On the Design of Optimal Zoning for Pattern Classification

Size: px

Start display at page:

Download "On the Design of Optimal Zoning for Pattern Classification"

Ilene Martin
6 years ago
Views:

1 On he Design of Opimal Zoning fo Paen Classificaion A. Feane, S. Impedovo, G. Pilo, C. Tullo Dipaimeno di Infomaica Univesià degli Sudi di Bai Via Oabona 4 Bai Ialy Ceno Ree Puglia - Univesià degli Sudi di Bai Via G. Peoni 5/F. Bai Ialy {segeeia@eepuglia.uniba.i} Absac Recenly, a new echnique fo zoning design has been poposed in which zoning is consideed as he esul of an opimizaion poblem. In his case he opimal zoning can be found as he one which minimizes he value of he cos funcion associaed o he classificaion. In his pape, on he basis of his echnique, he cadinaliy of he se of zones vesus he cadinaliy of he se of classes is expeimenally analyzed. The expeimenal ess, caied ou in he field of h-wien numeal chaace ecogniion, show ha he bes esuls ae obained when he numbe of zones is simila o he numbe of classes, whaeve feaue se is consideed. eywods: Classificaion, Feaue Exacion, Opical Chaace Recogniion, Zoning.. Inoducion Zoning is a diffuse saegy in hwiing ecogniion seveal zoning mehods have been poposed in he pas based on sad paiioning cieia of he paen image [-9]. Figue shows some sad zoning mehods Z nxm obained by supeimposing a (n x m) egula gid on he paen image (fo he case m=n, n=2,3,4,5). (a) Z 2x2 (b) Z 3x3 (c) Z 4x4 (d) Z 5x5 Figue. Tadiional Zoning ehods oe ecenly a new echnique fo zoning design has been poposed in which zoning design is consideed as an opimizaion poblem, zoning descipion is based on Voonoi Diagams a eal-coded geneic algoihm is used o find he opimal zoning [0]. Accoding o his echnique, his pape addesses he poblem concening he opimal numbe of zones fo a given classificaion poblem. In paicula, he mos effecive cadinaliy of he se of zones vesus he cadinaliy of he se of classes is expeimenally invesigaed. Fo his pupose, seveal expeimenal ess have been pefomed by consideing ses of h-wien numeals chaaces, consideing diffeen feaue ses. The esuls demonsae ha he bes pefomances can be obained when he cadinaliy of he se of zones is simila o he cadinaliy of he se of classes, whaeve feaue se is consideed. The pape is oganized as follows. Secion 2 pesens he poblem of classificaion by zoning. Secion 3 summaizes he new echnique fo zoning design, in which Voonoi Diagams ae used fo zoning epesenaion a geneic algoihm is used o find he opimal zoning. The expeimenal ess he esuls ae discussed in Secion 4. The conclusions ae epoed in Secion Zoning fo Paen Classificaion Le B be a paen image, a zoning Z ={z, z 2,..., z } of B is a paiion of B ino sub-images, named zones, each one poviding infomaion elaed o a specific pa of he paen [0]. Now, le be Ω ={C,..., C } a se of class labels; Z ={z, z 2,..., z } a zoning mehod; - F={f,...,f T } a se of geomeical feaues, he classificaion of a paen x can be consideed as a mapping D defined as [0,, 2]: D : S(x) Ω { } () wih: S(x)={(f iq,z q )} q=,2,,q : he descipion se of x (i q {,2,,T}, q {,2,,}, q=,2,,q), whee [2]: (f iq,z q ) indicaes ha he feaue f iq has been deeced in he zone z q ; Q is he oal numbe of feaues deeced in he paen x. : he eecion class (x is eeced if i is no possible o classify i wih high confidence). In ode o implemen he mapping descibed in (), a se X L of aining paens is consideed he feaue disibuion in each zone is evaluaed fo each class of paens duing he aining phase [0]: n i Γ C ( fi, z ) =, =,2,,; i=,2, T; =,2,,. (2) n

