1 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Numerical geometry of shapes Lecture II – Numerical Tools non-rigid Alex Bronstein

2 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Shape matching blueprints Isometric embedding A. Elad & R. Kimmel, 2003

3 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Shape matching blueprints Compute canonical forms EXTRINSIC SIMILARITY OF CANONICAL FORMS INTRINSIC SIMILARITY = INTRINSIC SIMILARITY A. Elad & R. Kimmel, 2003

4 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Numerical ingredients Shape discretization Metric discretization How to compute geodesic distances? Discrete embedding problem How to compute the canonical forms? Can we do better? EMBEDDING MATCHING

5 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Discrete Geodesics and Shortest Path Problems Discrete Geodesics and Shortest Path Problems

6 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Shapes as graphs Cloud of pointsEdges Undirected graph = +

7 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Discrete geodesic problem Local length function Path length Length metric in graph

8 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Dijkstra’s algorithm INPUT: source point Initialize and for the rest of the graph; Initialize queue of unprocessed vertices. While Find vertex with smallest value of, For each unprocessed adjacent vertex, Remove from. OUTPUT: distance map. E.W. Dijkstra, 1959

9 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Troubles with the metric Inconsistent metric approximation!

10 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Metrication error Graph induces inconsistent metric SOLUTION 1 Change the graph Both sampling & connectivity Sampling theorems guarantee consistency for some conditions SOLUTION 2 Change the algorithm Stick to same sampling Discrete surface rather than graph New shortest path algorithm SOLUTION 1 Discretized shape Discrete metric SOLUTION 2 Discretized metric

11 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Forest fire Fermat’s Principle (of Least Action): Fire chooses the quickest path to travel. Pierre de Fermat ( ) Fermat’s Principle (of Least Action): Fire chooses the shortest path to travel.

12 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Eikonal equation Source Equidistant contour Steepest distance growth direction Eikonal equation

13 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Fast marching methods (FMM) A family of numerical methods for solving eikonal equation Simulates wavefront propagation from a source set A continuous variant of Dijkstra’s algorithm Consistently discretized metric J.N. Tsitsiklis, 1995; J. Sethian, 1996, R. Kimmel & J. Sethian, 1998; A. Spira & R. Kimmel, 2004

14 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Fast marching algorithm Initialize and mark it as black. Initialize for other vertices and mark them as green. Initialize queue of red vertices. Repeat Mark green neighbors of black vertices as red (add to ) For each red vertex For each triangle sharing the vertex Update from the triangle. Mark with minimum value of as black (remove from ) Until there are no more green vertices. Return distance map.

15 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Fast marching Dijkstra’s update Vertex updated from adjacent vertex Distance computed from Path restricted to graph edges Fast marching update Vertex updated from triangle Distance computed from and Path can pass on mesh faces

16 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Fast marching update step Update from triangle Compute from and Model wave front propagating from planar source unit propagation direction source offset Front hits at time Hits at time When does the front arrive to ? Planar source

17 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Fast marching update step is given by the point-to-plane distance Solve for parameters and using the point-to-plane distance …after some algebra where

18 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools

19 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Uses of fast marching Geodesic distances Minimal geodesics Voronoi tessellation & sampling Offset curves

20 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Marching even faster Heap-based update Unknown grid visiting order Inefficient use of cache Inherently sequential Raster scan update Regular access to memory Can be parallelized Suitable only for regular grid

21 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Raster scan fast marching Parametric surface Parametrization domain sampled on Cartesian grid Four alternating scans

22 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Raster scan fast marching 4 scans=1 iteration 2 iterations3 iterations 4 iterations5 iterations6 iterations Several iterations required for non-Euclidean geometries

23 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Marching even faster Heap-based update Irregular use of memory Sequential Any grid Single pass, Raster scan update Regular access to memory Can be parallelized Only regular grids Data-dependent complexity

24 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools O. Weber, Y. Devir, A. Bronstein, M. Bronstein & R. Kimmel, 2008 Parallellization Rotate by 45 0 On NVIDIA GPU 50msec per distance map on 10M vertices 200M distances per second!

25 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Embedding Problems

26 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Isometric embedding

27 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Mapmaker’s problem

28 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Mapmaker’s problem A sphere has non-zero curvature, therefore, it is not isometric to the plane (a consequence of Theorema egregium) Karl Friedrich Gauss ( )

29 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Minimum distortion embedding A. Elad & R. Kimmel, 2003

30 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Discrete embedding problem Ingredients: Discretized shape Discretized metric Euclidean embedding space Embedding is a configuration of points in represented as a matrix with the Euclidean metric Embedding distortion (stress) function e.g., quadratic Numerical procedure to minimize

31 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Discrete embedding problem Multidimensional scaling (MDS)

32 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Minimization of quadratic stress Quadratic stress where Its gradient Non-linear non-convex function of variables

33 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Minimization of quadratic stess Start with some Repeat for Steepest descent step Until convergence OUTPUT: canonical form Can converge to local minimum Minimum defined modulo congruence

34 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Examples of canonical forms Canonical forms Near-isometric deformations of a shape

35 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Examples of canonical forms Embedding distortion limits discriminative power! J.N. Tsitsiklis, 1995; J. Sethian, 1996, R. Kimmel & J. Sethian, 1998; A. Spira & R. Kimmel, 2004

36 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Non-Euclidean Embedding Non-Euclidean Embedding

37 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Euclidean embedding

38 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Non-Euclidean embedding

39 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Spherical embedding Richer geometry than Euclidean (asymptotically Euclidean). Minimum embedding distortion obtained for shape-dependent radius.

40 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools The ultimate embedding space Embed one shape directly into the other If shapes are isometric, embedding is distortionless Otherwise, distortion is the measure of dissimilarity

41 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Generalized embedding problem A. Bronstein, M. Bronstein & R. Kimmel, 2006

42 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Generalized multidimensional scaling Representation How to represent arbitrary points on ? Metric approximation How to compute distance terms between arbitrary points on ? Numerical procedure How to minimize stress? A. Bronstein, M. Bronstein & R. Kimmel, 2006

43 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Representation Triangle index Convex combination Barycentric coordinates = + Triangular mesh How to represent an arbitrary point on ? A. Bronstein, M. Bronstein & R. Kimmel, 2006

44 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Metric approximation How to approximate distance between arbitrary points? Precompute distances between all pairs of vertices on A. Bronstein, M. Bronstein & R. Kimmel, 2006

45 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Generalized stress Fix all variables except for Quadratic in Convex in A. Bronstein, M. Bronstein & R. Kimmel, 2006

46 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Minimization algorithm Initialize Select corresponding to maximum gradient Compute minimizer If constraints are active translate to adjacent triangle Iterate until convergence… A. Bronstein, M. Bronstein & R. Kimmel, 2006

47 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools GMDS in action CANONICAL FORMS (MDS, 500 points) MINIMUM DISTORTION EMBEDDING (GMDS, 50 points) A. Bronstein, M. Bronstein & R. Kimmel, 2006

48 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Summary Discrete geodesic problem Dijkstra – inconsistent discrete metric Fast marching – consistently discretized metric Discrete embedding problem MDS – embedding distortion is an enemy Non-Euclidean embedding Generalized embedding GMDS – embed one surface into the other Embedding distortion is the measure ofdissimilarity