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Data Structures and Image Segmentation

Data Structures and Image Segmentation. Luc Brun L.E.R.I., Reims University, France and Walter Kropatsch Vienna Univ. of Technology, Austria. Segmentation. Segmentation: Partition of the image into homogeneous connected components. S 1. S 2. S 5. S 4. S 3. Segmentation. Problems

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Data Structures and Image Segmentation

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  1. Data Structures and Image Segmentation Luc Brun L.E.R.I., Reims University, France and Walter Kropatsch Vienna Univ. of Technology, Austria

  2. Segmentation • Segmentation: Partition of the image into homogeneous connected components S1 S2 S5 S4 S3

  3. Segmentation • Problems • Huge amount of data • Homogeneity: • Resolution/Context dependent • Needs • Massive parallelism • Hierarchy

  4. Contents of the talk • Hierarchical Data Structures • Combinatorial Maps • Combinatorial Pyramids

  5. Regular Pyramids

  6. Matrix-Pyramids • Stack of images with progressively reduced resolution 2x2/4 Pyramid Level 2 Level 3 Level 1 Level 0

  7. M-Pyramids • M-Pyramid NxN/q (Here 2x2/4) • NxN : Reduction window. Pixels used to compute father’s value (usually low pass filter) • q : Reduction factor. Ratio between the size of two successive images. • Receptive field: set of children in the base level

  8. M-Pyramids • NxN/q=1: Non overlapping pyramid without hole (eg. 2x2/4) • NxN/q<1: Holed Pyramid. • NxN/q>1: Overlapping pyramid

  9. Regular Pyramids • Advantages(Bister) • makes the processes independent of the resolution…. • Drawbacks(Bister) : Rigidity • Regular Grid • Fixed reduction window • Fixed decimation ratio

  10. Irregular Pyramids • Stack of successively reduced graphs

  11. Irregular Pyramids [Mee89,MMR91,JM92] • From G=(V,E) to the reduced graph G’=(V’,E’) • Selection of a set of surviving vertices V’V • Child Parent link  Partition of V • Definition of E’ • Selection of Roots

  12. Stochastic Pyramids • V’ : Maximum Independent Set • maximum of • a random variable • [Mee89,MMR91] • a criteria of interest • [JM92]

  13. 1 5 8 10 9 6 20 6 15 3 11 9 21 13 10 7 Stochastic Pyramids • Child-Parent link : • maximum of • a random variable • [Mee89,MMR91] • a similarity measure • [JM92]

  14. Stochastic Pyramids • Selection of surviving edges E’ • Two father are joint if they have adjacent children

  15. Stochastic Pyramids • Selection of Roots: • Restriction of the decimation process by a class membership function [MMR91] • contrast measure with legitimate father exceed a threshold [JM92]

  16. Stochastic Pyramid [MMR91] • Restriction of the decimation process : Class membership function

  17. Stochastic Pyramids • Advantages • Purely local Processes [Mee89] • Each root corresponds to a connected region[MMR91] • Drawback • Rough description of the partition

  18. Given an edge to be contracted Identify both vertices Remove the contracted edge Formal DefinitionsEdge Contraction

  19. Formal DefinitionDual Graphs • Two graphs encoding relationships between regions and segments

  20. Formal DefinitionDual Graphs • Two graphs encoding relationships between regions and segments

  21. Dual Graphs • Advantages (Kropatsch)[Kro96] • Encode features of both vertices and faces • Drawbacks [BK00] • Requires to store and to update two data structures • Contraction in G  Removal in G • Removal in G  Contraction in G

  22. Decimation parameter • Given G=(V,E), a decimation parameter (S,N) is defined by (Kropatsch)[WK94]: • a set of surviving vertices SV • a set of non surviving edges NE • Every non surviving vertex is connected to a surviving one in a unique way:

  23. Example of Decimation : S :N

  24. Decimation parameters • Characterisation of non relevant edges(1/2) d°f = 2

  25. Decimation parameters • Characterisation of non relevant edges(2/2) d°f = 1

  26. Decimation Parameter • Dual face contraction • remove all faces with a degree less than 3

  27. Decimation Parameter • Edge contraction: Decimation parameter (S,N) • Contractions in G • Removals in G • Dual face contraction : Dual Decimation parameter • Contractions in G • Removals in G

