ROTATION-INVARIANT TEXTURE CLASSIFICATION USING FEATURE DISTRIBUTIONS. Abstract

ROTATION-INVARIANT TEXTURE CLASSIFICATION USING FEATURE DISTRIBUTIONS M. PIETIKÄINEN, T. OJALA and Z. XU Machne Vson and Meda Processng Group, Infotech Oulu Unversty of Oulu, P.O. Box 4500, FIN-90401 Oulu, Fnland Abstract A dstrbuton-based classfcaton approach and a set of recently developed texture measures are appled to rotaton-nvarant texture classfcaton. The performance s compared to that obtaned wth the well-known crcular-symmetrc autoregressve random feld (CSAR) model approach. A dffcult classfcaton problem of 15 dfferent Brodatz textures and seven rotaton angles s used n experments. The results show much better performance for our approach than for the CSAR features. A detaled analyss of the confuson matrces and the rotaton angles of msclassfed samples produces several nterestng observatons about the classfcaton problem and the features used n ths study. Texture analyss Classfcaton Feature dstrbuton Rotaton nvarant Performance evaluaton 1. INTRODUCTION Texture analyss s mportant n many applcatons of computer mage analyss for classfcaton, detecton, or segmentaton of mages based on local spatal varatons of ntensty or color. Important applcatons nclude ndustral and bomedcal surface nspecton; for example for defects and dsease, ground classfcaton and segmentaton of satellte or aeral magery, segmentaton of textured regons n document analyss, and content-based access to mage databases. There are many applcatons for texture analyss n whch rotaton-nvarance s mportant, but a problem s that many of the exstng texture features are not nvarant wth respect to rotatons. Some nvarance for features derved from co-occurrence matrces or dfference hstograms can be obtaned, for example, by smply averagng the matrces (hstograms) or features computed for dfferent angles, e.g. for 0, 45, 90 and 135 degrees. Other early approaches proposed for rotatonnvarant texture classfcaton nclude the methods based on polarograms (1), generalzed co-occurrence matrces (2) and texture ansotropy (3). Kashyap and Khotanzad developed a method based on the crcular symmetrc autoregressve random feld (CSAR) model for rotaton-nvarant texture classfcaton (4). Mao and Jan proposed a multvarate rotaton-nvarant smultaneous autoregressve model (RISAR) that s based on the CSAR model, and extended t to a multresoluton SAR model MR-RISAR) (5). A method for classfcaton of rotated and scaled textures usng Gaussan Markov random feld models was ntroduced by Cohen et al. (6). Approaches based on Gabor flterng have been proposed by, among others, Leung and Peterson (7), Porat and Zeev (8), and Haley and Manjunath (9). A steerable orented pyramd was used to extract rotaton nvarant features by Greenspan et al. (10) and a covarance-based representaton to transform neghborhood about each pxel nto a set of nvarant descrptors was proposed by Madraju and Lu (11). You and Cohen extended Laws masks for rotaton-nvarant texture characterzaton n ther tuned mask scheme (12). Recently, we ntroduced new measures for texture classfcaton based on center-symmetrc auto-correlaton and local bnary patterns, usng Kullback dscrmnaton of sample and prototype dstrbutons, and conducted an extensve comparatve study of texture measures wth classfcaton based on feature dstrbutons (13,14). By usng a standard set of test mages, we showed that very good texture dscrmnaton can be obtaned by usng dstrbutons of smple texture measures, lke absolute gray level dfferences, local bnary patterns or center-symmetrc auto-correlaton. The performance s usually further mproved wth the use of two-dmensonal dstrbutons of jont pars of complementary features (14). In experments nvolvng varous applcatons, we have obtaned very good results wth ths dstrbuton-based classfcaton approach, see e.g. (15). Recently, we also extended our approach to unsupervsed