TEL-AVIV UNIVERSITY RAYMOND AND BEVERLY SACKLER FACULTY OF EXACT SCIENCES SCHOOL OF MATHEMATICAL SCIENCES An Algorithm for the Computation of the Metric Average of Two Simple Polygons and Extensions M.Sc. thesis presentation by Shay Kels The

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TEL-AVIV UNIVERSITYRAYMOND AND BEVERLY
SACKLERFACULTY OF EXACT SCIENCESSCHOOL OF
MATHEMATICAL SCIENCESAn Algorithm for the
Computation of the Metric Average of Two Simple
Polygons and ExtensionsM.Sc. thesis
presentation by Shay KelsThe research work has
been carried out under the supervision of Prof.
Nira Dyn
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Motivation Reconstruction of a 3D object from a
set of its 2D parallel cross-sections

• Applications
• Tomography (CT, MRI)

• Microscopy

• Computer Vision

• Some approaches

• Contour stitching

• Distance fields

• Mathematical morphology

• Wavelets

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Approximation of set-valued functions (N. Dyn,
E. Farkhi)

• N-dimensional body can be regarded as a
  univariate set-valued function with compact sets
  of dimension n-1 as images

• Given a binary operation between sets, the
  set-valued function
• can be approximated from the given samples
  using

• Spline subdivision schemes

• Bernstein operators via de Casteljau algorithm

• Schoenberg operators via de Boor algorithm

• The applied operation is termed the metric
  average (Z. Artstein )

• Repeated computations of the metric average
  are required.

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Research outline

• An algorithm that applies segment Voronoi
  diagrams and
• planar arrangements to the computation of
  the metric average
• of two simple polygons (based on an idea
  by E. Lipovetsky ).

• Implementation of the algorithm as a C
  program using
• CGAL.

• Connectedness and complexity of the metric
  average.

• The modified metric average.

• Extension to compact sets that are
  collections of simple
• polygons with holes.

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Preliminary definitions

• Let be the collection of nonempty compact
  subsets of .
• 
  
• 
  
• 
  

The Euclidean distance from a point p to a set
is
The Hausdorff distance between two sets
is
The set of all projections of a point p on a set
is
The linear Minkowksi combination of two sets
is
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The metric average
Example in
Let and the
one-sided t-weighted metric average of A and B
is
The metric average of A and B is
The metric property
or
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Conic polygons with holes
A conic segment is defined by

• its base conic

• its beginning point

• its end point

A simple conic polygon is a region of the plane
bounded by a single finite chain of conic
segments, that intersect only at their
endpoints.
A simple conic polygon with holes is a conic
polygon that contains holes, which are simple
conic polygons.
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Planar arrangements

• Given a collection C of curves in the plane, the
• arrangement of C is the subdivision of the plane
• into vertices, edges and faces induced by the
• curves in C.

The overlay of two arrangements and is the
arrangement produced by edges from
and .
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Segment Voronoi diagrams

• For a set S of n simple geometric objects
• (called sites) , the Voronoi diagram of S is
• the subdivision of the plane into regions
• (called faces), each region being associated
  with
• some site , and containing all points
  of
• the plane for which is closest among all
• the sites in S.

• A segment is represented as three objects
• an open segment and the endpoints.

• The diagram is bounded by a frame.

• All edges are conic segments.

• The diagram constitutes of an arrangement of
  conic segments.

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Computation of the metric average - the algorithm
The metric average can be written as
Computation of where A, B are
simple polygons
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Computation of the metric average - the metric
faces

• Let A, B be simple polygons and let F be
• a Voronoi face of VD?B , we call a connected
  component of as a metric face
  originating from F.

• The metric faces are faces of the overlay
• of the arrangements representing VD?B
• and A \ B, which are intersection of the
  bounded
• faces of the two original arrangements.

• Each metric face inherits the Voronoi site
  of
• the face of VD?B that contains it.

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Computation of the metric average - the metric
faces(1)

• By definition of the Voronoi diagram, for

thus

• The operation
  
• is a continuous and one-to-one function
  
• from F to .

is the region bounded by
We can compute the metric average only for
boundaries of the metric faces and only relative
to the corresponding Voronoi sites.
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Computation of the metric average - the algorithm
(1)
Computation of the one-sided metric average
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Computation of the metric average - the algorithm
(2)
Computation of for a
metric face F
S
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Computation of the metric average - the algorithm
(3)
Computation of for a conic
segment and the corresponding point Voronoi
site S
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Complexity bounds
Proposition Let A,B be simple polygons and let n
be the sum of the number of vertices in A and the
number of vertices in B. Let k be the
combinatorial complexity (the sum of the number
of vertices, the number of edges, and the number
of faces) of the overlay of the arrangements
representing the sets and
.