  28. Decimation parameter Characterisation of redundant edges requires the dual graph Dual graph data structure (G,G)

  29. Decimation Parameter • Advantages • Better description of the partition • Drawbacks • Low decimation Ratio

  30. Contraction Kernels Given G=(V,E), a Contraction kernel (S,N) is defined by: • a set of surviving vertices SV • a set of non surviving edges NE Such that: • (V,N) is a forest of (V,E) • Surviving vertices S are the roots of the trees

  31. Contraction kernels • Successive decimation parameters form a contraction kernel

  32. Example of Contraction Kernel , , : S :N

  33. Example of Contraction kernel • Removal of redundant edges: Dual contraction kernel

  34. Hierarchical Data Structures / Combinatorial Maps • M-Pyramids • Overlapping Pyramids • Stochastic Pyramids • Adaptive Pyramids • Decimation parameter • Contraction kernel

  35. - : edge encoding -2 -1 -6 6 -5 -4 5 4 -3 3 2 1 Combinatorial Maps Definition • G=(V,E)  G=(D,,) • decompose each edge into two half-edges(darts): D ={-6,…,-1,1,…,6}

  36. Combinatorial Maps Definition • G=(D,,) •  : vertex encoding -2 -1 -6 6 -5 -4 *(1)=(1, *(1)=(1,3 *(1)=(1,3,2) 5 4 -3 3 2 1

  37. Combinatorial MapsProperties • Computation of the dual graph : • G=(D,,)  G=(D, = , ) • The order defined on  induces an order on  1 -2 3 -1 2 -3 5 -5 -4 4 6 -6 *(-1)=(-1,3 *(-1)=(-1,3,4 *(-1)=(-1, *(-1)=(-1,3,4,6)

  38. Combinatorial MapsProperties • Computation of the dual graph : • G=(D,,)  G=(D, = , ) -2 -1 -6 6 -5 -4 *(-1)=(-1,3,4 *(-1)=(-1,3 *(-1)=(-1, *(-1)=(-1,3,4,6) 5 4 -3 3 2 1

  39. Combinatorial MapsProperties • Summary • The darts are ordered around each vertex and face • The boundary of each face is ordered  • The set of regions which surround an other one is ordered • The dual graph may be implicitly encoded • Combinatorial maps may be extended to higher dimensions (Lienhardt)[Lie89]

  40. Combinatorial Maps/Combinatorial Pyramids • Combinatorial Maps • Computation of Dual Graphs • Combinatorial Maps properties • Discrete Maps [Bru99] http://www.univ-st-etienne.fr/iupvis/color/Ecole-Ete/Brun.ppt

  41. Removal operation • G=(D,,) • dD such that d is not a bridge • G’=G\ *(d)=(D’, ’,) d -d

  42. -2 -1 -6 6 -5 -4 5 4 -3 3 2 1 Removal Operation • Example -2 -1 -6 6 -4 d=5 4 -3 3 2 1

  43. Contraction operations • G=(D,,) • dD such that d is not a self-loop • G’=G/*(d)=(D’, ’,) d -d

  44. Contraction operations • Preservation of the orientation d d 1 1 c c 2 2 b 3 3 b 4 4 a a

  45. -2 -2 -1 -6 3 6 -1 -4 2 -3 -3 4 3 4 -4 2 1 6 -6 -2 3 -1 2 -3 5 -5 -4 4 6 -6 Basic operationsImportant Property • The dual graph is implicitly updated -2 -1 -6 6 -5 -4 d=5 4 removal 5 -3 3 2 1 1 d=5 contraction

  46. Contraction Kernel • Given G=(D,,), KD is a contraction kernel iff: • K is a forest of G • Symmetric set of darts ((K)=K) • Each connected component is a tree • Some surviving darts must remain • SD=D-K

  47. Contraction Kernel • Example K= 1 2 3 13 14 15 16 4 5 6 17 18 19 20 7 8 9 21 22 23 24 10 11 12

  48. 1 2 3 13 14 15 16 4 5 6 17 18 19 20 7 8 9 21 22 23 24 10 11 12 Contraction Kernel • Example K=

  49. 1 -1 2 -2 2 -2 13 13 14 15 15 4 14 4 Contraction Kernel • How to compute the contracted combinatorial map ? • What is the value of ’(-2) ?

  50. 17 2 -2 15 4 14 7 Contraction Kernel • How to compute the contracted combinatorial map ? • What is the value of ’(-2) ? 1 -1 2 -2 13 14 15 -13 4 17 7

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