Texture Analysis and Indexing Using Gabor-like Hermite Filters

Transcription

1 Submitted to Image ad Visio Computig, Elsevier, 2004 Texture Aalysis ad Idexig Usig Gabor-like Hermite Filters Carlos Joel Rivero-Moreo Stéphae Bres LIRIS, FRE 2672 CNRS, Lab. d'iformatique e Images et Systèmes d'iformatio INSA de Lyo, Bât. Jules Vere, 7 av. Jea Capelle Villeurbae Cedex, 6962 FRANCE rivero@rfv.isa-lyo.fr stephae.bres@isa-lyo.fr ABSTRACT I this paper, we study texture discrimiatio based o two filter families, Gabor ad Hermite, which agree with the Gaussia derivative model of the huma visual system. I the first part, discrimiatio of differet textures, based o the output eergy of these filters, is compared usig the Fisher criterio ad classificatio result. Results show that the preseted filter bak is suitable for purposes of texture feature extractio ad discrimiatio. The secod part of this paper is dedicated to global ad local texture feature extractio for image idexig ad retrieval based o Hermite filters. Keywords Texture, Hermite trasform, Gaussia derivatives, Gabor filters, feature extractio, idexatio, image retrieval, discrimiatio, Fisher criterio.. INTRODUCTION A importat task i image processig is texture aalysis []. It allows to achieve oe of demaded applicatios i computer visio: image idexig ad retrieval based o textural iformatio [2] [3] [4] [5]. Without losig geerality, two stages are required. The first oe cosists of assigig a texture feature vector to each pixel i a image. The feature extractor, which is usually based o filterig the textured image [6] [7], must be the able to discrimiate amog several textures. The goal of the secod stage is idexig based o the extracted features vectors. I order to allow dimesioality reductio, the feature vectors are also combied to provide a ew idex of texture features as a quatitative descriptio which is useful for image searchig ad retrieval by meas of a similarity measure. Texture feature extractio lies essetially i image processig techiques. It is usually performed by liearly trasformig or filterig the textured image [8] [6] [9] [7] followed by some eergy measure or o-liear operator (e.g. rectificatio). I this work, we would like to use a texture feature extractor that approximates the huma visual system (HVS), i such a way that impulse resposes of the filters are equivalet to those of the receptive field profiles (RFPs) of simple cells i the visual cortex. I this paper, we focus o the multi-chael filterig (MCF) approach. It is ispired by the MCF theory for processig visual iformatio i the early stages of the HVS [7] [9], where RFPs of the visual cortex ca be modeled as a set of idepedet chaels, each oe with a particular orietatio ad frequecy tuig. It the ivolves the decompositio of a iput image ito multiple features images by filterig. Each such a image captures textural features occurrig i a arrow bad of spatial frequecy ad orietatio. This approach has the property of exploitig spatial iteractios betwee the pixels of a eighborhood at differet scales. Amog the MCF models havig the above properties, Gabor filters [0] have bee widely used i texture feature extractio [] [6] [7] ad image idexig ad retrieval [2] [3] [5]. Aother model correspods to Hermite filters of the Hermite trasform [2] that agree with the Gaussia derivative model of the HVS [3]. It has also bee show aalytically that Hermite ad Gabor filters are equivalet models of RFPs of the HVS [4] [5] ad they well match cortical data. However, Hermite filters have some advatages over Gabor oes, like beig a orthogoal basis leadig to perfect image recostructio after decompositio ad it has a discrete represetatio ad filter separability which allows efficiet implemetatio. Furthermore, they ca be implemeted as both scale-space ad pyramid represetatios because of their agreemet with Gaussia derivatives. Thus, they are also suitable for detectig shape features such as edges, bars, ad corers; somethig that differs from the Gabor multi-scale represetatio. Despite these advatages, Hermite filters are ot used as much as Gabor filters for texture feature extractio ad, as we will show, cartesia Hermite filters do ot perform a equivalet texture discrimiatio like Gabor filters. This fact is due to their defiitio i the frequecy domai. Therefore, i order to icrease texture discrimiatio ad maitai the equivalece of the two models i this aspect, we itroduce the Gabor-like Hermite filters, which correspod to a modified versio of steered Hermite filters [6]. They are desiged to have a frequecy coverage similar to Gabor filters. I this work, we use the output eergy of filters i the bak as texture features. These resposes are closely related to the local power spectrum. Texture discrimiatio is measured followig the methodology proposed by Grigorescu et al. []. It uses Fisher liear discrimiat aalysis [7]. The iterest of usig such a approach is that discrimiatio is decoupled from classificatio (which arises ofte i segmetatio) ad it is the purely measured o statistics of two projected clusters of texture feature