texture segmentaton wth excellent results (16). An overvew of our recent progress s presented n (17). In our earler experments, the problems related to rotaton nvarance have not been consdered. Ths paper nvestgates the effcency of dstrbuton-based classfcaton, and our feature set n rotaton-nvarant texture classfcaton. We are especally nterested n seeng the performances for relatvely small wndows (64x64 and 32x32 pxels) requred by many applcatons, whereas many of the exstng approaches have been tested wth larger wndows. Texture measures based on center-symmetrc autocorrelaton, gray level dfferences and a rotaton-nvarant verson of the Local Bnary Pattern operator, LBPROT, are used n experments. A smple method based on blnear gray level nterpolaton s used to mprove rota-

ton-nvarance when extractng texture features n a dscrete 3x3 neghborhood. The performance of LBPROT recently proposed by Ojala (18) s now expermentally evaluated for the frst tme. The use of jont pars of features to ncrease the classfcaton accuracy s also studed. The results for dstrbuton-based classfcaton are compared to those obtaned wth the well-known CSAR features. 2. TEXTURE MEASURES 2.1 Measures Based on Center-symmetrc Auto-correlaton In a recent study of Harwood et al. (13), a set of related measures was ntroduced, ncludng two local center-symmetrc auto-correlaton measures, wth lnear (SAC) and rank-order versons (SRAC), together wth a related covarance measure (SCOV). All of these are rotaton-nvarant robust measures and, apart from SCOV, they are locally gray-scale nvarant. These measures are abstract measures of texture pattern and gray-scale, provdng hghly dscrmnatng nformaton about the amount of local texture. A mathematcal descrpton of these measures computed for center-symmetrc pars of pxels n a 3x3 neghborhood (see Fg. 1) s presented n equatons (1) - (4). µ denotes the local mean and σ 2 the local varance n the equatons. x 2 x 3 x 4 x 1 x 1 x 4 x 3 x 2 Fg. 1. 3x3 neghborhood wth four center-symmetrc pars of pxels. SCOV s a measure of the pattern correlaton as well as the local pattern contrast. Snce t s not normalzed n respect to local gray-scale varaton, t provdes more texture nformaton than the normalzed auto-correlaton measures SAC and SRAC. SAC s an auto-correlaton measure, a normalzed, gray-scale nvarant verson of the texture covarance measure SCOV. SAC s nvarant under lnear gray-scale shfts such as correcton by mean and standard devaton. It should also be noted that values of SAC are bound between -1 and 1. SCOV 4 1 = -- x 4 ( µ )( x ' µ ) (1) SAC SCOV = --------------- (2) σ 2 Texture statstcs based drectly on gray values of an mage are senstve to nose and monotonc shfts n the gray scale. Wth SRAC, the local rank order of the gray values s used nstead of the gray values themselves. Hence, SRAC s nvarant under any monotonc transformaton ncludng correcton by mean and standard devaton and hstogram equalzaton. The amount of auto-correlaton n the ranked neghborhood s gven by Spearman s rank correlaton. It s defned for the nxn neghborhood wth 4 center-symmetrc pars of pxels as (3), where m s n 2, and each t s the number of tes at rank r n the ranked neghborhood. The values of SRAC are bound between -1 and 1. 4 12 ( r r ') 2 + T x SRAC = 1 ------------------------------------------------------ m 3 m (3)

