Evaluation of Segmentation in Magnetic Resonance Images Using k-means and Fuzzy c-means Clustering Algorithms

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1 ELEKTROTEHIŠKI VESTIK 79(3): , 2011 EGLISH EDITIO Evaluaton of Segmentaton n Magnet Resonane Images Usng k-means and Fuzzy -Means Clusterng Algorthms Tomaž Fnkšt Unverza v Lublan, Fakulteta za stronštvo, Aškerčeva 6, 1000 Lublana, Slovena E-mal: tomaz.fnkst@fs.un-l.s Abstrat. The purpose of luster analyss s to partton a data set nto a number of dsont groups or lusters. Members wthn a luster are more smlar to eah other than to members from dfferent lusters. Applablty of the entrod-based k-means and representatve obet-based fuzzy -means algorthms for study of the Magnet Resonane Images s analysed n the work. The two algorthms are mplemented and ther applablty for the analyss of the MRI s evaluated. The rteron s the qualty of the lusterng ompared to the lusters n the referene mages. The qualty of lusterng n both algorthms depends on the number of data ponts n the mage and on the number of the to-be-formed lusters. The algorthms are mplemented, they are appled to the referene two-dmensonal multspetral MR mages and the resultng mage segmentaton s analysed by the obetve rtera. The obet-based fuzzy -means algorthm outperforms the entrod-based k-means algorthm by all means. The advantage of the former stems from utlzaton of the luster membershp fuzzness. Keywords: mage segmentaton, fuzzy -means lusterng, k-means lusterng, magnet resonane magng 1 ITRODUCTIO Importane of medal magng s undsputable n modern medne. Imagng s the only soure of nformaton when adustng skeleton fratures; t s ombned wth other dagnost means for early deteton of tssue hanges that an develop to a fatal dsease when not suessfully ured. Progress n mromahnng, mehatrons, development of mro motors and real-tme graphs proessng has gven means to buld surgeon-ontrolled robot hands wth great geometr preson. Insons are smaller, less harm s made to the tssue and surgeons an perform operatons based on enlarged mages of the body nternals. Dfferent prnples of physs are appled n medal magng. Most tehnques produe mages that onsst of dfferent levels of grey. Image bakground represents the spae around the body and t s usually blak. Medal magng started wth the nventon of X rays. Inventon of the ultrasound dagnosts followed. Progress n omputer tehnology has gven means for ntense graph proessng that has evolved [1] X ray examnatons to the Computer Tomography. The most reent s Magnet Resonane Imagng (MRI). Medal mages are adequately analysed by a spealst a medal dotor that has been learnng readng the mages for years. Mahnes an help wth the readng. They an be programmed to read the dagnose for many of the medal ases. The proess, where tssues and strutures are lassfed nto areas representng dfferent obets, s mage segmentaton. Dfferent approahes are utlsed n the segmentaton proess: segmentaton rules an defne haratersts of the surfaes, homogenety of the surfae an be a segmentaton rteron, and edge deteton algorthms an be appled to defne the mage segments [2]. MRI s a medal magng tehnque for vsualzaton of detaled nternal strutures [3, 4, 5]. MRI makes use of the property of nulear magnet resonane to mage nule of atoms nsde the body. MRI provdes good ontrast between the dfferent soft tssues of the body, whh makes t espeally useful n magng the bran, musles, the heart, and aners ompared wth other medal magng tehnques suh as omputed tomography (CT) or X-rays. Unlke CT sans or tradtonal X-rays, MRI does not use onzng radaton. MRI mages are onstruted by many methods that are optmzed for vsualzaton of dfferent tssues and, more mportant, for dfferent ondtons of the same tssue. Searh rtera are grouped nto dfferent modaltes of mage onstruton. The most used modaltes are T1, T2 n PD. MR mages are desgned to have hgh ontrast between dfferent types of the tssue. The mages an be most detaled as suh they are utlzed even for the teahng purposes n the undergraduate anatomy ourses. MR s the de-fato most produtve magng method for the bran dagnosts. Tumours and other rregulartes an be deteted n an early phase of Reeved May 31, 2012 Aepted June 4, 2012

