1 Numerical geometry of non-rigid shapes Non-Euclidean Embedding Non-Euclidean Embedding Lecture 6 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

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

3 Numerical geometry of non-rigid shapes Non-Euclidean Embedding Why non-Euclidean? Reminder: prototype MDS problem with L p stress Minimizer is the canonical form of the shape. Minimum is the embedding distortion (in the L p 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.

4 Numerical geometry of non-rigid shapes Non-Euclidean Embedding Non-Euclidean MDS Prototype non-Euclidean MDS problem with L p 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.

5 Numerical 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

6 Numerical geometry of non-rigid shapes Non-Euclidean Embedding parametrized by where corresponding to parameters. Parametrization domain: Spherical geometry

7 Numerical 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

8 Numerical 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.

9 Numerical geometry of non-rigid shapes Non-Euclidean Embedding Spherical embedding Richer geometry than Euclidean (asymptotically Euclidean). Minimum embedding distortion obtained for shape-dependent radius.

10 Numerical 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.

11 Numerical 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 !

12 Numerical geometry of non-rigid shapes Non-Euclidean Embedding Generalized multidimensional scaling Multidimensional scaling (MDS)Generalized multidimensional scaling (GMDS)

13 Numerical geometry of non-rigid shapes Non-Euclidean Embedding GMDS: pro et contra Pro 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.

14 Numerical geometry of non-rigid shapes Non-Euclidean Embedding GMDS: pro et contra Contra: 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.

15 Numerical geometry of non-rigid shapes Non-Euclidean Embedding 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.

16 Numerical geometry of non-rigid shapes Non-Euclidean Embedding 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.

17 Numerical geometry of non-rigid shapes Non-Euclidean Embedding 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.

18 Numerical geometry of non-rigid shapes Non-Euclidean Embedding Geodesic distance approximation Compute for. falls into triangle and is represented as Particular case: Hence, we can precompute distances How to compute from ?

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

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

21 Numerical geometry of non-rigid shapes Non-Euclidean Embedding A four-step dance

22 Numerical geometry of non-rigid shapes Non-Euclidean Embedding Minimization algorithm How to minimize the generalized stress? Particular case: L 2 stress Fix all and all except for some. Stress as a function of only becomes quadratic

23 Numerical geometry of non-rigid shapes Non-Euclidean Embedding Quadratic stress is positive semi-definite. is convex in (but not necessarily in together).

24 Numerical 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.

25 Numerical geometry of non-rigid shapes Non-Euclidean Embedding 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…

26 Numerical geometry of non-rigid shapes Non-Euclidean Embedding How to move to adjacent triangles? Three cases All : inside triangle. : on edge opposite to. : on vertex. inside on edgeon vertex

27 Numerical geometry of non-rigid shapes Non-Euclidean Embedding 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.

28 Numerical 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.

29 Numerical 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

30 Numerical 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:

31 Numerical geometry of non-rigid shapes Non-Euclidean Embedding Multiresolution Initialize at the coarsest resolution in. For Starting at initialization, solve the GMDS problem Interpolate solution to next resolution level Return.

32 Numerical geometry of non-rigid shapes Non-Euclidean Embedding GMDS Interpolation GMDS

33 Numerical geometry of non-rigid shapes Non-Euclidean Embedding Multiresolution encore So far, we created a hierarchy of embedded spaces. One step further: create a hierarchy of embedding spaces.

34 Numerical geometry of non-rigid shapes Non-Euclidean Embedding MATLAB ® intermezzo GMDS