l 1 3 T x = ----- t 12 ( t ) (4) The symmetrc varance rato (rato between the wthn-par and between-par varances), SVR, s a statstc equvalent to the auto-correlaton measure SAC. SVR s also nvarant under lnear gray-scale shfts. SVR WVAR = ----------------- (5) BVAR Addtonally, the dscrmnaton nformaton provded by three local varance measures can be used. VAR (= BVAR + WVAR) and the two elements contrbutng to t are all measures of local gray-scale varaton, very senstve to nose and other local gray-scale transformatons. The between-par varance, BVAR, s mostly a measure of resdual texture varance and usually t s a very small part of VAR. The majorty of local varance s generally due to the wthn-par varance WVAR. In our experments, the classfcaton results for the VAR measure are reported. VAR BVAR 4 1 = -- x 2 8 ( + x ' 2 ) µ 2 (6) 4 1 = ----- x 16 ( + x ' ) 2 µ 2 (7) WVAR 4 1 = ----- x 16 ( x ') 2 (8) 2.2 Rotaton-Invarant Local Bnary Pattern In the Local Bnary Pattern (LBP) texture operator we ntroduced (14), the orgnal 3x3 neghborhood (Fg. 2a) s thresholded at the value of the center pxel. The values of the pxels n the thresholded neghborhood (Fg. 2b) are multpled by the bnomal weghts gven to the correspondng pxels (Fg. 2c). The result for ths example s shown n Fg. 2d. Fnally, the values of the eght pxels are summed to obtan the LBP number (169) of ths texture unt. By defnton LBP s nvarant to any monotonc gray scale transformaton. The texture contents of an mage regon are characterzed by the dstrbuton of LBP. 6 5 2 7 6 1 9 3 7 1 0 0 1 0 1 0 1 1 2 4 8 16 32 64 128 1 0 0 8 0 32 0 128 (a) (b) (c) (d) LBP = 1+8+32+128=169 Fg. 2. Computaton of Local Bnary Pattern (LBP). LBP s not rotaton nvarant, whch s undesrable n certan applcatons. It s possble to defne rotaton nvarant versons of LBP; one soluton s llustrated n Fg. 3 (18). The bnary values of the thresholded neghborhood (Fg. 3a) are mapped nto an 8-bt word n clockwse or counter-clockwse order (Fg. 3b). An arbtrary number of bnary shfts s then

made (Fg. 3c), untl the word matches one of the 36 dfferent patterns (Fg. 3d) of 0 and 1 an 8-bt word can form under rotaton. The ndex of the matchng pattern s used as the feature value, descrbng the rotaton nvarant LBP of ths partcular neghborhood. a b c h Fg. 3. d g f e 0 0 1 0 0 0 1 0 abcdefgh efghabcd cdefghab (a) (b) (c) (d) pattern ndex 00000000 0 00000001 1 00000011 2 00000101 3 00001001 4 00010001... 5 01111111 34 11111111 35 00100100 00001001 LBPROT = 4 Computaton of LBPROT, rotaton-nvarant verson of LBP. 2.3 Gray Level Dfference Method The method based on hstograms of absolute dfferences between pars of gray levels or of average gray levels has performed very well n some comparatve studes and applcatons, see e.g. (14,15). For any gven dsplacement d = (dx,dy), where dx and dy are ntegers, let f (x,y) = f(x,y) - f(x+dx,y+dy). Let P be the probablty densty functon of f. If the mage has m gray levels, ths has the form of an m-dmensonal vector whose th component s the probablty that f (x,y) wll have value. P can be easly computed by countng the number of tmes each value of f (x,y) occurs. For a small d, the dfference hstograms wll peak near zero, whle for a larger d they are more spread out. The rotaton nvarant feature DIFF4 s computed by accumulatng, n the same 1-dmensonal hstogram, the absolute gray level dfferences n all four prncpal drectons at the chosen dsplacement D. If D = 1, for example, the dsplacements d = (0,1), (1,1), (1,0) and (1,-1) are consdered. 3. CLASSIFICATION BASED ON FEATURE DISTRIBUTIONS Most of the approaches to texture classfcaton quantfy texture measures by sngle values (means, varances etc.), whch are then concatenated nto a feature vector. In ths way, much of the mportant nformaton contaned n the whole dstrbutons of feature values s lost. In ths paper, a log-lkelhood pseudo-metrc, the G statstc, s used for comparng feature dstrbutons n the classfcaton process. The value of the G statstc ndcates the probablty that the two sample dstrbutons come from the same populaton: the hgher the value, the lower the probablty that the two samples are from the same populaton. For a goodness-of-ft test the G statstc s: n s G = 2 s log----- m (9) = 1 where s and m are the sample and model dstrbutons, n s the number of bns and s, m are the respectve sample and model probabltes at bn. In the experments a texture class s represented by a number of model samples. When a partcular test sample s beng classfed, the model samples are ordered accordng to the probablty of them comng from the same populaton as the test sample. Ths probablty s measured by a two-way test-of-nteracton: n n n n G 2 f log f f log f f log s, m = 1 s, m = 1 = 1 s, m = 1 s, m f + s, m n f log = 1 s, m n f = 1 = (10)

