iastro/idha workshop strasbourg observatory 28-29 november 2002 vito di gesù, giosuè lo bosco dma...
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Vito Di Gesù, Giosuè Lo Bosco
DMA – University of Palermo, ITALY
THE COST-TIST 283
Image Segmentation based on Genetic Algorithms Combination
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Introduction
- Supervised Global Segmentation (SGS)
- Unsupervised Tree Segmentation (UTS)
The image segmentation problem as a GOP
(Global Optimization Problem)
Combined strategies
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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..
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Combined Genetic Segmentation (CGS)
Unsupervised Tree Segmentation
Supervised Global Segmentation
Maximal Connected Components
Relaxation procedure
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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 :
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Supervised Global Segmentation (SGS)
P={p1, p2,...,pk} partition
x
y
z 'x
'y
'z
't
t
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Unsupervised Tree Segmentation (UTS)
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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.
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
G.A. terminology
Population: set of individuals named chromosome
Chromosome: sequence of genes.
AABDCCDA
ABCD
Code symbols
Coded information
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Crossover operatorwith probability q
CCDAAADDDCDDAADD
DCDDAABDCCDAAABD
||
||
Cut point random
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Mutation operator
Binary alphabet: 0110
with probability p q
AABDDCDA
AABDCCDA
Mutation point random
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Clonation
To strength the survival of parents features in the chromosome population
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
cromF
cromF
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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.
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Genetic Solution Halting Condition :
tt VarVar 1
total variance
K
itt iVar
1
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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.
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Convergence of CGS
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Maximal connected component (MCC)
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Experimental result on syntetic images
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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)
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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.
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
CGS
C-means Single-link
GPS
Human
Corel Image Database
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
CGS
C-means Single-link
GPS
Human
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Range images
CGS YAR
http://marathon.csee.usf.edu/
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
CGS YAR
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Images from astronomy
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
Images from astronomy
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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.
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iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002
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