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

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

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

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

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

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

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

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

9Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs

Surface properties

Proposition 2 (Bernstein et al. 2000)

Let . Suppose

Then

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

11Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs

Proof

Since , condition

implies

Then, we have:

(straightforward)

(Proposition 1)

(condition )

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

13Numerical geometry of non-rigid shapes Consistent approximation of geodesics in graphs

Why both conditions are important?

Insufficient density Too long edges

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