Texture Characterization Based on a Chandrasekhar Fast Adaptive filter

Transcription

1 Texure Characerizaio Based o a Chadrasehar Fas Adapive filer Mouir Sayadi ad Farha Faiech Absrac I he framewor of adapive parameric modellig of images, we propose i his paper a ew echique based o he Chadrasehar fas adapive filer for exure characerizaio. A Auo-Regressive (AR) liear model of exure is obaied by scaig he image row by row ad modellig his daa wih a adapive Chadrasehar liear filer. The characerizaio efficiecy of he obaied model is compared wih he model adaped wih he Leas Mea Square (LMS) 2-D adapive algorihm ad wih he cooccurrece mehod feaures. The compariso crieria is based o he compuaio of a characerizaio degree usig he raio of "beweeclass" variaces wih respec o "wihi-class" variaces of he esimaed coefficies. Exesive experimes show ha he coefficies esimaed by he use of Chadrasehar adapive filer give beer resuls i exure discrimiaio ha hose esimaed by oher algorihms, eve i a oisy coex. Keywords Texure aalysis, Saisical feaures, Adapive filers, Chadrasehar algorihm. T I. INTRODUCTION EXTURE aalysis plays a impora role i several image processig ad paer recogiio applicaios such as remoe sesig, carography, robo visio, miliary surveillace ad medical imagig. I has log bee he opic of iese research [][6][3]. Texure ca be foud i he bacgroud of aural scees as well, as fillig elemes of surface images, hus he exural feaures are a impora paer elemes i image ierpreaio. Various mehods for exure feaures exracig for exure characerisaio have bee proposed durig he las wo decades. Oe such mehod characerizes exure by discree Wavele represeaio [][8][5]. Fracal based feaures have bee also used as feaures for exure characerizaio [6]. These feaures deped mosly o exural characerisics ha o iesiy iformaio. Several auhors have made a compariso of he performace of various feaures for exure characerisaio purpose. I addiio, he co-occurrece marix is a popular saisical echique for exracig exural feaures [5]. I [0], Ojala e al have compared four exure feaures: gray level differeces, Laws exure feaures, ceer-symeric covariace feaures, ad local biary paers. Oe of he mos promisig mehods for exure feaures exracio is he parameric modellig [7][2][3], where he coefficies of he AR parameric model are used for exure characerisaio ad syhesis. M. Sayadi (Correspodig auho is wih he Ui SICISI, High school of scieces ad echiques of Tuis (ESSTT), 5 Av. Taha Hussei, 008, Tuis, Tuisia. mouir.sayadi@ess.ru. F. Faiech is wih he direcor of he Ui SICISI, High school of scieces ad echiques of Tuis (ESSTT), 5 Av. Taha Hussei, 008, Tuis, Tuisia. farha.faiech@ess.ru. Various adapive parameric filers have bee proposed for exure characerizaio. I [2], he AR coefficies esimaed wih he 2-D Fas Laice RLS filer [7] are used for exure classificaio i a oiseless coex. I [3], a adapive approach i exure characerizaio wih he rasversal 2-D AR coefficies obaied from he 2-D OLRIV (Overdeermied Laice Recursive Isrumeal Variable) algorihm is preseed. This algorihm uses higher order saisics. I is compared i [2] o he 2-D Fas Laice RLS algorihm, based o secod order saisic. I has bee show ha alhough he effec of he addiive gaussia oise is more impora o he 2-D Fas Laice RLS filer ha he 2-D OLRIV filer, he 2-D FLRLS provides a good characerizaio of he exure model. I preses a large classificaio sesiiviy. O he oher had, oe of he approaches for decreasig he compuaio cos of he adapive filerig algorihms is he use of he Chadrasehar facorisaio echiques [9][]. The sregh of his approach derives from he fac ha i avoids he resoluio of he sadard Riccai equaio of he Kalma filer. The derivaio of fas adapive algorihms based o Chadrasehar fas equaios usig a sae space model was preseed i [3] ad [4] for MA ad ARMA liear filerig, respecively. I has bee exeded o he mulichael ad o-liear filerig i []. The mai coribuio of he prese wor is he use of he coefficies issued from he fas Chadrasehar adapive filer as feaures for exure characerizaio. We will show how much hese coefficies ca improve he exure characerizaio i compariso wih hose esimaed by oher adapive filers. II. THE IMAGE PARAMETRIC ADAPTIVE LINEAR MODEL A exure image of size (LL) ca be represeed by a 2-D AR parameric liear model wih quarer-plae suppor of order (p,q) i.e. R i, j : 0i p,0 j p wih (, i j) (0,0) as show i Fig.. The value of a pixel a posiio (, is represeed by he followig relaioship: p q y (, ( l, m) (0,0) alm. y( l, r m). () l0 m0 ad r are i he ierval 0..L-. I he saioary case, he 2-D AR filer coefficies a lm do' deped o he posiio of he pixel (,. 564

2 Sep 2: Adapaio of he 2-D AR coefficies: For l from 0 o p, For m from 0 o q- : alm alm μ d(, y(, y( l, r m). (3) I exure modellig, he desired oupu d (, is he gray level value of he pixel (, of he exure image o be modelled. μ is he sep size of he algorihm. Fig. The quarer-plae suppor 2-D scheme III. THE PROPOSED CHANDRASEKHAR ADAPTIVE FILTER To use he Chadrasehar adapive filer, we propose o sca he quarer plae model widow row by row as show i Fig. 3. Equaio () ca be he wrie i he followig form: pq x ( bi. i) (4) i0 Fig. 3 The rasformaio of he 2-D model Fig. 2 The adapive scaig mas from lef o righ ad op o boom To produce a approximae value for a give pixel, all he values of he pixel image which are covered by he slidig mas R, weighed by he 2-D AR filer coefficies ad summed. The mas is he moved o he followig locaio (Fig. 2). I he followig, we give a overview o he 2-D Leas Mea Square algorihm [5], based o he miimizaio of he mea square error bewee he filer oupu ad he desired oupu. This algorihm is he mos widely used adapive filer due o is implemeaio simpliciy. The seps of he 2-D LMS algorihm, whe i is used for modellig a exure d of size (LL), are [5]: Sep : Usig a slidig widow from lef o righ ad op o boom, calculae he adapive filer oupu: p q y (, ( l, m) (0,0) alm. y( l, r m) (2) l0 m0 where a lm are he AR coefficies a idex ime, beig expressed accordig o coordiaes of he pixel (, by L r. The filer coefficies i he firs ieraio. a lm are iiialised o zero I his model, wo liear scaig idexes i ad are respecively i=m+lq ad = L+r. The sample correspods o y(, ad -) o y(,r-). The coefficies b i correspodig o alm are se i a vecor b of size N=pq-. Accordig o he wors preseed i [3][4] ad [], he Chadrasehar fas adapive algorihm ca be applied o esimae he coefficies of he model (4). We summarize i he followig he seps of his algorihm applied i exure modellig: [] U( ) V( S(., where U( ) N)... ) (5) W( M( V( (6) M( ) M( W( ( R( ) W ( (7) R( ) R( V ( W( (8) ( ) ( 0 C K( ) C ) S( W( 0 (9) S( ) S( K( )( R( )) V ( (0) Predicio error: e( d( U( ) b( () Esimaed coefficie vecors: b ( ) b ( ) C ( )( R ( )) e ( ). (2) 565

3 d( represes he correspodig pixel of idex (, i he desired exure image. The algorihm is iiialised wih: 2 K ( ) 0 N ad R( ) d. By defiig a piig vecor p of legh p: i 0i 0 pi, he iiialisaio of M() ad S () are: M () 0, 0 N N S( ) d 0 d N N. The marix 0 is a vecors whose elemes are zeros. The iiial values of he coefficies b i are se o zero. For more deails abou he Chadrasehar algorihm, he reader is referred o []. IV. CHARACTERIZATION DEGREE OF THE FILTERS COEFFICIENTS Cosider a se of 25 differe gray-scale exures of (256256) pixels exraced from he Brodaz album [2] (Fig. 5). From each exure, 00 sample images of (3232) pixels are radomly chose. Boh Chadrasehar filer ad 2-D LMS AR coefficies are he evaluaed for each oe of he resulig 2500 sample images. A quarer-plae suppor wih a slidig (33) order widow. I order o chec wheher he use of he Chadrasehar adapive filer improves he exure characerizaio i compariso wih he 2-D LMS adapive filer, we propose o evaluae he characerizaio efficiecy of he coefficies esimaed by each adapive filer. Thus we defie a "characerizaio degree" J as he raio bewee "ier-class" ad "ira-class" deviaios of each exure class [4]. For a give esimaed coefficie, we defie he "ierclass" (bewee-class) deviaio as he sadard deviaio of his coefficie wih respec o he exure class variaio. The "ira-class" (wihi-class) deviaio is defied as he sadard deviaio of his coefficie i he same exure class wih respec o various realizaios. Toal ier-class ad ira-class deviaios are calculaed by averagig ou all he coefficie sadard deviaios obaied from all idepede realizaios. Large ier-class deviaios alog wih a small ira-class oe yield a high characerizaio degree. The greaer he characerizaio degree is, he more robus he classificaio process is. The compariso of he capabiliy of he filer coefficies will be preseed hrough he ex wo experimes. Experime : Compariso of he characerizaio degree i a oisy coex For each coefficie family class i.e. he Chadrasehar adapive coefficies ad he 2-D LMS filer oes, we compue he "characerizaio degree" J as follows [4]: Noe x, he h esimaed vecor of coefficies for he h exure class 25, 00. The mea of he h exure class vecors of coefficie is oed μ 00 x, ad oes he mea of c 25 all he coefficie vecors. The mea of he wihi-class (iraclass) dispersio marices is give by he marix: S ira x μ x, μ 2500, (3) which is he maximum lielihood esimaio of he covariace marix of he class. Complemeary o his is he mea of he bewee-class (ier-class) dispersio marices which describes he scaerig of he class sample meas ad is calculaed by he marix: 25 S ier μ μ μ μ. (4) 25 c c The "characerizaio degree" J proposed ad deailed i [4] is give by: J race ra er S i.s i. (5) The greaer he characerizaio degree is, he more robus he classificaio process is. The compariso of he abiliy of he sudied 2-D coefficies i presece of addiive oise will be preseed hrough he ex simulaio resuls. I Fig. 4, we plo he characerizaio degree J for boh Chadrasehar adapive filer ad 2-D LMS coefficies. For reaso of compariso, he mehod based o he cooccurrece marices feaures [5] is also applied. The elemes of he co-occurrece marix C i,j, represes how ofe pairs of pixels wih values i ad j separaed by a specific disace occurs. The horizoal direcio is used o obai he cooccurrece marices. A se of seve sadard feaures are exraced from hese marices for each sudied image for each direcio. These feaures are: he coras, he eergy, he eropy, he homogeeiy, he maximum probabiliy, he cluser shade ad he cluser promiece of he co-occurrece marix [5]. Four cases are cosidered: he oiseless case, ad wo Sigal o Noise Raio (SNR) values: 5 db ad 0 db. A addiive Gaussia oise ad a 2-D order of 33 quarer-plae suppor have bee used providig 8 AR coefficies. Clearly, he characerizaio degree of he Chadrasehar filer is greaer ha ha of he 2-D LMS filer ad he co-occurrece mehod. I all cases, he icrease of addiive oise variace causes aeuaio i he characerizaio degree. The addiive oise perurbs he classificaio process. Therefore, he Chadrasehar filer based coefficies are beer exure discrimiaors ha hose of he oher filers. 566