• Then
• k is .

• The combinatorial complexity of
  with is

• Then the run-time complexity of the computation
  of the
• metric average is
  .

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Examples
The metric average of two simple convex polygons
with
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Examples (1)
The metric average of two simple polygons with
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Connectedness of the metric average
The metric average of two intersecting simple
polygons can be a union of several disjoin conic
polygons.
The connectedness problem is model by a graph.
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Connectedness of the metric average (1)
There is an edge on the graph between each
two vertices corresponding to metric connected
elements.
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Connectedness of the metric average (2)
Several propositions considering metric
connectedness, for example
In terms of metric faces and the corresponding
Voronoi sites
condition 1
condition 2
condition 3
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The modified metric average
An artifact occurs near the common endpoint of
two adjacent segment Voronoi sites.
Points lying on the boundary between two
corresponding metric faces are equidistant from
both segment sites.
They are mapped toward both sites, creating a
"split" in the obtained set
1. The site corresponding to F is a segment.
3. The sites are not collinear and have a common
endpoint.
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The modified metric average (1)
Two metric faces , with the
corresponding adjacent segment Voronoi sites
, .
The problematic segment
By the metric average
The operation is defined
for a problematic segment by
a polyline trough 4 points
For all other edges
A twin of in is also problematic.
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The modified metric average (2)
For A,B simple polygons, the t-weighted modified
metric average is computed as the
metric average , with
replaced by
Properties
()
is not true in the general case.
For
For t1/2 we get ().
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The restricted modified metric average
Let and , the
compact set
is the Minkowski sum of A with the compact ball
of radius r centered at the origin.
Definition Let A, B be simple polygons. The
t-weighted restricted modified metric average of
A with B is
Proposition
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Examples
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The metric average of two simple polygonal sets
A set consisting of pairwise disjoint polygons
with holes is termed a simple polygonal set.
The segment Voronoi diagram induced by the
boundary of a simple polygonal set is well
defined.
Let be simple polygonal sets and F a
face of , a connected component of
is termed a metric face
originating from F.
The metric faces are conic polygons with holes.
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The metric average of two simple polygonal sets
(1)
Let the metric face F be a conic polygon P
with holes .
The operation
is a continuous and one-to-one function
from F to
is a conic polygon
with holes
The computation is similar to the computation of
the metric average and the modified metric
average of two simple polygons.
The implementation is supported by CGAL.
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Examples
, where A is a polygon and B is a simple
polygonal set
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Examples (1)
,where A, B are simple polygonal sets.
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Future work

• An algorithm for the computation of the metric
  average of two-dimensional compact sets with
  boundaries consisting of spline curves.

• An algorithm for the computation of the metric
  average of
• two polyhedra.

• Research for new set averaging operations with
  approximation properties relative to some metric
  similar to the approximation properties of the
  metric average relative to the Hausdorff
  distance, but with better geometry.

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References

• Z. Artstein, Piecewise linear approximation of
  set valued maps, Journal of Approximation
  Theory, vol. 56, pp. 41-47, 1989.
• F. Aurenhammer, R Klein, "Voronoi Diagrams" in
  Handbook of Computational Geometry, J. R. Sack,
  J. Urrutia, Eds., Amsterdam Elsevier, 2000, pp.
  201-290.
•  
• N. Dyn, E. Farkhi, A. Mokhov, Approximation of
  univariate set-valued functions - an overview,
  Serdica, vol. 33, pp. 495-514, 2007.
• D. Halperin, "Arrangements", in Handbook of
  Discrete and Computational Geometry, J. E.
  Goodman, J. ORourke, Eds., Chapman Hall/CRC,
  2nd edition, 2004, pp 529562.
• The CGAL project homepage. http//www.cgal.org/.

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Appendix A Computation of the metric average
with Voronoi diagrams the mathematics
Let A, B be simple polygons, the set A\ B can
be written as
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Appendix A Computation of the metric average
with Voronoi diagrams the mathematics (1)
For a site S(F) of the segment Voronoi diagram
and a point p in R2 the set is a
singleton.
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Appendix B does not have the metric
property
The lines are thin polygons.
p