FUZZY SEGMENTATION IN IMAGE PROCESSING

Transcription

1 FUZZY SEGMENTATION IN IMAGE PROESSING uevas J. Er,, Zaldívar N. Danel,, Roas Raúl Free Unverstät Berln, Insttut für Inforat Tausstr. 9, D-495 Berln, Gerany. Tel , Fax {uevas, zaldvar, Unversdad de Guadalaara, UEI Av. Revoluon No. 500,.P 44430, Guadalaara, Jal, Mexo. Tel , Fax ABSTRAT. In the area of pattern reognton and age proessng, unsupervsed lusterng s often used to perfor age segentaton. A oon proble n olor segentaton s the sensblty to hanges n lght ntensty, whh auses varatons n the pereved olor, that n fat orrespond to the sae obet. Fuzzy lusterng algorths fnd a soft partton of a gven dataset, where a data pont an partally belong to ultple lusters. In ths wor we propose the Fuzzy -Means algorth to segentng the fae s olors. The real-te pleentaton was ade n part usng a Matlab tool to fnd the luster s enters and nludng ths values n a ++ progra. The robustness showed by the algorth was qute good.. INTRODUTION. Pattern reognton tehnques an be lassfed nto two broad ategores: unsupervsed tehnques and supervsed tehnques. An unsupervsed tehnque does not use a gven set of unlassfed data ponts, whereas a supervsed tehnque uses a dataset wth nown lassfatons []. These two types of tehnques are opleentary. For exaple, unsupervsed lusterng an be used to produe lassfaton nforaton needed by a supervsed pattern reognton tehnque. In the area of pattern reognton and age proessng, unsupervsed lusterng s often used to perfor the tas of segentng the ages (.e., parttonng pxel on an age nto regons that orrespond to dfferent obets or dfferent faes of obets n the ages). Ths s beause age segentaton an be vewed as nd of data lusterng proble where eah data pont desrbe a set of age features (e.g., ntensty, olor, texture, et) assoated to eah pxel. Ths paper s organzed n the followng way: n seton a lusterng tehnques ntroduton s presented. In seton the unsupervsed lusterng defntons are desrbed. In seton 3 Fuzzy - eans algorth s presented. In seton 4 the pleentaton of the Fuzzy -Means algorth s used to aheve the fae segentaton. Fnally n seton 5 the results are presented and the possble proveents are proposed.. UNSUPERVISED LUSTERING. Unsupervsed lusterng s otvated by the need to fnd nterestng patterns or groupngs n a gven set of data. onventonal lusterng algorths fnd a hard partton of gven dataset based on ertan rtera that evaluate the goodness of a partton. By hard partton we ean that eah data belongs to exatly one luster of the partton. More forally, we an defne the onept hard partton as follows. Defnton. Let X be a set of datu and x be an eleent of X. A partton P={,,., L } of X s hard f and only f ) x X P suh that x ) x X x x where,, P The frst ondton n the defnton assures that the partton overs all data ponts n X, the seond ondton assures that all lusters n the partton are utually exlusve. In any real-world lusterng probles, however, soe data ponts partally belong to ultple lusters, rather than a sngle luster exlusvely. For exaple, a pxel n a agnet resonane

2 age ay orrespond to xture of a dfferent types of ssues. A soft lusterng algorths fnds a soft partton of a gven dataset based on ertan rtera. In soft partton, a data an partally belong to ultple lusters. We forally defne ths onept below. Defnton. Let X be a set a data, and x be an eleent of X. A partton P={,,., L } of X s soft f and only f the followng two ondton hold ) x X P 0 µ ( x ) ) x X P suh that µ ( ) > 0 x where µ x ) denotes the degree to whh x ( belongs to luster. A type of soft lusterng of speal nterest s one that ensures the ebershp degree of a pont x n all lusters addng up to one,.e., luster. A set of luster are well separated when any two ponts n a luster are loser than the shortest dstane between two lusters n dfferent lusters. Fgure 3 shows two lusters that are not well separated beause there are ponts n that are loser to a pont n than pont n. We forally defne well separated lusters bellow. Fg.. An Exaple of opat well separated lusters. µ ( x ) = x X A soft partton that satsfes ths addtonal ondton s alled a onstraned soft partton. The fuzzy -eans algorth, whh s best nown as fuzzy lusterng algorth, produes a onstraned soft partton. A onstraned soft partton an also be generated by a probablst lusterng algorth (e.g., axu lelhood estators). Even thought both fuzzy -eans and probablst lusterng produe a partton of slar propertes, the lusterng rtera underlyng these algorths are very dfferent. Whle we fous our dsusson n fuzzy lusterng n ths seton, we should pont out that probablst lusterng has also found suessful real-world applatons. Fuzzy lusterng and probablst lusterng are two dfferent approahes to the proble of lusterng. The fuzzy -eans algorth generalzes a hard lusterng algorth alled the -eans algorth, whh was ntrodued n the ISODATA lusterng ethod [4]. The (hard) -eans algorth as to dentfy opat, well-separated luster. Fgure shows a two-densonal dataset ontanng opat well separated lusters. In ontrast, the dataset shown n the fgure ontan lusters that are not opat and well separated. Inforally, a opat luster has a ball-le shape. The enter of the ball s alled the prototype of the Fg.. An exaple of two lusters that are not opat and well separated. Fg. 3. Two lusters that are opat, but not well separated. Defnton 3. A partton P={,,, } of de dataset X has opat separated luster f and only f any two ponts n a luster are loser than the dstane between two ponts n dfferent luster,.e, x, y d ( x, y) < d( z, w) where P z, w,, and d denotes a dstane q easure. r

