Irregular Pyramids with Combinatorial Maps

1
Irregular Pyramids with Combinatorial Maps

• Luc Brun and Walter Kropatsch
• L.E.R.I. - Reims University (France)
• P.R.I.P. - Vienna University of Technology
  (Austria)

2
Contents

• Problem and alternative solutions
• Combinatorial maps
• Irregular Pyramids
• Construction of Irregular Pyramids

3
Embedding in the image plane

• Encode topological relationships.

?
G(V,E) ?
4
Topological Structures

• Alternative solutions
• Store the coordinates of the objects
• Local deformation of digital curves Rosenfeld97
• Dual Graphs

5
Dual Graphs
6
Dual Graphs
G(V,E)
7
Dual Graphs
G ?
?
8
Dual Graphs
9
Dual Graphs
Boundary Graph
10
Combinatorial Maps
?)
?,
G (
D,
D Set of half edges (darts)
11
Combinatorial Maps Basic Properties

• Dual graph
• Bridge
• Self-loop

12
Basic Operations
Removal
Contraction
13
Basic OperationsDuality
14
Combinatorial Maps Advantages

• Efficient implementation
• Implicit encoding of the dual
• Explicit encoding of the orientation
• Same Formalism for 3D and 4D combinatorial maps
  Lienhardt91.

15
Simplifications of the initial Embedding
Image
Combinatorial map
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Simplifications of the initial Embedding
G0(D0,?0,?)
G1(D1,?1,?)
G2(D2,?2,?)
G3(D3,?3,?)
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Irregular Pyramids

• Stack of successively reduced graphs
• A set of Contraction or removal operations

• Applications
• Connected component labelling
• Segmentation
• Line images
• Matching

Construction
Gn(Dn,?n,?)
Dn-1,n
D2,3
G2(D2,?2,?)
D1,2
G1(D1,?1,?)
D0,1
G0(D0,?0,?)
18
Building one pyramid level

• Do not remove bridges nor contract self-loops.

19
Decimation Parameter

• a set of contracted darts
• A contracted dart link a surviving vertex to a
  non surviving one.
• One contracted edge is incident to each non
  surviving vertex.

20
Contracted MapG(D,?,?)/?(D)

• ? remains unchanged on SD D- ?(D)
• ? is modified

1
-1
4
-4
1
-1
4
-4
3
6
-2
-7
-3
5
3
2
-2
5
-5
-3
2
-5
-6
-7
6
7
6
7
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Construction of the contracted map
G(D,?,?)
G(D,?,?)

• For each surviving dart d, in parallel
• do
• iMin j?0,1,2 ?j(?(d)) survives
• ?(d) ?(d)
• ?(d) ?i(?(d))
• done

22
Conclusion

• Combinatorial Maps can build Irregular Pyramids
• Each level is a planar graph
• The base is the 4-neighbourhood graph of the
  pixel array
• Contraction preserves Topological Structures.

23
Future Work

• Extend the idea of Decimation Parameters
  (Contraction Kernels TR-57,TR-63)
• Study some applications
• Segmentation
• Structural matching
• Integration of moving objects

24
References

• Rosenfeld97 Azriel Rosenfeld and Akira
  Nakamura. Local deformations of digital curves.
  Pattern Recognition Letters, 18613-620, 1997.
• Lienhardt91 Pascal Lienhardt. Topological
  models for boundary representation a comparison
  with n-dimensional generalised maps.
  Computer-aided design, 23(1)59-82, 1991
• Technical Reports
• http//www.prip.tuwien.ac.at/