where f s the frequency at bn. For a detaled dervaton of the formula, see Sokal and Rohlf (19). After the model samples have been ordered, the test sample s classfed usng the k-nearest neghbor prncple,.e. the test sample s assgned to the class of the majorty among ts k nearest models. In our experments, a value of 3 was used for k. The feature dstrbuton for each sample s obtaned by scannng the texture mage wth the local texture operator. The dstrbutons of local statstcs are dvded nto hstograms havng a fxed number of bns; hence, the G tests for all parngs of a sample and a model have the same number of degrees-of-freedom. The feature space s quantzed by addng together feature dstrbutons for every sngle model mage n a total dstrbuton whch s dvded nto 32 bns havng an equal number of entres. Hence, the cut values of the bns of the hstograms correspond to 3.125 (100 / 32) percentle of combned data. Dervng the cut values from the total dstrbuton, and allocatng every bn the same amount of the combned data guarantees that the hghest resoluton of the quantzaton s used where the number of entres s largest and vce versa. It should be noted that the quantzaton of feature space s only requred for texture operators wth a contnuous-valued output. Output of dscrete operators lke LBP or LBPROT, where two successve values can have totally dfferent meanng, does not requre any further processng; operator outputs are just accumulated nto a hstogram. To compare dstrbutons of complementary feature pars, metrc G s extended n a straghtforward manner to scan through the two-dmensonal hstograms. If quantzaton of the feature space s requred, t s done separately for both features usng the same approach as wth sngle features. 4. EXPERIMENTAL DESIGN 4.1. Texture Images In the experments, 15 classes of Brodatz (20) textures - pressed cork (D4), grass lawn (D9), herrngbone weave (D16), woolen cloth (D19), french canvas (D21), calf leather (D24), beach sand (D29), pressed cork (D32), straw cloth (D53), handmade paper (D57), wood gran (D68), cotton canvas (D77), raffa (D84), pgskn (D92) and calf fur (D93) - were used. The orgnal 600x450 mages were globally gray-scale corrected by Gaussan match (21). D4 D9 D16 D19 D21 D24 D29 D32 D53 D57 D68 D77 D84 D92 D93 Fg. 4. Brodatz textures.

Frst, each mage was rotated by 11 degrees around ts center usng bcubc nterpolaton, n order to have a unform behavor wth respect to nterpolaton effects for both rotated and unrotated samples n the experments. The operator ncluded n the MATLAB Image Processng Toolbox was used (22). In further dscusson, we call these processed mages reference mages and they served as tranng data n the classfcaton experments. Then, the rotated mages used for testng the texture classfer were generated by rotatng each mage counter-clockwse around ts center wth the same bcubc nterpolaton method. We used the same set of sx dfferent rotaton angles that was used by Kashyap and Khotanzad (4) : 30, 60, 90, 120, 150, and 200 degrees. In other words ths s a true test of rotaton-nvarant texture classfcaton, for the classfer sees only nstances of reference textures, and t s tested wth nstances of rotated textures t has not seen before. Fg. 5. Extracton of 64x64 samples from a rotated 600 x 450 mage. In order to analyze the effect of the wndow sze, samples of two dfferent szes were used: 64x64 and 32x32 pxels. The samples were extracted nsde a 256x256 rectangle located n the center of each texture mage. Fg. 4 depcts the 256x256 mages of each texture and Fg. 5 llustrates the extracton of sxteen 64x64 samples for a texture wth a rotaton of 30 degrees (+ orgnal rotaton of 11 degrees). Hence, wth the wndow sze of 64x64 pxels each texture class contaned 16 reference samples for tranng the classfer and 6x16=96 rotated samples for testng t. Smlarly, each texture class contaned 64 tranng samples and 6x64=384 testng samples when the wndow sze of 32x32 pxels was used. Even though the orgnal texture mages were globally corrected by Gaussan match, each ndvdual sample mage was also hstogram equalzed pror to feature extracton, to mnmze the effect of possble local varatons wthn the mages. 