3d(W, start_point, options);
% display the results using vol3d

 Exploration of the distance function using vol3d.
• In order to extract a geodesic, we need to select an ending point and perform a descent of the distance function D from this starting point. The selection is done by choosing a point of low distance value in the slice D(:,:,end-delta).
• % extract a slice
d = D(:,:,n-delta);
% select the point (x,y) of minimum value of d
% hint: use functions ‘min’ and ‘ind2sub’

[x,y] = …;
end_point = [x;y;n-delta];

% extract the geodesic by discrete descent
path = compute_discrete_geodesic(D,end_point);
% draw the path
Dend = D(end_point(1),end_point(2),end_point(3));
D1 = double( D<=Dend );
clf;
plot_fast_marching_3d(M,D1,path,start_point,end_point);

 Exploration of the distance function using vol3d. The red surface indicates the region of the volume that has been explored before hitting the ending point.

## Voronoi segmentation and geodesic Delaunay triangulation in 2D.

• With the same code as above, one can use multiple starting points. The function perform_fast_marching_2d returns a segmentation map Q that contains the Voronoi segmentation.
• n=300;
name = ‘constant'; % other possibility is ‘mountain’
W = load_potential_map(name, n);
m = 20; % number of center points
% compute the starting point at random.
start_points = floor( rand(2,m)*(n-1) ) +1;
[D,Z,Q] = perform_fast_marching_2d(W, start_points);
% display the sampling with the distance function
clf; hold on;
imagesc(D’); axis image; axis off;
plot(start_points(1,:), start_points(2,:), ‘.’);
hold off;
colormap gray(256);
% display the segmentation
figure; clf; hold on;
imagesc(Q’); axis image; axis off;
plot(start_points(1,:), start_points(2,:), ‘.’);
hold off;
colormap gray(256);

 Example of Voronoi cells (distance functions) obtained with a constant speed W (left) and the mountain map (right). Note how the cell on the left have polygonal boundaries whereas cells on the right have curvy boundaries.
• A geodesic Delaunay triangulation is obtained by linking starting points whose Voronoi cells touch. This is the dual of the original Voronoi segmentation.
• faces = compute_voronoi_triangulation(Q);
hold on;
imagesc(Q’); axis image; axis off;
plot(start_points(1,:), start_points(2,:), ‘b.’, ‘MarkerSize’, 20);
plot_edges(compute_edges(faces), start_points, ‘k’);
hold off;
axis tight; axis image; axis off;
colormap jet(256);

 Examples of triangulations. Notice how the canonical euclidean Delaunay triangulation (left) differs from the geodesic one (right) when the metric is not constant.

## Farthest point sampling.

• We are now back to computations in 2D. The function farthest_point_sampling iteratively compute the distance to the already computed set of points, and add to the list (initialized as empty) the farthest point. You should read the function so that you fully understand what it is doing. You can also try different speed functions to see the resulting sampling.
• n=300;
name = ‘mountain';
W = load_potential_map(name, n);
% perform sampling
landmark = [];
landmark = farthest_point_sampling( W, landmark, nbr_landmarks-size(landmark,2) );
% display
hold on;
imagesc(M’);
plot(landmark(1,:), landmark(2,:), ‘b.’, ‘MarkerSize’, 20);
hold off;
axis tight; axis image; axis off;
colormap gray(256);
% try with other metric W, like ‘constant’

 Farthest point sampling with 50, 100, 150, 200, 250 and 300 points respectively.
• Now you can compute the corresponding triangulation using the already given code.
•  Farthest point triangulations.

