1. 1 Numerical geometry of non-rigid shapes In the Rigid Kingdom In the Rigid Kingdom Lecture 4 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

2. 2 Numerical geometry of non-rigid shapes In the Rigid Kingdom Imagine a glamorous ball…

3. 3 Numerical geometry of non-rigid shapes In the Rigid Kingdom A fairy tale shape similarity problem

4. 4 Numerical geometry of non-rigid shapes In the Rigid Kingdom Extrinsic shape similarity Given two shapes and, find the degree of their incongruence. Compare and as subsets of the Euclidean space. Invariance to rigid motion: rotation, translation, (reflection): is a rotation matrix, is a translation vector

5. 5 Numerical geometry of non-rigid shapes In the Rigid Kingdom How to get rid of Euclidean isometries? How to remove translation and rotation ambiguity? Find some “canonical” placement of the shape in. Extrinsic centroid (a.k.a. center of mass, or center of gravity): Set to resolve translation ambiguity. Three degrees of freedom remaining…

6. 6 Numerical geometry of non-rigid shapes In the Rigid Kingdom How to get rid of the rotation ambiguity? Find the direction in which the surface has maximum extent. Maximize variance of projection of onto is the covariance matrix Second-order geometric moments of : is the first principal direction

7. 7 Numerical geometry of non-rigid shapes In the Rigid Kingdom How to get rid of the rotation ambiguity? Project on the plane orthogonal to. Repeat the process to find second and third principal directions.

8. 8 Numerical geometry of non-rigid shapes In the Rigid Kingdom Canonical basis span a canonical orthogonal basis for in.

9. 9 Numerical geometry of non-rigid shapes In the Rigid Kingdom How to get rid of the rotation ambiguity? Direction maximizing = largest eigenvector of. and correspond to the second and third eigenvectors of. admits unitary diagonalization. Setting aligns with the standard basis axes. Principal component analysis (PCA), a.k.a. Karhunen-Loéve transform (KLT), or Hotelling transform. Bottom line: the transformation brings the shape into a canonical configuration in.

10. 10 Numerical geometry of non-rigid shapes In the Rigid Kingdom Second-order geometric moments Eigenvalues of are second-order moments of. In the canonical basis, mixed moments vanish. Ratio describe eccentricity of. Magnitudes of express shape scale.

11. 11 Numerical geometry of non-rigid shapes In the Rigid Kingdom Higher-order geometric moments Second-order moments allow some discrimination. Use higher-order moments gives more discrimination. -th order moment Computed in the canonical basis. Invariant to rigid motion. Signature of moments A fingerprint of the extrinsic geometry of.

12. 12 Numerical geometry of non-rigid shapes In the Rigid Kingdom A signal decomposition intuition Moments are decomposition coefficients in the monomial basis is a Dirac delta function for and elsewhere. span.

13. 13 Numerical geometry of non-rigid shapes In the Rigid Kingdom A signal decomposition intuition uniquely identify a shape (up to a rigid motion). can be reconstructed exactly from is the bi-orthonormal basis, i.e. The monomial basis is not orthogonal. The bi-orthonormal basis is ugly, but we do not need to reconstruct.

14. 14 Numerical geometry of non-rigid shapes In the Rigid Kingdom Truncated signatures of moments Compute the truncated moment signature Construct a moments distance function, e.g. A distance function on the shape of spaces. Quantifies the extrinsic dissimilarity of and.

15. 15 Numerical geometry of non-rigid shapes In the Rigid Kingdom Moments distance is small for nearly congruent and. is large for strongly non-congruent and. If and are truly congruent,. However, does not imply that and are congruent (unless ). Which shapes are indistinguishable by ? Ideally, congruent at a coarse resolution (“low frequency”) and differing in fine details (“high frequency”). Degree of coarseness is controlled by the moments order. Geometric moments do not satisfy this requirement.

16. 16 Numerical geometry of non-rigid shapes In the Rigid Kingdom Other moments Instead of the monomial basis, other bases can be chosen Fourier basis Spherical harmonics, Zernike polynomials, wavelets, etc, etc.

17. 17 Numerical geometry of non-rigid shapes In the Rigid Kingdom Moments of joy, moments of sorrow Joy: Shape similarity is translated to similarity of moment signatures. Comparison of moments signatures is fast (e.g. Euclidean distance). Sorrow: Do not allow for partial similarity!

18. 18 Numerical geometry of non-rigid shapes In the Rigid Kingdom Given two shapes and, find the best rigid motion bringing as close as possible to : is some shape-to-shape distance. Minimum = extrinsic dissimilarity of and. Minimizer = best rigid alignment between and. ICP is a family of algorithms differing in The choice of the shape-to-shape distance. The choice of the numerical minimization algorithm. Iterative closest point (ICP) algorithms

19. 19 Numerical geometry of non-rigid shapes In the Rigid Kingdom Shape-to-shape distance The Hausdorff distance is the distance between a point and the shape. is the distance between a point and the shape.

