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

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. Formal Definitions Edge Contraction
Given an edge to be contracted
Identify both vertices
Remove the contracted edge

19. Formal Definition Dual Graphs
Two graphs encoding relationships between regions and segments

20. Formal Definition Dual 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

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

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

37. Combinatorial Maps Properties
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
*(-1)=(-1,
*(-1)=(-1,3
*(-1)=(-1,3,4,6)
*(-1)=(-1,3,4 -6

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

39. Combinatorial Maps Properties
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
Computation of Dual Graphs
Combinatorial Maps properties
Discrete Maps
[Bru99]

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

42. Removal Operation Example -2 -1 -1 3 -3 4 -4 5 -5 -2 6 -6 1 2 -6 6 -4
d=5 -3 4 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 1 d 1 c c 2 2 b 3 3 b 4 4 a a

45. Basic operations Important Property
The dual graph is implicitly updated -1 3 -3 4 -4 -2 6 -6 1 2 -2 -1 -6 -5 6 -4
d=5 4 5
removal -3 3 2 1 5 -5 -4 -3 -6 6 2 -2 -1 4 3 1 -4 -3 -6 6 2 -2 -1 4 3
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 7 821 22 23 24 10 11 12

48. Contraction Kernel Example K= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1617 18 19 20 21 22 23 24

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

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

51. Contraction Kernel Connecting Walk Given G=(D,, ) , KD and SD=D-K
If d SD, CW(d) is the minimal sequence of non surviving darts between d and a surviving one.
The connecting walks connect the surviving darts.

52. Contraction Kernel Connecting Walk CW(-2)=-2. -1. 13. 17. 21. 10 1 -114 15 16 4 5 6
CW(-2)=-2.
-1.
13.
17.
21. 10 17 18 19 20 7 8 9 21 22 23 24 10 11 12

53. Contraction Kernel CW(-2)=-2,-1,13,17,21,10 1 -1 2 -2 3 -2 3 2 13 1415 16 14 15 16 4 5 6 5 6 4 17 18 19 20 18 19 20 7 8 9 7 8 9 22 21 22 23 24 23 24 10 11
-11 12 12 11
-11

54. Contraction Kernel Construction of the contracted combinatorial map.
For each d in SD
compute d’: last dart of CW(-d)
’(-d)=(d’)’(d)=’(-d) = (d’) 3 11 2 5 6 8 9 12 15 16 19 20 23 24 -2 18 14 22 7 4

55. Extensions (1/2) Dual contraction kernel Replace  by 
Successive Contraction kernels with a same type
Concatenation of connecting walks

56. Extensions (2/2) Successive contraction kernels with different types
From connecting walks to Connecting Dart Sequences
Label Pyramids:
for each dart encode
Its maximum level in the pyramid (life time)
How its disappear (contracted or removed)

57.  Combinatorial Pyramids
Conclusion
Graphs encode efficiently topological features. Combinatorial maps:
Encode the orientation
Provide an implicit encoding of the dual
May be generalised to higher dimension
Irregular Pyramids overcome the limitations of their regular ancestors
 Combinatorial Pyramids