## Heuristically driven front propagation.

• The function perform_fast_marching_2d can be used with a fourth argument that gives an estimation of the distance to the end point. It helps the algorithm to reduce the number of explored pixels. This remaining distance H should be estimated quickly (hence the name « heuristic »). Here we propose to cheat and use directly the true remaining distance using a classical propagation. You have to test this code with various values of weight.
• [start_points,end_points] = pick_start_end_point(W);
% compute the heuristic
[H,S] = perform_fast_marching_2d(W, end_points);
% perform the propagation
options.end_points = end_points;
weight = 0.5; % should be between 0 and 1.
[D,S] = perform_fast_marching_2d(W, start_points, options, H*weight);
% compute the path
p = extract_path_2d(D,end_points, options);
% display
clf;
plot_fast_marching_2d(W,S,p,start_points,end_points);
colormap jet(256);
saveas(gcf, [rep name ‘-heuristic’ num2str(weight) ‘.jpg’], ‘jpg’);

• As a final question, try to devise a fast way to estimate the remaining distance function. If you have no idea, you can use the function perform_fmstar_2d which implement two methods to achieve this.
•  Heuristic front propagation with a weight parameter of 0, 0.5 and 0.9 respectively. Notice how the explored area shrinks around the true path.

# Lecture 3 – Geodesic Computations on 3D Meshes

Abstract : The goals of this lecture is to manipulate the fast marching algorithm on triangulated meshes. Application to geodesic extraction, Voronoi segmentation, remeshing and bending invariants are presented.

## Setting up Matlab.

• First download the Matlab toolbox toolbox_fast_marching.zip and toolbox_graph.zip. Unzip it into your working directory. You should have directories toolbox_fast_marching/ and toolbox_graph/ in your path.
• The first thing to do is to install these toolboxes in your path.
• path(path, ‘toolbox_fast_marching/’);
path(path, ‘toolbox_fast_marching/toolbox/’);
path(path, ‘toolbox_graph/’);
path(path, ‘toolbox_graph/off/’);

• Recompile the mex files for your machine (this can produce some warning). If it does not work, either use the already compiled mex file (they should be available in toolbox_fast_marching/ for MacOs and Unix) or try to set up matlab with a C compiler (e.g. gcc) using ‘mex -setup‘.
• cd toolbox_fast_marching
compile_mex;
cd ..

## Distance Computation on Triangulated Surface.

• Using the fast marching on a triangulated surface, one can compute the distance from a set of input points. This function also returns the segmentation of the surface into geodesic Voronoi cells.
• name = ‘elephant'; % other choices includes ‘skull’ or ‘bunny’
[vertex,faces] = read_mesh([name ‘.off’]);
nverts = max(size(vertex)); % number of vertices
nstart = 8; % number of starting points
start_points = floor(rand(nstart,1)*nverts)+1;
options.end_points = end_points(:);
% perform the front propagation
[D,S,Q] = perform_fast_marching_mesh(vertex, faces, start_points, options);
% display the distance function
col = D; col(col==Inf) = 0; % the color of the vertices
clf; hold on;
options.face_vertex_color = col;
plot_mesh(vertex, faces, options);
h = plot3(vertex(1,start_points),vertex(2,start_points), vertex(3,start_points), ‘r.’);
set(h, ‘MarkerSize’, 25);
hold off;
colormap jet(256);
axis tight; shading interp;
% you can now display the segmentation function Q

 Example of distance function from a set of random starting points (upper row), and the corresponding Voronoi segmentation (bottom row).
• You can even stop the propagation after a fixed number of steps and see the resulting partially propagated front.
• options.nb_iter_max = …; % trying with a varying number of iterations
[D,S,Q] = perform_fast_marching_mesh(vertex, faces, start_points, options);
% display the distance

 Example of front propagation.

## Geodesic Remeshing.

• A regular sampling of the surface can be performed by seeding in a greedy farthest fashion samples. These points can be linked according to the Voronoi cells adjacency which gives a powerful but yet simple remeshing scheme.
• nbr_landmarks = …; % number of points, eg. 400
W = ones(nverts,1); % the speed function, for now constant speed to perform uniform remeshing
% perform the sampling of the surface
landmark = farthest_point_sampling_mesh( vertex,faces, [], nbr_landmarks, options );
% compute the associated triangulation
[D,Z,Q] = perform_fast_marching_mesh(vertex, faces, landmark);
[vertex_voronoi,faces_voronoi] = compute_voronoi_triangulation_mesh(Q,vertex,faces);
% display the distance function (same as before)

