Internatonal Journal of Scentfc & Engneerng Research Volume 3, Issue, May-2012 1 An Optmal teratve Mnmal Spannng tree Clusterng Algorthm for mages S. Senthl, A. Sathya, Dr.R.Davd Chandrakumar Abstract:-Lmted Spatal resoluton, poor contrast, overlappng ntenstes, nose and ntensty n homogenetes varaton make the assgnment of segmentaton of medcal mages s greatly dffcult. In recent days, mathematcal algorthm supported automatc segmentaton system plays an mportant role n clusterng of magng. The mnmal spannng tree algorthm s capable of detectng clusterng wth rregular boundares. In ths paper we propose an optmal teratve mnmal spannng tree clusterng algorthm (OPIMSTCA).At each herarchcal level, t optmzes the number of cluster, from whch the proper herarchcal structure of underlyng data set can be found. The algorthm uses a new cluster valdaton crteron based on the geometrc property of data partton of the data set n order to fnd the proper number of clusters at each level. The center and standard devaton of the cluster are computed to fnd the tghtness of the ndvdual clusters. In ths paper we compute tghtness of clusters, whch reflects good measure of the effcacy of clusterng. The algorthm works n two phases. The frst phase of the algorthm produces sub trees. The second phase creates objectve functon usng optmal number of clusters. The performance of proposed method has been shown wth random data and then the new Cluster separaton approach to optmal number of clusterng. The expermental results demonstrate that our proposed method s a promsng technque for effectve optmal clusters. Key words: Eucldean mnmum spannng tree, clusterng, eccentrcty, center herarchcal clusterng, sub tree, standard devaton, cluster separaton. 1. INTRODUCTION Cluster analyss s playng an mportant role n solvng many problems n medcal feld, psychology, bology, socology, pattern recognton and mage processng. Due to the lmtatons n mage equpments n MRI, the mage has manly three consderable dffcultes: Nose, partal volume and ntensty n-homogenety.also the mage sgnals have hghly affected by shackng of patent s body and patent s moton. So the medcal MRI s serously affected and t has mproper nformaton about the anatomc structure. Hence the segmentaton of medcal mages s an mportant one before t to go for treatment plannng for proper dagnoss. Automated segmentaton methods based on artfcal ntellgence A spannng tree s an acyclc sub graph of a graph G, whch contans all vertces from G. The mnmum spannng tree (MST) of a weghted graph s mnmum weght spannng tree of that graph wth the classcal MST algorthms [2, 3, 4] the cost of constructng a mnmum spannng tree s O (mlogn), where m s the number of edges n the graph and n s the number of vertces. More effcent algorthm for constructng MSTs have also been extensvely researched [, 1 Department of Mathematcs, Vckram College of Engneerng, Enath, Svaganga, Taml nadu, Inda. 2 Department of Mathematcs, Natonal Insttute of Technology Goa, Goa Engneerng college campus, Inda. 3 Department of Mathematcs, Vckram College of Engneerng, Enath, Svaganga, Taml nadu, Inda. http://www.jser.org technques were proposed n (Clark etal.[1]).the mage segmentaton vewed as partton a gven mage nto regons(or) segments such that pxels belongng to a regon are more smlar to each other than pxel belongng to dfferent regons. We also requre that these regons be connected so regons consst of neghborng pxels. Image segmentaton used to partton a gven mage nto a number of regons. So that each regon corresponds to an object(ntensty, color, texture ).Here we address the problem of segmentng a dgtal mage nto a set of dsjont regons such that each regon s composed of nearby pxels wth smlar colors (or) ntenstes (or) spatal locaton. 