Discrete geometryLecture 2

© Alexander & Michael Bronsteintosca.cs.technion.ac.il/book

Numerical geometry of non-rigid shapesStanford University, Winter 2009

The world is continuous,

but the mind is discrete

David Mumford

‘‘‘‘

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Discretization

Surface

Metric

Topology

Sampling

Discrete metric (matrix of

distances)

Discrete topology

(connectivity)

Continuous world Discrete world

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How good is a sampling?

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Sampling density

How to quantify density of sampling?

is an -covering of if

Alternatively:

for all , where

is the point-to-set distance.

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Sampling efficiency

Also an r-covering!

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.

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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 ?

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Farthest point sampling

Farthest point

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Farthest point sampling

Start with some .

Determine sampling radius

If stop.

Find the farthest point from

Add to

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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 .

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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).

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Voronoi decomposition

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 ).

Voronoi region Voronoi edge Voronoi vertex

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Voronoi decomposition

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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).

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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.

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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.

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Sufficient sampling density conditions

Invalid tessellation Valid tessellation

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Voronoi tessellations in Nature

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Farthest point samplingand Voronoi decomposition

MATLAB® intermezzo

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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

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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

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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 .

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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.

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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)

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Farthest point sampling encore

Start with some ,

For

Find the farthest point

Compute the sampling radius

Lemma

.

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Proof

For any

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Proof (cont)

Since , we have

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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.

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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

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Proof (first case)

Assume one of the clusters contains

two or more of the points ,

e.g.

triangle inequality

Hence:

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Proof (second case)

Assume each of the clusters

contains exactly one of the points

, e.g.

triangle inequality

Then, for any point

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Proof (second case, cont)

We have: for any , for any point

In particular, for