% display the remeshed triangulation (same but with vertex_voronoi and faces_voronoi)

 Farthest point sampling with an increasing number of points (upper row) and corresponding remeshing (bottom row).
• One can use a non-constant speed function in order to have an adapted remeshing. The sampling will be denser in area where the speed function is low.
• % first kind of speed: low on the left, high on the right
v = rescale(vertex(1,:));
options.W = rescale(v>0.5, 1, 3); options.W = options.W(:);
% do the remeshing

% second kind of speed: continuously increasing
v = rescale(-vertex(1,:),1,8);
options.W = v(:);
% do the remeshing

 Rows 1&2: uniform sampling and remeshing. Rows 3&4: adapted (split left/right) remeshing. Rows 5&6: adapted (continuously increasing) remeshing.

## Bending Invariants of a Surface.

• One can use the Isomap procedure in order to modify the surface to get bending invariant signature, useful to perform articulation-invariant recognition of 3D surfaces. One first need to compute the pairwise geodesic distances between points on the surfaces. One then look for new 3D positions of the vertices so that the new Euclidean distances matches closely the geodesic distances. In order to speed up computation, the geodesic distances are computed only on a reduced set of landmarks points. The 3D new locations are then interpolated.
• nlandmarks = 300; % number of landmarks (low to speed up)
landmarks = floor(rand(nlandmarks,1)*nverts)+1; % samples landmarks at random
% compute the distance between landmarks and the rest of the vertices
Dland = zeros(nverts,nlandmarks);
for i=1:nlandmarks
fprintf(‘.’);
[d,S,Q] = perform_fast_marching_mesh(vertex, faces, landmarks(i));
Dland(:,i) = d(:);
end
fprintf(‘\n’);
% perform isomap on the reduced set of points
D1 = Dland(landmarks,:); % reduced pairwise distances
D1 = (D1+D1′)/2; % force symmetry
J = eye(nlandmarks) – ones(nlandmarks)/nlandmarks; % centering matrix
K = -1/2 * J*D1*J; % inner product matrix
% compute the rank-3 approximation of the inner product to compute embedding
opt.disp = 0;
[xy, val] = eigs(K, 3, ‘LR’, opt);
xy = xy .* repmat(1./sqrt(diag(val))’, [nlandmarks 1]);% interpolation on the full set of points
% extend the embeding using geodesic interpolation
vertex1 = zeros(nverts,3);
deltan = mean(Dland,1);
for x=1:nverts
deltax = Dland(x,:);
vertex1(x,:) = 1/2 * ( xy’ * ( deltan-deltax )’ )';
end
% display both the original mesh and the embedding

 Original mesh (left) and bending invariant (right).

# Lecture 4 – Mesh Processing

Abstract : The goal of this lecture is to manipulate a 3D mesh. This includes the loading and display of a 3D mesh and then the processing of the mesh. This processing is based on computations involving various kinds of Laplacians. These Laplacians are extensions of the classical second order derivatives to 3D meshes. They can be used to perform heat diffusion (smoothing), compression and parameterization of the mesh.

## Setting up Matlab.

• First download the Matlab toolbox toolbox_graph.zip. Unzip it into your working directory. You should have a directory toolbox_graph/ in your path. Download also the set of additional meshes in .off format: meshes.zip and unzip this file into your directory.
• The first thing to do is to install this toolbox in your path.
• path(path, ‘toolbox_graph/’);
path(path, ‘toolbox_graph/off/’);
path(path, ‘toolbox_graph/toolbox/’);
path(path, ‘meshes/’);

• Recompile the mex file for your machine (this can produce some warning). If it does not work, either use the already compiled mex file (they should be available in toolbox_graph/ for MacOs and Unix) or try to set up matlab with a C compiler (e.g. gcc) using ‘mex -setup‘.
• cd toolbox_graph
compile_mex;
cd ..

