Integral equation mei applied to three-dimensional arbitrary surfaces

In Table 1, we show he aul co e age as he numbe o unscanned lip- lips in ci cui S5378 inc eases. As shown in Table 1, he aul co e age does no d op signi ican ly un il as many as 50 lip- lops a e scanned. Table 2 Faul co e age wi h unscanned lip- lops To al O iginal Sel Unscanned Unscanned FFs co e age loop FFs a io Ci cui Faul co e age I S27 13 S298 114 I 100 3 2 66.7 100 100 14 2 14.3 93.57 In Table 2, we show he aul co e age o each ci cui wi h he gi en numbe o lip- lops unscanned. The hi d column gi es he ull scan aul co e age using ou implemen a ion o he o iginal FAN algo i hm. The i h column gi es he numbe o unscanned lip- lops. The se en h column gi es he aul co e age. We s opped unscanning when he aul co e age d opped signi ican ly. When he ci cui con ains enough sel -loop ee lip- lops, he p o- posed pa ial scan selec ion me hod allows mo e han 20% lip- lops o be unscanned wi h li le dec ease in he aul co e age. Fo he 838 ci cui , wi h only one unscanned lip- lop, he aul co e age d opped om 100 o 91.74%, bu when we assumed he p ese signal a he unscanned lip- lop, we we e able o ob ain 100% aul co e age. S382 s444 Conclusion: We ha e in oduced an in e es ing pa ial scan es me hod wi h no da a holding o e head. The expe imen al esul s show ha he p oposed me hod can achie e almos ull scan aul co e age while ha ing a signi ican numbe o lip- lops unscanned. The p oposed es me hod will be mos e ec i ely used o ci cui s in which he a ea occupied by memo y elemen s domi- na es. These kinds o ci cui s a e p e alen in mul imedia applica- ions. We will s udy he pa ial scan selec ion algo i hm mo e and will imp o e he ATPG algo i hm by adop ing a ious echniques o mo e ad anced ATPGs [l, 21. 21 100 15 4 19.0 95.32 21 97.34 15 3 14.3 90.11 0 IEE 1997 Elec onics Le e s Online No: 19971414 Dong Ho Lee and Hyun Chul Noh (School o ’ Elec onics and Elec ical Enginee ing, KyungPook Na ional Uni e si y, Tuegu, 701- 702, Ko ea) 26 Augus 1997 Re e ences POMERANZ, I., REDDY, L.N., and REDDY, s.M.: ‘COMPACTEST: A me hod o gene a e compac es se s o combina ional ci cui s’. P oc. In . Tes Con ., 1991, pp. 194203 SCHULZ, M., TRISCHLER, E., and SARFERT, T.: ‘SOCRATES: A highly e icien au oma ic es pa e n gene a ion sys em’. P oc. In . Tes Con ., 1987, pp. 101&1026 CHENG, K.T., and AGRAWAL, .D.: ‘A pa ial scan me hod o sequen ial ci cui s wi h eedback‘, IEEE T ans., 1990, C-39, pp. 546548 LEE, D.H., and REDDY, s.M.: ‘On de e mining scan lip- lops in pa ial scan designs’. P oc. In . Con . Compu e -Aided Design, No embe 1990, pp. 322-325 PARIKH, P.s., and ABRAMOVICI, M.: ‘Tes abili y-based pa ial scan analysis’, J. Elec on. Tes ., 1995, 7, pp. 61-70 FUJIWARA, H., and SHIMONO, T.: ‘On he accele a ion o es gene a ion algo i hms’. P oc. 13 h In . Symp. Faul Tole an Compu ing, 1983, pp. 98-105 In eg al equa ion ME1 apiplied o h ee- dimensional a bi a y su aces J.M. Rius, J. Pa on, E. Ubeda and J.R. Mosig Indexing e m: Me hod o mom m s, Bounda y elemen me hod, Nume ical me hods, Elec omagne ic wa e sca e ing The au ho s p esen a new o mula ion o he in eg al equa ion o he measu ed equa ion o in a iance as a con ned ield in eg al equa ion disc e ised by he me hod o momen , in which he use o nume ically de i ed es ing unc ions esul s in an app oxima ely spa se linea sys em