Unsupervised Image Segmentation Method based on Finite Generalized Gaussian Distribution with EM & K-Means Algorithm

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1 IJCSS Intrnatonal Journal of Computr Scnc and twork Scurty, VOL.7 o.4, Aprl Unuprvd Imag Sgmntaton bad on Fnt Gnralzd Gauan Dtrbuton wth EM & -Man Algorthm raad Rddy.V.G.D, Srnva Rao. 2, Srnva Yarramall 3, Dpartmnt of Computr Scnc & Sytm Engnrng, Andhra Unvrty (IDIA 2 Dpartmnt of Stattc, Andhra Unvrty (IDIA 3 Dpartmnt of Informaton Tchnology, Collg of Engnrng GITAM (IDIA Summary In Imag rocng Modl Bad Imag Sgmntaton play a domnant rol n Imag Analy and Imag Rtrval. Rcntly much work ha bn rportd rgardng Imag Sgmntaton bad on Fnt Gauan Mxtur Modl ung EM algorthm. (Ymng Wu t al (2003, (Yamazak.T (998. Howvr, n om mag th pxl ntnt nd th mag rgon may not b Mo- urtc or Bll Shapd, bcau of th on- Gauan natur. Hnc thr ar om tuaton whr Imag Sgmntaton to b don wth a mor Gnralzd Fnt Mxtur Dtrbuton. In th artcl w dvlop and analyz an mag gmntaton mthod bad on Fnt Gnralzd Gauan Mxtur Modl ung EM and -Man algorthm. Th -Man algorthm utlzd to obtan th numbr of rgon and th ntal tmat of th modl paramtr. Th updat quaton of th modl paramtr ar obtand by ung th EM algorthm. Th gmntaton of th pxl n th mag don by maxmzng th componnt lklhood functon. Th prformanc of th mthod valuatd through ral tm data on 3 mag by calculatng mclafcaton rat and mag qualty mtrc. It obrvd that th propod mthod prform much upror to th arlr mag gmntaton mthod. y word: Imag Sgmntaton, EM algorthm, -Man algorthm, Fnt Gauan Mxtur Modl, Fnt Gnralzd Gauan Mxtur modl.. Introducton Imag rocng a growng ara of Computr Scnc and Engnrng. Th man objctv of mag procng to undrtand th componnt of an mag and ntrprt t mantc manng. To analyz th fatur nd th mag, Modl bad Sgmntaton algorthm wll b mor ffcnt compard to non-paramtrc mthod. Th ffcncy of th gmntaton algorthm bad on th utabl probablty dtrbuton acrbd to th pxl ntnt n th ntr mag. In many mag gmntaton tchnqu t aumd that th pxl ntnt n ach mag rgon follow a Gauan dtrbuton and th ntr mag aumd a a charactrzaton of Fnt Gauan Mxtur Modl. Th mxtur modl ar mor utabl for mag gmntaton only whn th pxl ntnt nd th mag rgon ar ymmtrc and havng mo-kurtc natur. Howvr, n many practcal tuaton arng at plac lk Mdcal Imagng, Robotc, hoto Copr tc., th pxl ntnt nd th mag rgon may not b kwd or mo-kurtc. To hav a clo approxmaton to th raltc tuaton t ndd to gnralz th mag gmntaton algorthm wth a mor gnral mxtur dtrbuton whch nclud th Fnt Gauan Mxtur modl a a partcular ca. Th Gnralzd Gauan Dtrbuton nclud th Gauan dtrbuton a a partcular ca and t can b paramtrzd n uch a mannr that t man μ and Varanc σ2 concd wth th Gauan dtrbuton. In addton to locaton and calng paramtr, th Gnralzd Gauan Dtrbuton havng a Shap paramtr whch th maur of pakd n of th dtrbuton.. Th Gnralzd Gauan Dtrbuton wa ud by Sharf. t al (995 for modlng th atmophrc no ub band ncodng of Audo and Vdo Sgnal, Cho S t al (2000 ha ud th dtrbuton for mpulv no drcton of arrval and ndpndnt