2 130 FIKŠT dsease where treatment an result n mprovements or ure of the ondton. To grade effetveness of the MR mage lusterng by dfferent omputerzed methods one needs dotors' hand-drawn lusterfaton of the mage frst. Ths makes referene results. Mahne-generated results are then ompared to the referene. A omparson of the results defnes rankng among dfferent mahne lusterng methods. Segmentaton of the mage nto lusters that represent areas of dfferent types and ondtons of a tssue, and lassfaton of the ondtons an be performed by an algorthm to a sgnfant extent [6]. Charatersts of eah luster (representng the type and ondton of the tssue) n the desgned set of lusters, and the mages themselves are nput to the lusterng proess. Pxels wth propertes that adhere to lassfaton ondtons for partular luster, beome members of that luster. Results of automat mage segmentaton are mages where dfferent tssues and dfferent ondtons of the tssue are well dfferentated among themselves. A medal dotor, spealzed n readng the mages, examnes orgnal and segmented mages for nadequate segmentaton and approves flawless segmentaton results. A data base of referene pars of the orgnal and segmented mage s reated. The data base serves for teahng new spealsts readng the mages, and for onstruton of automat dagnost proedures. In ths work we are examnng effeny of two lusterng methods, namely k-means (km) and Fuzzy - Means (FM) lusterng algorthms, for the MR bran mage segmentaton nto the grey matter (GM), whte matter (WM) and the erebrospnal flud (CSF). Indvdual members (pxels, groups of pxels) of eah luster are to be smlar to eah other by a set of predefned rtera. Smlarty defnes them as members of a partular luster. They are more smlar to eah other than to the members of other lusters. Smlarty s measured by an obetve rteron that produes a smlarty ndex, a real number, as a measure of smlarty between the two dfferent surfaes. Indes are alulated for dfferent mathng rtera a smlarty vetor s a result of the smlarty evaluaton. To ondense the amount of data, the Euldan dstane among smlarty vetors s alulated from ther omponents. The sze of the Euldan dstane s a measure of mathng a smaller number adheres to better mathng. In both, the k-means and fuzzy -means lusterng algorthms one frst sets ntal propertes for the luster, the entre vetor [7]. Propertes of ndvdual anddates for membershp of the luster are tested for mathng to the entre vetor. The resultng Euldan dstane between the entre and the anddate vetor s a measure of mathng. A threshold mathng value defnes nluson or exluson of the anddate to/from the luster. The entre vetor s adusted by weghted averagng wth the vetors of the nluded surfae n the proess of mathng. The lusterng proess gets stable when addton of new vetors does not affet the value of the entral vetor. Fuzzy -means unsupervsed lusterng s reported as effetve lusterng tehnque for segmentaton of dfferent medal mages [7, 8, 9, 10]. It has even wder applablty. Reports on fuzzy -means segmentaton n astronomy, geology and war ndustry for target reognton are gven n [11, 12, 13]. Among dfferent approahes to segmentaton, the fuzzy -means method has an advantage of retanng more nformaton from the orgnal mage than the lassal segmentaton methods [11]. Intal values of the entre vetors of the lusters an be hosen by many dfferent approahes: the values an be set randomly; they an be alulated from the frst k samples of the data ponts, or from some other mageharaterst data ponts. Startng the segmentaton by evaluaton of nfluene of dfferent ntal values for entre vetors on the outome of lusterng s not pratal, espeally for a large number of lusters [9]. Therefore, dfferent methods have been proposed n the lterature for the lusterng setup [7, 14, 15, 16]. Sne mage lusterng s an effetve tool for segmentaton of the medal mages, the medal ondton treatment depends on readng the mage, whether by a spealst only or by a dotor mahne tandem. The bran tssue s a omplex struture and hene proper dagnoss of many bran dsorders sgnfantly depends upon aurate segmentaton of the three bran tssues GM, WM and CSF. 2 IMAGE SEGMETATIO Image segmentaton s the most essental proess of any mahne or man-made mage analyss. Mahne made mage segmentaton an be performed off lne, or t an be desgned for real-tme applatons, e.g., for mahne vson of autonomous robots, qualty ontrol of ndustral proesses, management and ontrol of produton and smlar. In medne, segmentaton of the mage an take some tme sne dagnosts s not onsdered to be n the doman of real-tme applatons. That takes burden from the onstrants on hardware and from onstrants on speed optmzaton n the frst pass of the algorthm desgn. Mahne-made dagnosts has potental n early deteton of tssue hanges n preventve hek-ups, whh need to be ost and tme effetve sne preventon nvolves perodal hek-ups of a sgnfant number of people.