• You can load a mesh from a file and display it. Remember that a mesh is given by two matrix: vertex store the 3D vertex positions and face store 3-tuples giving the number of the vertices forming faces.
• name = ‘mushroom'; % other possibilities include ‘venus’
[vertex,face] = read_mesh([name ‘.off’]);
% display the mesh
clf;
plot_mesh(vertex,face);
% remove the display of the triangle

 Two examples of rendered triangle meshes.
• A mesh can also be built by starting from a coarse triangulation and then subdividing it.
• name = ‘sphere'; % another choice could be ‘L1′
j = 2; % number of subdivision levels
[vertex,face] = gen_base_mesh(name, 0); % the initial mesh
[vertex,face] = gen_base_mesh(name, j);
clf;
plot_mesh(vertex,face);

 Two example of meshes computed by regular subdivision.

## Heat Diffusion on 3D Meshes.

• A Laplacian on a 3D mesh is a (n,n) matrix L, where n is the number of vertices, that generalize the classical laplacian of image processing to a 3D mesh. There are several forms of laplacian depending whether it is symmetric (L’=L) and normalized (1 on the diagonal) :
 L0 = D + W (symmetric, normalized) L1 = D^{-1}*L0 = Id + D^{-1}*W (non-symmetric, non-normalized) L2 = D^{-1/2}*L0*D^{-1/2} = Id + D^{-1/2}*W*D^{-1/2} (symmetric, normalized)

Where W is a weight matrix, W(i,j)=0 if (i,j) is not an edge of the graph. There are several kinds of such weights

 W(i,j)=1 (combinatorial) W(i,j)=1/|vi-vj|^2 (distance) W(i,j)=cot(alpha_ij)+cot(beta_ij) (harmonic)

where {vi}_i are the vertex of the mesh, i.e. vi=vertex(:,i), and (alpha_ij,beta_ij) is the pair of adjacent angles to the edge (vi,vj). A gradient matrix G associated to the laplacian L is an (m,n) matrix where m is the number of edges in the mesh, that satisfies L=G’*G. It can be computed as G((i,j),k)=+sqrt(W(i,j)) if k=j and G((i,j),k)=-sqrt(W(i,j)) if k=i, and G((i,j),k)=0 otherwise.

• In the following, you can compute gradient, weights and laplacian using the compute_mesh_xxx functions.
• % kind of laplacian, can be ‘combinatorial’, ‘distance’ or ‘conformal’ (slow)
laplacian_type = …;
% load two different kind of laplacian and check the gradient factorization
options.symmetrize = 1;
options.normalize = 0;
L0 = compute_mesh_laplacian(vertex,face,laplacian_type,options);
disp([‘Error (should be 0): ‘ num2str(norm(L0-G0’*G0, ‘fro’)) ‘.’]);
options.normalize = 1;
L1 = compute_mesh_laplacian(vertex,face,laplacian_type,options);
disp([‘Error (should be 0): ‘ num2str(norm(L1-G1’*G1, ‘fro’)) ‘.’]);
% these matrices are stored as sparse matrix
spy(L0);

• In order to smooth a vector f of size n (i.e. a function defined on each vertex of the mesh), one can perform a heat diffusion by solving the following PDE
d F / dt = -L*f       with       F(x,t=0)=f
until some stopping time t.
When this diffusion is applied to each component of the positions of the vertices f=vertex(i,:), this smoothes the 3D mesh. Implement this PDE using an explicit discretization in time.
• % the time step should be small enough
dt = 0.1;
% stopping time
Tmax = 10;
% number of steps
niter = round(Tmax/dt);
% initialize the 3 vectors at time t=0
vertex1 = vertex;
% solve the diffusion
for i=1:niter
% update the position by solving the PDE
vertex1 = vertex1 + …;
end

