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Discrete geometry. © Alexander & Michael Bronstein, 2006-2009 © Michael Bronstein, 2010 tosca.cs.technion.ac.il/book. 048921 Advanced topics in vision Processing and Analysis of Geometric Shapes EE Technion , Spring 2010. ‘. ‘. ‘. ‘. The world is continuous,.

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slide1

Discrete geometry

© Alexander & Michael Bronstein, 2006-2009

© Michael Bronstein, 2010

tosca.cs.technion.ac.il/book

048921 Advanced topics in vision

Processing and Analysis of Geometric Shapes

EE Technion, Spring 2010

slide2

The world is continuous,

but the mind is discrete

David Mumford

slide3

Discretization

Continuous world

Discrete world

  • Surface
  • Metric
  • Topology
  • Sampling
  • Discrete metric (matrix of

distances)

  • Discrete topology (connectivity)
slide5

Sampling density

  • How to quantify density of sampling?
  • is an -covering ofif

Alternatively:

for all , where

is the point-to-set distance.

slide6

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!

slide7

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 ?
slide9

Farthest point sampling

  • Start with some .
  • Determine sampling radius
  • If stop.
  • Find the farthest point from
  • Add to
slide10

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 .
slide11

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 samemetric ball.

(Euclidean case: no four cocircular points).

slide12

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 ).

slide14

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).

slide15

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.

slide16

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.
slide17

Sufficient sampling density conditions

Invalid tessellation

Valid tessellation

slide19

MATLAB® intermezzo

Farthest point sampling

and Voronoi decomposition

slide20

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
slide21

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

slide22

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 .

slide23

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.

slide24

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)

slide25

Farthest point sampling encore

  • Start with some ,
  • For
  • Find the farthest point
  • Compute the sampling radius

Lemma

  • .
slide26

Proof

For any

slide27

Proof (cont)

Since , we have

slide28

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.

slide29

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

slide30

Proof (first case)

Assume one of the clusters contains

two or more of the points ,

e.g.

triangle inequality

Hence:

slide31

Proof (second case)

Assume each of the clusters contains exactly one of the points

, e.g.

Then, for any point

triangle inequality

slide32

Proof (second case, cont)

We have: for any , for any point

In particular, for

slide33

Connectivity

  • Neighborhood is a topological definition

independent of a metric

  • Two points are adjacent (directly connected)

if they belong to the same neighborhood

  • The connectivity structure can be

represented as an undirected graph

with vertices and

edges

Vertices

Edges

  • Connectivity graph can be represented as a matrix
slide34

Shape representation

Graph

Cloud of points

slide35

Connectivity in the plane

Four-neighbor

Six-neighbor

Eight-neighbor

slide36

Delaunay tessellation

Define connectivity as follows: a pair of points whose Voronoi cells are adjacent are connected

The obtained connectivity graph is dual to the Voronoi diagram and is called Delaunay tesselation

Boris Delaunay

(1890-1980)

Voronoi regions

Connectivity

Delaunay tesselation

slide37

Delaunay tessellation

For surfaces, the triangles are replaced by geodesic triangles

[Leibon & Letscher]: under conditions that guarantee the existence of Voronoi tessellation, Delaunay triangles form a valid tessellation

Replacing geodesic triangles by planar ones gives Delaunay triangulation

Geodesic triangles

Euclidean triangles

slide38

Shape representation

Graph

Cloud of points

Triangular mesh

slide39

Triangular meshes

A structure of the form consisting of

  • Vertices
  • Edges
  • Faces

is called a triangular mesh

The mesh is a purely topological object and does not contain any geometric properties

The faces can be represented as an matrix of indices, where each row is a vector of the form , and

slide40

Triangular meshes

The geometric realization of the mesh is defined by specifying the coordinates of the vertices for all

The coordinates can be represented as an matrix

The mesh is a piece-wise planar approximation obtained by gluing the triangular faces together,

Triangular face

slide41

Example of a triangular mesh

Vertices

Coordinates

Edges

Faces

Topological

Geometric

slide42

Barycentric coordinates

Any point on the mesh can be represented providing

  • index of the triangle enclosing it;
  • coefficients of the convex combination of the triangle vertices

Vector is called barycentric coordinates

slide43

Manifold meshes

  • is a manifold
  • Neighborhood of each interior

vertex is homeomorphic to a disc

  • Neighborhood of each boundary

vertex is homeomorphic to a half-disc

  • Each interior edge belongs to two triangles
  • Each boundary edge belongs to one triangle
slide44

Non-manifold meshes

Edge shared by

four triangles

Non-manifold mesh

slide45

Geometry images

Surface

Geometry image

Global parametrization

Sampling of parametrization domain on a Cartesian grid

slide46

Geometry images

Six-neighbor

Eight-neighbor

Manifold mesh

Non-manifold mesh

slide47

Geometric validity

Topologically valid Geometrically invalid

Topological validity (manifold mesh) is insufficient!

Geometric validity means that the realization of the triangular mesh does not contain self-intersections

slide48

Skeleton

For a smooth compact surface , there exists an envelope (open set in

containing ) such that every point is continuously mappable to a unique point on

The mapping is realized as the closest point on from

Problem when is equidistant from two points on

(such points are called medial axis or skeleton of )

If the mesh is contained in the envelope (does not intersect the medial axis), it is valid

slide49

Skeleton

Points equidistant from the boundary

form the skeleton of a shape

slide50

Local feature size

Distance from point on to the medial axis of is called the local feature size, denoted

Local feature size related to curvature

(not an intrinsic property!)

[Amenta&Bern, Leibon&Letscher]: if the surface is sampled such that for every an open ball of radius contains a point of , it is guaranteed that does not intersect the medial axis of

Conclusion: there exists sufficiently dense sampling guaranteeing that

is geometrically valid

slide51

Geometric validity

Insufficient density

Invalid mesh

Sufficient density

Valid mesh

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