3 EVALUATIO OF SEGMETATIO I MAGETIC RESOACE IMAGES USIG K-MEAS AD FUZZY C-MEAS 131 The fnal knowledge, obtaned by analyss of the mage, s rually dependant on the qualty of segmentaton where the mage s parttoned nto nonoverlappng surfaes that adhere to the homogenety rteron [17]. The rteron s defned as follows: Let f ( x, y) be a set of haraterst values for the dfferent spots n the mage. Let P() be a homogenety predate whh s defned for the surfae of the mage. Clusterng s segmentaton of the mage f ( x, y ) nto n onneted subsets of the mage ( S1, S2,, Sn), where n 1 S f ( x, y) ; S S 0,. (1) The value of the homogenety predate s 1 for any subset of the mage - PS ( ) 1, and t s 0 for two neghbourng subsets - P( S S ) 0. Suh s the defnton of the homogenety predate for all mage types. In grey mages, the PS ( ) s about levels of grey; n olour mages, the PS ( ) s about olour ntensty; n vetor mages, the PS ( ) s about homogenety of olour or texture. One fnds n the lterature many dfferent mage segmentaton proedures. one of them gves optmal results for all dfferent types of mages, and one mage annot be segmented n dent surfaes by dfferent segmentaton proedures. [17] gves a survey of proedures for mage segmentaton. Algorthms that are developed for segmentaton of one type of an mage (e.g. mage onsstng of surfaes havng many grey levels) are not produtve n segmentaton of other type of an mage (e.g. MRI). ot surprsngly, one should look nto mehansms that were appled n the proess of mage formaton to dede on a most approprate segmentaton algorthm. The frst task n segmentaton s to hoose or develop a most suessful segmentaton approah. Ths s a multdoman problem. For example, Pavlds [18] understands mage segmentaton as a problem n the doman of psyhophysal perepton. Beng so, mathematal algorthms are n many oasons supported by heursts to mprove the overall segmentaton effeny. A logal development n mage reognton s nluson of fuzzy tehnques nto the makng of the algorthms. At present, two approahes ompete n mage reognton: determnst and fuzzy [17]. There are not many methods to aurately assess the qualty of segmentaton. For the ase of range mages, an aurate assessment method s reported n [19], where there s a speally prepared referene mage. We perform the bran MRI segmentaton by lusterng [6], not by edge deteton. Two methods were mplemented and evaluated: k-means (km) and fuzzy - means (FM) lusterng. Other anddate lusterng methods nlude knowledge about spefs of partular mages or/and some heursts. Beng so, we exluded them from our study of the segmentaton algorthms. The lusterng algorthms [6] are lassfed by dfferent rtera: approah - obetve, theory of graphs, herarhal: type - determnst, statst, fuzzy. km and FM are obetve lusterng algorthms. Usablty of the two algorthms s extensve, whh s due to the theoretal and mplemental progress n the last ouple of years [12]. 2.1 k-means segmentaton k-means segmentaton s a parametr type of segmentaton. Crteron funton J s ntrodued, (2). Segmentaton s performed n teratve steps. The mean ntensty s alulated n eah teraton for eah luster, and eah part of the mage s ategorsed nto the luster wth a most smlar mean ntensty. Funton J (2) represents the sum of squares of the dstanes among luster members. The mean luster ntensty s alulated by (3). J 2 x (2) x (3) 1 x s an n dmensonal sample, s the number of lusters, ndex n S stands for the -th luster, s the mean ntensty value of the -th luster, s the number of samples n the luster S. The followng proedure apples n alulaton of the k-means: (1) One frst hooses, where stands for the number of lusters havng mean ntensty values 1(1), 2(1),, (1). (2) In the k-th teraton, the samples are grouped nto the lusters. Sample x beomes a -th luster member f x m ( k) x m ( k) 1,2,,. (3) ext, new mean ntensty values ( k 1), 1,2,,. are alulated. (4) When ( k1) ( k) for all the 1,2,,, k-means teratons stop. To apply the k-means algorthm to a partular segmentaton problem, one frst needs to defne the number of lusters and the ntal luster ntensty values. We defned the latter ones by takng nto

