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1Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Spectral MethodsTutorial 6
© Maks Ovsjanikovtosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapesStanford University, Winter 2009
2Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Outline
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
1.Classic MDS and PCA review.
2.Metric MDS.
3.Kernel PCA, kernel trick, relation to Metric MDS.
4.Summary.
Articulated Shape Matching by Robust Alignment of Embedded
Representations Mateus D. et al., Workshop on 3DRR, 2007
Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape
Representation Rustamov R., SGP, 2007
3Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
Classic MDS (classical scaling) recap.
1. Given a dissimilarity matrix arising from a normed vector space:
2. We want to find the coordinates of points that would give rise to
E.g. given pairwise distances between cities on a map, find the locations:
Can only hope to find up to rotation, translation
4Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
Classic MDS (classical scaling).
1. Centering matrix H:
2. Define , where
Attention: Only works for normed vector spaces!
5Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
Classic MDS (classical scaling).
2. Define ,
3. Express to obtain
Note that if , then for any orthonormal
4. Since is symmetric, can find its eigendecomposition:
and
6Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Multivariate Analysis Mardia K.V. et al., Academic Press., 1979
Classic MDS (classical scaling).
1. Although is a matrix, it has only non-zero eigenvalues if was
sampled from .
2. Can project on the first eigenvectors, by taking:
7Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Multivariate Analysis Mardia K.V. et al., Academic Press., 1979
Classic MDS (classical scaling).
1. Although is a matrix, it has only non-zero eigenvalues if was
sampled from .
2. Can project on the first eigenvectors, by taking:
Optimality condition of classic MDS
Theorem: If is a set of points in with distances:
For any k-dimensional orthonormal projection , the distortion
is minimized when is projected onto its principal directions,
8Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Classic MDS – Relation to PCA.
1. During standard Principal Component Analysis, one performs
eigendecomposition of the covariance matrix:
2. Try to find a more natural basis to express the points in.
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
9Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Classic MDS – Relation to PCA.
1. During standard Principal Component Analysis, one performs
eigendecomposition of the covariance matrix:
2. Try to find a more natural basis to express the points in.
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
10Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Classic MDS – Relation to PCA.
1. During standard Principal Component Analysis, one performs
eigendecomposition of the covariance matrix:
2. Try to find a more natural basis to express the points in.
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
11Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Classic MDS – Relation to PCA.
1. During standard Principal Component Analysis, one performs
eigendecomposition of the covariance matrix:
2. Using the centering matrix, we can express:
3. For any eigenvalue of we have:
which implies:
4. The eigenvalues of and are the same and the eigenvectors are given by
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
12Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Classic MDS – Relation to PCA.
1. The eigenvalues of and are the same and the eigenvectors are given by:
2. has the advantage that its size is and it is positive definite
rather than positive-semidefinite. Eigendecomposition more stable.
3. If we’re only given pairwise distances, cannot construct directly. Solving
different problems!
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
13Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Metric MDS.
1. Suppose instead of minimizing distortion (stress), we want to minimize derived
stress. Given pairwise distances , find a set of points to minimize:
2. Even if come from a Euclidean space, the problem is much more difficult.
3. Resort to numerical optimization. Differentiate w.r.t. to to get the gradient.
4. Alternative: perform classical MDS on derived distances. Eigensystem.
Problem: The matrix is no longer guaranteed to be positive semi-definite.
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
Critchley F., Multidimensional Scaling: a short critique and a new algorithm, COMPSTAT, 1978
14Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Kernel PCA.
1. Basic Idea: represent a point by its image in a feature space:
2. Domains can be completely different!
3. Kernel Trick: In many applications we do not need to know explicitly, we
only need to operate if the kernel can be
computed efficiently (e.g. can be infinite dimensional)
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
15Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Kernel PCA.
1. Basic Idea: represent a point by its image in a feature space:
2. Domains can be completely different!
3. Kernel Trick: In many applications we do not need to know explicitly, we
only need to operate if the kernel can be
computed efficiently (e.g. can be infinite dimensional)
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
16Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Kernel PCA.
1. Could do PCA in the feature space: compute covariance matrix of feature
vectors, and perform its eigen-decomposition.
2. However, instead of , could use
If the dimension of feature vectors > , this is more efficient!
3. To center the data, so that can use the centering matrix and
find eigenvalues of
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
Schölkopf, B., et al., Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation 1998
17Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Kernel PCA and Metric MDS.
1. Spherical (isotropic) kernel. Depends only on the distance between points:
2. If we assume that then:
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
18Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Kernel PCA and Metric MDS.
1. Suppose we’re given a matrix of pairwise distances:
2. If we set then
In matrix form: , and moreover:
3. Thus, performing Classical MDS on is equivalent to performing it on A.
4. Classical MDS on attempts to approximate: which is a
nonlinear function of distance. So classical MDS on is metric MDS on
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
19Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Kernel PCA and Metric MDS.
1. Thus, performing Classical MDS on is equivalent to performing it on A.
2. Classical MDS on attempts to approximate: which is a
nonlinear function of distance. Classical MDS on is metric MDS on .
3. Since , it is positive semi-definite if the kernel is chosen
appropriately. This is not the case for arbitrary Metric MDS functions.