4.2. Use of Gray Scale Interpolaton and Jont Pars of Complementary Features A problem wth computng rotaton-nvarant texture features n a local neghborhood s that the dagonal neghbors are farther from the center pxel than the horzontal and vertcal neghbors, respectvely. To reduce the effects of ths on the classfcaton performance, the pxel values for vrtual dagonal neghbors, located at the same dstance from the center pxel than the horzontal and vertcal neghbors, can be computed from the orgnal pxel values by nterpolaton. In our experments, we used a smple blnear gray scale nterpolaton method for ths purpose. In most cases, a sngle texture measure cannot provde enough nformaton about the amount and spatal structure of local texture. Better dscrmnaton of textures should be obtaned by consderng the jont occurrence of two or more features. In (14), we showed that the use of jont dstrbutons of such pars of features whch provde complementary nforma-

ton about textures, usually mproves the classfcaton accuracy. In ths study, we perform experments wth varous pars of center-symmetrc features and LBPROT. Our goal s not to fnd an optmal feature par for ths task, but to see how much mprovement a jont feature par can provde n the rotatonnvarant classfcaton problem. 5. RESULTS AND DISCUSSION 5.1. Sngle Feature Performance Table 1 shows the results of rotaton-nvarant classfcaton for sngle features for wndow szes of 64x64 and 32x32 pxels, wthout and wth the nterpolaton of dagonal neghbors when features are extracted. In the case of 64x64 samples, VAR and SCOV features wth nterpolaton provde the best error rates of 14.2% and 16.3%, respectvely. DIFF4 also performs reasonably well wth an error rate of 27.2%, whereas the worst results are obtaned wth the gray-scale nvarant features (LBPROT, SVR, SAC, SRAC), ndcatng that nformaton about local gray-scale contrast s useful n dscrmnatng these Brodatz textures. We see that the use of gray level nterpolaton usually mproves the performance. Understandably, error rates for 32x32 samples are much hgher, and the performance of e.g. VAR deterorates to 36.4%. The 32x32 samples contan only one fourth of the pxel data of the 64x64 samples, and consequently the feature hstograms possess less dscrmnatve power. A closer examnaton of confuson matrces reveals that the center-symmetrc features have most trouble n dscrmnatng dsordered textures. For example, n the case of 64x64 samples, textures D4, D9, D24, and D32 contrbute almost 70% of the msclassfed samples when feature VAR s used. Interestngly, LBPROT has no trouble separatng D9 nor D32, mssng only three of the 192 samples belongng to these two classes. Ths suggests that a better result could be obtaned by carryng out classfcaton n stages, selectng features whch best dscrmnate among remanng alternatves. The local gray-scale contrast seems to be partcularly useful n separatng textures D21, D57, and D77, for n these cases SCOV and VAR provde almost perfect results whle ther gray-scale nvarant counterparts LBPROT, SVR, SAC, SRAC fal mserably. All features are able to dscrmnate textures D16 and D93, whch both exhbt strong local orentaton. Smlarly, a closer look at the rotaton angles of msclassfed samples reveals several nterestng detals. As expected, of the sx rotaton angles all features have the fewest msclassfed