wi h s o age memo y equi emen s and a CPU ime o compu ing he ma ix coe icien s p opo ional o he numbe o unknowns. In oduc ion: Bounda y elemen me hods (BEMs) a e widely used o he nume ical analysis o elec omagne ic adia ion and sca - e ing. Howe e , hei applica ion is limi ed o elec ically small o esonan size objec s due o he ac ha he compu a ional equi emen s inc ease apidly wi h 1.he elec ical size. One ecen app oach o achie ing mo e e icien BEMs is he in eg al equa- ion o mula ion o he measu ed equa ion o in a iance (IE-MEI) [l - 31. This Le e p esen s he o mula ion o he IE-ME1 o 3D a bi- a y sca e ing su aces as a special case o a combined ield in e- g al equa ion (CFIE), disc e ised Iby he me hod o momen s (MOM), in which he choice o di e en es ing unc ions o he elec ic and magne ic ields esul s i i an app oxima ely spa se lin- ea sys em o sol e o he induced cu en , whe e mos o he ma ix elemen s can be neglec ed. These new es ing unc ions a e nume ically de i ed by a p ocedu e bo owed om he measu ed equa ion o in a iance (MEI) me hod [4] o iginally de eloped o nume ically ind he unca ion bounda y coe icien s o ini e di - e ence and ini e elemen meshes [5]. A signi ican ea u e o nume ically de i ed es ing unc ions is ha hey a e adap i e, i.e. speci ic o he pa icula shape o he sca e e bounda y and o he loca ion o he unc ion in he bounda y. Fo mula ion: The elec ic ield in eg al equa ion (EFIE) and he magne ic ield in eg al equa ion (MIFIE) disc e ised by he MOM may each be exp essed in ma ix o m, espec i ely, as I di a e he (subdomain) basis unc ions, J’, = Zi c, 2, and Gm a e he (subdomain) es ing unc ions, he ma ix elemen s in eqn. 1 a e em = (Gm,Ei) zEi = (Gm,lA~~C ) d,i = (Gm,.i;Z) whe e LEJ and LHJ a e, espec i ely, he linea ope a o s ha ob ain he elec ic and magne ic ields due o an elec ic cu en . We le -mul iply he EFIE and he MFIE in eqn. 1 by wo a bi- a y ma ices [A] and [B], espec i ely, and add he wo equa ions o o m a CFIE -[E9 = [ZE][C] - [W] = ([ZH] - [D]) [C] (I) +. +. hk = (Gm, i x E L) z,”; = (Gm, i x LHJGZ) (2) -[AI[@] - [Bl[Hil = ([Al[ZEl + [Bl[ZHl - [BI[Dl) [Cl (3) Le -mul iplica ion o he EFIE o he MFIE by a ma ix is equi - alen o changing he es ing unc ions. Fo ins ance: m m I sugges s ha he new es ing unc ions a e Zm a,,, $, o he EFIE and Zm b,,, G, o he MFIE. The ows o [A] and [B] ma ices a e he coe icien s o he new es ing unc ions expanded in he old ones w,. Spa se [A], [B] ma ices ep esen subdomain es ing unc ions: he ze os co espond o o iginal es ing unc ions (col- LUIUI index) ha do no o e lap wi h he new ones ( ow index). We de me he ‘combined impedance ma ix’ as [Zq = [A][Zq + [B][Zq. Now, eqn. 3 becomes -- -[AlP 1 - PI[W = ELECTRONICS LETTERS 20 h No embe 1997 Vol. 33 No. 24 2029 whe e ma ix [Zc] is ull, [D] is spa se (because 2 and 2, a e sub- domain unc ions) and [A] and [B] a e a bi a ily chosen as spa se and such ha [Zq[Cl = 0. The e o e, eqn. 5 can be app oxima ed by he spa se sys em o equa ions whe e [Cj is he unknown ec o . The abo e is he 3D o mula ion o he IE-MEI, equi alen o a CFIE- ype equa ion wi h di e en es ing o he EFIE and he MFIE, such ha he elec ic and magne ic impedance ma ices cancel each o he . The 2D o mula ion o he IE-ME1 [I, 21, [q = [B]-' [A][EI + [w, is a special case o eqn. 6 wi h [D] = [I] (iden i y ma ix), due o he use o pulse basis and del a es