1 Numerical geometry of non-rigid shapes A journey to non-rigid world objects Numerical methods non-rigid Alexander Bronstein Michael Bronstein Numerical geometry of

2 Numerical geometry of non-rigid shapes A journey to non-rigid world 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

3 Numerical geometry of non-rigid shapes A journey to non-rigid world Minimization of quadratic stress Quadratic stress where Its gradient Non-linear non-convex function of variables

4 Numerical geometry of non-rigid shapes A journey to non-rigid world 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

5 Numerical geometry of non-rigid shapes A journey to non-rigid world Shapes as graphs Cloud of pointsEdges Undirected graph = +

6 Numerical geometry of non-rigid shapes A journey to non-rigid world Discrete geodesic problem Local length function Path length Length metric in graph

7 Numerical geometry of non-rigid shapes A journey to non-rigid world 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

8 Numerical geometry of non-rigid shapes A journey to non-rigid world Troubles with the metric Inconsistent metric approximation!

9 Numerical geometry of non-rigid shapes A journey to non-rigid world 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

10 Numerical geometry of non-rigid shapes A journey to non-rigid world 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.

11 Numerical geometry of non-rigid shapes A journey to non-rigid world Eikonal equation Source Equidistant contour Steepest distance growth direction Eikonal equation

12 Numerical geometry of non-rigid shapes A journey to non-rigid world 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

13 Numerical geometry of non-rigid shapes A journey to non-rigid world 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.

14 Numerical geometry of non-rigid shapes A journey to non-rigid world 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

15 Numerical geometry of non-rigid shapes A journey to non-rigid world 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

16 Numerical geometry of non-rigid shapes A journey to non-rigid world 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

17 Numerical geometry of non-rigid shapes A journey to non-rigid world

18 Numerical geometry of non-rigid shapes A journey to non-rigid world 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

19 Numerical geometry of non-rigid shapes A journey to non-rigid world Raster scan fast marching Parametric surface Parametrization domain sampled on Cartesian grid Four alternating scans

20 Numerical geometry of non-rigid shapes A journey to non-rigid world Raster scan fast marching 4 scans=1 iteration 2 iterations3 iterations 4 iterations5 iterations6 iterations Several iterations required for non-Euclidean geometries

21 Numerical geometry of non-rigid shapes A journey to non-rigid world 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

22 Numerical geometry of non-rigid shapes A journey to non-rigid world 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!

23 Numerical geometry of non-rigid shapes A journey to non-rigid world Fast marching MATLAB ® intermezzo