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This tour explores the use of farthest point sampling to compute bending invariant with classical MDS (strain minimization).
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('toolbox_graph')
addpath('solutions/shapes_3_bendinginv_landmarks')
For large mesh, computing all the pairwise distances is intractable. It is possible to speed up the computation by restricting the computation to a small subset of landmarks.
This seeding strategy was used for surface remeshing in:
Geodesic Remeshing Using Front Propagation, Gabriel Peyr and Laurent Cohen, International Journal on Computer Vision, Vol. 69(1), p.145-156, Aug. 2006.
Load a mesh.
name = 'elephant-50kv';
options.name = name;
[vertex,faces] = read_mesh(name);
nverts = size(vertex,2);
Display it.
clf;
plot_mesh(vertex,faces, options);
Compute a sparse set of landmarks to speed up the geodesic computations. The landmarks are computed using farthest point sampling.
First landmarks, at random.
landmarks = 23057;
Dland = [];
Perform Fast Marching to compute the geodesic distance, and record it.
[Dland(:,end+1),S,Q] = perform_fast_marching_mesh(vertex, faces, landmarks(end));
Select farthest point. Here, |min(Dland,[],2)| is the distance to the set of seed points.
[tmp,landmarks(end+1)] = max( min(Dland,[],2) );
Update distance function.
[Dland(:,end+1),S,Q] = perform_fast_marching_mesh(vertex, faces, landmarks(end));
Display distances.
clf;
options.start_points = landmarks;
plot_fast_marching_mesh(vertex,faces, min(Dland,[],2) , [], options);
Exercise 1
Compute a set of |n = 300| vertex by iterating this farthest point sampling. Display the progression of the sampling.
exo1()
%% Insert your code here.
Compute the distance matrix restricted to the landmarks.
D = Dland(landmarks,:);
D = (D+D')/2;
One can compute the bending invariant of the set of landmarks, and then apply it to the whole mesh using interpolation.
Compute a centered kernel for the Landmarks, that should be approximately a matrix of inner products.
J = eye(n) - ones(n)/n;
K = -1/2 * J*(D.^2)*J;
Perform classical MDS on the reduced set of points, to obtain new positions in 3D.
opt.disp = 0;
[Xstrain, val] = eigs(K, 3, 'LR', opt);
Xstrain = Xstrain .* repmat(sqrt(diag(val))', [n 1]);
Xstrain = Xstrain';
Interpolate the locations to the whole mesh by Nystrom eigen-extrapolation, as detailed in
Sparse multidimensional scaling using landmark points V. de Silva, J.B. Tenenbaum, Preprint.
vertex1 = zeros(nverts,3);
deltan = mean(Dland.^2,1);
for i=1:nverts
deltax = Dland(i,:).^2;
vertex1(i,:) = 1/2 * ( Xstrain * ( deltan-deltax )' )';
end
vertex1 = vertex1';
Display the bending invariant mesh.
clf;
plot_mesh(vertex1,faces,options);
The proposed interpolation method is valid only for the Strain minimizer (spectral Nistrom interpolation). One thus needs to use another interpolation method.
See for instance this work for a method to do such an interpolation:
A. M. Bronstein, M. M. Bronstein, R. Kimmel, Efficient computation of isometry-invariant distances between surfaces, SIAM J. Scientific Computing, Vol. 28/5, pp. 1812-1836, 2006.
Exercise 2
Create an interpolation scheme to interpolate the result of MDS dimensionality reduction with Stree minimization (SMACOF algorithm).
exo2()
%% Insert your code here.