samples at 90 degrees. Ths attrbutes to the pseudo rotaton-nvarant nature of our features. For example, both the center-symmetrc features and DIFF4 are truly rotaton-nvarant only n rotatons that are multples of 45 degrees, when computed n a 3x3 neghborhood, and n the chosen set of rotaton angles 90 degrees happens to be the only one of ths type. A related observaton, smlarly attrbuted to the pseudo rotaton-nvarant nature of the features, s that the results obtaned at a partcular rotaton α and at ts orthogonal counterpart (α+90 o ) are very smlar for all features. In other words, almost dentcal results are obtaned at 30 and 120 degrees, just lke s the case wth 60 and 150 degrees. Ths suggests that the set of sx rotaton angles we adopted from Kashyap and Khotanzad (4) s suboptmal, at least when 3x3 operators are used. A more comprehensve test could be obtaned by choosng a set of rotaton angles that does not contan any two angles dfferng a multple of 45 degrees. The rotaton-nvarance of the features can be mproved by utlzng ther generc nature. The features are not restrcted to a 3x3 neghborhood, but they all can be generalzed to scale. For example, the center-symmetrc measures can be computed for sutably symmetrcal dgtal neghborhoods of any sze, such as dsks or boxes of odd or even szes. Ths allows for obtanng a fner quantzaton of orentatons for example wth a 5x5 box whch contans eght center-symmetrc pars of pxels. In earler studes LBP, the rotaton-varant ancestor of LBPROT, has proven to be very powerful n dscrmnatng unrotated homogeneous Brodatz textures. However, n ths study LBPROT provdes farly poor results wth rotated Brodatz textures. By defnton LBPROT s rotaton-nvarant only n dgtal doman, and consequently, of the sx rotaton angles, the nterpretaton of rotated bnary patterns works properly only at 90 degrees, where LBPROT provdes clearly the best result of all features. In other angles, however, rotated bnary patterns are obvously not mapped properly. Ths s partcularly apparent n the case of strongly ordered textures (e.g. D21, D53, D68, and D77) where LBPROT provdes perfect results at 90 degrees, but fals completely at other rotaton angles. Ths suggests that the current mappng of rotated bnary patterns s probably far too strct, and could be relaxed by groupng together rotated patterns that have a specfc Hammng dstance, for example.

Table 1: Error rates (%) obtaned wth sngle features. wndow sze nterpolaton DIFF4 LBPROT SCOV SAC SRAC SVR VAR 64x64 no 28.5 38.5 27.6 37.1 41.2 36.9 24.0 yes 27.2 39.2 16.3 32.4 48.1 31.8 14.2 32x32 no 44.9 50.5 39.0 53.6 48.4 53.9 40.7 yes 39.6 47.7 35.1 50.0 59.4 50.3 36.4 5.2. Results for Jont Pars of Features The results obtaned wth jont pars of features are presented n Tables 2 and 3. The gray scale nterpolaton was used n feature extracton and the hstograms were quantzed nto 8x8 bns (36x8 for pars ncludng LBPROT). As expected, the use of feature pars clearly mproves the classfcaton compared to the case of sngle features. We see that the feature pars LBPROT/VAR and LBPROT/SCOV provde the best performance wth error rates of 10.1% and 10.8% for 64x64 samples and 24.1% and 24.0% for 32x32 samples, respectvely. Many other feature pars also acheve error rates close to 10% for 64x64 samples, ncludng SCOV/VAR, SCOV/SVR, SAC/SCOV and DIFF4/SAC. All these pars have one thng n common: they nclude a feature that ncorporates local gray-scale contrast. Ths emphaszes the mportance of contrast n dscrmnatng Brodatz textures. Consequently, the pars of gray-scale nvarant features fal. Table 2: Error rates (%) obtaned wth pars of features for 64x64 samples. LBPROT SCOV SAC SRAC SVR VAR DIFF4 17.6 12.8 12.6 30.6 12.9 16.3 LBPROT 10.8 29.2 38.1 29.2 10.1 SCOV 11.6 31.7 11.5 10.6 SAC 43.9 32.0 14.6 SRAC 44.2 23.4 SVR 14.4 Table 3: Error rates (%) obtaned wth pars of features for 32x32 samples. LBPROT SCOV SAC SRAC SVR VAR DIFF4 30.2 25.5 28.7 40.0 28.5 30.7 LBPROT 24.0 41.2 45.5 41.0 24.1 SCOV 28.7 38.1 28.6 29.8 SAC 54.5 50.9 28.4 SRAC 54.2 35.2 SVR 28.4

