FORMALIZATION OF THE OBJECT CLASSIFICATION ALGORITHM

Transcription

1 DOI.7/s Cybereics ad Sysems Aalysis, Vol. 5, No. 5, Sepember, 25 FORMALIZATION OF THE OBJECT CLASSIFICATION ALGORITHM T. B. Maryiuk, A. V. Kozhemiako, ad L. M. Kupershei 2 UDC 4.93' Absrac. A algorihm for objec classificaio usig he crierio of he maximum of discrimia fucios is cosidered. A special feaure of his algorihm is parallel processig over he colums of he marix ha cosiss of elemes of discrimia fucios. This algorihm is represeed i erms of Glushkov s sysem of algorihmic algebras. Keywords: objec classificaio, Glushkov s sysem of algorihmic algebras, processig by differece slices, discrimia fucio. INTRODUCTION A umber of fudameal sudies explicily describe o oly he basic priciples of Glushkov s sysems of algorihmic algebras (SAA) [ 5], bu also he heoreical fudameals of he formaio of heir varieies ad cloes. They also propose varias for he use of he basis of Glushkov s SAA o represe specific iformaio processig algorihms [6 ]. These publicaios coiue he cycle of sudies i daa arrays sorig [, 2 5] ad i oher ypes of muliprocessig such as muliprocessig of vecor daa arrays by differece slices (DS) [6]. I [7], i he basis of a modified Glushkov s SAA, varias of mulioperad summaio (covoluio) of elemes of a vecor array wih respec o DS are represeed, ad (as a exesio of such processig), varias of hreshold processig of a vecor array by DS for implemeaio of hreshold euro model [8, 9]. I he paper, we propose a DS-based objec classificaio algorihm, which assumes parallel processig of elemes of he marix of weighed compoes of he ipu aribue vecor, which expads he fucioal capabiliies of he classificaio process due o formig he vecor of raks of discrimia fucios [2] simulaeously wih he classificaio vecor. We prese a improved algorihm for objec classificaio i erms of Glushkov s SAA. The purpose of he sudy is o reveal he fucioal capabiliies of he compac descripio, i erms of Glushkov s SAA, of he objec classificaio algorihm o he basis of DS. SPECIAL FEATURES OF PROCESSING OF ELEMENTS OF DISCRIMINANT FUNCTIONS BY DS Discrimia fucios (DF) are fucios separaig domais (cofiguraios) of classes i aribue space of objecs (images) durig heir classificaio [2]. Based o he aribue of he maximum of oe of m DFs, where m is he umber of classes, he membership of a image, represeed by he ipu aribue vecor, o a specific class is deermied [22]. Thus, if m DFs are formed, g( X ) wx w2x2 wx, () gm ( X ) wmx wm2x2 wmx, Viisa Naioal Techical Uiversiy, maryiuk..b.@gmail.com; kvaro@gmail.com. 2 Viisa Fiacial-Ecoomic Uiversiy, kuperok@mail.ru. Traslaed from Kibereika i Sisemyi Aaliz, No. 5, Sepember Ocober, 25, pp. 95. Origial aricle submied December 22, /5/ Spriger Sciece+Busiess Media New York 75

2 or g ( X ) w x, (2) i ij j j where x j is he jh compoe of he ipu vecor X of image aribues, w ij is he weigh of he membership of he jh compoe i he ih class Ñ i, i, m, he i is possible o represe he classificaio decisio rule by he maximum DF crierio like i [22, 23], amely, X if g ( X ) max g ( X ), k, m. C k I [2], i is proposed o o form all he m DF () wih he subseque deermiaio of he maximum oe, bu o form he marix A from elemes (erms) of m DF () followed by heir parallel processig wih respec o colums sice hey cosis of similar elemes: erms of DF (). Such parallel processig wih respec o colums of marix A is eligible sice ay sum (2) possesses commuaiviy ad associaiviy [24]. Thus, he maximum from m sums of he form () ca be deermied durig heir simulaeous reducio by he value of heir commo compoe formed by he miimum elemes i each colum of he curre marix A i he h processig cycle, where, N. I he course of sequeial zeroig of he respecive rows of he origial marix A i is possible o deermie he raks of DFs, from he firs o he mh oe. As a resul, he objec classificaio algorihm usig such DF elemes processig priciple ca be represeed as follows. The iiial daa are: vecor of ipu sigals X { x j }; weigh marix W { w ij }; ad classificaio vecor P { p i }; classes C { C i }, where j is he dimesio of vecor Õ { j, N} ad i is he umber of classes Ñ { i, m}. Sep. Form he marix A of weighed ipu sigals x j of he form a a a A A 2 a a a A m m2 m m k i i, (3) hese ipu sigals as elemes a ij of he marix A ca be calculaed as aij wij xj ; assig ui value o all he elemes p i of he classificaio vecor P, i.e., P (... ) T. Sep 2. Selec he miimum elemes q j simulaeously i all he colums of he curre marix A j q mi { a },, N, (4) i ij where N is he umber of classificaio cycles. Sep 3. Calculae differece slices A j simulaeously i all he colums of he curre marix A ad form a disordered marix A as follows: ad A 2 a a a, (5) a a a m m 2 m a a q, j, ; (6) ij ij j es he codiio of equaliy o zero of a leas oe of he rows of he disordered marix A of he form k A, k, m, (7) ad he codiio of equaliy o zero of all he rows of he disordered marix A Ai, i, m. (8) If codiio (7) is saisfied, go o Sep 4; if (8) is saisfied, go o Sep