3 Assung that a dataset ontans opat, wellseparated lusters, the goal of hard -eans algorth s twofold: () To fnd the enters of these lusters, and () To deterne the lusters (.e., labels) of eah pont n the dataset. In fat, the seond goal an easly be aheved one we aoplsh the frst goal, based on the assupton that lusters are opat and well separated. Gven luster s enters, a pont n the dataset belongs to luster whose enter s losest,.e., x f x v < x v =,...,, () where v denotes the enter of the luster. In order to arhve the frst goal (.e., fndng the luster s enters), we need to establsh a rteron that an be used to searh for these luster enters. One suh rtera s the su of the dstane between ponts n eah luster and ther enter. J ( P, V ) = x v = x where V s a vetor of luster enter to be dentfed. Ths rteron s useful beause a set of true luster enters wll gve a nal J value for a gven database. Based on these observatons, the hard -eans algorth tres to fnd the luster s enters V than nze J. However, J s also a funton of partton P, whh s deterned by the luster s enters V aordng to equaton. Therefore, the hard -eans algorth (HM) [] searhes for the true luster enter by teratng the followng two step: () alulatng the urrent partton based on the urrent luster. () Modfyng the urrent luster enters usng a gradent deent ethod to nze the J funton. The yle ternates when the dfferene between luster enters n two yles s saller than a threshold. Ths eans that the algorth has onverged to a loal nu of J. 3. FUZZY MEANS ALGORITHM. The fuzzy -Means algorth (FM) generalzes the hard -ans algorth to allow a pont to partally belong to ultple lusters. Therefore, t produes a soft partton for a gven dataset. In fat, t produes a onstraned soft partton. To do ths, the obetve funton J of hard -eans has been extended n two ways: () The fuzzy ebershp degrees n lusters were norporated nto the forula, and () An addtonal paraeter was ntrodued as a weght exponent n the fuzzy ebershp. The extended obetve funton [3], denoted J, s: J ( P, V ) = ( ( x )) x v = x µ () where P s a fuzzy partton of the dataset X fored by,,,. The paraeter s a weght that deternes the degree to whh partal ebers of a lusters affet the lusterng result. Le hard -eans, fuzzy -eans also tres to fnd a good partton by searhng for prototypes v that nze the obetve funton J. Unle hard -eans, however, the fuzzy -eans algorth also needs to searh for ebershp funtons µ that nze J. To aoplsh these two obetves, a neessary ondton for loal nu of J was derved fro J. Ths ondton, whh s forally stated below, serves as the foundaton of the fuzzy -eans algorth. 3. FM Theore. A onstraned fuzzy partton {,,, } an be a loal nu of the obetve funton J only f the followng ondtons are satsfed: µ ( x) = v = = x v x v x X n ( µ ( x)) x x X ( µ ( x)), x X () Based on ths theore, FM updates the prototypes and the ebershp funton teratvely usng equatons and 3 untl a onvergene rteron s reahed. We desrbe the algorth below. (3)