In the prevous secton we made a remark about all features havng the fewest msclassfed samples at the rotaton angle of 90 degrees. Ths phenomenon s much stronger when jont pars of features are used. For example, n the case of LBPROT/VAR only 5 of the 145 msclassfed samples occur at 90 degrees, whle each other rotaton angle contrbutes at least 24 msclassfed samples. Interestngly, even though LBPROT does farly poorly by tself, LBPROT combned wth a gray-scale varant feature (SCOV or VAR) provdes the best results of all parngs. Ths ndcates that the combnaton of just crude pattern shape/ code and pattern contrast s a useful descrpton of rotated Brodatz textures. The shortcomngs of LBPROT are stll apparent, though, for n the case of 64x64 samples, strongly ordered texture D21 contrbutes 80 of the 145 samples msclassfed by the LBPROT/VAR par. These 80 msclassfed D21 samples, whch all are ncdentally assgned to class D9, correspond to samples of all other rotaton angles but those of 90 degrees whch are classfed correctly. Obvously, a sgnfcant mprovement n the classfcaton accuracy can be expected, when the shapes of rotated bnary patterns are descrbed effectvely. 6. QUANTITATIVE COMPARISON TO CIRCULAR-SYMMETRIC AUTOREGRESSIVE RANDOM FIELD (CSAR) MODEL 6.1. CSAR Features For comparson purposes we used the crcular-symmetrc autoregressve random feld (CSAR) model whch was proposed for rotaton nvarant texture classfcaton by Kashyap and Khotanzad (4). The general expresson of the model s descrbed as: where {y(s), s Ω, Ω = ( 0 s 1, s 2 M 1) } s a set of pxel ntensty values of a gven MxM dgtzed dscrete mage; s are the pxel coordnates; N c s the neghborhood pxel set; r are the neghborhood pxel coordnates; ν(s) s a correlated sequence wth zero mean and unt varance; α and β are the coeffcents of the CSAR model; and denotes modulo M addton. The model from ths equaton yelds two parameters, α and β. Parameter α measures a certan sotropc property of the mage and β a roughness property. The rotaton nvarant characterstcs of α and β are contrbuted by the choce of the nterpolated crcular neghbourhood of mage pxels. There s also a thrd parameter ζ whch s a measure of drectonalty and s determned by fttng other SAR models over the mage (4). 6.2. Classfcaton Procedure Texture classfcaton s performed by extractng parameters α, β and ζ from the sample mages, and feedng them to a feature vector classfer. In other words, a texture sample s represented by three numercal CSAR features, n contrast to our approach where features are descrbed by hstograms of 32 bns. The texture classfer s traned wth the features of the reference samples, and tested wth the rotated samples. Both a multvarate Gaussan (quadratc) classfer and a 3-NN (nearest neghbor) classfer were used. When the 3-NN classfer was used, the features were frst normalzed nto the range 0-1 by dvdng each feature wth ts maxmum value over the entre tranng data. 6.3. Results and Dscusson y( s) = α g r y( s r) + βv( s) (11) r N c Snce the quadratc classfer provded slghtly better results than the 3-NN classfer, the dscusson s based on the results obtaned wth the quadratc classfer (Table 4). Even though none of the ndvdual features s very powerful by tself (the best feature ( β ) provdes error rates of 51.9% and 60.0% for the 64x64 and 32x32 samples, respectvely), the three features combned offer a reasonable texture dscrmnaton wth error rates of 15.3% and 29.3%, respectvely. However, we see that the dstrbuton-based classfcaton wth sngle features (e.g. VAR and SCOV) performs about as well as the three CSAR features combned n the case of 64x64 samples, and several pars of jont features provde better performance than the combned CSAR features for both the 64x64 and 32x32 samples.