3 TABLE Cycles Sep 2 Sep 3 Sep 4 Curre marix A [ 2 2] Disordered marix A Ordered marix A Vecor P [ 2 4 ] [ ] [ 2 ] Sep 4. Move wih exchage (raspose) o he righmos colums of zero elemes a ij i each row of he disordered marix A (5) ad form ordered marix A as follows: A Tr ( A ), i, m; (9) i 2 i simulaeously se o zero he kh eleme p k of vecor P, correspodig o he zeroed row A k (7); mask all he zero elemes a kj of he kh zeroed row of he ordered marix A ad go o Sep 2. Sep 5. Save he ui value of he lh eleme p l of vecor P, correspodig o he las zeroed row A N l i he N h cycle; ed of he classificaio process. Thus, he ui value of he lh eleme p l of vecor P meas ha he ipu objec specified by he vecor X of is aribues belogs o he lh class C l, i.e., ( X pl, l, m) Cl. A cycle of classificaio process is carried ou i Seps 2 4. Table gives a example of he implemeaio of he preseed classificaio algorihm begiig wih Sep 2, for he origial 33 marix À of he form À A Sep 2, alog wih he curre marix À, he vecor q j of he miimum elemes is specified for each is colum for he h cycle. Dashes (see Table ) deoe zero elemes of compleely zeroed row of he marix, which are he o processed. The ui value of eleme ð 3 of vecor Ð correspods i his case o he maximum DF g3 ( X ). EXTENSION OF THE BASIS OF GLUSHKOV S SAA Parallel processig wih respec o rows ad colums of he marix A eeds wo marked arrays o be iroduced. Le us cosider hem: marked array of elemes ( a, a 2..., a ) i he form of row vecor Mr PLa a 2 a *, where PL ad * are special markers ha deoe he lef ad righ boudary of his array, respecively, is a poier, ad is array dimesio; 753

4 marked array of elemes ( a, a 2..., a m ) i he form of colum vecor Mñ PLa a 2 am*. The followig saemes are ake as basis oes [, 2, 3]: EST( Z) is he saeme of esablishig he sequece Z of poiers V ad markers W, where Z( V W); IAP is he saeme of iiial arrageme of poiers; FIN is he saeme of edig he operaio of he regular scheme; OUT( R ) is he saeme of oupu of he resul; C is he saeme of shifig he poier by oe eleme o he righ; TRANSP( lr, ) is rasposiio of adjace elemes l ad r of he array M r. Moreover, he followig logical operaios are used [7,, 2, 3]: disjucio, cojucio, egaio, composiio A B (i.e., sequeial execuio of saemes A ad B), aleraive [ ]( A B) (i.e., if, he A; oherwise, B), ad cycle [ ]{ A }(i.e., if is false, he A, if is rue, he ed of he cycle). The followig basis codiios are also used: l ris rue if he specified relaio holds for adjace elemes l ad r of he array ad d(*) is rue whe he poier reaches marker *. A special feaure of he proposed processig of elemes of he marix wih respec o DS is lef-side processig of elemes of arrays M r ad M c as a resul of cyclic shif of poier from lef o righ. Wih regard for he special feaures of parallel processig of daa arrays wih respec o DS, he followig supplemes o basis saemes ad codiios are subsaiaed: SUBT( M, q) is he saeme of parallel subracio of eleme q from all elemes of he array M ; NUL( a, a ) is he saeme of zeroig of a eleme of he array M ; is rue if zero elemes are a he umos righ posiios of he array M r ; k is rue if codiio (7) is saisfied; is rue if codiio (8) is saisfied. NOTATION OF THE OBJECT CLASSIFICATION ALGORITHM IN TERMS OF GLUSHKOV S SAA The followig operaios are used as he basic oes for he proposed classificaio algorihm: selecio of he miimum eleme q j (4) i he vecor array A j, where j, ; calculaio of DS A j wih elemes a ij (6); moio wih exchage (rasposiio) o he righ o he exreme posiios of zero elemes i he vecor array A i (9), where i, m. These basic operaios ca be wrie as compoud saemes as follows: he compoud saeme of selecig he miimum eleme amog elemes ( a, am ) A of DS j, deoed as a array M cj : MIN j( a, am ):: [ d(*)] {[ lr]( EST(mi l) EST(mi r) C ) [mi r]( EST(mi mi) EST (mi r) C )}, () used o impleme sequeial selecio of he miimum eleme i each pair (mi, r) of adjace elemes of he array ( a, a ), begiig wih he firs pair (, lr, ) wih he shif by oe eleme o he righ alog he array ad wih esablishig (assigig) oe of wo values o he eleme mi ; he compoud saeme of calculaig DS A j, which we ca also deoe as array M cj : SLICE ( a, a ):: IAPMIN ( a, a ) SUBT j ( M,mi) () j m j m akig io accou oe saeme of sequeial execuio MIN j ( a, a m ) () ad oe saeme of parallel execuio SUBT j ( M,mi) i each array M cj ; 754