2 vectors. Ideed, the Fisher criterio measures the maximum separability of two cocered clusters i the reduced space. We also apply a supervised classificatio to segmet some textured images i order to compare the similarity discrimiatio betwee Gabor ad Gabor-like Hermite filters from a visual viewpoit. Almost all proposed texture image retrieval systems i the literature, where idexig is based o extracted texture features, deal with uiform texture images, i.e., each of the images i the database is supposed to represet oly oe texture. A query texture patter ca be ay texture patter, belogig or ot to the database. Such idexes represet globally the texture i the whole image. It thus correspods to a global texture feature extractio ad idexig. I cases where the aalyzed image is composed of several textures formig uiform regios, the this approach for image browsig ad retrieval will ot work. Sice i such cases texture depeds o distict spatial locatios, the a local texture feature extractio ad idexig is required. A similarity matchig betwee extracted features at differet spatial locatios ad features of the query patter ca therefore be applied to search for similar lookig regios. We preset i this paper both global ad local approaches of texture feature extractio ad idexig. Our cotributios ca be summarized as follows: we itroduce a frequecy desig ito the defiitio of cartesia Hermite filters to achieve a steerable versio that yields Gabor-like Hermite filters ad we show how this filter bak improves texture discrimiatio. We preset a ormalized recurrece relatio for the Krawtchouk filters that allows fast computatios. We preset a dimesioality reductio approach based o relevat statistic parameters, which represets a compact texture model, by exploitig the joit statistics of Hermite coefficiets. Fially, we itroduce a local approach, based o the same texture feature extractio method, for images with several textured regios. This approach preserves the dimesioality reductio requiremet ad it is useful for texture image retrieval ad locatio without correspodece. The paper is orgaized as follows. I sectio 2, we show defiitios of Gabor ad cartesia Hermite filters, ad we preset the Gabor-like Hermite filter bak. I sectio 3, we describe the Fisher criterio ad give results o texture discrimiatio. I sectio 4, we apply filters for supervised texture segmetatio ad discuss results. I sectio 5, we explai the parametric texture model used for idexig. I sectio 6, we preset results ad discussio of the global approach. Sectio 7, cocers the local approach. Sectio 8 cotais fial coclusios. 2. GABOR-LIKE HERMITE FILTER BANK 2. Cartesia Hermite Filters Hermite filters d -m,m (x,y) decompose a localized sigal l v (x-p,y-q) = v 2 (x-p,y-q) l(x,y) by a Gaussia widow v(x,y) with spread σ ad uit eergy, ito a set of Hermite orthogoal polyomials H - m,m(x/σ, y/σ). Coefficiets l -m,m (p,q) at lattice positios (p,q) P are the derived from the sigal l(x,y) by covolvig with the Hermite filters. These filters are equal to Gaussia derivatives where m ad m are respectively the derivative orders i x- ad y-directios, for =0,,D ad m=0,,. Thus, the two parameters of Hermite filters are the maximum derivative order D (or polyomial degree) ad the scale σ. Hermite filters are separable both i spatial ad polar coordiates, so they ca be implemeted very efficietly. Thus, d -m,m (x,y) = d -m (x) d m (y), where each -D filter is: ( ) 2 2 x / σ πσ σ d ( x) = ( ) ( 2! ) H ( x / ) e () where Hermite polyomials H (x), which are orthogoal with respect to the weightig fuctio exp(-x 2 ), are defied by Rodrigues formula [8] as: 2 2 x d x H ( x) = ( ) e e (2) dx I the frequecy domai, these filters are Gaussia-like bad-pass filters with extreme value for (ωσ) 2 = 2 [4] [5], ad hece filters of icreasig order aalyze successively higher frequecies i the sigal. 2.2 Krawtchouk Filters Krawtchouk filters are the discrete equivalet of Hermite filters. They are equal to Krawtchouk polyomials multiplied by a biomial widow v 2 x N (x) = C / 2, which is the discrete N couterpart of a Gaussia widow. These polyomials are orthoormal with respect to this widow ad they are defied as [8] : K x C C (3) τ τ τ ( ) = ( ) N x x C τ = 0 N for x=0,,n ad =0,,D with D N. It ca be show that the Krawtchouk filters of legth N approximates the Hermite filters of spread σ = N / 2. I order to achieve fast computatios, we preset a ormalized recurrece relatio to compute these filters: K ( x) = [(2 x N) K ( x) + ( N )( + ) ( N + ) K ( x), (4) with iitial coditios K ( x ) = 0, 2 N K( x) = x N Steered Hermite filters I order to have a MCF approach based o Hermite filters, they must be adapted to orietatio selectivity ad multi-scale selectio. For doig this, we apply their property of steerability [6]. The resultig filters the may be iterpreted as directioal derivatives of a Gaussia (i.e. the low-pass kerel). Sice all Hermite filters are polyomials times a radially symmetric widow fuctio (i.e. a Gaussia), it is easy to prove that the + Hermite filters of order form a steerable basis [9] for every idividual filter of order. More specifically, rotated versios of a filter of order ca be costructed by takig liear combiatios of the filter of order. The Fourier trasform of Hermite filters d -m,m (x,y) ca be expressed i polar coordiates 2

