1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent...
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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
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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: 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © Alexander.](https://reader030.fdocuments.in/reader030/viewer/2022033107/56649d2a5503460f949ff97f/html5/thumbnails/3.jpg)
3Numerical 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
![Page 4: 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © Alexander.](https://reader030.fdocuments.in/reader030/viewer/2022033107/56649d2a5503460f949ff97f/html5/thumbnails/4.jpg)
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: 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © Alexander.](https://reader030.fdocuments.in/reader030/viewer/2022033107/56649d2a5503460f949ff97f/html5/thumbnails/5.jpg)
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: 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © Alexander.](https://reader030.fdocuments.in/reader030/viewer/2022033107/56649d2a5503460f949ff97f/html5/thumbnails/6.jpg)
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
![Page 7: 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © Alexander.](https://reader030.fdocuments.in/reader030/viewer/2022033107/56649d2a5503460f949ff97f/html5/thumbnails/7.jpg)
7Numerical 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
![Page 8: 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © Alexander.](https://reader030.fdocuments.in/reader030/viewer/2022033107/56649d2a5503460f949ff97f/html5/thumbnails/8.jpg)
8Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Surface properties
Minimum curvature radius (“local feature size”)
Minimum branch separation:
for all
![Page 9: 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © Alexander.](https://reader030.fdocuments.in/reader030/viewer/2022033107/56649d2a5503460f949ff97f/html5/thumbnails/9.jpg)
9Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Surface properties
Proposition 2 (Bernstein et al. 2000)
Let . Suppose
Then
![Page 10: 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © Alexander.](https://reader030.fdocuments.in/reader030/viewer/2022033107/56649d2a5503460f949ff97f/html5/thumbnails/10.jpg)
10Numerical 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
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11Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Proof
Since , condition
implies
Then, we have:
(straightforward)
(Proposition 1)
(condition )
![Page 12: 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © Alexander.](https://reader030.fdocuments.in/reader030/viewer/2022033107/56649d2a5503460f949ff97f/html5/thumbnails/12.jpg)
12Numerical 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 13: 1 Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs Consistent approximation of geodesics in graphs Tutorial 3 © Alexander.](https://reader030.fdocuments.in/reader030/viewer/2022033107/56649d2a5503460f949ff97f/html5/thumbnails/13.jpg)
13Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs
Why both conditions are important?
Insufficient density Too long edges
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14Numerical 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