componnt analy. Varana M.. t al (987 dcud th paramtr tmaton for th Gnralzd Gauan Dtrbuton by ung th mthod of Momnt and Maxmum Lklhood. J.Armando Domnguz t al (2003 dvlopd a procdur to tmat th hap paramtr n Gnralzd Gauan Dtrbuton. Howvr, vry lttl work ha bn rportd rgardng Imag Sgmntaton bad on Gnralzd Gauan Dtrbuton. In th papr w dvlop and analyz an mag gmntaton mthod bad on Fnt Gnralzd Gauan Mxtur Dtrbuton. Th numbr of mag rgon (Componnt tmatd by utlzng th - Man algorthm. Th tmaton of th modl paramtr carrd by EM algorthm. Th gmntaton algorthm dvlopd bad on pxl allocaton to th rgon whch maxmz th componnt lklhood functon. Th prformanc of th dvlopd gmntaton algorthm Manucrpt rcvd Aprl 5, 2007 Manucrpt rvd Aprl 25, 2007

2 38 IJCSS Intrnatonal Journal of Computr Scnc and twork Scurty, VOL.7 o.4, Aprl 2007 valuatd by obtanng th Imag Qualty Mtrc lk Man Squar Error, Sgnal to o Rato and Imag Qualty Indx. Th accuracy of th clafr ud n th gmntaton alo tudd wth rpct to th mclafcaton rat. Th gmntaton algorthm nclud vral of th arlr xtng algorthm lk Imag Sgmntaton algorthm bad on Fnt Gauan Mxtur Dtrbuton, Laplac Dtrbuton tc., for dffrnt valu of th hap paramtr. 2. Gnralzd Gaulan Dtrbuton In th cton w brfly dcu th probablty dtrbuton and t proprt ud n th mag gmntaton algorthm. Lt th pxl ntnt n th ntr mag obtand by pxl ntnt (Obtand through pxl grabbr undr JAVA nvronmnt a Random Varabl and follow a Fnt Gnralzd Gauan Mxtur Dtrbuton. It alo aumd that th ntr mag a collcton of mag rgon, thn th pxl ntnt n ach mag rgon follow a Gnralzd Gauan Dtrbuton. Th probablty dnty functon f( z μσ,, 2 Γ ( + A (, σ whr 2 2 ( Z μ A (, σ σ Γ( σ > 0, A (, σ Γ ( 3. Th paramtr μ th man, th functon Aσ (, an calng factor whch allow that th 2 Var(Z σ, and th hap paramtr. Whn, th corrpondng Gnralzd Gauan corrpond to a Laplacan or Doubly Exponntal Dtrbuton, Whn 2, th corrpondng Gnralzd Gauan corrpond to a Gauan dtrbuton. In lmtng ca +, quaton (4.3. convrg to a unform dtrbuton n ( μ 3 σ, μ+ 3 σ and whn 0 +, th dtrbuton bcom a dgnrat on n Z μ. Th man valu of th Gnralzd Gauan dtrbuton μ. Th 2 Varanc σ. 3. -Man Algorthm Th mot prplxng u for Imag Clafcaton dtrmnng th fnt numbr of rgon ( to b formd. Many tattcal crtra ar ud for dtrmnng th numbr of cla, on uch mthod th -Man algorthm. Stp. Bgn wth a dcon on th valu of numbr of gmnt Stp2. ut any ntal partton that claf th pxl nto gmnt. W can arrang th tranng ampl randomly, or ytmatcally a follow:. Tak th frt tranng ampl a a ngl-lmnt Sgmnt 2. Agn ach of th rmanng (- tranng ampl to th gmnt wth th nart cntrod. Lt thr b xactly gmnt (C, C 2 C and n pattrn to b clafd uch that, ach pattrn clafd nto xactly on gmnt.. Aftr ach agnmnt, rcomput th cntrod of th ganng gmnt. Stp3. Tak ach ampl n qunc and comput t dtanc from th cntrod of ach of th gmnt. If th ampl not currntly n th clutr wth th clot cntrod wtch th ampl to that gmnt and updat th cntrod of th gmnt ganng th nw ampl and clutr long th ampl Stp4. Rpat tp 3 untl convrgnc achvd, that untl a pa through th tranng ampl cau no nw agnmnt. Aftr dtrmnng th fnal valu of th (numbr of rgon, w obtan th tmat th paramtr µ, σ and for th th rgon ung th gmntd rgon pxl ntnt wth th mthod gvn by J.Armando t al (2003. Subttutng th valu a th ntal tmat, w rfn th tmat of th paramtr by ung th EM algorthm. 