2 whee: o n is he numbe of aining paens belonging o he class C (i.e. n = cad{ x L X x C }); o n i is he numbe of insances of he pai (f i,z ) fo he aining paens of he class C (i.e. n = cad x X L x C, f, z S x ). i { ( ) ( )} i Successively, he lielihood ha feaue f i is deeced in zone z, fo paens of he class C is esimaed by he funcion Ψ C (f i,z ) (=,2,, ; i=,2, T ; =,2,,) defined as: ( fi, z ) ( f, z ) ΓC if ΓC ( fi, z ) > 0 ΓC = i = ΨC. (3) ( fi, z ) = 0 ohewise In he classificaion phase, he confidence value ha an unnown inpu paen x belongs o he class C (=,2,,), is given by [2, 3]: C C Q ( ) C x = Ψ ( fi, z ) = Ψ ( f, z ) S ( x ) q= i ( fi q, z q ), (4) whee S(x )={(f i q,z q )} q=,2,,q (i q {,2,,T}, q {,2,,}, q=,2,,q ) is he descipion se of x, obained by using he feaue se F he zoning Z. Now, le be [0] ( x ) = max ( ) {,2,..., } C x (5) C* ( x ) = max ( ) {,2,..., } { * } C x, (6) C* he classificaion decision D (S(x )) fo he paen x is given by [0]: C * if C* (x ) - C** (x ) > ε D (S(x )) = (7) whee ε is a suiable heshold value. ohewise, 3. A New echnique fo Zoning Design In his Secion, a new echnique ha has been ecenly poposed fo zoning design is summaized [0]. In paicula, his echnique consides zoning design as an opimizaion poblem he opimal zoning Z* ={z*, z* 2,..., z* } is found as he zoning fo which he cos funcion CF(Z ) associaed o classificaion is minimum, wih [2, 4]: CF(Z ) = η E(Z ) + Re(Z ) (8) whee E(Z ), Re(Z ) η ae especively he subsiuion eo ae, he eecion ae he cos value associaed o he eamen of an eo wih espec o a eecion. Fuhemoe, a well-suied zoning epesenaion is consideed based on Voonoi Diagams using he Euclidean Disance (ED) [5, 6]. Le B be a paen image P={p, p 2,, p } a se of disinc poins in B. The Voonoi Diagam deemined by P is he paiion of B ino zones {z, z 2,..., z }, wih he popey ha each zone z i coesponding o p i P, conains all he poins p fo which i esuls: ED (p, p i ) < ED (p, p ), p P, p p i. (9) oeove, le p be a poin fo which i esuls: ED(p, p i ) = min{ed (p, p ) p P}, i is hee assumed ha [2]: i I, I { 2,, }, p z min{ i i I }. (0) When Voonoi Diagams ae used fo zoning descipion, he zoning design opimizaion poblem can be wien as follows [0]: Find he se of poins {p*, p* 2,..., p* } so ha: CF(Z* )=min {p, p 2,..., p }CF(Z ) () wih: Z* ={z*, z* 2,, z* }, z* is he Voonoi egion coesponding o p*, =,2,, ; Z ={z, z 2,, z }, z is he Voonoi egion coesponding o p, =,2,,. Fo he opimizaion poblem () a geneic algoihm has been consideed [0, 7, 8]. The iniial populaion Pop={Φ, Φ 2,...,Φ i,...,φ Νpop } is ceaed by geneaing N pop om individuals (N pop even). Each individual is a veco Φ i = p,p 2,...,p,...,p Μ (whee each elemen p is a poin defined as p =(x,y )) ha coesponds o he zoning Z ={z, z 2,..., z }, whee each z is he Voonoi egion coesponding o p, =,2,,. Consequenly, he finess value of he individual Φ i = p,p 2,...,p,...,p Μ is aen as he classificaion cos CF(Z ), obained by (8), whee Z ={z, z 2,..., z } z is he Voonoi egion coesponding o p, =,2,,. Fom he iniial - populaion, he following geneic opeaions ae used o geneae he new populaions of individuals:

