Vito Di Gesù, Giosuè Lo Bosco DMA – University of Palermo, ITALY [email protected]

iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

Vito Di Gesù, Giosuè Lo Bosco

DMA – University of Palermo, ITALY

[email protected]

THE COST-TIST 283

Image Segmentation based on Genetic Algorithms Combination


Introduction

- Supervised Global Segmentation (SGS)

- Unsupervised Tree Segmentation (UTS)

The image segmentation problem as a GOP

(Global Optimization Problem)

Combined strategies


Related works

•Shi, Malik, Normalized Cuts and Image Segmentation, 2000.

•V.Di Gesù A Clustering Approach to Texture Classification, 1988.

•Jain and Flynn, Image Segmentation Using Clustering, 1996,

•Ridder, Kittler, Lemmers, an Duin. The adaptive subspace map for texture segmentation, 2000..


Combined Genetic Segmentation (CGS)

Unsupervised Tree Segmentation

Supervised Global Segmentation

Maximal Connected Components

Relaxation procedure


Graphs and perception

G <X ,>

is a distance (similarity) function:

x

x

i+1

m-1 x

1

x i

0 x

1

i

m-1

(x,y)

x

y

RX X :


Image Segmentation and

Graph Partitioning Problem Input: A (weighted) graph G=(V,E Integers j, k, and m. Problem: Partition the vertices into m subsets such that each subset has size at most j, while the cost of the edges spanning subsets is bounded by k.

a

b

c

d

e

3

1

4

6

2

5

211

2

k

m


Bipartition,, EVG XBABA partition

The optimal bi-partition is the one that minimize( similarity function) or maximize ( distance function)

Problem: disjoin A and B removing edges connecting the two parts. The cut of A and B is defined:

A B

y,xB,ABy,Ax

cut


A weighted graph G is associated to the image X

A pixel x X is represented with (ix, jx , gx )

1,0,0

,,,

babawhere

gxdbggdayx yEyxg

xCz yz

yz

ryE

yx

yx

yxg

rgg

gg

xCgxd

gg

ggggd

),max(

1,

),max(,

yxyxXyx ,~, 0


Supervised Global Segmentation (SGS)

P={p1, p2,...,pk} partition

x

y

z 'x

'y

'z

't

t


SGS Algorithm Procedure SGS (X,Kmax) choose at random pk, k=1,2 …. , Kmax classes; repeat for x X ,,1 , ,δ min,δ maxKiigxkgx pp if then

update (k,k)

compute

until (F reaches the minimum)

assign (x,pk)

end

max

1

2

K

k

kF

Genetic computation

Fitness function

Optimization


Unsupervised Tree Segmentation (UTS)


UTS Algorithm

Procedure UTS (A) if not(uniform (A)) then (Al , Ar) SGS(A,2); UTS(Al ); UTS(Ar ); else return (A); end

The function uniform(A) returns the growing condition and it is based un a uniformity test.


G.A. terminology

Population: set of individuals named chromosome

Chromosome: sequence of genes.

AABDCCDA

ABCD

Code symbols

Coded information


Crossover operatorwith probability q

CCDAAADDDCDDAADD

DCDDAABDCCDAAABD

||

||

Cut point random


Mutation operator

Binary alphabet: 0110

with probability p q

AABDDCDA

AABDCCDA

Mutation point random


Clonation

To strength the survival of parents features in the chromosome population


Evolutionary computing (EC)

EC are optimization procedures in the space o events

Fitness function RPopulationF :

The fitness function depends on the problem to be solved

The goal of EC is to maximize the fitness function


cromF

cromF


Genetic Solution for the segmentation problem

GA Data coding : The generic pixel x is coded by a 32 bit binary string that codes the pixel-label, x in the 8 less significant bits and the pixel position (ix,jx) in the 24 most significant bits. Here, x identifies the cluster to which the pixel belongs.

xxx kmn

kSandK

L

1212 248

kx=ix*m+jx and K is the maximum number of clusters.