4. An advantage of doing Kernel PCA is that a new point can be quickly projected
onto a pre-computed basis. Difficult with numerical optimization.
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
20Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Summary:
1. If the distance matrix comes from points in a normed vector space, MDS
reduces to an Eigenvalue Problem – classical scaling.
2. This classical MDS is also closely related to PCA, which computes the optimal
basis when positions are known.
3. Kernel PCA transforms points to a feature space and uses the kernel trick to
compute PCA in this space.
4. Metric MDS approximates derived distances , for some given function .
5. If the kernel is spherical, then Kernel PCA is a special case of Metric MDS, for
the function
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
21Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Outline
On a Connection between Kernel PCA and Metric Multidimensional
Scaling Williams C., Advances in Neural Information Proc. Sys., 2001
1.Classic MDS and PCA review.
2.Metric MDS.
3.Kernel PCA, kernel trick, relation to Metric MDS.
4.Summary.
Articulated Shape Matching by Robust Alignment of Embedded
Representations Mateus D. et al., Workshop on 3DRR, 2007
Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape
Representation Rustamov R., SGP, 2007
22Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Problem:
1. Given 2 articulated shapes in different poses, find point correspondences :
2. Many degrees of freedom, cannot apply rigid alignment.
Articulated Shape Matching by Robust Alignment of Embedded
Representations Mateus D. et al., Workshop on 3DRR, 2007
Images by Q.-X. Huang et al. 08
23Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Approach:
1. Embed each shape into a feature space, defined by the Laplacian.
2. The embedding is isometry invariant:
for any isometric deformation .
3. The embedding is only defined up to a rigid transform in the feature space.
4. Find the optimal rigid transform in the feature space to find the
correspondences.
Articulated Shape Matching by Robust Alignment of Embedded
Representations Mateus D. et al., Workshop on 3DRR, 2007
24Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Approach:
1. The shape is defined as a point cloud. Approximate the Laplacian:
2. Solve the generalized eigenvalue problem:
3. Find the most significant eigenvalues/vectors.
4. For each data point , let
Where is the i-th eigenvector of
Articulated Shape Matching by Robust Alignment of Embedded
Representations Mateus D. et al., Workshop on 3DRR, 2007
25Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Approach:
1. For each data point , let
Where is the i-th eigenvector of .
2. Would like to have for corresponding points. However, each
eigenvector is only defined up to a sign. Reflection:
3. If correspond to the same eigenvalue, then for any
is also an eigenvector. Rotation:
4. Points from the two point sets can be aligned using:
where is orthogonal.
Articulated Shape Matching by Robust Alignment of Embedded
Representations Mateus D. et al., Workshop on 3DRR, 2007
26Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Approach:
1. Given point correspondences it is easy to obtain the optimal orthogonal matrix:
SVD approach from optimal rigid alignment.
2. Let , and compute its singular value decomposition:
3. The optimal solution is given by:
4. With this step, can perform ICP in the feature space to find the optimal
correspondeces.
Articulated Shape Matching by Robust Alignment of Embedded
Representations Mateus D. et al., Workshop on 3DRR, 2007
27Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Results:
Articulated Shape Matching by Robust Alignment of Embedded
Representations Mateus D. et al., Workshop on 3DRR, 2007
28Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Main Goal: Find a good, isometry-invariant shape descriptor.
Good: Efficient, Easily Computable, Insensitive to local topology changes
(unlike MDS)
Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape
Representation Rustamov R., SGP, 2007
29Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Main Idea: For every point define a Global Point Signature
Where is an eigenvector of the Laplace-Beltrami operator.
GPS is a mapping of the surface onto an infinite dimensional space. Each
point gets a signature.
Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape
Representation Rustamov R., SGP, 2007
30Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Properties of GPS:
1. If .
2. GPS is isometry invariant (since Laplace-Beltrami is)
3. Given all eigenfunctions and eigenvalues, can recover the shape up to
isometry (not true if only eigenvalues are known).
4. Euclidean distances in the GPS embedding are meaningful:
K-means done on the embedding provides a segmentation.
Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape
Representation Rustamov R., SGP, 2007
31Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Comparing GPS:
1. Given a shape, determine its GPS embedding.
2. Construct a histogram of pairwise GPS distances (note that GPS is
defined up to sign flips, distances are preserved)
3. For any 2 shapes, compute the -norm difference between their
histograms.
4. For refined comparisons use more than one histogram.
Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape
Representation Rustamov R., SGP, 2007
32Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
Results:
Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape
Representation Rustamov R., SGP, 2007
33Numerical geometry of non-rigid shapes Spectral Methods Tutorial.
1. Kernel methods attempt to embed the shape into a feature space, that
can be manipulated more easily.
2. Laplacian embedding is useful because of its isometry-invariance. Can
be used for comparing non-rigid shapes under isometric deformations.
3. Sign flipping and repeated eigenvalues can cause difficulties (no
canonical way to chose them).
Limitations:
1. Embeddings are not necessarily stable or mesh independent.
2. Difficult to compute for large meshes (millions of points)
3. Both topological and geometric stability is not well understood.
Conclusions