1Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Non-Euclidean Embedding

Out of nothing I have created a strange new universe.

J. Bolyai, on the creation of non-Euclidean geometry.

Alexander Bronstein, Michael Bronstein© 2008 All rights reserved. Web: tosca.cs.technion.ac.il

2Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Non-rigid shape similarity

3Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Why non-Euclidean?

Reminder: prototype MDS problem with Lp stress

Minimizer is the canonical form of the shape .

Minimum is the embedding distortion (in the Lp sense).

Practically, distortion is non-zero.

Sets a data-dependent limit to the discriminative power of the canonical

form-based shape similarity.

Can the distortion be reduced?

Yes, by finding a better embedding space.

4Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Non-Euclidean MDS

Prototype non-Euclidean MDS problem with Lp stress

Desired properties of :

Convenient representation of points .

Preferably global system of coordinates.

Efficient computation of the metric .

Preferably closed-form expression.

Simple isometry group .

Practical algorithm for “rigid” shape matching in .

5Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Spherical geometry

-dimensional unit sphere: geometric location of the unit vectors

Sphere of radius obtained by scaling by .

circle, parametrized by

sphere, parametrized by

6Numerical geometry of non-rigid shapes Non-Euclidean Embedding

parametrized by

parametrized by

where corresponding to parameters .

Parametrization domain:

Spherical geometry

7Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Spherical geometry

Minimal geodesics on the sphere are great circles.

Section of the sphere with the plane passing through origin.

Geodesic distance: length of the arc

Geodesic distance on sphere

of radius

Great circle

8Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Spherical embedding

Spherical MDS problem:

Embedding into a sphere of radius :

In the limit , we obtain Euclidean embedding into .

9Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Spherical embedding

Richer geometry than Euclidean (asymptotically Euclidean).

Minimum embedding distortion obtained for shape-dependent radius.

10Numerical geometry of non-rigid shapes Non-Euclidean Embedding

What we found?

Global system of coordinates for representing points on the sphere.

Closed-form expression for the metric

Simple isometry group

Orthogonal group.

Origin-preserving rotations in

Smaller embedding distortion.

Complexity similar to Euclidean MDS.

11Numerical geometry of non-rigid shapes Non-Euclidean Embedding

What we are still missing?

Embedding distortion still non-zero and depends on data.

No straightforward (ICP-like) algorithm for comparing canonical forms.

Way out:

Embed directly into !

12Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Generalized multidimensional scaling

Multidimensional scaling (MDS)Generalized multidimensional scaling (GMDS)

13Numerical geometry of non-rigid shapes Non-Euclidean Embedding

GMDS: croce e delizia

Delizia:

Embedding distortion is no more our enemy

Before, it limited the sensitivity of our method.

Now, it quantifies intrinsic dissimilarity of and .

Measures how much needs to be changed to fit into .

If and are isometric, embedding is distortionless.

No more need to compare canonical forms

Dissimilarity is obtained directly from the embedding distortion.

14Numerical geometry of non-rigid shapes Non-Euclidean Embedding

GMDS: croce e delizia

Croce:

How to represent points on ?

Global parametrization is not always available.

Some local representation is required in general case.

No more closed-form expression for .

Metric needs to be approximated.

Minimization algorithm.

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Local representation

is sampled at and represented as a

triangular mesh .

Any point falls into one of the triangles .

Within the triangle, it can be represented as convex combination

of triangle vertices ,

Barycentric coordinates .

We will need to handle discrete indices in minimization algorithm.

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Geodesic distances

Distance terms can be precomputed, since are fixed.

How to compute distance terms ?

No more closed-form expression.

Cannot be precomputed, since are minimization variables.

can fall anywhere on the mesh.

Precompute for all .

Approximate

for any .

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Geodesic distance approximation

Approximation from .

First order accurate:

Consistent with data:

Symmetric:

Smoothness: is and a closed-form expression

for its derivatives is available to minimization algorithm.