4 FM (X,,, ε) X: an unlabeled data set. : the nuber the lusters. : the paraeter n the obetve funton. ε: a threshold for the onvergene rtera. Intalze prototype V={v,v,,v } Repeat V Prevous V opute ebershp funtons usng equatons 3. Update the prototype, v n V usng equaton. Untl = Pr evous v v ε These are portant ponts regardng the FM algorth: - FM s guaranteed to onverge for >. Ths portant onvergene theore was establshed n 980 [4]. - FM fnds a loal nu (or saddle pont) of the obetve funton J. Ths s beause the FM theore s derved fro the ondton that the gradent of the obetve funton J should be 0 at an FM soluton, whh s satsfed by all loal na and saddle ponts. The result of applyng FM to a gven dataset depends not only on the hoe of paraeters and, but also on the hoe of ntal prototypes. 4. IMPLEMENTATION. The Matlab Fuzzy Log Toolbox s equpped wth soe algorths that allow to fnd lusters n nput-output tranng data. We an use the luster nforaton to generate a Sugeno-type fuzzy nferene syste that best odels the data behavour usng a nu nuber of rules [5]. The FM algorth pleented n Matlab starts wth an ntal guess for the lusters enters, whh are ntended to ar the ean loaton of eah luster. The ntal guess for these lusters enters s ost lely norret. Addtonally, f assgns to every data pont a ebershp grade for eah luster. By teratvely updatng the lusters enters and the ebershp grades for eah data pont, f teratvely oves the lusters enters to the orret loaton wthn a data set. Ths teraton s based on nzng an obetve funton that represents the dstane fro any gven data pont to a lusters enter weghted by that data pont s ebershp grade. To pleent the segentaton syste t s neessary to use as data, an age of the obet to be segent (n our ase a person fae). Eah pxel of the age s oded n three oponents represented respetvely by red, green and blue olors. The next ode assgn to eah pxel ts respetve olor oponent dataset represented by VP wth the f funton forat (that eans the pxel data s presented n row for). Soethng that one ust not forget s that the age dataset s obtaned n nteger forat but to wor wth t wll be neessary to hange t to double forat. R=I(:,:,); G=I(:,:,); B=I(:,:,3); [,n]=sze(r); nde=*n; er=0; for a=: for an=:n end end data=r(a,an); data=g(a,an); data=b(a,an); nu=nu+; VR(nu)=data; VG(nu)=data; VB(nu)=data; VP=[VR;VG;VB]; VP=double(VP); There s an portant f paraeter, ths s the luster nuber n whh one wants to dvde the presented dataset, ths paraeter should be founded heurstally. For ths wor ts value was 7. If ths value s bg, then the syste generalzaton s not good enough and f s very sall then the neghbor olors an be onfused. The atlab ode to fnd the age lusters s: [enter,u,of]=f(vpt,7); After used ths funton we have n the varable enter the lusters enters, whh wll be used to lassfy the pxels belongng to the nterest lass. In our ase the nterest lass s the lass that represent the flesh olor. In ths wor the lassfaton s aheved alulatng the nu dstane fro eah pxel to the luster entrod (ths entrod was prevously obtaned wth the f funton). The ode n ++ to aheve that n real te s:

5 for(nt =;<=szeiage;++) { b=*pbuffer; g=*pbuffer; r=*pbuffer; dst=sqrt((abs(r )*abs(r ))+(abs(g-5.489)*abs(g ))+(abs(b )*abs(b ))); f (dst<45) tep=55; else tep=0; *pbuffer=tep; *pbuffer=tep; *pbuffer=tep; pbuffer=pxel; } Fg. 4 luster dstrbuton. The prevous ode onsders that szeiage s the age sze and also that the flesh olor lass entrod values are for red, for green and for blue and a slarty rtera nor to RESULTS. The obtaned results usng the fuzzy -Means as a segentaton ethod was qute good. A fast tranng s an portant advantage obtaned wth the use of Fuzzy -Means atlab tools as well as the easy hange of ts paraeters. Ths allows to experent wth dfferent ondtons le hangng the lass nuber untl the syste robustness s satsfed. The fgure 4 shows the luster dstrbuton obtaned by tranng the f funton. Whle the fgure 5 shows an age and ther respetve segentaton usng the followng luster enter values for the lass flesh olor: red=76.448, green=5.489 and blue = In ths wor we used as a lassfy rtera the entrod dstane but for future wor we proporse the use of the lass dsperson as a lassfy rtera (dstane of Mahalanobs) plus the entrod dstane. Fg. 5. (left) orgnal age, (rgth) segented age. 4. REFERENES. [] Yen J. and Langar R., Fuzzy log, ntellgene, ontrol and nforaton, Prente Hall, New Yor, 000. [] J.. Bezde and S. Pal (eds.) Fuzzy Models for Pattern Reognton, IEEE Press, 99. [3] J. Bezde and L. Hall, and L.P. lar. Revew of MR Iage segentaton tehnques usng pattern reognton, Medal Physs, Vol. 67, pp , 980. [4] J. Dunn. A fuzzy relatve of the ISODATA proess and ts use detetng opat wellseparated lusters, J. ybernets, Vol. 8, pp. 3-57, 983. [5] E. uevas, D. Zaldívar, and R. Roas. Fuzzy segentaton appled to fae segentaton, Tehnal Report B 04-09, Free Unverstät Berln, Fahbereh Matheat und Inforat, Berln, June, 004.