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Discrete geometry
Lecture 2
© Alexander & Michael Bronstein
tosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapes
Stanford University, Winter 2009
The world is continuous,
but the mind is discrete
David Mumford
‘‘‘‘
3Numerical geometry of non-rigid shapes Discrete geometry
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
14Numerical geometry of non-rigid shapes Discrete geometry
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 sampling
and 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