6, 7].These algorthms promse close to lnear tme complexty under dfferent assumptons. A Eucldean mnmum spannng tree (EMST) s a spannng tree of a set of n ponts n a metrc n space ( E ).where the length of an edge s the Eucldean dstance between a par of ponts n the pont set. MSTs have been used for data Classfcaton n the feld of pattern recognton (8) and mage processng (9, 10, 11).we have also seen some lmted applcatons n bologcal data analyss (12).One popular form these MST applcatons s called the sngle-lnkage cluster analyss (13, 14, 1, 16). Our study on these methods has led us to beleve that all these applcatons have used the MSTs n some heurstc ways;eg.cuttng long edges to separate clusters wthout fully explorng ther power and understandng ther rch propertes related to clusterng. Geometrc noton of centralty are closely lnked to faclty locaton problem. The dstance matrx D
Internatonal Journal of Scentfc & Engneerng Research Volume 3, Issue, May-2012 2 (data set) can Computed rather effcently usng Djkstra s algorthm wth tme complexty O ( V 2 ln V ) (17). The eccentrcty of a vertex n G and radus ( G), respectvely are defned as e(x) max d(x, y) and (G) y V The center of G s the set C (G) {x V / e(x) (G)} mn e(x) x V The length of the longest path n the graph s called dameter of the graph G, D(G) max e(x). x V The dameter set of G s Da (G) {x V / e(x) D(G)}. An mage pxels represents a mode on vertces and an edge reflects par wse smlartes between the pxels. we take a graph-based approach to segmentaton. Let G (V, E) be an undrected graph wth vertces be segmented and ( j v V, the set of elements to v, v ) E correspondng to pars of neghborng vertces, each edge ( v, v j) E has a correspondng weght w(v, v ), whch s a non-negatve j measure of the dssmlarty between neghborng elements v and weghts of the edges are computed by a smlarty v j functon locaton, brghtness and color. Wth ths representaton, the segmentaton task can be solved by mnmum spannng tree clusterng methods. In ths paper, we wll provde n-depth studes for MST based clusterng. our major contrbutons nclude a rgorous formulaton for Optmal teratve mnmal spannng tree clusterng algorthm(opimstca).we beleve t s a good dea to allow users to defne ther desred smlarty wthn a cluster and allow them to have some flexblty to adjust the smlarty f the adjustment s needed. In ths paper we propose optmal teratve mnmal spannng tree clusterng algorthm for mage segmentaton algorthm to address the ssues of undesred clusterng structure and unnecessary large number of clusters. Our algorthm works n two phases. The frst phase construct the Eucldean dstance based MST from the pxels of nput mage data, then creates subtree (cluster/regons) from mnmum spannng tree (MST) by removes the nconsstent edges that satsfy the predefned nconsstence measure. The second phase optmal teratve mnmal spannng tree algorthm, whch produces optmal (or) best number of clusters wth segmentaton. The performance of proposed method has been shown wth random data for mages. Fnally expermental results and concluson we summarze the strength of our methods and possble mprovements. 2. MINIMAL SPANNING TREE-BASED CLUSTERING ALGORITHMS We wll use a MST to represent a set of expresson data and ther sgnfcant nter-data relatonshps to facltate fast rgorous clusterng algorthm. Gven a pont set D n E n,the herarchcal methods stats by constructng a mnmal spannng tree(mst) from the ponts n D.Each edge weght represents the dstance ((or)dssmlarty), u v between u and v,whch could be defned as the Eucldean dstance,so we named ths MST as EMST1.Next the average weght w of the edges n the entre EMST1 and ts standard devaton are computed; any edge wth w w (or)longest edge s e removed from the tree. Ths leads to a set of dsjont subtrees ST {T1,T2,T3...