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,.
© 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
for all , where
is the point-to-set distance.
dense (may even not be a discrete set).
samples are separated.
for all .
set is finite if is compact.
Also an r-covering!
number of points in dense sampling, number of points in .
polytope or polygon, Wigner-Seitz zone, domain of action).
(Euclidean case: no three collinear points).
(Euclidean case: no four cocircular points).
three points ).
covers the entire .
a collection of tiles .
(are homeomorphic to a disk).
a finite collection of disjoint open topological
disks, whose closure cover the entire .
decomposition is always a tessellation.
not be topological disks.
sampling is sufficiently dense.
closed ball is convex in , i.e., minimal geodesics between
every lie in .
An -separated -covering of with convexity radius of
is guaranteed to produce a valid Voronoi tessellation.
Farthest point sampling
and Voronoi decomposition
(mean squared error).
Alternatively: Given a fixed representation error , minimize sampling
by in which each is the intrinsic centroid of .
honeycomb shape – the densest possible tessellation.
quantization, k-means, etc.
Partition the space into clusters with centers
to minimize some cost function
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)
Since , we have
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.
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
Assume one of the clusters contains
two or more of the points ,
Assume each of the clusters contains exactly one of the points
Then, for any point
We have: for any , for any point
In particular, for
independent of a metric
if they belong to the same neighborhood
represented as an undirected graph
with vertices and
Cloud of points
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
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
Cloud of points
A structure of the form consisting of
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
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,
Any point on the mesh can be represented providing
Vector is called barycentric coordinates
vertex is homeomorphic to a disc
vertex is homeomorphic to a half-disc
Edge shared by
Sampling of parametrization domain on a Cartesian grid
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
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