4. EM Algorthm Th tmator of th modl functon ar obtand by ung EM algorthm. For obtanng th EM algorthm, w condr that a ampl of th xl ntnt z z z..., 2, ar drawn form an mag wth probablty dnty functon hz ( θ f( z, θ, Thn th lklhood functon of th pxl ntnt ar

3 IJCSS Intrnatonal Journal of Computr Scnc and twork Scurty, VOL.7 o.4, Aprl L( θ π ( f ( z, θ l π 2 Γ ( + A (, σ whr Th mpl ( Z μ A (, σ 2 2 σ Γ( A(, σ Γ ( 3 L( θ log hz (, θ, l hz ( θ log f( z, θ log 2 Γ ( + A (, σ ( Z μ A(, σ ( ( ( j, J Q( θ, θ log( f ( z θ Th updat quaton of th EM algorthm ar μ σ f ( z, θ (, f z θ k ( l ( l ( l γ (, z γ (, Γ(3/ z μ Γ(/ ( l 4. Intalzaton of aramtr ( l W hav to fnd th paramtr, μ and σ for,2,---, maxmzng th lklhood functon (or Log lklhood functon. Hr th hap paramtr tmatd by th procdur gvn by J.Armando Domnguz t al (2003 and alo w aum that hap paramtr am for all mag rgon of an mag undr condraton. For obtanng th tmat of th paramtr w utlz th EM algorthm. Whr t( k z; θ ( k zj; θ f z θ hz ( j, θ (, Th xpctd valu of L(θ Followng th hurtc argumnt of Jff A. Blm (998, Th ffcncy of th EM algorthm n tmatng th paramtr havly dpndnt on th numbr of Sgmnt (Clutr ( and th ntal tmat of th modl paramtr µ, σ and (,---. Uually n EM algorthm th mxng paramtr and th rgon paramtr µ, σ ar known a pror. A commonly ud mthod n ntalzaton by drawng a random ampl n th ntr mag data (mxtur data (Mclanchan and T.rhnan(997, G.Mclanchan and D.l(2000. Th mthod can b prformd wll whn th ampl z larg, but th computaton tm alo havly ncrad, whn th ampl z mall t lkly that om mall rgon may not b ampld. To ovrcom th problm, w u -Man algorthm. Th numbr of mxtur componnt ntally takn for -Man algorthm by th htogram of th pxl ntnt of th ntr mag. Aftr dtrmnng th fnal valu of th (numbr of rgon, w obtan th ntal tmat th paramtr p µ, σ and for th th rgon ung th gmntd rgon pxl ntnt wth th mthod gvn byj.armando tal (2003

4 320 IJCSS Intrnatonal Journal of Computr Scnc and twork Scurty, VOL.7 o.4, Aprl Th Sgmntaton Algorthm Aftr rfnng th paramtr th prm tp mag rcontructon by allocatng th pxl to th gmnt. Th opraton prformd by Sgmntaton Algorthm. Th mag gmntaton algorthm cont of 3 tp Stp obtanng th ntal tmat of th Fnt Gnralzd Gauan Mxtur Modl wth -Man algorthm orgnal and th rcontructd mag of LEA, MA and MA ar hown n Fgur-. Th comparatv prformanc of varou algorthm wth rfrnc to Imag Qualty Mtrc ar gvn n Tabl- Fgur: Imag Orgnal F.Gauan F.Gnralzd Imag Mxtur Modl Mxtur Modl Stp 2 wth th ntal tmat obtand n tp, th EM algorthm tratvly carrd wth th updat quaton, th EM algorthm convrg whn th dffrnc of th old tmat and th nw tmat ar l than om thrhold valu (0.00, and th fnal tmat of th Fnt Gnralzd Gauan Mxtur Modl ar obtand. Stp 3 th mag gmntaton carrd out by agnng ach pxl nto a propr rgon (gmnt accordng to th Maxmum lklhood