}.Each of these subtrees T s treated as cluster. 3. OUR ALGORITHM,OPTIMAL ITERATIVE MINIMAL SPANNING TREE CLUSTERING ALGORITHM The MST T nto k subtrees objectve functon s gven by k J (U) v c, 1 v T k { T } 1 to optmze a more general where c s the center of T, 1,2... k. that s to optmze the k-clusterng so that the total dstance between the center of each cluster and ts data ponts s mnmzed-objectve functon for data clusterng. The centers of clusters are dentfed usng eccentrcty of ponts. These ponts are a representatve pont for the each cluster (or) subtrees. A pont c s assgned to a cluster f c T, 1,2...k.The group of center ponts c1, c2... ck are connected and agan mnmum spannng tree EMST2 s constructed. To each T calculate standard devaton, dstance between the pont of T and clusters center c.thus the problem of fndng the optmal number of clusters of a data set can be transformed nto problem of fndng the proper regon (clusters) of the data set. Here we use the MST as a crteron to test the nter cluster property based on ths observaton, we use a cluster valdaton crteron, called cluster separaton (CS) n OPIMST clusterng algorthm. Cluster separaton :(CS) s defned as the rato between mnmum and maxmum standard devaton of clusters (subtrees), http://www.jser.org
Internatonal Journal of Scentfc & Engneerng Research Volume 3, Issue, May-2012 3 where CS clusters and, 1,2... k, s the maxmum value of standard devaton of s the mnmum value of standard devaton clusters. Then the CS represents the relatve separaton of centrods.the value of CS ranges from 0 to 1.A low value of CS means that the two centrods are too close to each other and the correspondng partton s not vald. A hgh CS value means the parttons of the data s even and vald. In practce, we predefne a threshold to test the CS.If the CS s greater than the threshold; the partton of the data set s vald. Then agan parttons the data set by creatng subtree (cluster/regon).ths process contnuous untl the CS s smaller than the threshold The CS crteron fnds the proper bnary relatonshp among clusters n the data space. The value settng of the threshold for the CS wll be practcal and s dependent on the dataset. The hgh the value of the threshold the smaller the number of clusters would be Generally, the value of the threshold wll be 0.8 [18].The gven clusters the CS value 0. 8 and our OPIMST algorthm processng the results, the proper number of clusters/regons for the data set (pxel) s 2.Further more, the computatonal cost of CS s much lghter because the number of sub clusters s small. The created clusters/regons are well separated. Algorthm: OPIMSTCA Input : Image data (pxel value) Output : optmal number of clusters Let e1 be an edge n the EMST1 constructed from mage data. Let e2 be an edge n the EMST2 constructed form C. Let w be the weght of e1. e Let be the standard devaton of the edge weghts n EMST1. Let ST be the set of dsjont subtrees of EMST1. 1. Create a node v, for each pxel of an mage I. 2. Compute the edge weght usng Eucldean dstance from mage data. 3. Construct an EMST1 from 2 4. Compute the average weght of w of all the edges from EMST1.. Compute the standard devaton of the edges from EMST1. 6. S, n 1, C T c. 7. Repeat. 8. For each e 1 EMST 1. 9. If (w w ) (or) current longest edge e remove e1 10. e from EMST1. T T ' S S {T }// T s new dsjont subtrees (regons). 11. n n 1 c c. ' 12. Compute the center c of T usng eccentrcty of ponts. 