1. Discrete geometry Lecture 2 1 © Alexander & Michael Bronstein
tosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapes
Stanford University, Winter 2009 1

2. ‘
The world is continuous, ‘
but the mind is discrete
David Mumford

3. Discretization Continuous world Discrete world Surface Sampling Metric
Topology
Sampling
Discrete metric (matrix of
distances)
Discrete topology (connectivity)

4. How good is a sampling?

5. Sampling density How to quantify density of sampling?
is an -covering of if
Alternatively:
for all , where
is the point-to-set distance.

6. Sampling efficiency Are all points necessary?
An -covering may be unnecessarily
dense (may even not be a discrete set).
Quantify how well the
samples are separated.
is -separated if
for all
For , an -separated
set is finite if is compact.
Also an r-covering!

7. Farthest point sampling
Good sampling has to be dense and efficient at the same time.
Find and -separated -covering of
Achieved using farthest point sampling.
We defer the discussion on
How to select ?
How to compute ?

8. Farthest point sampling

9. Farthest point sampling
Start with some
Determine sampling radius
If stop.
Find the farthest point from
Add to

10. Farthest point sampling
Outcome: -separated -covering of
Produces sampling with progressively increasing density.
A greedy algorithm: previously added points remain in
There might be another -separated -covering containing less points.
In practice used to sub-sample a densely sampled shape.
Straightforward time complexity:
number of points in dense sampling, number of points in
Using efficient data structures can be reduced to

11. Sampling as representation
Sampling represents a region on as a single point
Region of points on closer to than to any other
Voronoi region (a.k.a. Dirichlet or Voronoi-Dirichlet region, Thiessen
polytope or polygon, Wigner-Seitz zone, domain of action).
To avoid degenerate cases, assume points in in general position:
No three points lie on the same geodesic.
(Euclidean case: no three collinear points).
No four points lie on the boundary of the same metric ball.
(Euclidean case: no four cocircular points).

12. Voronoi decomposition
Voronoi region
Voronoi edge
Voronoi vertex
A point can belong to one of the following
Voronoi region ( is closer to than to any other ).
Voronoi edge ( is equidistant from and ).
Voronoi vertex ( is equidistant from
three points ).

13. Voronoi decomposition

14. Voronoi decomposition
Voronoi regions are disjoint.
Their closure
covers the entire .
Cutting along Voronoi edges produces
a collection of tiles
In the Euclidean case, the tiles are
convex polygons.
Hence, the tiles are topological disks
(are homeomorphic to a disk).

15. Voronoi tesellation Tessellation of (a.k.a. cell complex):
a finite collection of disjoint open topological
disks, whose closure cover the entire
In the Euclidean case, Voronoi
decomposition is always a tessellation.
In the general case, Voronoi regions might
not be topological disks.
A valid tessellation is obtained is the
sampling is sufficiently dense.

16. Non-Euclidean Voronoi tessellations
Convexity radius at a point is the largest for which the
closed ball is convex in , i.e., minimal geodesics between
every lie in
Convexity radius of = infimum of convexity radii over all
Theorem (Leibon & Letscher, 2000):
An -separated -covering of with convexity radius of
is guaranteed to produce a valid Voronoi tessellation.
Gives sufficient sampling density conditions.

17. Sufficient sampling density conditions
Invalid tessellation
Valid tessellation

18. Voronoi tessellations in Nature

19. Farthest point sampling and Voronoi decomposition
MATLAB® intermezzo
Farthest point sampling
and Voronoi decomposition

20. Representation error
Voronoi decomposition replaces with the closest point
Mapping copying each into
Quantify the representation error introduced by
Let be picked randomly with uniform distribution on
Representation error = variance of

21. Optimal sampling In the Euclidean case: (mean squared error).
Optimal sampling: given a fixed sampling size , minimize error
Alternatively: Given a fixed representation error , minimize sampling
size

22. Centroidal Voronoi tessellation
In a sampling minimizing , each has to satisfy
(intrinsic centroid)
In the Euclidean case – center of mass
In general case: intrinsic centroid of
Centroidal Voronoi tessellation (CVT): Voronoi tessellation generated
by in which each is the intrinsic centroid of

23. Lloyd-Max algorithm Start with some sampling (e.g., produced by FPS)
Construct Voronoi decomposition
For each , compute intrinsic centroids
Update
In the limit , approaches the hexagonal
honeycomb shape – the densest possible tessellation.
Lloyd-Max algorithms is known under many other names: vector
quantization, k-means, etc.

24. Sampling as clustering
Partition the space into clusters with centers
to minimize some cost function
Maximum cluster radius
Average cluster radius
In the discrete setting, both problems are NP-hard
Lloyd-Max algorithm, a.k.a. k-means is a heuristic, sometimes minimizing average cluster radius (if converges globally – not guaranteed)

25. Farthest point sampling encore
Start with some ,
For
Find the farthest point
Compute the sampling radius
Lemma .

26. Proof
For any

27. Proof (cont)
Since , we have

28. Almost optimal sampling
Theorem (Hochbaum & Shmoys, 1985)
Let be the result of the FPS algorithm. Then
In other words: FPS is worse than optimal sampling by at most 2.

29. Idea of the proof Let denote the optimal clusters, with centers
Distinguish between two cases
One of the clusters contains two or more of the points
Each cluster contains exactly one of the points

30. Proof (first case) Assume one of the clusters contains
two or more of the points ,
e.g.
triangle inequality
Hence:

31. Proof (second case)
Assume each of the clusters contains exactly one of the points
, e.g.
Then, for any point
triangle inequality

32. Proof (second case, cont)
We have: for any , for any point
In particular, for