4 132 FIKŠT aount the herarhal nature of lusterng. The lowest level of herarhy s represented by ndvdual samples, and the hghest herarhal level s represented by a luster that nludes all samples [9]. On eah herarhal level of lusterng, one makes a luster from two most smlar lusters n a lower herarhal level. Suh a luster stays put; t s not dvded nto sub lusters as the proedure ontnues. The teratve lusterng proedure s ended when three lusters, representng the GM, WM and CSF, are obtaned. The mean ntensty values of these lusters are the ntal mean values when searhng for the k-th mean values. 2.2 Fuzzy -means proedure In fuzzy lusterng, eah pont has a degree of belongng to lusters, as n fuzzy log, rather than belongng ompletely to ust one luster. Thus, ponts on the edge of a luster may be n the luster to a lesser degree than ponts n the entre of the luster. Besdes, the ponts an belong to more than one luster, wth dfferent levels of belongng. The method s based on mnmzaton of the followng obetve funton (4): J m 2 MR ( ) x (4) 1 1 where 1 m and m s a real number. s the number of samples, s the number of lusters, s the degree of the sample x membershp to the luster, x s the th of the d-dmensonal measured data, s the d-dmenson entre of the luster, and x s any norm expressng the smlarty between any measured data and the entre. The Euldan dstane (5) s a most used measure on smlarty: T x x x (5) Fuzzy parttonng s arred out through an teratve optmzaton of the obetve funton (4), wth the update of membershp and the luster entres by (6) and (7): 1 x k 1 x k C 1 for eah {1,, } k m m x 2 m1, for. (6) (7) In equatons (6) and (7), s the ndex of the ndvdual sample, and k are ndes of the ndvdual soft luster spefally of ts entre. The m value s a measure of softness / sharpness n the dstrbuton of membershp funtons n the spae. In the extreme ase of m 1 are the membershp funtons, whh are defned above the spae, degenerated nto a sharp level of membershp. The membershp funtons an therefore have only one of 0,1 the values n the Equaton (6) shows that the 1 f the norm of the x, for the -th sample and the entre of the -th luster, s the smallest regardng the entres of all lusters. 0 for the other lusters where k {1,,C}. In the other extreme ase of m are the membershp funtons degenerated nto a ompletely soft level of membershp. The membershp funtons therefore have the same values of 1/ C n the whole spae, for eah {1,,C}. Fgure 1: Referene MR mage and the referene segmentaton of the referene mage. In pratal alulatons, authors usually report hoe of the m values beng 1.25 or 2.00 [20]. The proedure for soft lusterng the -means s: (1) One hooses the number of lusters C and the parameter m. Then the membershp matrx (0) [ ] s defned. (2) In the k-th teraton, one defnes the entres of lusters, for 1,, C, aordng to ( k). (3) One then defnes a new membershp matrx ( k 1). (4) If ( k1) ( k), the proedure s fnshed, else one returns to (2). 3 RESULTS The lusterng proess depends on the ntal randomly set ondtons thus makng the lusterng results to be slghtly dfferent n eah reteraton of the method.