13. C S {c }. T T 14. Construct an EMST2 T from C. 1. Compute the standard devaton ( T ). 16. ( T ) =get-mn standard devaton. 17. ( T ) =get-max standard devaton. (T ) 18. CS=, 1,2... k (T ) 19. Untl CS 0. 8. 20. Merge the closest neghbor from EMST2. 21. Update the clusters ponts, repeat step 12 to step 20. 22. The followng optmze the k-clusterng objectve functon mnmzed, termnaton crteron s satsfed J t (U) J t 1(U), where t s the teraton count and s a thresholdng value les between 0 and 1. 4. EXPERIMENTAL RESULTS Ths secton descrbes some expermental results on random data to the segmentaton Performance of proposed method, TABLE1: RANDOM DATA http://www.jser.org
Internatonal Journal of Scentfc & Engneerng Research Volume 3, Issue, May-2012 4 TABLE2: DISSIMILARITY MATRIX Table3: MINIMUM SPANNING TREE EDGES from the EMST1,vertces(data ponts) n the EMST1 parttoned nto four sets (four subtrees (or) clusters) T,T,T and namely 1 2 3 T4 T 1 {1,2,3,4,,6,7,8,9,10,11,12,13,18,20}, T 2 {14,1,16,17}, T 3 {19} and {20} s show n the fgure 2.center pont T 4 (vertex) for each of the each subtree s fnd usng eccentrcty of ponts (vertces).these center pont (or) vertex s connected and agan another mnmum spannng tree EMST2(fgure3) s constructed. To calculate the standard devaton each subtree usng center. The maxmum value of standard devaton of clusters and the mnmum value of standard devaton of clusters, s fnd to compute cluster separaton value. If the CS s greater than 0.8, then to remove the mnmum edge weght of EMST2.To update the clusters(subtree) vertces then to compute the center usng eccentrcty ponts(vertces).fnally the optmal teratve mnmal spannng tree clusterng algorthm produce, the optmal number of clusters 2. T 4 12 8 7 13 10 1 12 11 8 6 7 13 3 10 1 2 11 4 9 18 6 3 2 9 18 T 2 4 17 16 Fgure1: Clusters connected through ponts-emst1 14 Our OPIMSTCA constructs EMST1 from the dssmlarty matrx s shown n the fgure1.the mean w and standard devaton of the edges from the EMST1 are computed respectvely as 2.38 and 4.66.The sum of the mean w and standard devaton s computed as 7.04.Ths value s used to dentfy the nconsstence edges n the EMST1 to generate subtrees(clusters).based on the above value the edge havng weght 8.06,9.21 and 10.63.By removng nconsstent edges 1 T 3 T4 19 20 http://www.jser.org
Internatonal Journal of Scentfc & Engneerng Research Volume 3, Issue, May-2012 Fgure2: Clusters connected through ponts- (number of clusters 4) EMST1 Fgure3:EMST2 usng four clusters center ponts. Center ponts:,14,19,20 Our algorthm OPIMST clusterng algorthm, processng the optmal number of clusters Two. 14 16 1 19 19 17 13 3 10 14 Fgure: 4 OPIMST clusterng algorthm, center and16, 2.94and 3. 60 CS 0.82 0. 8. CONCLUSION 1 In ths paper proposed an optmal teratve MST clusterng algorthm and appled to random data segmentaton. The algorthm was formulated by ntroducng Eucldean dstance functon, eccentrcty, and standard devaton of each cluster, wth basc Objectve functon of the optmal teratve mnmal spannng tree clusterng algorthm to have proper effectve segmentaton n mage. Our algorthm uses new cluster valdaton crteron based on the geometrc property of parttoned clusters to produce optmal number of true clusters. The test result shows that the proposed method outperformed the base lne methods. In future we wll explore and test our proposed clusterng algorthm n varous domans and ths work hopes that the proposed method can also be used to mprove the performance of other clusterng algorthm based on Eucldean dstance functons. References: [1].Clark,M.C.,Hall,L.O.,Goldgof,D.B.,Velthuzen,R. Murtagh,F.R.,Slbger,M.S:- Automatc tumorsegmentaton usng knowledge-based technque. IEEE