4 Characerizaio Degree Chadrasehar 2D LMS Co-occurece Noiseless 5 db 0 db SNR Fig. 4 Characerizaio degree of he Chadrasehar filer coefficies, he 2-D LMS filer oes ad he co-occurrece mehod feaures for various SNR values (Filer order of 33). Experime 2: Ifluece of he filer order I his experime, he characerizaio degree is calculaed for differe filer orders ragig from 33 o 66 wihou ay addiive oise. Fig. 6 depics he characerizaio degree wih respec o he 2-D filer orders for boh Chadrasehar adapive filer ad 2-D LMS coefficies. I should be oed ha for ay order, he characerizaio degree provided by he Chadrasehar adapive filer coefficies is greaer ha he oe provided by he 2-D LMS filer oes. Furhermore, he coefficies seem o give beer characerizaio degree for a high filer order. Characerizaio Degree Chadrasehar 2D LMS 3x3 4x4 5x5 6x6 2D filer order Fig. 6 Plo of he characerizaio degree wih respec o he filer orders for he Chadrasehar filer coefficies (*) ad he 2-D LMS filer oes (o) V. CONCLUSION I his paper, we have proposed for he firs ime he adapive Chadrasehar filer coefficies as ew parameric feaures for exure characerisaio. The exure image is scaed row by row ad filered wih he adapive Chadrasehar moo dimesioal filer. We have show how much he use of he Chadrasehar adapive filer i exure modellig ca improve he exure characerizaio i compariso wih he classical 2-D LMS adapive filer. The esimaed coefficies by he Chadrasehar adapive filer give beer resuls i exure discrimiaio ha hose esimaed by oher classical algorihms, eve i a oisy coex. Furher wor aig io accou he effec of he exure orieaio ad he scalig of he exure images o he esimaed coefficies sill remai o be doe. REFERENCES [] M. Acharyya ad M. K. Kudu, "A adapive approach o usupervised exure segmeaio usig M-Bad wavele rasform," Sigal Processig, vol. 8, pp , 200. [2] P. Brodaz, Texures: a phoographics album for aris ad desigers, Dovers, New Yor, 966. [3] G. Demome ad R. Reyaud, "Fas RLS adapive algorihms ad Chadrasehar equaios," Proceedigs SPIE Adapive Sigal Processig, S. Hayi ed., vol P-565, pp , July 99. [4] X. C. Du, A. Brella ad R. Logchamp, "Fas algorihm of Chadrasehar ype for ARMA model ideificaio," Auomaica, vol. 28, o. 5, pp , 992. [5] M. Hadhoud, D W. Thomas. "The Two-dimesioal adapive LMS (TDLMS) algorihm" IEEE Trasacios o Circuis ad Sysems, vol. 35, o. 5, pp , May 988. [6] T. Kasparis, D. Charalampidis, M. Georgiopoulos ad J. Rollad, "Segmeaio of exure images based o fracals ad image filerig," Paer Recogiio, vol. 34, pp , 200. [7] X. Liu ad M. Najim, "A wo dimesioal fas laice recursive leas squares algorihm", IEEE Trasacios o Sigal Processig, vol. 44, o. 0, Ocober 996. [8] C. S. Lu, P. C. Chug ad C. F. Che, "Usupervised exure segmeaio via wavele rasform," Paer Recogiio, vol. 30, o. 5, pp , 997. [9] M. Morf, G. S. Sidhu ad T. Kailah, "Some ew algorihms for recursive esimaio i cosa, liear sysems," IEEE Trasacios o Auomaic Corol, vol. 9, pp , Augus 974. [0] T. Ojala, K. Valealahi, E. Oja, M. Pieiaie. "Texure Discrimiaio wih muli-dimesioal disribuios of siged gray-level differeces," Paer Recogiio, vol. 34, o. 3, pp , 200. [] M. Sayadi, F. Faiech ad M. Najim, "Mulichael liear ad quadraic adapive filerig based o he Chadrasehar fas algorihm," IEEE Trasacios o Sigal Processig, vol. 47, pp , March 999. [2] M. Sayadi ad M. Najim, "Compariso of secod ad hird order saisics based adapive filers for exure characerizaio," Proceedigs of ICASSP'99, Phoeix, Arizoa, USA, pp , March 999. [3] M. Sayadi, V. Buzeac ad M. Najim, "Texure characerizaio usig 2-D cumula-based laice adapive filerig," Proceedigs of ICASSP'98, Seale, Washigo, USA, pp , 2-5 May 998. [4] F. Va Heijder, Image based measureme sysem, Joh Wiley ad sos Ediio, UK, 995. [5] G. Va de Wouwer, P. Scheuders, ad D. V. Dyc, "Saisical Texure Characerizaio from Discree Wavele Represeaios," IEEE Trasacios o Image Processig, vol. 8, pp , Aug

5 Bulle Neig Wove alumium wire Pressed cor Wood Fur Wool Clohs Frech cavas Woolle cloh Repile si Bubbles Raffia Orieal sraw cloh Orieal sraw cloh Grass Sraw maig Radom fibber Pressed calf leaher Cheese cloh Bric wall Orieal fibber Coo cavas Orieal grass cloh Waer Fig. 5 The 25 Brodaz exures used i he sudy 568