
Wavelet Transform on 3D Meshes

# Wavelet Transform on 3D Meshes

This tour explores multiscale computation on 3D meshes using the lifting wavelet transform.

## Installing toolboxes and setting up the path.

You need to download the following files: signal toolbox, general toolbox, graph toolbox and wavelet_meshes toolbox.

You need to unzip these toolboxes in your working directory, so that you have toolbox_signal, toolbox_general, toolbox_graph and toolbox_wavelet_meshes 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/');
getd('toolbox_wavelet_meshes/');


## Functions Defined on Surfaces

One can define a function on a discrete 3D mesh that assigns a value to each vertex. One can then perform processing of the function according to the geometry of the surface. Here we use a simple sphere.

First compute a multiresolution sphere.

options.base_mesh = 'ico';
options.relaxation = 1;
options.keep_subdivision = 1;
J = 6;
[vertex,face] = compute_semiregular_sphere(J,options);


Options for the display.

options.use_color = 1;
options.rho = .3;
options.color = 'rescale';
options.use_elevation = 0;


Then define a function on the sphere. Here the function is loaded from an image of the earth.

f = load_spherical_function('earth', vertex{end}, options);


Display the function.

clf;
plot_spherical_function(vertex,face,f, options);
colormap gray(256);


## Wavelet Transform of Functions Defined on Surfaces

A wavelet transform can be used to compress a function defined on a surface. Here we take the example of a 3D sphere. The wavelet transform is implemented with the Lifting Scheme of Sweldens, extended to triangulated meshes by Sweldens and Schroder in a SIGGRAPH 1995 paper.

Perform the wavelet transform.

fw = perform_wavelet_mesh_transform(vertex,face, f, +1, options);


Threshold (remove) most of the coefficient.

r = .1;
fwT = perform_thresholding( fw, round(r*length(fw)), 'largest' );


Backward transform.

f1 = perform_wavelet_mesh_transform(vertex,face, fwT, -1, options);


Display it.

clf;
subplot(1,2,1);
plot_spherical_function(vertex,face,f, options);
title('Original function');
subplot(1,2,2);
plot_spherical_function(vertex,face,f1, options);
title('Approximated function');
colormap gray(256);


Exercice 1: (check the solution) Plot the approximation curve error as a function of the number of coefficient.

exo1;


Exercice 2: (check the solution) Perform denoising of spherical function by thresholding. Study the evolution of the optimal threshold as a function of the noise level.

exo2;


Exercice 3: (check the solution) Display a dual wavelet that is used for the reconstruction by taking the inverse transform of a dirac.

exo3;


## Spherical Geometry Images

A simple way to store a mesh is using a geometry images. This will be usefull to create a semi-regular mesh.

Firs we load a geometry image, which is a (n,n,3) array M where each M(:,:,i) encode a X,Y or Z component of the surface. The concept of geometry images was introduced by Hoppe and collaborators.

name = 'bunny';
n = size(M,1);


A geometry image can be displayed as a color image.

clf;
imageplot(M);


But it can be displayed as a surface. The red curves are the seams in the surface to map it onto a sphere.

clf;
plot_geometry_image(M, 1,1);
view(20,88);


One can compute the normal to the surface, which is the cross product of the tangent.

Compute the tangents.

options.order = 2;
u = zeros(n,n,3); v = zeros(n,n,3);
for i=1:3
end


Compute normal.

v = cat(3, u(:,:,2).*v(:,:,3)-u(:,:,3).*v(:,:,2), ...
u(:,:,3).*v(:,:,1)-u(:,:,1).*v(:,:,3), ...
u(:,:,1).*v(:,:,2)-u(:,:,2).*v(:,:,1) );


Compute lighting with an inner product with the lighting vector.

L = [1 2 -1]; L = reshape(L/norm(L), [1 1 3]);
A1 = max( sum( v .* repmat(L, [n n]), 3 ), 0 );
L = [-1 -2 -1]; L = reshape(L/norm(L), [1 1 3]);
A2 = max( sum( v .* repmat(L, [n n]), 3 ), 0 );


Display.

clf;
imageplot(A1, '', 1,2,1);
imageplot(A2, '', 1,2,2);


## Semi-regular Meshes

To be able to perform computation on arbitrary mesh, this surface mesh should be represented as a semi-regular mesh, which is obtained by regular 1:4 subdivision of a base mesh.

Create the semi regular mesh from the Spherical GIM.

J = 6;
[vertex,face,vertex0] = compute_semiregular_gim(M,J,options);


Options for display.

options.func = 'mesh';
options.name = name;
options.use_elevation = 0;
options.use_color = 0;


We can display the semi-regular mesh.

selj = J-3:J;
clf;
for j=1:length(selj)
subplot(2,2,j);
plot_mesh(vertex{selj(j)},face{selj(j)}, options);
% title(['Subdivision level ' num2str(selj(j))]);
end
colormap gray(256);


## Wavelet Transform of a Surface

A wavelet transform can be used to compress a suface itself. The surface is viewed as a 3 independent functions (X,Y,Z coordinates) and there are three wavelet coefficients per vertex of the mesh.

The function to process, the positions of the vertices.

f = vertex{end}';


Forward wavelet tranform.

fw = perform_wavelet_mesh_transform(vertex,face, f, +1, options);


Threshold (remove) most of the coefficient.

r = .1;
fwT = perform_thresholding( fw, round(r*length(fw)), 'largest' );


Backward transform.

f1 = perform_wavelet_mesh_transform(vertex,face, fwT, -1, options);


Display the approximated surface.

clf;
subplot(1,2,1);