3 . Individual Selecion. In he selecion pocedue N pop /2 om pais of individuals ae seleced, accoding o a oulee-wheel saegy [9]. 2. Cossove. An aihmeic cossove is used o exchanges infomaion beween wo seleced individuals [20]. In paicula, le be p a,pa 2,...,pa, s- pa s,...,pa (2a) p b,pb 2,...,pb, s- pb s,...,pb, (2b) wo paen individuals, he wo offsping individuals of he nex geneaion ae: p a,pa 2,...,pa, s- pa s,...,pa (3a) p b,pb 2,...,pb, s- pb s,...,pb ; (3b) whee p a =α s p a s+(- α) p b (4a) s p b =α s p b s+(- α) p a ; (4b) s being α a om value in he ange [0,]. 3. uaion. A non-unifom muaion opeao is consideed, based on he muaion pobabiliy u_pob [20]. In paicula, le Φ i = p,p 2,...,p Μ be an individual p =(x,y ) an elemen of Φ i seleced fo muaion. In he fis sep, he muaion opeaos geneae a om value β [0,] accoding o a unifom disibuion selec a diecion veco d=(d x,d y ) as follows: + d = d x = (,0) (Eas), if β [0,0.25[; - d = d x = (-,0) (Wes), if β [0.25,0.50[; + d = d y = (0,) (Noh), if β [0.50,0.75[; d = d - y = (0,-) (Souh), if β [0.75,]. In he second sep, he displacemen fo p is deemined accoding o a Non-unifom uaion. In his case he value of p becomes equal o p, wih [20]: b ie = + N ie p' p d max_ displ δ, (5) whee: δ is a om value geneaed in he ange [0, ], accoding o a unifom disibuion; max_displ is he maximum displacemen allowed; ie is he coune of he geneaions pefomed; N ie is he maximum numbe of geneaions; b is a sysem paamee deemining he degee of nonunifomiy. 4. Eliis Saegy.Fom he N pop individuals geneaed by he above opeaions, one individual is omly emoved he individual wih he minimum cos in he pevious populaion is added o he cuen populaion [9, 20]. Seps fom () o (4) ae epeaed unil N ie ve populaions of individuals ae geneaed. When he pocess sops, he opimal zoning is obained by he bes individual of he lasgeneaed populaion. (a) Feaue Se F (b) Feaue Se F2 Conou Pofiles: LP(): Lef pofile (-h ow) ; RP(): Righ pofile (-h ow) LAX, LIN: maximum minimum in he lef pofiles RAX, RIN: maximum minimum in he igh pofiles RPEAAX, RPEAIN: maximum minimum peas in he igh pofiles WAX, WIN: maximum minimum in weigh Inesecion wih lines Exema Poins Coss Poins Figue 2. Feaue Ses 4. Expeimenal Resuls Fou goups of expeimens have been caied ou, using diffeen ses of paen classes feaues. Two ses of paen classes ae consideed: Ω ={0,,2,3,4,5,6,7,8,9}: he se of 0 numeal digis. The daa ses of he CEDAR daabase ae used in his case [2]: 8467 paens fo aining (BR diecoy) 289 paens fo esing (BS diecoy). Ω 2 ={A,B,C,D,E,F,G,H,I,J,,L,,N,O,P,Q,R,S,T,U,V,W,X,Y,Z}: he se of 26 English chaaces. Fo he expeimens wih h-wien chaaces he daa ses of he ETL daabase ae used [22]. oe pecisely, paens fo aining (45 samples fo each class) 7800 paens fo esing (300 samples fo each class) have been consideed. Two feaue ses, F F 2, ae exaced fom he nomalized paen images (72x54 pixel images). The feaues ae vey common well-nown in he lieaue [2, 3, 7, 9, 0, 23], hus only a bief descipion is given hee (see Fig. 2): F ={f,...,f 9 }, whee: f : holes;