Table 4: Error rates (%) obtaned wth the CSAR features. wndow sze α β ζ α+β α+ζ β+ζ α+β+ζ 64x64 56.4 51.9 68.5 24.9 34.2 27.6 15.3 32x32 67.0 60.0 72.5 44.5 48.2 37.0 29.3 Confuson matrces reveal that the CSAR features have dffcultes n separatng textures D4, D24, D57, and D68. When all three features are used, these four classes contrbute almost 80% of the 221 msclassfed 64x64 samples. The confuson s partcularly severe between classes D4 (pressed cork) and D24 (pressed calf leather). Examnaton of the rotaton angles of the msclassfed samples verfes the observaton of the pseudo rotaton-nvarant nature of 3x3 operators made n Secton 5.1., for agan by far the fewest msclassfed samples occur at 90 degrees (only 10, whle each other rotaton angle contrbutes at least 31), and agan the results obtaned at two orthogonal rotaton angles are almost dentcal. 7. CONCLUSION In ths paper, a dstrbuton-based classfcaton approach and a set of texture measures based on center-symmetrc autocorrelaton and local bnary patterns were appled to rotaton-nvarant texture classfcaton. The performance of the proposed approach was compared to that of crcular-symmetrc autoregressve random feld (CSAR) model wth a dffcult classfcaton problem nvolvng 15 dfferent Brodatz textures and seven rotaton angles. The error rates of the best sngle features (SCOV, VAR) were comparable to those obtaned wth the three CSAR features combned, and better results were acheved wth dstrbutons of jont pars of features. It was also shown that the rotaton nvarance of texture measures n a dscrete 3x3 neghborhood can be mproved by gray level nterpolaton. The expermental results also emphasze the mportance of local pattern contrast n dscrmnatng Brodatz textures, for even though the samples were corrected aganst global gray-scale varatons, the features measurng local gray-scale varatons clearly outperformed ther gray-scale nvarant counterparts. Another observaton, verfed both wth our operators and wth the CSAR features, was the pseudo rotaton-nvarant nature of the features whch should be taken nto account when desgnng new operators and new studes on rotaton-nvarant texture classfcaton. The shortcomngs of LBPROT, the rotaton-nvarant verson of the powerful LBP operator, were exposed, and a sgnfcant mprovement n classfcaton accuracy can be expected, once the shapes of rotated bnary patterns are descrbed effectvely. Acknowledgements - The fnancal support provded by the Academy of Fnland and Technology Development Center s gratefully acknowledged. The authors also wsh to thank the anonymous referee for hs helpful comments. REFERENCES 1. L.S. Davs, Polarograms: a new tool for texture analyss, Pattern Recognton 13, 219-223 (1981). 2. L.S. Davs, S.A. Johns and J.K. Aggarwal, Texture analyss usng generalzed co-occurrence matrces, IEEE Transactons on Pattern Analyss and Machne Intellgence 1, 251-259 (1979). 3. P. Chetverkov, Experments n the rotaton-nvarant texture dscrmnaton usng ansotropy features, n Proc. 6th Internatonal Conference on Pattern Recognton, Munch, Germany, 1071-1073 (1982). 4. R.L. Kashyap and A. Khotanzad, A model-based method for rotaton nvarant texture classfcaton, IEEE Transactons on Pattern Analyss and Machne Intellgence 8, 472-481 (1986). 5. J. Mao and A.K. Jan, Texture classfcaton and segmentaton usng multresoluton smultaneous autoregressve models, Pattern Recognton 25, 173-188 (1992). 6. F.S. Cohen, Z. Fan and M.A. Patel, Classfcaton of rotated and scale textured mages usng Gaussan Markov random feld models, IEEE Transactons on Pattern Analyss and Machne Intellgence 13, 192-202 (1991). 7. M.M. Leung and A.M. Peterson, Scale and rotaton nvarant texture classfcaton, n Proc. 26th Aslomar Conference on Sgnals, Systems and Computers, Pacfc Grove, CA (1992). 8. M. Porat and Y. Zeev, Localzed texture processng n vson: analyss and synthess n the Gaboran space, IEEE Transactons

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