 Heat diffusion on a 3D mesh, at times t=0, t=10, t=40, t=200.
• Another way to smooth a mesh is to perform the following quadratic regularization for each componnent f=vertex(i,:)
F = argmin_g |f-g|^2 + t * |G*f|^2
The solution of this optimization is given in closed form using the Laplacian L=G’*G as the solution of the following linear system:
(Id+t*L)*F = f
Solve this problem for various t on a 3D mesh. You can use the operator \ to solve a system. How does this method compares with the heat diffusion ?
• % solve the equation
vertex1 = …;
% display

## Combinatorial Laplacian: Spectral Decomposition and Compression.

• The combinatorial laplacian is a linear operator (thus a NxN matrix where N is the number of vertices). It depends only on the connectivity of the mesh, thus on face only. The eigenvector of this matrix (which is symmetric, thus can be decomposed by SVD) forms an orthogonal basis of the vector space of signal of NxN values (one real value per vertex). Those functions are the extension of the Fourier oscillating functions to surfaces.
• % combinatorial laplacian computation
options.symmetrize = 1;
options.normalize = 0;
L0 = compute_mesh_laplacian(vertex,face,’combinatorial’,options);
%% Performing eigendecomposition
[U,S,V] = svd(L0);
% extract one of the eigenvectors
c = U(:,end-10); % you can try with other vector with higher frequencies
% assign a color to each vertex
options.face_vertex_color = rescale(c, 0,255);
% display
clf;
plot_mesh(vertex,face, options);
shading interp; lighting none;

 Eigenvectors of the combinatorial laplacian with increasing frequencies from left to right.
• Like the Fourier basis, the laplacian eigenvector basis can be used to perform an orthogonal decomposition of a function. In order to perform mesh compression, we decompose each coordinate X/Y/Z of the mesh into this basis. Once this decomposition has been performed, a compression is achieved by keeping only the biggest coefficients (in magnitude).
• % Warning : vertex should be of size 3 x nvert
keep = 5; % number of percents of coefficients kept
vertex2 = (U’*vertex’)'; % projection of the vector in the laplacian basis
% set threshold to remove coefficients
vnorm = sum(vertex2.^2, 1);
vnorms = sort(vnorm); vnorms = vnorms(end:-1:1);
nvert = size(vertex,2);
thresh = vnorms( round(keep/100*nvert) );
% remove small coefs by thresholding
vertex2 = vertex2 .* repmat( vnorm>=thresh, [3 1] );
% reconstruction
vertex2 = (U*vertex2′)';
% display
clf;
plot_mesh(vertex2,face);
axis tight;

 Spectral mesh compression performed by decomposition on the eigenvectors of the laplacian.

## Mesh Flattening and Parameterization.

• The eigenvector of the combinatorial laplacian can also be used to perform mesh flattening. Flattening means finding a 2D location for each vertex of the mesh. These two coordinates are composed by the eigenvectors n°2 and n°3 of the laplacian.
• % load a mesh
name = ‘nefertiti';
[vertex,face] = read_mesh([name ‘.off’]);
% perform the embedding using the combinatorial eigendecomposition
xy = perform_spectral_embedding(2,A);
% display the flattened mesh
gplot(A,xy,’k.-‘);
axis tight; axis square; axis off;

• This combinatorial flattening does not use geometric information since it only use the connectivity of the mesh. So any mesh with the same connectivity will have the same 2D embedding. In order to improve the quality of the embedding, one can use a conformal laplacian, who approximate the Laplace Beltrami operator of the continuous underlying surface.
• % this time, we use the information from vertex to compute flattening
xy = perform_spectral_embedding(2,A,vertex,face);
% display
gplot(A,xy,’k.-‘);
axis tight; axis square; axis off;

• Another way to compute the flattening is to use the Isomap algorithm. This algorithm is not based on local differential operator such as laplacian. Instead, the geodesic distance between points on the mesh graph is first computed (see the course 4 for example of geodesic computations on graphs). Then the 2D layout of point is computed in order to match the geodesic distance with the distance in the plane.
• % the embedding is now computed with isomap
xy = perform_spectral_embedding(2,A,vertex);
% display
gplot(A,xy,’k.-‘);
axis tight; axis square; axis off;