Genetic Solution

Fitness Function : the inverse function of L and S, L-

1(and S-1(return the label L-1(of a pixel in position (i,j)= S-1(The fitness function f is defined on the basis of the similarity function computed between a given chromosome a and the corresponding segment P

)(

11),()(

L

mvSf

j

P

jP

SXmv j

)( 1


Genetic Solution

Genetic operator : the application of the classical single point crossover and the bit mutation.

Selection process :

}),....,(),({ ttttP mn

rrmn wherettttP }),....,(),({

otherwise f if fthatsuch

tP

rrrr

r

mn

ααββγ

}γ,....,γ,γ{1 21


Genetic Solution Halting Condition :

tt VarVar 1

total variance

K

itt iVar

1


Genetic Algorithm 1. (Input) - Read image X of size nxm;

2. (Initial condition) - Set up a population of chromosomes

}0),....,0(),0({0 )(ααα 21 mnP

and assign at random a label to each i(0);

3. (Genetic process) - Apply the genetic operators (sinlge point crossover and bit mutation) to current population P(t);

4. (Selection process) - Build population P(t+1) choosing by selecting the best chromosome from P(t) and ( P(t));

5. (Set iteration) - t t + 1;

6. (Halting condition) – if |Vart-1- Vart| goto 3; else stop.


Convergence of CGS


Maximal connected component (MCC)


Experimental result on syntetic images


Experimental result and comparison

The results of the application of the CGS on real data is compared with three methods :

•C-means (Bezdek, 1981)

•Single-Link (EPRI, 1999)

•Graph partition segmentation (Malik, 2000)


Normalized cut criterionShi, Malik 1999

y,xV,B

y,xV,A

V,BB,A

V,AB,A

B,A

Vy,Bx

Vy,Ax

assoc

assoc

assoccut

assoccut

Ncut

B,AB,A

V,BB,B

V,AA,A

B,A

assoccut

assoc

N2N

assocassoc

assocassoc

N

Min-cut procedure


Normalized cut criterion

A partition of the image into regions such that there is high similarity within a region and low similarity across regions is obtained by solving a generalized eigenvalue problem.

Minimizing normalized CUT is NP-Complete even for graph on grid (Papadimitriou 1999)

The resulting eigenvectors provide a hierarchical partitioning of the image into regions ordered according to salience.

Brightness, color, texture, motion similarity, proximity and good continuation can all be encoded into this framework.


1. From an image X built G=(V,E2. Solve (D-W)x=Dxx for eigenvectors with the smallest

eigenvalues.3. Use the eigenvector with the second smallest eigenvalue

to bipartition the graph.4. Decide if the current partition should be subdivided and

recursively repartition the segmented parts if necessary.

The grouping algorithm

1. G is only locally connected the eigensystem is sparse2. Only the top few eigenvectors are needed.3. The precision requirement is low Lanczos method

Time complexity 3nO n=number of nodes

Time complexity nmMOmnO where: m=maximum number of matrix-vector computationsM(n)= the cost of a matrix-vector computations


Experimental result and comparison

Corel Image Database

http://elib.cs.berkeley.edu/photos/corel

Range Images

http://marathon.csee.usf.edu/range/DataBase.html

Astronomical images

Miscellanea


CGS

C-means Single-link

GPS

Human

Corel Image Database


CGS

C-means Single-link

GPS

Human


Range images

CGS YAR

http://marathon.csee.usf.edu/


CGS YAR


Images from astronomy


Evaluation of CGS

The comparison has been performed between the automatic segmentation and the segmentation deriving from the evaluation of an odd number (5) of persons.

SSeg

agr

K k

kK

k ,max

#1

1

Segk denotes the k-th segment retrieved by humans

S denotes the k-th segment retrieved by the machine

| Segk | and |S| denote the corresponding size

#agrk is the largest pixel intersection between Segk and S.


Comparision

The CPU times are referred to an INTEL PENTIUM III 1GHz.

Method Cpu Time(seconds)

CGS 0,74 154C-means 0,7 97

Single-link 0,77 220GPS 0,73 300

Vito Di Gesù, Giosuè Lo Bosco DMA – University of Palermo, ITALY [email protected]

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