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This tour explores multiscale computation on 3D meshes using the lifting wavelet transform.
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('toolbox_graph')
addpath('toolbox_wavelet_meshes')
addpath('solutions/meshwav_5_wavelets')
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);
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);
Exercise 1
Plot the approximation curve error as a function of the number of coefficient.
exo1()
%% Insert your code here.
Exercise 2
Perform denoising of spherical function by thresholding. Study the evolution of the optimal threshold as a function of the noise level.
exo2()
%% Insert your code here.
Exercise 3
Display a dual wavelet that is used for the reconstruction by taking the inverse transform of a dirac.
exo3()
%% Insert your code here.
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);
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);
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');