Etmat of th jth lmnt L j accordng to th followng quaton EM z μ EM A (, σ xp L max 2 Γ ( + A (, σ Whr μ, z ar th nput data (pxl ntnt and σ ar th tmatd paramtr rpctvly. Tabl - Imag Qualty Mtrc Man Squar Error Sgnal to o Rato ImagQualty Indx 6. Exprmntal Rult In ordr to valuat th propod mthod, w dmontratd our mag gmntaton algorthm wth Fnt Gnralzd Gauan Mxtur modl by applyng t to 3 mag namly, LEA, MA & MA. Th prformanc of th dvlopd algorthm compard by valuatng dffrnt Imag Qualty Mtrc uch a Man Squar Error, Sgnal to o Rato and Qualty Indx. Th pxl ntnt of th whol mag ar takn a nput for mag rcontructon. Th pxl ntnt n th mag ar aumd to b th mxtur of vral componnt (gmnt of th mag. In ach mag rgon th pxl ntnt follow a Fnt Gnralzd Gauan dtrbuton wth dffrnt paramtr. Th numbr of Sgmnt nd th mag dtrmnd by ung -Man algorthm. For dtrmnng th ntal valu of n th -Man algorthm, w hav obtand th htogram of th pxl ntnt. Th am of th mag F.G.M. wth -Man Modl wth -Man F.G.M. wth -Man Modl wth -Man F.G.M wth - Man LEA MA MA Modl wth - Man From th tabl-, and Fgur-, t can b obrvd that th dvlopd prform much upror to th xtng algorthm wth rpct to th mag Qualty Mtrc. Th prformanc of th Imag Sgmntaton Modl alo tud through clafr accuracy by computng th mclafcaton rat. Th mclafcaton rat of

5 IJCSS Intrnatonal Journal of Computr Scnc and twork Scurty, VOL.7 o.4, Aprl dffrnt mag namly LEA, MA and MA wth rfrnc to th dvlopd gmntaton algorthm and th Fnt Gauan Mxtur Modl wth -Man ar computd and gvn n Tabl-2 am of th Imag Tabl-2 Clafr Accuracy F.G.M. wth -Man Modl wth - Man [6] Ymng WU t al, -(2003. Unuprvd Color Imag Sgmntaton bad on Gauan Mxtur Modl rocdng of 2003 Jont Confrnc [7] Mclanchlan G. And rhnan T.(997, Th EM Algorthm and Extnon, John Wly and Son, w York [8] Mclanchlan G. tal,(2000 Th EM Algorthm For aramtr Etmaton, John Wly and Son, w York LEA MA MA From th abov tabl t can b obrvd that th accuracy of th dvlopd algorthm upror to that of th Fnt Gauan Mxtur Modl wth -Man. 7. Concluon In th papr w propod Unuprvd Imag Sgmntaton bad on Fnt Gnralzd Gauan Mxtur Modl wth EM & -Man algorthm. Th mag wa condrd a a mxtur of -Componnt Gnralzd Gauan Dnt, ung -Man algorthm th numbr of componnt n th mag ar tmatd and through EM algorthm th fnal tmat of th paramtr ar obtand. Exprmntal rult how that th propod algorthm ha bttr rtrval capablty compard to th Fnt Gauan Mxtur Modl. Rfrnc [] Armando Domnguz J t al (2003, A ractcal rocdur to Etmat th Shap aramtr n th Gnralzd Gauan Dtrbuton, IEEE Tranacton on Imag rocng [2] Cho S t al (2000 Local tablty analy of flxbl ndpndnt componnt analy algorthm. rocdng of 2000 IEEE Intrnatonal Confrnc on Acoutc, Spch and Sgnal rocng, ICASS 2000, pp , [3] Sharf. t al (995 Etmaton of hap paramtr for gnralzd Gauan dtrbuton n ubband dcompoton of vdo. IEEE Tran. On Crcut and Sytm for Vdo Tchnology Vol 5, o., pp [4] Varana M. t al (989 aramtrc gnralzd Gauan dnty tmaton, J.Acout Socty of Amrca 86(4 pp 404. [5] Yamazak T., (998 Introducton of EM algorthm nto color Imag Sgmntaton, rocdng of ICIRS 98, pp