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content.m
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%% Toolbox Wavelets on Meshes - A Toolbox for Multiscale Processing of Triangulated Meshes
%
% Copyright (c) 2008 Gabriel Peyre
%
%%
% The toolbox can be downloaded from Matlab Central
% http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=17577&objectType=FILE
%%
% First add to the path additional scripts.
path(path, 'toolbox/');
path(path, 'gim/');
path(path, 'data/');
clear options;
%% Curve Subdivision
% Starting from an initial set of control points (which can be seen as a
% coarse curve), subdivision produces a smooth 2D curve.
%%
% We initialize the filter (subdivision kernel) and the control points.
options.h = [1 4 6 4 1];
% this is an example of points, you can try other sets of points
f0 = [0.11 0.18 0.26 0.36 0.59 0.64 0.80 0.89 0.58 0.22 0.18 0.30 0.58 0.43 0.42; ...
0.91 0.55 0.91 0.58 0.78 0.51 0.81 0.56 0.10 0.16 0.35 0.42 0.40 0.24 0.31];
for i=1:2
f0(i,:) = rescale(f0(i,:),.01,.99);
end
%%
% Perform the subdivision.
Jmax = 5; ms = 20; lw = 1.5;
clf;
for j=0:Jmax
f = perform_curve_subdivision(f0, j, options);
x = linspace(0,1,size(f,2)+1);
subplot(2,3,j+1);
hold on;
h = plot([f(1,:) f(1,1)], [f(2,:) f(2,1)], 'k.-');
set(h, 'MarkerSize', ms);
set(h, 'LineWidth', lw);
% plot control polyhedron
h = plot([f0(1,:) f0(1,1)],[f0(2,:) f0(2,1)], 'r.--');
set(h, 'LineWidth', lw);
title([num2str(j) ' levels of subdivisons']);
hold off;
axis([0 1 0 1]); axis off;
end
%% Triangulated Mesh Subdivision
% You can subdivide a triangulated mesh (2D or 3D) using various rules for
% the subdivision of the connectivity (1:4, 1:3, dual, etc) and various
% rules for the subdivision of the positions of the vertices.
%%
% You can tests subdivision of a regular poyhedra.
% At each step, the positions are projected on the sphere, and the position
% are smoothed to enhance the distribution.
[vertex,face] = compute_base_mesh('ico');
options.spherical = 1;
options.relaxation = 3;
clf;
for s=0:3
subplot(2,2,s+1);
plot_mesh(vertex,face); axis tight;
title([num2str(s) ' levels of subdivisons']);
lighting flat;
if s~=3
[vertex,face] = perform_mesh_subdivision(vertex,face,1,options);
end
end
%%
% You can subdivide an arbitrary mesh.
% load coarse mesh
name = 'mannequin';
vertex = {}; face = {};
[vertex{1},face{1}] = read_mesh(name);
options.name = name;
% you can also try with 'sqrt3', 'butterfly', 'linear4'
options.sub_type = 'loop';
options.spherical = 0;
options.verb = 0;
clf;
for j=2:3
if j>1
[vertex{j},face{j}] = perform_mesh_subdivision(vertex{j-1}, face{j-1}, 1, options);
end
subplot(1,2,j-1);
hold on;
plot_mesh(vertex{j},face{j},options); axis tight;
title([num2str(j) ' levels of subdivisons']);
shading faceted;
% display control polyhedron
h = plot3(vertex{1}(1,:), vertex{1}(2,:), vertex{1}(3,:), 'r.'); % control mesh
set(h, 'MarkerSize', 20);
hold off;
end
%% 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.
%%
% First compute a multiresolution sphere.
% options for the display
options.use_color = 1;
options.rho = .3;
options.color = 'rescale';
options.use_elevation = 0;
% options for the multiresolution mesh
options.base_mesh = 'ico';
options.relaxation = 1;
options.keep_subdivision = 1;
J = 6;
[vertex,face] = compute_semiregular_sphere(J,options);
%%
% 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);
%%
% Perform the wavelet transform and remove small coefficients.
fw = perform_wavelet_mesh_transform(vertex,face, f, +1, options);
% threshold (remove) most of the coefficient
r = .1;
fwT = keep_biggest( fw, round(r*length(fw)) );
% backward transform
f1 = perform_wavelet_mesh_transform(vertex,face, fwT, -1, options);
% display
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);
%%
% By taking the inverse transform of a dirac, you can display a dual
% wavelet that is used for the reconstruction.
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 = [];
%% Wavelet Transform of a Surface
% A wavelet transform can be used to compress a suface itself.
% This surface should be represented as a semi-regular mesh, which is
% obtained by regular 1:4 subdivision of a base mesh. The surface is viewed
% as a 3 independent functions (X,Y,Z coordinates) and there are three
% wavelet coefficients per vertex of the 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';
% Load the geometry image
M = read_gim([name '-sph.gim']);
%%
% Next we create the semi regular mesh from the Spherical GIM.
% option for the load and display
options.func = 'mesh';
options.name = name;
options.use_elevation = 0;
options.use_color = 0;
J = 6;
% creation of the mesh
[vertex,face,vertex0] = compute_semiregular_gim(M,J,options);
%%
% 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; axis tight;
title(['Subdivision level ' num2str(selj(j))]);
end
colormap gray(256);
%%
% Now we compress a the finest mesh as 3 functions defined on the mesh.
% The function to compress.
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 = keep_biggest( fw, round(r*length(fw)) );
% backward transform
f1 = perform_wavelet_mesh_transform(vertex,face, fwT, -1, options);
% display
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');