Information about Numerical geometry of non-rigid shapes (Stanford, winter 2009) -...

Consistent metric approximation - Main idea - Sampling conditions - Surface properties - Sufficient conditions for consistency (Bernstein-de Silva-Langford-Tenenbaum theorem) - Why both conditions are important? - Probabilistic version

Troubles with the metric Inconsistent Consistent Geodesic approximation consistency depends on the graph

Consistent metric approximation Find a bound of the form Sampling quality Graph connectivity Surface properties where , depend on

Sampling quality

Graph connectivity

Surface properties

Main idea Sampling Connectivity graph Geodesic metric Length metric Sampled metric Main idea: show

Sampling conditions Proposition 1 (Bernstein et al. 2000) Let and . Suppose -neighborhood connectivity is a -covering Then

-neighborhood connectivity

is a -covering

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 The length of the path

Surface properties Minimum curvature radius Minimum branch separation :

Surface properties Proposition 2 (Bernstein et al. 2000) Let . Suppose Then

Sufficient conditions for consistency Theorem (Bernstein et al. 2000) Let , and . Suppose Connectivity is a -covering The length of edges is bounded Then

Connectivity

is a -covering

The length of edges is bounded

Proof Since , condition implies Then, we have: (straightforward) (Proposition 1)

Proof (cont) Let be the shortest graph path between and Condition allows to apply Proposition 2 for each of the path segments which gives

Why both conditions are important? Insufficient density Too long edges

Probabilistic version Suppose the sampling is chosen randomly with density function Given , for sufficiently large holds with probability at least

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