Page 1
1Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Consistent approximation of geodesics in graphs
Tutorial 3
© Alexander & Michael Bronsteintosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapesStanford University, Winter 2009
Page 2
2Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Troubles with the metric
Inconsistent Consistent
Geodesic approximation consistency depends on the graph
Page 3
3Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Consistent metric approximation
Find a bound of the form
Sampling qualityGraph connectivitySurface properties
where , depend on
Page 4
4Numerical 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
Page 5
5Numerical 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
Page 6
6Numerical 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
The length of the path
Page 7
7Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Surface properties
Minimum curvature radius
Minimum branch separation:
Page 8
8Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Surface properties
Proposition 2 (Bernstein et al. 2000)
Let . Suppose
Then
Page 9
9Numerical 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
Page 10
10Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Proof
Since , condition
implies
Then, we have:
(straightforward)
(Proposition 1)
Page 11
11Numerical 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
Page 12
12Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Why both conditions are important?
Insufficient density Too long edges
Page 13
13Numerical 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