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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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Discrete geometry

© Alexander & Michael Bronstein, 2006-2009

© Michael Bronstein, 2010


048921 Advanced topics in vision

Processing and Analysis of Geometric Shapes

EE Technion, Spring 2010

The world is continuous,

but the mind is discrete

David Mumford


Continuous world

Discrete world

  • Surface

  • Metric

  • Topology

  • Sampling

  • Discrete metric (matrix of


  • Discrete topology (connectivity)

Sampling density

  • How to quantify density of sampling?

  • is an -covering ofif


    for all , where

    is the point-to-set distance.

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!

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 ?

Farthest point sampling

Farthest point

Farthest point sampling

  • Start with some .

  • Determine sampling radius

  • If stop.

  • Find the farthest point from

  • Add to

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 .

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).

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 ).

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).

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.

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.

Sufficient sampling density conditions

Invalid tessellation

Valid tessellation

MATLAB® intermezzo

Farthest point sampling

and Voronoi decomposition

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

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


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 .

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.

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)

Farthest point sampling encore

  • Start with some ,

  • For

  • Find the farthest point

  • Compute the sampling radius


  • .


For any

Proof (cont)

Since , we have

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.

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

Proof (first case)

Assume one of the clusters contains

two or more of the points ,


triangle inequality


Proof (second case)

Assume each of the clusters contains exactly one of the points

, e.g.

Then, for any point

triangle inequality

Proof (second case, cont)

We have: for any , for any point

In particular, for


  • 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




  • Connectivity graph can be represented as a matrix

Shape representation


Cloud of points

Connectivity in the plane




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


Voronoi regions


Delaunay tesselation

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

Shape representation


Cloud of points

Triangular mesh

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

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

Example of a triangular mesh







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

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

Non-manifold meshes

Edge shared by

four triangles

Non-manifold mesh

Geometry images


Geometry image

Global parametrization

Sampling of parametrization domain on a Cartesian grid

Geometry images



Manifold mesh

Non-manifold mesh

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


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


Points equidistant from the boundary

form the skeleton of a shape

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

Geometric validity

Insufficient density

Invalid mesh

Sufficient density

Valid mesh