4 f 2, f 3,, f 4, f 5 : veical-up, veical-down, hoizonal-igh, hoizonal-lef caviies; f 6, f 7,, f 8, f 9 : veical-up, veical-down, hoizonal-igh, hoizonal-lef end-poins. F 2 = {f,...,f 57 }, whee: f, f 32 : Conou pofiles. Fom he fou main pofiles of he paens (op, boom, lef, igh pofiles) he following feaues ae exaced (see [23] fo he complee descipion): f,, f 8 : he locaions of maxima minima in he pofiles f 9,,,f 24 : he locaions of maximum minimum peas in he pofiles f 25, f 28 : he locaions of maxima minima in he heigh f 29, f 32 : he locaions of maxima minima in he widh; f 33,,f 52 : Inesecion wih saigh lines: f 33,,f 37 : inesecions wih five veical lines f 38,,f 42 : inesecions wih five hoizonal lines f 43,,f 47 : inesecions wih five diagonal lines (+45 ) f 48,,f 52 : inesecions wih five diagonal lines (-45 ); f 53, f 54, f 55, f 56: Top, boom, lef, igh exema poins ; f 57 : Coss Poins. Noe ha feaues f,f 2,,f 9 of F f 33, f 34,,f 57 of F 2 ae exaced fom he seleon of he paens, obained by he Safe Poin Thinning Algoihm [24], wheeas feaues f,f 2,,f 32 of F 2 ae exaced diecly fom he binay image of he paens. Naually, befoe conducing he expeimens, he mos suiable geneic opeaos he fee-paamee values of he opimizaion poblem have been pe-esimaed by pefoming some pilo ess [25, 26]. The seleced paamees ae: N Pop =0, N ie =000, u_pob=0.05, max_displ=3, b=.0. Figue 3 shows an example of how finess impoves wih successive geneaions. Cos Funcion 0,4 0,3 0,2 0, Figue 3. Finess impovemen vs successive geneaions Using hese paamees, he opimal zonings Z* fo diffeen values of (=4, 9,6,25) have been obained, as Figue 4 shows. The pefomances of hese zoning mehods on he aining se on he es se ae also epoed in Figue 5, in which hey ae compaed o he adiional zoning mehods in Figue (Z 2x2, Z 3x3, Z 4x4, Z 5x5 ). The esuls poin ou ha he opimal zonings always oupefom adiional zoning mehods. Fuhemoe, when he ie se of 0 h-wien numeal digis is consideed, he bes esuls ae obained by paiioning he paen image ino =9 zones. In his case, in fac, he opimal zoning Z* 9 leads o a ecogniion ae up o 97% (96%) an eo ae up o 2% (3%) when he aining se (es se) is used. Convesely, when he se of 26 h-wien chaaces is consideed, he bes esuls ae achieved fo =25. In his case, in fac, he opimal zoning Z* 25 leads o a ecogniion ae up o 93% (92%) an eo ae up o 2% (2%) when he aining se (es se) is used. (A) (B) (C) (D) Figue 4. The Opimal Zoning Z * 9 fo: (A) Class Se: Ω - Feaue Se: F (B) Class Se: Ω - Feaue Se: F 2 (C )Class Se: Ω 2 - Feaue Se: F (D) Class Se: Ω 2 - Feaue Se: F 2 5. Conclusion This pape addesses he poblem of he opimal numbe of zones o be used fo a specific classificaion poblem. Fo his pupose, a new echnique is consideed fo zoning design ha consides zoning as he esul of an opimizaion poblem. The expeimenal esuls, obained in he field of hwien numeal chaace ecogniion, have been achieved by using diffeen feaue ses. They show ha bes esuls ae obained when he cadinaliy of he se of zones is simila o he cadinaliy of he se of classes, whaeve feaue se is used. Refeences [] S. oi, C.Y.Suen,.Yamamoo, Hisoical Review of OCR eseach developmen, Poc. IEEE, Vol. 80, pp , 992. [2] O.D.Tie, A..Jain, T.Tax, Feaue Exacion ehods Fo Chaace Recogniion A Suvey, Paen Recogniion, Vol. 29, n.4, pp , 996. [3]. Bose, Omnidocumen Technologies, Poc. IEEE, Vol. 80, 992, pp [4] R. Plamondon, S.N. Sihai, On-line Off-line Hwiing Recogniion: A compehensive suvey, IEEE Tans. Paen Analysis achine Inelligence, Vol. 22, n., pp , [5] C.Y.Suen, C. Nadal, R.Legaul, T.A.ai, L.Lam, Compue Recogniion of unconsained hwien numeals, Poc. IEEE, Vol. 80, pp , 992. [6] C.Y.Suen, J.Guo, Z.C.Li, Analysis Recogniion of Alphanumeic Hpins by Pas, IEEE Tans. Sysems, an Cybeneics, Vol. 24, n. 4, pp , 994. [7] F. imua,. Shidha, Hwien Numeical Recogniion Based on uliple Algoihms, Paen Recogniion, Vol. 24, n. 0, pp , 99. [8] G. Dimauo, S. Impedovo, R. odugno, G. Pilo, Numeal ecogniion by weighing local decisions, Poc. In l. Documen Analysis Recogniion 2003, pp , Edinbugh, U, Aug [9] G.Bapisa,..ulani, A high accuacy algoihm fo ecogniion of h-wien numeals, Paen Recogniion, Vol.4, pp , 988.