 Comparison of the flattening obtained with the combinatorial laplacian, the conformal laplacian and isomap.
• Mesh parameterization is similar to flattening, except that we fix the boundary of the mesh to be flattened onto some given convex 2D curve. The flattening can then be proved to be 1:1 (no triangle flip) and long as the curve is convex and Id+laplacian is positive. While flattening involve spectral computation (eigenvectors extraction), which is very slow, mesh parameterization involve the solution of a sparse linear system, which is quite fast (even if the Laplacian matrix is ill-conditioned).
• % pre-compute 1-ring, i.e. the neighbors of each triangle.
ring = compute_1_ring(face);
lap_type = ‘combinatorial'; % the other choice is ‘conformal’
boundary_type = ‘circle'; % the other choices are ‘square’ and ‘triangle’
% compute the parameterization by solving a linear system
xy = compute_parametrization(vertex,face,lap_type,boundary_type,ring);
% display
clf;
gplot(A,xy,’k.-‘);
axis tight; axis square; axis off;

 The first row shows parameterization using the combinatorial laplacian (with various boundary conditions). This assumes that the edge lengths is 1. To take into account the geometry of the mesh, the second row uses the conformal laplacian.

# Lecture 5 – Graph-based data processing

Abstract : The goal of this lecture is to manipulate data in arbitrary dimensions using graph-based method. The points is the data are linked together in a graph structure. Geodesic computations can be performed to compute distance on the dataset.

## Setting up Matlab.

• First download the Matlab toolbox toolbox_dimreduc.zip and toolbox_graph.zip. Unzip it into your working directory. You should have directories toolbox_dimreduc/ and toolbox_graph/ in your path.
• The first thing to do is to install this toolbox in your path.
• path(path, ‘toolbox_dimreduc’);
path(path, ‘toolbox_dimreduc/toolbox/’);
path(path, ‘toolbox_dimreduc/data/’);
path(path, ‘toolbox_graph’);

• Recompile the mex file for your machine (this can produce some warning). If it does not work, either use the already compiled mex file (they should be available for MacOs and/or Unix) or try to set up matlab with a C compiler (e.g. gcc) using ‘mex -setup‘.
• cd toolbox_graph
compile_mex;
cd ..

## Distance Computation on Graphs.

• You can load synthetic and real datasets (only frey_rawface is included in the distribution however)
• name = ‘swissroll'; % you can also try with ‘scurve’
n = 800; % number of points
[X,col] = load_points_set( name, n );
clf; plot_scattered(X,col);
axis equal; axis off;

• From points in arbitrary dimension, you can create a graph data structure using the nearest-neighbor rule (you can also try the alternative epsilon rule).
• options.use_nntools = 0; % set to 1 if you have compile TStool for your machine, this will speed up computations
options.nn_nbr = 7; % number of nearest neighbor
% compute NN graph and the distance between nodes
[D,A] = compute_nn_graph(X,options);
% display the graph
clf; plot_graph(A,X, col);
axis tight; axis off;

• You can now compute the geodesic distance on this graph using the Dijkstra algorithm.
• % set up the length of the edges (0 for no edges)
D(D==Inf) = 0; % weight on the graph
% find some cool location for starting point
[tmp,start_point] = min( abs(col(:)-mean(col(:)))); % starting point
% compute the geodesic distance
[d,S] = perform_dijkstra(D, start_point);
% display the graph with distance colors
clf; hold on;
plot_scattered(X, d);
h = plot3(X(1,start_point), X(2,start_point), X(3,start_point), ‘k.’);
set(h, ‘MarkerSize’, 25);
axis tight; axis off; hold off;

 Two examples of 3D point clouds (left) ; corresponding NN-graph (center) ; geodesic distance to the point in black (right), blue means close.

