Spherical Mesh Parameterization

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This tour explores parameterization of 3D surfaces onto a sphere.

We use a simple minimization of the Dirichlet energy under spherical constraints. There is no theoritical guarantee, but for some meshes, it seems to work correctly.

In [2]:
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('toolbox_graph')
addpath('toolbox_wavelet_meshes')
addpath('solutions/meshdeform_2_parameterization_sphere')

Smoothing Operator

We start by creating a smoothing operator.

First load a mesh.

In [3]:
name = 'horse';
[vertex,faces] = read_mesh(name);
n = size(vertex,2);
m = size(faces,2);
clear options; options.name = name;

Display the mesh.

In [4]:
clf;
options.lighting = 1;
plot_mesh(vertex,faces,options);
shading faceted;

Compute the weights. The weights should be positive for the method to work.

In [5]:
weight = 'conformal';
weight = 'combinatorial';
switch weight
    case 'conformal'
        W = make_sparse(n,n);
        for i=1:3
            i1 = mod(i-1,3)+1;
            i2 = mod(i  ,3)+1;
            i3 = mod(i+1,3)+1;
            pp = vertex(:,faces(i2,:)) - vertex(:,faces(i1,:));
            qq = vertex(:,faces(i3,:)) - vertex(:,faces(i1,:));
            % normalize the vectors
            pp = pp ./ repmat( sqrt(sum(pp.^2,1)), [3 1] );
            qq = qq ./ repmat( sqrt(sum(qq.^2,1)), [3 1] );
            % compute angles
            ang = acos(sum(pp.*qq,1));
            a = max(1 ./ tan(ang),1e-1); % this is *very* important
            W = W + make_sparse(faces(i2,:),faces(i3,:), a, n, n );
            W = W + make_sparse(faces(i3,:),faces(i2,:), a, n, n );
        end
    case 'combinatorial'
        E = [faces([1 2],:) faces([2 3],:) faces([3 1],:)];
        E = unique_rows([E E(2:-1:1,:)]')';
        W = make_sparse( E(1,:), E(2,:), ones(size(E,2),1) );
end

Compute the normalized weight matrix |tW| such that its rows sums to 1.

In [6]:
d = full( sum(W,1) );
D = spdiags(d(:), 0, n,n);
iD = spdiags(d(:).^(-1), 0, n,n);
tW = iD * W;

Spherical Relaxation

It is possible to smooth the positions of the mesh on the sphere by performing an averaging according to |W|, and projecting back on the sphere.

Compute an initial mapping on the sphere. This simply a radial projection.

In [7]:
vertex1 = vertex;
vertex1 = vertex1 - repmat( mean(vertex1,2), [1 n] );
vertex1 = vertex1 ./ repmat( sqrt(sum(vertex1.^2,1)), [3 1] );

Check which faces have the correct orientation.

normal to faces

In [8]:
[normal,normalf] = compute_normal(vertex1,faces);

center of faces

In [9]:
C = squeeze(mean(reshape(vertex1(:,faces),[3 3 m]), 2));

inner product

In [10]:
I = sum(C.*normalf);

Ratio of inverted triangles. For a bijective mapping, there should not be any inverted triangle.

In [11]:
disp(['Ratio of inverted triangles:' num2str(sum(I(:)<0)/m, 3) '%']);
Ratio of inverted triangles:0.224%

Display on the sphere.

In [12]:
options.name = 'none';
clf;
options.face_vertex_color = double(I(:)>0);
plot_mesh(vertex1,faces,options);
colormap gray(256); axis tight;
shading faceted;

Perform smoothing and projection.

In [13]:
vertex1 = vertex1*tW';
vertex1 = vertex1 ./ repmat( sqrt(sum(vertex1.^2,1)), [3 1] );

Exercise 1

Perform iterative smoothing and projection. Record the evolution of the number of inverted triangle in |ninvert|. Record also the evolution of the Dirichlet energy in |Edir|.

In [14]:
exo1()