5 EVALUATIO OF SEGMETATIO I MAGETIC RESOACE IMAGES USIG K-MEAS AD FUZZY C-MEAS 133 The bran mages analysed n our ase, were made n dfferent modaltes. Image segmentaton was amed at struturng the mage nto the three types of tssue: GM, WM and CSF. Fgure 1 shows the referene MR mage and the referene segmented mage. In our study we analysed 30 MRIs of the bran n the T1 modalty and n the axal proeton [21]. 3.1 Gradng the effeny of the bran MR mage segmentaton by the two algorthms The obetve gradng rtera are needed frst. Sne the lusterng s an teratve proess, the gradng metrs needs to be bult nto teratve loops of the proess. ot only to montor onvergene of the lusterng proess but prmarly to ntervene at the onset of ambguty about how to proeed. Gradng metrs needs to be smple [21]. Ideally, t should be as smple as evaluaton of the students' knowledge. An nteger number (n ontnental Europe) or a letter (n the UK and n the US) represents the quantty and qualty of the students' knowledge. To evaluate mage segmentaton performane, one needs nput referene mages and output referene segmented mages. The latter are usually made by spealsts a ertan amount of freedom n nterpretaton of the orgnal mage s possble. To mnmze the subetve fator, the followng two approahes are reommended [22]: a) Many spealsts should segment the same mage. Then, a referene segmented mage needs to be produed by a statstal method that takes nformaton from work of all the ndvdual spealsts. b) Generaton of pars of mages onsstng of a mokup referene mage and a orrespondng mok-up referene segmented mage [22]. As a measure of gradng we took the expresson for p n equaton (8), where f ( x, y) s the number of mage segments n the mage whh s segmented by the proedure, and f ( x, y) s the number of mage segments n the lusterng norm. n p 3.2 Quanttve results f ( x, y) fn( x, y) f ( x, y) f ( x, y) n (8) The MR mages were produed by three dfferent modaltes. The 20 mages used n our expermentaton were segmented by both algorthms. The algorthms were operatng n the auto mode, wth no human nterventon. The ntal values for the proess varables were produed by a random value generator. For the - means method, m = 2 resulted n best results. Table 1 shows the quanttatve results. Fgure 2: Results of lusterng for the MR mage n T1 modalty: k-means a) whte tssue, ) grey matter, e) bran lquor; fuzzy -means b) whte tssue, d) grey matter, e) bran lquor Applaton of the fuzzy -means algorthm results n better mathng to the referene segmented mages than the applaton of the k-means algorthm, Mathng for the WM s better than mathng for the GM. CSF mathng has the lowest value CSF s manfested n many levels of grey n the orgnal MR mages. Table 1: Effeny of the k-means and fuzzy -means algorthms n segmentaton of the MR mages nto GM, WM and CSF Tssue/ algorthm WM GM CSF k-means Fuzzy -means

6 134 FIKŠT 4 DISCUSSIO Segmentaton methods are essental omponents n mahne-derved medal dagnosts. Segmentaton of medal mages dffers from mage segmentaton n most of other tehnal felds by the followng: a) The maorty of medal mages are made of dfferent levels of grey only. The mplaton s there s less nformaton n suh an mage than n the olour mages. b) The MR mages are produed by algorthms; many other mages are produed by a amera only. The mplaton s there s more researh spae n magng reognton for the MR mages than for the amera-taken mages. ) Medal dagnosts and therapy are an rreversble proess. Wrong or nomplete dagnosts s a legal wrongdong. In lne wth the ), the results not ompletely adherng to requrements annot be aepted. Consequentally s the