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 S1 S2 S5 S4 S3

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

Contents of the talk Hierarchical Data Structures Combinatorial Maps Combinatorial Pyramids

Regular Pyramids

Matrix-Pyramids Stack of images with progressively reduced resolution Level 2 Level 3 Level 1 Level 0

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

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

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

Irregular Pyramids Stack of successively reduced graphs

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

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

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

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

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]

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

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

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

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

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

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

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:

Example of Decimation : S :N

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

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

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

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

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

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

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

Contraction kernels Successive decimation parameters form a contraction kernel

Example of Contraction Kernel , , : S :N

Example of Contraction kernel Removal of redundant edges: Dual contraction kernel

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

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

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

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

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

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]

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

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

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

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

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

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

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

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

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

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

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

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.

Contraction Kernel Connecting Walk CW(-2)=-2. -1. 13. 17. 21. 10 1 -1 14 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

Contraction Kernel CW(-2)=-2,-1,13,17,21,10 1 -1 2 -2 3 -2 3 2 13 14 15 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

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

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

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)

 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