Might be only at some points or along some lines.

Efficiently computed: constant complexity independent of .

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Geodesic distance approximation

Compute for .

falls into triangle and is represented as

Particular case:

Hence, we can precompute distances

How to compute from ?

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Geodesic distance approximation

We have already encountered this problem in fast marching.

Wavefront arrives at triangle vertex at time .

When does it arrive to ?

Adopt planar wavefront model.

Distance map is linear in the triangle (hence, linear in )

Solve for coefficients and obtain a linear interpolant

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Geodesic distance approximation

General case: falls into triangle and is

represented

as

Apply previous steps in triangle to obtain

Apply once again in triangle to obtain

21Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Un ballo a quattro passi

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Minimization algorithm

How to minimize the generalized stress?

Particular case: L2 stress

Fix all and all except for some .

Stress as a function of only becomes quadratic

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Quadratic stress

Quadratic stress

is positive semi-definite.

is convex in (but not necessarily in together).

24Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Quadratic stress

Closed-form solution for minimizer of

Problem: solution might be outside the triangle.

Solution: find constrained minimizer

Closed-form solution still exists.

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Minimization algorithm

Initialize

For each

Fix and compute gradient

Select corresponding to maximum .

Compute minimizer

If constraints are active

translate to adjacent triangle.

Iterate until convergence…

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How to move to adjacent triangles?

Three cases

All : inside triangle.

: on edge opposite to .

: on vertex .

inside on edge on vertex

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Point on edge

on edge opposite to .

If edge is not shared by any other triangle

we are on the boundary – no translation.

Otherwise, express the point as in triangle .

contains same values as .

May be permuted due to different vertex

ordering in .

Complication: is not on the edge.

Evaluate gradient in .

If points inside triangle, update to .

28Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Point on vertex

on vertex .

For each triangle sharing vertex

Express point as in .

Evaluate gradient in .

Reject triangles with pointing

outside.

Select triangle with maximum

.

Update to .

29Numerical geometry of non-rigid shapes Non-Euclidean Embedding

MDS vs GMDS

Stress

Generalized MDSMDS

Generalized stress

Analytic expression for

Nonconvex problem

Variables: Euclidean

coordinates

of the points

must be interpolated

Nonconvex problem

Variables: points on in

barycentric coordinates

30Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Multiresolution

Stress is non convex – many small local minima.

Straightforward minimization gives poor results.

How to initialize GMDS?

Multiresolution:

Create a hierarchy of grids in ,

Each grid comprises

Sampling:

Geodesic distance matrix:

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Multiresolution

Initialize at the coarsest resolution in .

For

Starting at initialization , solve the GMDS problem

Interpolate solution to next resolution level

Return .

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GMDS

Interpolation

GMDS

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Multiresolution encore

So far, we created a hierarchy of embedded spaces .

One step further: create a hierarchy of embedding spaces

.

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MATLAB® intermezzoGMDS

35Numerical geometry of non-rigid shapes Non-Euclidean Embedding

Summary and suggested reading

Spherical embedding

T. F. Cox and M. A. A. Cox, Multidimensional scaling on a sphere, Communications in Statistics: Theory and Methods 20 (1991), 2943–2953.

T. F. Cox and M. A. A. Cox, Multidimensional scaling, Chapman and Hall, 1994.

Generalized multidimensional scaling

A.M. Bronstein, M.M. Bronstein, and R. Kimmel, Efficient computation ofisometry-invariant distances between surfaces, SIAM J. Scientific Computing 28 (2006), no. 5, 1812–1836.

A.M. Bronstein, M.M. Bronstein, and R. Kimmel, Generalized multidimensionalscaling: a framework for isometry-invariant partial surface matching, PNAS 103(2006), no. 5, 1168–1172.

Initialization issues

D. Raviv, A. M. Bronstein, M. M. Bronstein, and R. Kimmel, Symmetries of non-rigid shapes, Proc. NRTL, 2007.