progress of mahne-made medal dagnosts relatvely slow, ompared to a smlar development n other tehnal felds. In ths work, applablty of the k-means and fuzzy -means algorthms for segmentaton of the MR bran mages s explored. The study shows that the fuzzy - means algorthm s better suted to the requrements of the MRI segmentaton than the k-means algorthm. Suh a result ould be ntutvely expeted sne ature behaves n many ourrenes more fuzzly than determnstally. Fuzzy tehnal feld was nvented to fll the vaany where determnst approahes showed less than optmal results. It s the author s understandng that mahne-made mage lusterng and derved dagnosts has onsderable potental n mahne-supported dagnosts, and less potental n buldng-up an autonomous dagnost system. 5 COCLUSIO The lusterng methods have potental to support dagnosts and to ontrbute to lassfaton of many mages for dfferent dagnost purposes. Sne lots of parameters are to be set for most suessful lusterng n eah type of mages, medal and engneerng ompetenes are needed for set-ups of bath obs. The lusterng methods for medal magng are smlar to those developed n other felds, where mage analyss s mportant. But there s one mportant dfferene: sne human lves and lfe qualty are nvolved, the error rate n deteton of dfferent obets has to lmt toward zero. Ths s ahevable by team workng of medal spealsts and mage-proessng engneers. REFERECES [1] P. Suetens. Fundamentals of Medal Imagng, Cambrdge Unversty Press, ew York, (2009). [2] J. C. Russ. The Image Proessng Handbook, 6ed, CRC Press, Boa Raton, [3] D. Hoa, A. Mheau. Magnet Resonane Imagng physs and tehnque ourse on the web. Courses/e-MRI, zadn dostop: 15/05/2012. [4] J. P. Hornak. The Bass of MRI. on-lne book: zadn dostop: 15/05/2012. [5] C. Guy, D. Ffythe. Introduton to the Prnples of Medal Imagng. Imperal College Press, London, [6]. Pavešć. Razpoznavane vzorev: Uvod v analzo n razumevane vdnh n slušnh sgnalov, Fakulteta za elektrotehnko, Lublana, [7] S. Deelers, S. 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Pattern Reognton wth Fuzzy Obetve Funton Algorthms, ew York, Plenum Press, [13] M. S. Yang, Y. J. Hu, K. C. R. Ln.»Segmentaton tehnques for tssue dfferentaton n MRI of Ophthalmology usng fuzzy lusterng algorthms«, Magnet Resonane Imagng, Vol. 20, p , [14] L.-H. Juanga, M.-. Wub,»MRI bran leson mage deteton based on olor-onverted K-means«, Measurement, Vol. 43, p , [15] J. C. Bezdek, L. Hall, L. Clarke.»Revew of MR mage segmentaton usng pattern reognton«, Medal Physs, Vol. 20, p , [16] W. Chen, L. G. Maryellen, U. Bk.»A fuzzy C-means (FCM)- based approah for omputerzed segmentaton of breast lesons n dynam ontrast-enhaned MR mages«, Aadem Radology, Vol. 13, o. 1, p , [17]. R. Pal, S. K. Pal.»A revew on mage segmentaton tehnques«, Pattern Reognton, Vol. 26, o. 9, p , [18] T. Pavlds, Strutural Pattern Reognton, Sprnger, ew York, [19] A. Hoover, G. Jean-Baptste, X. Jang, P. J. Flynn, H. Bunke, D. B. Goldgof, K. Bowyer, D. W. Eggert, A. Ftzgbbon, R.B. Fsher.»An expermental omparson of range mage segmentaton algorthms«, IEEE Transatons on Pattern Analyss and Mahne Intellgene, Vol. 18, o.7, p , [20] FMRIB's Automated Segmentaton Tool Bran Segmentaton and Bas Feld Correton. zadn dostop: 15/05/2012. [21] V. Chalana, K. Yongmn. A methodology for evaluaton of boundry deteton algorthms on medal mages, IEEE Transaton on medal magng, vol. 16 no.5, Tomaž Fnkšt s an assstant to professor for eletral engneerng at the Faulty of Mehanal Engneerng of the Unversty of Lublana, Slovena. Hs researh s foused on mage analyss, dstrbuted applatons and ontrol systems.