ing. [AI[E"l + PI[ 7 = [BI[DI[Cl (6) Coe Jicien compu a ion (MEIme hod): In eqn. 5, we need [Zq[q 0. Since [E] = [Zq[q and [PI = [Zq[q, whe e [E] and [PI a e, espec i ely, he disc e isa ion o elec ic and magne ic sca - e ed ields, i ollows ha [Zq[q = [A][E] + [BI[Hsl = 0. To nume ically ind he coe icien s o ma ices [A] and [B], he ME1 me hod en o ces [A][E;] + [B][H;] = 0, whe e [E;] = [Z7[opl,, [Hi] = [Zq[o] a e he disc e isa ion o he ields due o P a bi a y cu en s %p called me ons. [CY J is a ma ix whose columns a e he disc e isa ion o he me ons 2 o p = 1, ..., P. The esul is a se o linea sys ems, one o each ow o he ma ix equa ion [Al[ZE1[%l + [Bl[aHI[%l = [ZC1[.Pl = 0 (7) whe e he unknowns a e he elemen s o [A] and [B] ma ices. Ob iously, eqn. 7 does no gua an ee ha [Zc} [q 0, especially I he sys ems (eqn. 7) a e o e de e mined and canno be exac ly sa is ied. Howe e , i has been obse ed ha in he 2D case, some me on se s p oduce a negligible esidual [Zq[q in he IE-ME1 eqn. 6 [1, 2, 6, 71 o in he equi alen unca ion bounda y condi- ion [4, 81. Me ons: Fo 3D a bi a y sca e e su aces, he easies me ons o implemen a e del a me ons. Each del a me on is equal o a basis unc ion, .', = zp, and hus [CYJ = [I]. Wi h his se o me - ons, eqn. 7 becomes [Zq = 0. Row n o eqn. 7 co esponds o he linea sys em CL,,^:, + bnmzii) = z,", = 0 i = 1, ..., N (8) m whe e he non-ze os o a,, and b,, a e he unknowns. To a oid he i ial solu ion, one o he coe icien s mus be a bi a ily se , o ins ance a,, = 1. Since i is desi able ha ma ices [A] and [B] a e spa se, he numbe o unknowns in eqn. 8 is much smalle han N. Con e sely, he e a e N del a me ons, and hus N equa- ions in he sys em eqn. 8. This makes he sys em o e de e mined and, in gene al, i canno be sol ed exac ly, [Z,q # 0. The mini- mum ~~[Zn~~~* is ob ained by he leas squa es p ocedu e. I mus be no ed ha del a me ons a e equi alen o he G* me ons o Jb ic and Lee [SI, aking in o accoun he ac ha he combined impedance ma ix Zc plays he ole o he null- ield En",, in [8]. 2.5 al m 0. - 2 1.5 c ._ I . 7x 1 .o ' c 0 0.5 . 0' I -200 -1 00 0 100 200 e 187411 Fig. 1 Induced cu en in a sphe e o diame e 2h o inciden plane wa e IE-ME1 esul s compa ed wi h analy ical solu ion analy ical IE-ME1 ____ Wi h del a me ons, he ope a ion coun o ob ain [A] , [B] ma ices is p opo ional o Nz: Fo each ow o [A] and [B], a sys- em o N simul aneous equa ions (eqn. 8) mus be sol ed by leas squa es wi h an ope a ion coun o he o de o aN x 1Mz + /3M, whe e M is he numbe o a,,, On, coe icien s o compu e. Howe e , we ha e ound ha o cing z,", = 0 only o he P - 2M basis unc ions i closes o es ing unc ion n leads o he same alues o coe icien s a,,, ba n. Tha is, o cing z,"I = 0 only o nea ield mu ual impedances makes i also ue o a ield ones. Now, he numbe o simul aneous equa ions P does no inc ease wi h he numbe o unknowns N: we need aP x Mz + pM3 = AF ope a ions o each ow o [A], [B] and he o al ope a ion coun is hus p opo ional o N. Resul s: The p oposed IE-ME1 echnique has been applied o se - e al sca e ing 3D p oblems. I has been ound ha he accu acy ob ained wi h ou cu en o mula ion is p oblem-dependen , which con iis ha he selec ion o he me on se is he c i ical s ep. As a ypical-case example, Fig. 1 shows he cu en induced in a sphe e o diame e 2 h by a plane wa e inciden along he z axis, wi h he elec ic ield pola ised