## PCA and Isomap Flattening.

• The principal component analysis realize a linear dimensionality reduction by projecting the data set on the axes of largest variance.
• % dimension reduction using PCA
[Y,xy] = pca(X,2);
% display
clf; plot_scattered(xy,col);
axis equal; axis off;

• In order to better capture the manifold geometry of the data, Isomap compute the geodesic distance between pair of points, and find a low-dimensional layout that best respect these geodesic distance.
• % dimension reduction using Isomap
options.nn_nbr = 7; % number of NN for the graph
xy = isomap(X,2, options);
% display
clf; plot_scattered(xy,col);
axis equal; axis off;

 Original 3D data set (left) ; 2D flattening using PCA (center) and using Isomap (right). Note how the PCA does not recover the simple 2D parameterization of the manifold since it is a linear process. In contrast, Isomap is able to « unfold » the curvy surface.

## Dimension Reduction for Image Libraries.

• We can use the same process of flattening for data of arbitrary dimension. We first use a simple library of translating disks.
• name = ‘disks';
options.nbr = 1000;
M = load_images_dataset(name, options);
% turn it into a set of points
a = size(M,1);b = size(M,2);n = size(M,3);
X = reshape(M, a*b, n);
% perform isomap
options.nn_nbr = 7;
options.use_nntools = 0;
xy = isomap(X,2, options);
k = 30;
clf; plot_flattened_dataset(xy,M,k);
% perform pca
[tmp,xy] = pca(X,2);
clf; plot_flattened_dataset(xy,M,k);

• We can do the same computation on a more complex library of faces.
• name = ‘frey_rawface';
options.nbr = 2000;
M = load_images_dataset(name, options);
% subsample at random
n = 1000;
sel = randperm(size(M,3));
M = M(:,:,sel(1:n));
M = permute(M,[2 1 3]); % fix x/y orientation of the faces
% turn it into a set of points
a = size(M,1);b = size(M,2);n = size(M,3);
X = reshape(M, a*b, n);
% perform isomap
options.nn_nbr = 7;
options.use_nntools = 0;
xy = isomap(X,2, options);
k = 30;
clf; plot_flattened_dataset(xy,M,k);
% perform pca
[tmp,xy] = pca(X,2);
clf; plot_flattened_dataset(xy,M,k);

 Dimension reduction for two different libraries of images. Left: translating disks, right: face images. Although the disks data set depends on 2D translation, this is not a flat (euclidean) manifold (it is a bit curvy due to the disk shape). This is why the PCA mapping does not recover exactly a 2D square. The face database exhibits a more complex embedding.

## conclusion

 0 – Basic Matlab instructions. Matlab 1 – Active contour and level sets. Matlab PDF PPT 2 – Front propagation in 2D and 3D. Matlab PDF PPT 3 – Geodesic computation on 3D meshes. Matlab PDF PPT 4 – Differential Calculus on 3D meshes. Matlab PDF PPT 5 – High Dimensional Data Processing. Matlab PDF PPT

## Data Processing Using Manifold Methods

This course is an introduction to the computational theory of manifolds. Manifold models arise in various area of mathematics, image processing, data mining or computer science. Surfaces of arbitrary dimension can be used to model non-linear datasets that one encounters in modern data processing. Numerical methods allow to exploit this geometric non-linear prior in order to extract relevant information from the data. These methods include in particular local differential computations (related to the Laplacian operator and its variants) and global distance methods (related to geodesic computations). In this course, you will learn how to perform differential and geodesic computations on images, volumes, surfaces and high dimensional graphs.

The course includes a set of Matlab experiments. These experiments give an overview of various tasks in computer vision, image processing, learning theory and mesh processing. This includes computation of shortest paths, Voronoi segmentations, geodesic Delaunay triangulations, surface flattening, dimensionality reduction and mesh processing.

One should copy/paste the provided code into a file named e.g. tp.m, and launch the script directly from Matlab command line > tp;. Some of the scripts contain « holes » that you should try to fill on your own.

#### 3 Responses to “matlab .m .ppt course : introduction to the theory of manifolds; mesh generation ; mesh processing”

1. 2 esfahlan 28 septembre 2009 à 4:50

Merci beaucoup

2. 3 mojovski 16 novembre 2010 à 12:41

wow!
Thanks a lot!

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