Transactons on Medcal Imagng 117,187-201(1998). [2].Prm.R. Shortest connecton networks and some 12 2 20 11 4 20 9 8 18 6 7 generalzaton, Bell systems techncal journal 36:1389-1401(197). [3].Kruskal.J. On the Shortest spannng subtree and the travellng salesman problem In proceedngs of the Amercan Mathematcal Socety, Pages 48-0(196). [4].Nesetrl.J, Mlkova.E and Nesetrlova.H.otakar boruvka On mnmal Spannng tree problem: Translaton of both the 1926 papers, comments, hstory.dmath.dscrete Mathematcs, 233(2001). [].Karger.D, Klen.P and Tarjan.R A randomzed lnear-tme algorthm to fnd mnmum spannng tree, Journal of the ACM, 42(2):321-328(199). [6].Fredman.M and Wllard.D Trans-dchotomous algorthms for mnmum spannng trees and shortest paths,in proceedngs of the 31 st annual IEEE symposum on Foundatons of computer scence,pages 719-72(1990). [7].Gabow.h,Spencer.T and Rarjan.R, Effcent algorthms for fndng mnmal spannng trees n undrected and drected graphs, Combnatorca 6(2):109-122(1986). [8].Duda.R.O and hart.p.e. pattern classfcaton and scene analyss wley-nter Scence, New York (1973). [9].Gonzalez.R.C and wntz.p Dgtal mage processng, 2nd edn, Addson-wesley., Readng MA (1987). [10].Xu, Y.Olman.V and Uberbacher.E. A segmentaton algorthm for nosy mages; desgn and evaluaton, patt.recogn.lett19, 1213-1224 C (1998). [11].Xu.y and Uberbacher.E. 2D mage segmentaton usng mnmum spannng trees,mage Vs.comput 1, 47-7(1997). [12].States.D.JHarrs, N.L.and Hunter, Computatonally effcent cluster representaton n molecula Sequence megaclassfcaton, Ismb, 1,387-394(1993). [13].Gower J.C and Ross.G.J.S mnmum spannng trees and sngle lnkage analyss, Appl.stat.18, 4-64(1969). [14].Aho.A.V, Hopcroff.J.E and Ullman.J.D, The Desgn and Analyss of computer algorthms, Addson- wesley, Readng MA (1974). [1].A.k and Dubes.R. Algorthms for clusterng Data, prentce hall, New Jersey (1988). http://www.jser.org
Internatonal Journal of Scentfc & Engneerng Research Volume 3, Issue, May-2012 6 [16].Mrkn.B, Mathematcal classfcaton and clusterng DIMACS, Rutgers Unversty, Pscataway.Nj (1996). [7].Stefan wuchty and peter.f.stadler, Centers of complex networks (2006). [18].FengLuo,Latfur kahn,farokh Bastan,T-lng yen and Jzhong zhon, A dynamcally growng selforganzng tree(dgost)for herarchcal gene expresson profle,bo nformatcs,vol20,no 16,PP260-2617,(2004). Authors: 1. S.Senthl s workng as Assstant professor n Mathematcs, Vckram College of Engneerng, Enath, Svaganga.He earned hs M.Sc degree from Saraswath Narayanan College, Madura Kamaraj Unversty, Madura, He also earned hs M.Phl from Saraswath Narayanan College, Madura Kamaraj Unversty, Madura.Now he s dong Ph.D n Mathematcs at Anna Unversty of Technology Madura, Madura.Emal:senthl.lmec@gmal.com. 2. A.sathya s workng as Assstant professor n Mathematcs, Natonal Insttute of Technology Goa; Goa.she earned her M.Sc degree from Saraswath Narayanan College, Madura Kamaraj Unversty, and Madura. She also earned her M.Phl from Vnayaka mssons Unversty Selam. She was publshed research papers on clusterng algorthm n varous nternatonal journals. She was submtted Ph.D thess on Fuzzy Clusterng Analyss n Medcal mages at Gandhgram Unversty, Dndgul. Emal:sathyaarumugam.gru@gmal.com 3. R.Davd Chandrakumar receved hs graduate degree n Mathematcs from M.D.T. Hndu College, Trunelvel n 1969.Post graduate degree n Mathematcs from St.Xaver's College, Trunelvel n 1972.He receved M.Phl n Mathematcs from Unversty of Jammu n 1980.He also Receved Ph.D n Mathematcs from Unversty of Jammu n 1986.Presently he s workng as a Professor n Mathematcs department of Vckram College of Engneerng, Enath, Svaganga Emal:mathsvce@gmal.com. http://www.jser.org