along he 2 di ec ion. The IF- ME1 esul s a e compa ed wi h he analy ical solu ion. The plo co esponds o a cu o J, in he yz plane, wi h he o igin a he cen e o he sphe e. The su ace has been disc e ised in 2048 i- angles wi h sides anging om 0.1 h o 0.15 h. The numbe o unknowns o Rao, Wil on & Glisson basis unc ions [9] is N = 3072. The spa si y o [A], [B] ma ices is 1.2%. The o al amoun o memo y used is 6MB. The accu acy o he IE-ME1 esul is easonably good, bu poin wise accu acy could no be enough o some angles. Fo hese cases, a subs an ial imp o emen could ce ainly be ob ained by using a di e en se o me ons ha p oduces a smalle esidual [ZTCl. Acknowledgmen s: The au ho is indeb ed o K.K. Mei o Ci y Uni e si y o Hong Kong o he aluable discussions on he ME1 me hod. This wo k was pa ially suppo ed by he Spanish 'Comisibn In e minis e ial de Ciencia y Tecnologia' (CICYT) unde p ojec TIC 95-0983. Pa o he wo k was done a Ci y Uni e si y o Hong Kong, whe e J.M. Rius was a isi ing ellow om Janua y 4 o Feb ua y 3, 1997. G adua e s uden s J. Pa bn and E. Ubeda a e suppo ed by he Gene ali a de Ca alunya, Comissiona pe a Uni e si a s i Rece ca, unde g an s 1997 FI 00679 and 1997 FI 00747, espec i ely. 0 IEE 1997 Elec onics Le e s Online Nol: 19971419 J.M. Rius, J. Pa on and E. Ubeda (Depa men o TSC, Ed$ci 0-3, Uni e si a Poli &cnica de Ca alunya, Jo di Gi ona 1-3, 08034 Ba celona, Spain) 5 Augus 1997 J.R. Mosig (LEMA, ELB, Ecole Poly echnique FkdP ale de Lausanne (EPFL), CH-IO15 Lausanne, Swi ze land) Re e ences 1 RIUS, J.M.: 'In eg al o mula ion o he measu ed equa ion o in a iance', Elec on. Le ., 1996, 32, (l), pp. 23-25 2 RIUS, J.M., POUS, R., and CARDAMA, A.: 'In eg al o mula ion o he measu ed equa ion o in a iance: a no el spa se-ma ix bounda y elemen me hod', ZEEE T ans. Magn., 1996, 32, (3), pp. 962-967 3 RIUS, J.M., CARPINTERO, c.P., CARDAMA, A., and MOSIG, J.R.: 'F quency ex apola ion in he in eg al equa ion MEI', Elec on. Le ., 1996, 32, (25), pp. 2324-2326 4 MEI, KK., POUS, R., CHEN, z., LIU, Y.W, and PROUTY, M.D.: 'The measu ed equa ion o in a iance: a new concep in ield compu a ions', IEEE T ans. An ennas P opag., 1994, 42, (3), pp. 320-328 abso bing bounda y condi ions', IEEE T ans. An ennas P opag., 1995, 43, (5), pp. 478437 6 RIUS, J.M., CARPINTERO, c.P., CARDAMA, A., and MOSIG, J.R.: 'The heo e ical e o in he in eg al equa ion MEI', Elec on. Le ., 1996, 32, (23), pp. 2131-2132 7 RIUS, J M., CARPINTERO, c P , CARDAMA, A., and MICHALSKI, K.A.: 'Analysis o elec ically la ge conca e sca e e s wi h he in eg al equa ion MEI', Mic ow. Op . Technol. Le ., 1997, 14, (5), pp. 287- 289 5 STUPFEL, B., and MITTRA, R.: 'A heo e ical s udy O nume ical 2030 ELECTRONICS LETTERS 20 h No embe 7997 Vol. 33 No. 24 8 JEVTIC, J.O., and LEE, R.: 'A heo e ical and nume ical analysis o he measu ed equa ion o in a iance', IEEE T ans. An ennas P opag., 9 RAO, s.M., WILTON, D.R., and GLISSON, A.w.: 'Elec omagne ic sca e ing by su aces o a bi a y shape', IEEE T ans. An ennas P opag., 1982, 30, (3), pp. 409418 1994, 42, (8), pp. 1097-1105 Measu ing image edge de ec o accu acy using ealis ically simula ed edges R.C. S aun on Indexing e ms: Edge de ec ion, Image p ocessing Edge de ec o es s using edges de i ed om he acquisi ion sys em's edge sp ead unc ion a e desc ibed. The es edges used a e he mos di icul ha could exis in a p ac ical