Analysis of Geometric, Zernike and United Moment Invariants Techniques Based on Intra-class Evaluation

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1 0 Ffth Interntonl Conference on Intellgent Systes, odellng nd Sulton Anlyss of Geoetrc, ernke nd Unted oent Invrnts Technques Bsed on Intr-clss Evluton ohd Wf srudn *, Shhrul z Ykob, Roze Rzf Othn, Iszdy Isl, ohd Iln Js 6, School of Coputer nd Councton Engneerng Unverst lys Perls, Perls, lys wffy88@yhoo.co, ysnz@gl.co, roze@unp.edu.y, szdysl@gl.co, ln_kun@gl.co 6 A Slhh Abdul sr School of echtronc Engneerng Unverst lys Perls Perls, lys _slhh@yhoo.co Abstrct In ths pper, three oent nvrnts technques hve been used to extrct the shpe propertes of the ge. There re geoetrc oent, zernke oent nd unted oent nvrnts. These oent nvrnts hve been used to nlyze the ge due to ts nvrnt fetures of n ge bsed on sclng fctor nd rotton. A set of equtons known s ntr-clss nlyss hs been ppled to esure the slrty of feture vector tht represent the se object. The results obtned n ths study hve been nlyzed nd copred n ters of ntr-clss nlyss n order to fnd the best technque ong the three dfferent types of oents. Bsed on the results tht hve been obtned by usng the slr ge, t s found tht the geoetrc nd unted oent nvrnts technques re better wth sll vlues of totl percentge en bsolute error (TPAE) s copred to zernke oent nvrnts. Keywords-geoetrc oent nvrnt; zernke oent nvrnt; unted oent nvrnt; shpe nlyss; ntr-clss nlyss by Yunn n [8]. The UI s n extenson of GI tht tends to reduce the effects of sclng fctors n oent nvrnt functon becuse of the ffecton of ny fctors. In UI, the rotton, trnslton nd sclng cn be dscretely kept nvrnt to regon, closed nd unclosed boundry. The clculton of UI s qute slr to GI. enwhle, the zernke oent nvrnt (I) ws produced by Tegue n 980 [0,]. It s lso clled s zernke polynols becuse t s bsed on contnuous functons [9,]. Ths ethod hs been proposed to prove the conventonl geoetrc nvrnts for rotton purposed. Intr-clss evluton ndctes the study done wthn the se group but under the nfluence of dfferent sclng nd rottons fctors of the representng ges [,7]. A seres of equtons hs been ntroduced to nlyze between the orgnl object wth new vrous rottons nd scle object by esurng the slrtes between the. Ths evluton cn lso be used to fnd the best technque ong the vrous technques wthn the se group[]. I. ITRODUCTIO Shpe nlyss refers to the process of nlyzng rw dt nd tkng n cton bsed on the clsses of the shpe. The jor coponent n shpe nlyss s feture extrcton whch t s used for extrctng the fetures n n ge ether the structurl pproch or globl pproch. Shpe nlyss s lso prt of pttern recognton technques whch s to extrct the unque vlues fro n object tht dfferenttes t fro the other objects. For exple, n order to perfor the fce recognton, the feture extrcton s used to retreve the vlues tht represent chrcterstcs of the object. An object feture extrcton cn be done by vrous ethods such s geoetrc oent nvrnt, unted oent nvrnt nd zernke oent nvrnt. The geoetrc oent nvrnt (GI) ws frst ntroduced by Hu n 96 []. Hu ntroduced the seven functons usng lgebrc nvrnts []. Ths technque hs been chosen to perfor the feture extrcton for ge chrcterstcs becuse the rotton scle trnslton (RST) nvrnt or chrcterstc produced by ths ethod does not chnge the tretent of rotton, sclng nd trnslton [,]. The unted oent nvrnt (UI) ws ntroduced Fgure. Descrpton of the ntr-clss Fg. represents the bref descrpton of the ntr-clss. Bsed on Fg., f the vlue of totl percentge en bsolute error (TPAE) s lrge, t ens the technque s bd copred to the other technques. And f the vlue of TPAE s verge, the technque s good. enwhle, f the vlue of TPAE s sll, the technque s better thn other technques. II. THEORY OF OET IVARIATS A. Geoetrc oent Invrnt The geoetrc oent nvrnt (GI) s used for functon. The order (p+q) of f(x, y) for GI s defned n (). Bsed on (), p, q re the nteger of 0,,,., x nd y re the coordnte of the ge nd x p y p s the bsc functon [,6] / $. 0 IEEE DOI 0.09/ISS.0.9 7

