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This tour explores multiscale computation on 3D meshes using the lifting wavelet transform.
from __future__ import division
import nt_toolbox as nt
from nt_solutions import meshwav_5_wavelets as solutions
%matplotlib inline
%load_ext autoreload
%autoreload 2
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.
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.
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.
solutions.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.
solutions.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.
solutions.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.
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.
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 in 1: 3:
[u(: , : , i), v(: , : , i)] = grad(M(: , : , i), options)
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.
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
for j in 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))])
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.
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')