sys em. The esul s we e signi ican ly di e en o hose ob ained by adi ional me hods, indica ing ha simple de ec o s may su ice o some applica ions. In oduc ion: The analogue on end componen s o a digi al imaging sys em lowpass il e he signals hey p ocess. The cam- e a's lens and i s cha ge coupled de ice (CCD) a ay a e wo- dimensional (2D) il e s, whe eas he came a elec onics and he digi ise 's an ialiasing d e a e one-dimensional (ID). Thei com- bined e ec , he on end modula ion ans e unc ion (MTF), can be measu ed in 2D [I] and has been used he e o enable he simula ion o ealis ic edges on which o es and compa e de ec- o s. Edge model: S ep edges a e conside ed o be he mos di icul edges o de ec accu a ely [2] and, a e modi ica ion o allow o a pixel shaped windowing o he da a while sampling, ha e been used o es de ec o s [3, 41. He e, he windowing is eplaced by a ealis ic physical model o modi ying he s ep edge based on he sys em's edge sp ead unc ion (ESF). The ESF can be measu ed uniquely o each sys em and is one s ep in he calcula ion o he MTF. Resul s om he wo edge models ha e been compa ed he e, as ha e he g adien and angula accu acy o a numbe o commonly used i s -o de de ec o s. The high esolu ion ESF was scanned e ically h ough a 2D space o p oduce an e ec i ely con inuous image o an edge. A sampling g id was hen applied a en housand andom posi ions and o ien a ions wi h espec o ~ A 60 50 40 30 20 signal- o-noise a io, dB Fig. I E ec o noise on SD o g adien es ima ion (each mean alue scaled o 100) (i) s anda d cubic ace 5 x 5 (ii) Sobel 3 X 3 (iii) P ewi 5 X 5 (i ) Sobel 5 X 5 ( ) In eg a ed di ec ional de i a i e (IDD) 5 X 5 ( i) IDD 7 X 7 Edge con as = 255 uni s ELECTRONICS LETTERS 20 h No embe 1997 Vol. his edge. The esampled images we e cons ained so ha he edge always passed h ough he cen al pixel, and we e used o p o ide accu a e es da a. The new edge model was ob ained om he ID ESF by com- bining he esul s om many CCD elemen s, and he es da a we e easonably noise ee. Gaussian dis ibu ed andom noise was added o he sampled in ensi y alues. The signal- o-noise a io was de ined as SNR = 2010g,, (5) [dB] CT whe e c = 255 is he edge con as and CJ he s anda d de ia ion (SD) o he noise. L ioo .- I .- a l I ( i) _,., ," I ," 60 50 40 30 20 signal- o-mise m io 1764/2/ Fig. 2 E ec o noise on SD o angle ewe T aces labelled as in Fig. 1. Edge con as = 255 uni s Ope a o pe o mance: Fo i s de i a i e de ec o s, e o s occu in g adien and angle measu emen s. Pai s o de ec o s we e used o e u n ho izon al and e ical g adlien ec o s. These we e ana- lysed in he usual way o p o ide he magni ude o he edge g adi- en and he angle in o ma ion. The 2D MTF o he acquisi ion sys em [I] was consul ed in choosing he mos sui able ESF, he sha pes and hence mos di icul o de ec , on which o es he ope a o s. The ope a o empla es aned in size om 3 x 3 o 7 x 7 pixels and a e desc ibed in [4]. A 1,a ge empla e should ha e a highe noise immuni y and p oduce mo e accu a e esul s, bu may ake longe o compu e. 10' a ?i C .- I '5 U E U m c L OI - ioo 60 50 4CI 30 20 signal- o-noise a io, dB /7Mi3] Pig. 3 Compa ison o e ec o noise on SD o g adien es ima ion o wo edge models (each mean alue scaled o 100) Pixel a e age model: (i) Sobel 3 x 3 (ii) IDD 5 x 5 (iii) IDD 7 x 7 ESF model: (i ) Sobel 3 X 3 ( ) IDD 5 X 5 ( i) IDD 7 X 7 33 No. 24 203 1