2 p q x y f x, () Equton () shows the generl defnton of n ge n the sze pxels for dgtl ge where h(x,y) s pxel vlue for n ge of sze x. x y The centrl oents cn be defned by substtutng () nd () nto (). x () () () Equton (6) s used to norlzed the centrl oent n order to produce the nvrnt propertes. (6) pq The sx functons of GI cn be deterned fro the norlzed centrl oents nd re shown n (7). = (η 0 + η 0 ) = (η 0 η 0 ) + η = (η 0 η ) + ( η η 0 ) = (η 0 + η ) + (η + η 0 ) = (η 0 η )( η 0 + η )[( η 0 + η ) -(η + η 0 ) ] + ( η + η 0 )( η + η 0 )[(η 0 + η ) -( η + η 0 ) ] 6 = (η 0 - η 0 )[( η 0 + η ) -( η + η 0 ) ] + η (η 0 + η )( η + η 0 ) (7) B. Unted oent Invrnts The clcultons of UI re qute slr to GI. The defnton of sclng equton norl, dscrete nd boundry s gven n (8), (9), nd (0) respectvely re the three condtons tht relte to GI wth the effect of sclng fctors [8]. pq (8) pq ' pq pq (9) " pq (0) y dxdy p q x y h x, y x 0 y 0 y ( x x) p q ( y y) h x, y () The sx foruls for unted oent nvrnts (UI) re shown n (). () C. ernke oent Invrnts ernke oent hs been ntroduced bsed on contnung orthogonl functon clled ernke polynols. The zernke oent fro dgtl ge cn be coputed by usng () [,]. The vlue of f x, y s referrng to the pxel densty ge sze. Where n 6 n n k B k / x y x y f x, y nk x y nk n k ( )! B nk n k k k!!! () () The equtons for zernke nd geoetrc oents for rotton nd sclng fctors cn be derved fro () by substtutng fro (). Thus, the gntude n of the zernke oent cn be tken s rotton nvrnt feture of the underlyng ge functon s () [7] () III. EXPERIETATIO In ths study, the selected ge hs been used to get the rnge of the nvrnt. The ge tht hs been used for the nlyss s shown n Fg.. Ths ge hs been reszed nto the stndrd sze of 8 6 pxels. Before the ge hs been resze nto the stndrd sze, the ge hs been converted nto bnry level s shown n Fg.. Ths process s perfored n order to ese the extrcton of feture ge s the pxels of the ge regon re ssgned vlue of one nd the rest re s ssgned s zero. Then, ech ge wll llustrte nto vrous rottons nd scle s shown n Tble I. 8

3 o. b TABLE I. Orgnl ge DIFFERET ORIETATIO OF IAGE Ige Orentton Ige reszed to hlf of the orgnl ge c Ige rotted to d Ige reszed to hlf of the orgnl ge nd rotted to e Ige enlrged.x nd rotted to 0 PAE TPAE PAE PAE (8) (9) By usng the equtons for ntr-clss nlyss, the vlue of bsolute error (AE), percentge bsolute error (PAE), percentge en bsolute error (PAE) nd totl percentge en bsolute error (TPAE) of the ge clss re coputed. For ths nlyss, the vlues of AE, PAE, PAE nd TPAE for ech denson re clculted by usng () untl (9). IV. RESULTS AD DISCUSSIOS Ths secton descrbes the sets of feture vectors tht hve been produced by the three types of oent nvrnts nd ntr-clss nlyss n order to fnd the best type of oent tht cn gve the sgnfcnt results. The sx eleents of oent nvrnts of ech ethod hve been coputed to fnd feture vectors. Tbles II, III nd IV tbulted the feture vectors tht hve been produced by these three oent nvrnt technques. Fro the tble, t cn be seen tht dsslr of feture vectors hve been produced due to dfferent oent technques tht hve been used. Fgure. The result of bnry ge TABLE II. GEOETRIC OET IVARIAT FEATURE VECTOR Ø Ø Ø Ø Ø Ø6 Orgnl X X X Fgure. Bnry ge wth dfference sclng nd rotton : () Orgnl ge, (b) Ige reszed to hlf of the orgnl ge, (c) Ige rotted to, (d) Ige reszed to hlf of the orgnl ge nd rotted to, (e) Ige enlrged.x nd rotted to 0 Then, fetures vectors re coputed by usng the three oent nvrnts s entoned prevously. In order to fnd the best result, n ntr-clss nlyss whch re bsed on the shpe nd rotton hve been used. Intr-clss nlyss s the process tht defnes the slr ges wth the sllest vlue of TPAE. The seres of equtons tht re used to esure the nvrnt chrcterstcs cn be descrbed s follows: AE H F % PAE H I PAE PAE I () (6) (7) TABLE III. UITED OET IVARIAT FEATURE VECTOR Ө Ө Ө Ө Ө Ө6 Orgnl X X X TABLE IV. ERIKE OET IVARIAT FEATURE VECTOR 0 0 Orgnl X X X