5 Taining Se Tes Se (A) Class Se: Ω Feaue Se: F 2 2 Z 2x2 Z* 2 4 Z 3x3 3 Z* 4 9 Z 4x4 5 Z* 6 6 Z 5x5 7 Z* 8 25 Z 2x2 Z* 2 4 Z 3x3 3 Z* 4 9 Z 4x4 5 Z* 6 6 Z7 5x5 Z* 8 25 (B) Class Se: Ω Feaue Se: F2 2 Z 2x2 Z* 2 4 Z 3x3 3 Z* 4 9 Z 4x4 5 Z* 6 6 Z7 5x5 Z* Z 2x2 Z* 2 4 Z 3x3 3 Z* 4 9 Z 4x4 5 Z* 6 6 Z7 5x5 Z* 8 25 (C) Class Se: Ω2 Feaue Se: F 2 2 Z 2x2 Z* 2 4 Z 3x3 3 Z* 4 9 Z 4x4 5 Z* 6 6 Z7 5x5 Z* 8 25 Z 2x2 Z* 2 4 Z 3x3 3 Z* 4 9 Z 4x4 5 Z* 6 6 Z7 5x5 Z* 8 25 (D) Class Se: Ω2 Feaue Se: F2 2 2 Z 2x2 Z* 2 4 Z 3x3 3 Z* 4 9 Z 4x4 5 Z* 6 6 Z7 5x5 Z* 8 25 Z 2x2 Z* 2 4 Z 3x3 3 Z* 4 9 Z 4x4 5 Z* 6 6 Z7 5x5 Z* 8 25 Figue 5. Expeimenal Resuls [0] Lucchese.G., Impedovo, S. Pilo, G., Opimal Zoning Design by Geneic Algoihms, IEEE Tans. Sys. en Cyben. - Pa A, Vol. 36, Issue: 5, Sep. 2006, pp [] L.I.uncheva L.C.Jain, Designing Classifie Fusion Sysems by Geneic Algoihms, IEEE Tans. Evolu. Compu., Vol. 4, No.4, [2] V. Di Lecce, G. Dimauo, A. Gueieo, S. Impedovo, G. Pilo, A.Salzo, Zoning Design fo Hwien Numeal Recogniion, Poc. IWFHR 7, pp , [3] L.Xu, A. zyza C.Y.Suen, ehods of Combining uliple Classifies Thei Applicaions o Hwiing Recogniion, IEEE Tans. Sysems, an Cybeneics, Vol.22, No.3, pp , ay/june 992. [4] L. Lam, Y.-S. Huang, C.Y. Suen, Combinaion of uliple Classifie Decision fo Opical Chaace Recogniion, in Hboo of Chaace Recogniion Documen Image Analysis, H. Bune P.S.P.Wang (eds.), Wold Scienific Publ., Singapoe, 997, pp [5] F. Auenhamme, Voonoi Diagams: A Suvey of a Fundamenal Geomeic Daa Sucue, AC Compuing Suveys, Vol. 3, n. 3, pp , 99. [6]. de Beg, O. Schwazopf,. van eveld,. Ovemas, Compuaional Geomey: Algoihms Applicaions, Spinge-Velag, Belin, [7] D.Beasley, D.R.Bull, R.R.ain, An Oveview of Geneic Algoihms: Pa, Fundamenals, Univesiy Compu.,Vol 5,n.2,pp , 993. [8] D. Beasley, D.R.Bull, R.R.ain, An Oveview of Geneic Algoihms: Pa 2, Reseach Topics, Univesiy Compuing, Vol. 5, n. 2, pp. 70-8, 993. [9] T. Baec, Evoluionay Algoihms in Theoy Pacice: Evoluion Saegies, Evoluion Pogamming, Geneic Algoihms, New Yo: Oxfod Univ. Pess, 996. [20] Z. ichalewicz, Geneic Algoihms + Daa Sucue=Evoluion Pogams, Spinge Velag, Belin, Gemany, 996. [2] J. Hull, A daabase fo hwien ex ecogniion eseach, IEEE Tans. PAI, Vol. 6, n. 5, pp , 994. [22] T.Saio, H.Yamada,.Yamamoo, On he daa base ETL 9 of hpined chaaces in JIS chinese chaaces is analysis, IEICE Tans. Fund. Elec. Comm. Comp. Sciences, Vol. J68-D(4), pp , 985. [23] L. Heue, T. Paque, J.V. oeau, Y. Lecouie, C. Olivie, A Sucual / Saisical Feaues Based Veco fo Hwien Chaace Recogniion, Paen Recogniion Lees, Vol.9, pp , 998. [24] N.J. Naccache, R. Shinghal, SPTA: A poposed algoihm fo hinning binay paens, IEEE Tans. SC, Vol. 4, n.3, pp , 994. [25] J.J.Gefensee, Opimizaion of conol paamees fo geneic algoihms, IEEE IEEE Tans. SC, Vol. 6, n., pp , 986. [26] T. Bac, D. Fogel, Z. ichalewicz (Eds), Hboo of Evoluionay Compuaion, New Yo: Insiue of Physics Publishing Ld., Bisol Oxfod Univesiy Pess, 997.

Computer Propagation Analysis Tools

Computer Propagation Analysis Tools Compue Popagaion Analysis Tools. Compue Popagaion Analysis Tools Inoducion By now you ae pobably geing he idea ha pedicing eceived signal sengh is a eally impoan as in he design of a wieless communicaion