5 he compoud saeme of rasposiio of zero elemes i he array À i, deoed as array M ri : TRANS i( a, a ):: [ ] {TRANS i( l, r) C}, (2) which is used o impleme sequeial moio wih exchage i adjace pairs (rasposiio) of zero elemes o he righ o exreme posiios i each array M ri. Thus, he DS-based classificaio algorihm described above, begiig wih Sep 2, ca be wrie as follows: CLASS ( x, x ):: [ ]{ SLICE( a, a ) [ ]( NUL ( p, p ) j m k m m TRANS ( a, a ))} OUT( P) FIN. i (3) I he oaio (3), akig io accou he ecessiy o execue he operaio of formig DS i parallel wih respec o all he colums of marix A ad he operaio of rasposiio i parallel wih respec o all m rows of marix A, he followig represeaio of processes of parallel processig is used: SLICE( a, a m ) is parallel execuio of he saeme SLICE j ( a, a m ) () over all colums of M cj ; j m i TRANS ( a, a ) is parallel execuio of he saeme TRANS i ( a, a ) (2) over all m rows of M ri. Moreover, i he oaio (3) saeme OUT( P ) is used o oupu he resul of he classificaio process, amely, vecor P wih oe ui eleme p l accordig o Sep 5 of he algorihm. I he oaio of he classificaio algorihm i he form (3), i is supposed ha he origial marix A of he form (3) has bee formed before he begiig of he process. A aalysis of he compoud saeme of he calculaio of DS of he form SLICE j ( a, a m ) () allows us o suppose is modificaio due o combiig he execuio of saemes MIN j ( a, a m ) ad SUBT j ( M,mi), for example, wih he use of he operaio of decreme over all elemes ( a, a m ), bu i each colum M cj i parallel. This will cosiderably accelerae he implemeaio of he saeme SLICE j ( a, a m ). CONCLUSIONS A aalysis of he oaio of DS-based classificaio algorihm i erms of Glushkov s SAA has cofirmed he compacess of he represeaio i such approach ad he possibiliy of furher developme of he mehod of processig wo-dimesioal (marix) daa arrays wih respec o DS, ad also he fucioal cardialiy of Glushkov s SAA basis, which allows usig is erms o describe complex algorihms, i his case, he objec classificaio algorihm. REFERENCES. G. E. Tseili, A Iroducio o Algorihmics [i Russia], Sfera, Kyiv (998). 2. G. E. Tseili, Algebraic algorihmics: Theory ad applicaios, Cyber. Sys. Aalysis, 39, No., 6 5 (23). 3. F. I. Ado, A. E. Dorosheko, G. E. Tseili, ad E. A. Yaseko, Algebraic Algorihmic Models ad Mehods of Parallel Programmig [i Russia], Akademperiodika, Kyiv (27). 4. P. I. Ado, A. Yu. Dorosheko, ad K. A. Zhereb, Programmig high-performace parallel compuaios: Formal models ad graphics processig uis, Cyber. Sys. Aalysis, 47, No. 4, (2). 5. F. I. Ado, A. E. Dorosheko, A. G. Bekeov, V. A. Iovchev, ad E. A. Yaseko, Sofware ools for auomaio of parallel programmig o he basis of algebra of algorihms, Cyber. Sys. Aalysis, 5, No., (25). 6. G. E. Tseili, A. A. Amos, O. V. Golovi, ad A. Yu. Zubsov, Iegraed ools for desig ad syhesis of classes of algorihms ad programs, Cyber. Sys. Aalysis, 36, No. 3, (2). 755

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