4 In ths study, AE were used to copute the nvrnt chrcterstc of feture vectors. Ths s becuse AE cn defne the dfference between n orgnl dt wth new dt. Tble V nd VI hve shown the AE nd PAE for the ge wth nd dfferent types of oent. The nuber on the top colun n the tble refers to the feture denson nd t cn be seen clerly the dvntge of PAE gnst the AE. For exple, t cn be seen the AE for GI t the frst denson s sller thn UI. But when PAE s tbulted n Tble VI, the PAE for UI s sller thn GI. Bsed on ths, the UI s ore nvrnt s copred to GI. evertheless, PAE s not brngng lot of nforton needed becuse PAE s used n the nlyss perfornce oent technques under dfference perturbtons. TABLE V. AE FOR IAGE WITH AD DIFFERET TYPES OF OET 6 GI UI I Fgure. Grph of percentge en bsolute error (PAE) The results of PAE tht hve been obtned by clcultng usng (8) re shown n Fg.. The dvntge of usng the PAE s t cn deterne the error sctterng wthn the denson of feture vectors. Fro the grph, t cn be seen the GI nd UI produced sller error s copred to I. TABLE VI. PAE FOR IAGE WITH AD DIFFERET TYPES OF OET 6 GI UI I Fgure 6. Totl Percentge n Absolute Error (TPAE) Fgure. Grph of percentge en bsolute error (PAE) Fg. shows the grph of PAE tht hs been coputed by usng (7). There re two types of PAE for ths nlyss whch re PAE nd PAE. The gol of PAE s to clculte the error occurred n one object ong dfference vrtons. The grph n Fg. deonstrtes the vlue of PAE versus ge vrton. It descrbes the I genertes the hghest error ong other oents. enwhle, Fg. 6 represents the grph of totl percentge n bsolute error. The TPAE cn be coputed fro ether PAE or PAE becuse t wll produce the slr result. The TPAE s fnl process to detect the error for n ge of dfferent types of oent. Fro the grph, t cn be seen tht I hs produced hgher error s copred to GI nd UI. V. COCLUSIOS In ths pper, the feture vectors tht were receved fro three dfferent types of oent nvrnt hve been coputed for shpe nlyss n order to dentfy the technque tht hs the sllest error or better wth others ethod by usng ntr-clss nlyss. For the feture extrcton of three technques s no proble where s ll ethods cn be used to perfor feture extrcton of dgtl ge tht s ble to dstngush the chrcterstcs of the ge even f ths ge de chnges rotton scle 0

5 trnslton (RST). It cn be concluded tht the geoetrc nd unted oents re better thn zernke oent bsed on the ntr-clss nlyss. Ths hs been proven bsed on the results n Fg. for where the grph shows tht the zernke oent hs produced the hghest TPAE vlues s copred to the geoetrc nd unted oents nvrnts. REFERECES []. K. Hu, Vsul pttern recognton by oents nvrnts, IRE Trns. Inforton Theory, 8, 96, pp [] S. tr, oent nvrnts, Proc. of the IEEE, vol. 67, 979, pp [] S.. Ykob, P. Sd, nd. F. Jlos, "On Anlyss of Invrnt Chrcterstc for oent Invrnt Technques," Journl of Engneerng Reserch & Educton, vol., (6) pp. 9-. [] S.. Ykob, P. Sd nd A.H. Abdullh, Insert recognton usng fuzzy ARTAP, Proc. Intl. Conf. Robotcs, Vson, Inforton nd Sgnl Processng ROVISP,, pp []. Rzon, H. Yzd, P. Sd, A. Y.. Shkff, A.R. Sd,.R. t, S. Ycob, H. Des nd. Krthgyn, Object Detecton usng geoetrc nvrnt oent, Aercn Journl of Appled Scences, vol. 6, 6, pp [6] Belks, S. O. Shrdhr nd. Ahd, Pttern Recognton wth oent Invrnts, A Coprtve Study nd ew Results, Pttern Recognton, vol., no., 99, pp [7] P. Sd, Feture extrcton of trderk ges usng geoetrc nvrnt oent nd zernke oent coprson, Chng J. Sc.,,, pp 7-. [8] S. Ynn, L. Wejun nd W. Yuecho, Unted oent nvrnts for shpe dscrnton, IEEE Proceedngs on Interntonl Conference on Robotcs, Intellgent Systes nd Sgnl processng, vol,, pp [9] S. K. Hwng nd W. Y. K, A novel pproch to the fst coputton of ernke oents, Pttern Recognton, 9, 6, pp [0] A. Khotnzd, H. Yw Hu, Invrnt ge Recognton by ernke oents, IEEE Trnsctons on Pttern Anlyss nd chne Intellgence, Vol., y 990, pp [] J. Hnje nd. Hongqng, Degrded ge nlyss usng ernke oent nvrnts, n Acoustcs, Speech nd Sgnl Processng, 9. ICASSP 9. IEEE Interntonl Conference on, 9, pp [] Khotnzd nd H. Y. Hu, Rotton nvrnt pttern recognton usng ernke oents, n Pttern Recognton, 9th Interntonl Conference on, 988, vol., pp []. Hu, D. hfng, nd S. Huzhong, Object recognton by coplete set of pseudo-ernke oent nvrnts, n Acoustcs Speech nd Sgnl Processng (ICASSP), IEEE Interntonl Conference on,, pp [] S. J. Perntons nd P. J. G. Lsbo, Trnslton, rotton, nd scle nvrnt pttern recognton by hgh-order neurl networks nd oent clssfers, eurl etworks, IEEE Trnsctons on, vol., 99, pp. -.