20. 20 Numerical geometry of non-rigid shapes In the Rigid Kingdom Shape-to-shape distance A non-symmetric version is preferred to allow for partial similarity The (max-min) formulation is sensitive to outliers. Use the variant is a point-to-shape distance. Different possibilities to define.

21. 21 Numerical geometry of non-rigid shapes In the Rigid Kingdom Point-to-point distance Treat as a cloud of points. Find the closest point to on. Define the distance as

22. 22 Numerical geometry of non-rigid shapes In the Rigid Kingdom Point-to-plane distance Treat as a plane, and define the point-to-plane distance is the normal to the surface at point. Can be approximated as

23. 23 Numerical geometry of non-rigid shapes In the Rigid Kingdom Point-to-plane distance is a first-order approximation of the true point-to-shape distance. Construct a second-order approximation are the principal curvature radii at. are the principal directions. is the signed distance to the closest point. Second-order point-to-shape distance

24. 24 Numerical geometry of non-rigid shapes In the Rigid Kingdom Second-order point-to-shape distance The second-order distance approximant may become negative for some values of. Use a non-negative quadratic approximant

25. 25 Numerical geometry of non-rigid shapes In the Rigid Kingdom Second-order point-to-shape distance “Near-field” case – point-to-plane distance “Far-field” case – point-to-point distance

26. 26 Numerical geometry of non-rigid shapes In the Rigid Kingdom Second-order point-to-shape distance Second-order distance generalizes the point-to-point and the point-to- plane distances. Gives more accurate alignment between shapes. Requires principal curvatures and directions (second-order quantities).

27. 27 Numerical geometry of non-rigid shapes In the Rigid Kingdom Iterative closest point algorithm Initialize Find the closest point correspondence Minimize the misalignment between corresponding points Update Iterate until convergence…

28. 28 Numerical geometry of non-rigid shapes In the Rigid Kingdom Closest points How to find closest points efficiently? Straightforward complexity: number of points on, number of points on. divides the space into Voronoi cells Given a query point, determine to which cell it belongs.

29. 29 Numerical geometry of non-rigid shapes In the Rigid Kingdom Closest points

30. 30 Numerical geometry of non-rigid shapes In the Rigid Kingdom Approximate nearest neighbors To reduce search complexity, approximate Voronoi cells. Use binary space partition trees (e.g. kd-trees or octrees). Approximate nearest neighbor search complexity:.

31. 31 Numerical geometry of non-rigid shapes In the Rigid Kingdom Given two sets and of corresponding points. Find best alignment A numerical minimization algorithm can be used. For some point-to-shape distances, a closed-form solution exists. Best alignment

32. 32 Numerical geometry of non-rigid shapes In the Rigid Kingdom MATLAB ® intermezzo Iterative closest point algorithm

33. 33 Numerical geometry of non-rigid shapes In the Rigid Kingdom Until convergence… ICP should find the solution of Instead, it solves Correspondence fixed to instead of. Not guaranteed to produce a monotonically decreasing sequence of values of. Not guaranteed to converge!

34. 34 Numerical geometry of non-rigid shapes In the Rigid Kingdom Enter numerical optimization Treat as a numerical minimization problem. Express the distance terms as a quadratic function is a 3×3 symmetric positive definite matrix, is 3×1 vector, and is a scalar.

35. 35 Numerical geometry of non-rigid shapes In the Rigid Kingdom Local quadratic approximant Point-to-point distance: Point-to-plane distance:

36. 36 Numerical geometry of non-rigid shapes In the Rigid Kingdom Local quadratic approximant Minimize over. Dependence of and on might be complicated. For small motion, hence

37. 37 Numerical geometry of non-rigid shapes In the Rigid Kingdom Minimization variables is required to be unitary (orthonormal). Enforcing orthonormality is cumbersome. Minimization w.r.t. to the rotation angles involves nonlinear functions. Under small motion assumption, Linearize rotation matrix

38. 38 Numerical geometry of non-rigid shapes In the Rigid Kingdom Let Newton be! Linearized rotation yields a quadratic objective w.r.t. Use a Newton step to find the steepest descent direction. Approximation is valid only for small steps. Use Armijo rule to find a fractional step ensuring sufficient decrease of objective function. What is a fractional step?

39. 39 Numerical geometry of non-rigid shapes In the Rigid Kingdom Fractional step Let be a small transformation, which applied times gives. is a rotation by. Hence

40. 40 Numerical geometry of non-rigid shapes In the Rigid Kingdom Iterative closest point algorithm revisited Initialize Find closest point correspondence Construct local quadratic approximant of Find Newton direction Use Armijo rule to find such that Update Iterate until convergence…

41. 41 Numerical geometry of non-rigid shapes In the Rigid Kingdom Iterative closest point algorithm revisited Coefficients of the quadratic approximant can be computed on demand using efficient nearest neighbor search. Alternative: approximate the values of in the space using a space partition tree.