meshwav_5_wavelets.m

%% Wavelet Transform on 3D Meshes
% This tour explores multiscale computation on 3D meshes
% using the lifting wavelet transform.

perform_toolbox_installation('signal', 'general', 'graph', '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);

%EXO
%% Plot the approximation curve error as a function of the number of
%% coefficient.
%EXO

%EXO
%% Perform denoising of spherical function by thresholding. Study the
%% evolution of the optimal threshold as a function of the noise level.
%EXO

%EXO
%% Display a dual
%% wavelet that is used for the reconstruction 
%% by taking the inverse transform of a dirac.
clf;
nverts = size(vertex{end}, 2);
i = 0;
for j = [J-3 J-2]
    i = i+1;
    nj = size(vertex{j},2); nj1 = size(vertex{j+1},2);
    sel = nj+1:nj1-1;
    d = sum( abs(vertex{end}(:,sel)) );
    [tmp,k] = min(d); k = sel(k);
    fw2 = zeros(nverts,1); fw2(k) = 1;
    f2 = perform_wavelet_mesh_transform(vertex,face, fw2, -1, options);
    options.color = 'wavelets';
    options.use_color = 1;
    options.rho = .6;
    options.use_elevation = 1;
    options.view_param = [104,-40];
    subplot(1,2,i);
    plot_spherical_function(-vertex{end},face{end},f2, options); axis tight;
    title(['Wavelet, scale ' num2str(j)]);
end
options.view_param = [];
%EXO

%% 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';
M = read_gim([name '-sph.gim']);
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
    [u(:,:,i),v(:,:,i)] = grad(M(:,:,i), options);
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);
    shading('faceted'); lighting('flat'); axis tight;
    % 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);
plot_mesh(f,face{end},options); shading('interp'); axis('tight');
title('Original surface');
subplot(1,2,2);
plot_mesh(f1,face{end},options); shading('interp'); axis('tight');
title('Wavelet approximation');