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Vito Di Gesù, Giosuè Lo Bosco DMA – University of Palermo, ITALY digesu@math.unipa.it

THE COST-TIST 283  Image Segmentation based on Genetic Algorithms Combination. Vito Di Gesù, Giosuè Lo Bosco DMA – University of Palermo, ITALY digesu@math.unipa.it. Introduction. The image segmentation problem as a GOP (Global Optimization Problem). Combined strategies.

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Vito Di Gesù, Giosuè Lo Bosco DMA – University of Palermo, ITALY digesu@math.unipa.it

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  1. THE COST-TIST 283  Image Segmentation based on Genetic Algorithms Combination Vito Di Gesù, Giosuè Lo Bosco DMA – University of Palermo, ITALY digesu@math.unipa.it iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  2. Introduction The image segmentation problem as a GOP (Global Optimization Problem) Combined strategies - Supervised Global Segmentation (SGS) - Unsupervised Tree Segmentation (UTS) iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  3. Related works • 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.. • Shi, Malik, Normalized Cuts and Image Segmentation, 2000. iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  4. Relaxation procedure Maximal Connected Components Supervised Global Segmentation Unsupervised Tree Segmentation Combined GeneticSegmentation (CGS) iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  5. x (x,y) y Graphs and perception G  <X , d> dis a distance (similarity) function: iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  6. b a d c e Image Segmentation and Graph Partitioning Problem Input: A (weighted) graph G=(V,E,d). 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. iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  7. partition Problem: disjoin A and B removing edges connecting the two parts. The cut of A and B is defined: A B Bipartition The optimal bi-partition is the one that minimize (d similarity function) or maximize ( distance function) iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  8. A weighted graph G is associated to the image X A pixelx Î Xis represented with (ix, jx , gx ) iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  9. Supervised Global Segmentation (SGS) P={p1, p2,...,pk} partition iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  10. then if Genetic computation Fitness function Optimization SGS Algorithm ProcedureSGS (X,Kmax) choose at random pk, k=1,2 …. , Kmax classes; repeat forxÎ X assign (x,pk) update (mk,sk) end compute until (F reaches the minimum) iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  11. Unsupervised Tree Segmentation (UTS) iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  12. 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. iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  13. Code symbols A B C D Coded information G.A. terminology Population: set of individuals named chromosome Chromosome: sequence of genes. iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  14. Cut point random Crossover operator with probability q iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  15. Mutation point random Binary alphabet: Mutation operator with probability p  q iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  16. Clonation To strength the survival of parents features in the chromosome population iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  17. Fitness function Evolutionary computing (EC) ECare optimization procedures in the space o events The fitness function depends on the problem to be solved The goal of EC is to maximize the fitness function iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  18. iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  19. 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,lxin the 8 less significant bits and the pixel position (ix,jx) in the 24 most significant bits. Here,lxidentifies the cluster to which the pixel belongs. kx=ix*m+jxand K is the maximum number of clusters. iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  20. Genetic Solution Fitness Function : the inverse function of L and S,L-1(a) andS-1(a) return the labell=L-1(a) of a pixel in position (i,j)= S-1(a). The fitness functionf is defined on the basis of the similarity function computed between a given chromosome a and the corresponding segmentPl iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  21. Genetic Solution Genetic operator G :the application of the classical single point crossover and the bit mutation. Selection process : iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  22. Genetic Solution Halting Condition : total variance iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  23. Genetic Algorithm 1. (Input) - Read image X of size nxm; 2.(Initial condition) - Set up a population of chromosomes and assign at random a label to eachai(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 G( P(t)); 5. (Set iteration) - t¬t + 1; 6.(Halting condition)– if|Vart-1- Vart|> fgoto 3; else stop. iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  24. Convergence of CGS iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  25. Maximal connected component (MCC) iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  26. Experimental result on syntetic images iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  27. 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) iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  28. Min-cut procedure Normalized cut criterionShi, Malik 1999 iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  29. Normalized cut criterion Minimizing normalized CUT is NP-Complete even for graph on grid (Papadimitriou 1999) 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. 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. iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  30. Time complexity n=number of nodes Time complexity where: m=maximum number of matrix-vector computations M(n)= the cost of a matrix-vector computations The grouping algorithm • From an image X built G=(V,E,d) • Solve (D-W)x=Dxfor eigenvectors with the smallest eigenvalues. • Use the eigenvector with the second smallest eigenvalue to bipartition the graph. • Decide if the current partition should be subdivided and recursively repartition the segmented parts if necessary. • G is only locally connected  the eigensystem is sparse • Only the top few eigenvectors are needed. • The precision requirement is low  Lanczos method iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  31. 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 iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  32. Corel Image Database CGS GPS Human C-means Single-link iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  33. CGS GPS Human Single-link C-means iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  34. CGS YAR Range images http://marathon.csee.usf.edu/ iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  35. CGS YAR iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  36. Images from astronomy iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  37. Images from astronomy iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  38. 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. 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. iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

  39. Comparision The CPU times are referred to an INTEL PENTIUM III 1GHz. iAstro/IDHA Workshop Strasbourg Observatory 28-29 November 2002

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