Wavelet Transform on 3D Meshes¶
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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')
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);