1. 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © 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 Consistent approximation of geodesics in graphs Troubles with the metric Inconsistent Consistent Geodesic approximation consistency depends on the graph

3. 3 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent metric approximation Find a bound of the form Sampling quality Graph connectivity Surface properties where, depend on

4. 4 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Main idea Sampling Connectivity graph Geodesic metric Length metric Sampled metric Main idea: show

5. 5 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Sampling conditions Proposition 1 (Bernstein et al. 2000) Let and. Suppose -neighborhood connectivity is a -covering Then

6. 6 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Sketch of the proof is straightforward Let be the geodesic between and of length Divide the geodesic into segments of length at points Due to sampling density, there exist at most -distant from By triangle inequality hence

7. 7 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Sketch of the proof (cont) Thus, we have the poly-geodesic path whose length is

8. 8 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Surface properties Minimum curvature radius (“local feature size”) Minimum branch separation: for all

9. 9 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Surface properties Proposition 2 (Bernstein et al. 2000) Let. Suppose Then

10. 10 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Sufficient conditions for consistency Theorem (Bernstein et al. 2000) Let, and. Suppose Connectivity is a -covering The length of edges is bounded Then

11. 11 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Proof Since, condition implies Then, we have: (straightforward) (Proposition 1) (condition )

12. 12 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Proof (cont) Let be the shortest graph path between and Condition allows to apply Proposition 2 for each of the path segments which gives

13. 13 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Why both conditions are important? Insufficient density Too long edges

14. 14 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Probabilistic version Suppose the sampling is chosen randomly with density function Given, for sufficiently large holds with probability at least