3 ω x =ω cosθ ad ω y =ω siθ as dˆ ˆ m, m ( ωx, ωy ) = d ( ω) α m, m ( θ ) where d ˆ ( ω ), which expresses radial frequecy selectivity, is the -D Fourier trasform of the th Gaussia derivative i () but with radial coordiate r istead of x. The cartesia agular fuctios of order for m=0,,, are give as m m m α m, m ( θ ) = C cos θ si θ (5) which express the directioal selectivity of the filter. Steered coefficiets l (θ) resultig of filterig the sigal l(x,y) with these steered filters ca be directly obtaied by steerig the cartesia Hermite coefficiets l -m,m as: l ( θ ) = l α ( θ ) (6) m, m m, m m= Gabor-like Hermite Filters I order to tur the steered Hermite filters ito a MCF bak, we costruct a multi-scale represetatio that fulfils the desired costraits i the frequecy domai, which are maily the umber of scales S (radial frequecies ω 0 ) ad the umber of orietatios R i the filter bak. Sice previous works have bee doe essetially with Gabor filters, we have the adopted a similar multi-chael desig. Moreover, both Hermite ad Gabor filters are similar models of the RFPs of the HVS [4] [5]. For these reasos, we have amed the resultig filters as Gabor-like Hermite filters. The strategy desig i the frequecy domai is the same as that preseted i [2] [3]. It is to esure that the half peak magitude supports of the filter resposes i the frequecy spectrum touch each other. Let g(x,y) be a Gabor-like Hermite filter. The, its scaled ad orieted versios g s,r (x,y) are give by: s gs, r ( x, y) = a g( x, y ), a >, s, r ` (7) s s x = a ( x cosθ + y si θ ) ad y = a ( xsiθ + y cos θ ) where θ=rπ/r, r=0,,r, s=0,,s. The scale factor a s i (7) is meat to esure that the eergy is idepedet of scale s. Let σ x ad σ u be, respectively, the spatial ad frequecy spreads of a -D Hermite filter as defied i () which has radial frequecy selectivity ω 0. Let f l ad f h deote the lower ad upper ormalized ceter frequecies (betwee 0 ad ½) of iterest for the MCF bak. The, for each scale, parameters a, σ x, σ u, ad ω 0 of each chael are computed as: s ω0 = 2π f0, σ x = /(2 πσ u ), f0 = a f h ( a ) f a ( fh / fl ) S 0 =, σ u = ( a + ) 2l 2 Gabor-like Hermite filters already have zero mea (ull DC). The (discrete) Krawtchouk filters are liked to Hermite filters by these parameters as: N = ad 2 2σ x 2 ( σ xω0 ) / 2 (8) D = (9) where [] rouds to the earest iteger whereas does it too but towards ifiity. Notice that for each scale there is a set of parameters (N,D). I summary, costructio of a Gabor-like Hermite filter bak requires the followig procedure. First of all, set the umber of desired scales S ad orietatios R ad for each of the scales s=0,,s compute : the radial cetral frequecy ω 0 ad the spatial spread σ x of respective filters through (8). Krawtchouk s parameters such as widow legth N ad filter order D through (9). Krawtchouk filters: get the correspodig Krawtchouk polyomials through (4) ad multiply them by a biomial widow of legth N. Covolutios of iput image with Krawtchouk filters to obtai cartesia coefficiets. Steerig coefficiets to desired orietatios through (6) ad (5) to obtai the equivalet multi-chael outputs. 2.5 Real Gabor Filters We use real Gabor filters to maitai the equivalece preseted i [4] [5]. Nevertheless, complex Gabor filters ca be used sice the results i this paper remai valid. The desig of the Gabor filter bak is the same as that preseted i the previous subsectio for the Gabor-like Hermite filter bak, ad expressios (7) ad (8) are applied. The oly differece is the defiitio of g(x,y) i (7). It correspods to a real Gabor filter which is defied as: ( πσ xσ y ) ( x / σ x + y / σ y ) / 2 g( x, y) = /(2 ) e cos( ω x) (0) ad the y-coordiate is ivolved ito (8) with: σ = /(2 πσ ), y v 2 z = (2l 2) σ u π σ v = ta [ + ] 2l 2 / σ 2R 2 2 / 2 f0 z z u f 0 0 () For simplicity, the three filter baks, Gabor, Gabor-like Hermite ad cartesia Hermite, are deoted respectively by G, G2 ad H. 3. FISHER CRITERION-BASED DISCRIMINATION The Fisher criterio measures the maximum separability of two projected clusters ito a -D space, i.e. ito the lie which liks the ceter of the cocered clusters i the multidimesioal feature space. This oe is achieved by a liear trasform, W, applied to each of the feature vectors, x, belogig to the multidimesioal space, ad is called the Fisher liear discrimiat (FLD) fuctio. It is defied as: y T = W x, W S [ µ µ 2 ] = (2) where y are the projected vectors ito the -D space (resultig i scalars), µ ad µ 2 are the meas of the two clusters, S is the iverse of the pooled covariace matrix, S, of the two clusters. 3

4 This approach supposes that there are two clusters associated to two textures (i.e. two classes of textures to be discrimiated). S, the withi-class scatter matrix, is computed as the sum of two covariace matrices, S=S +S 2, oe for each texture class. They are computed as: L T ( µ )( µ ) (3) S = x x k i k i k i= where k=,2; x i is the associated vector to pixel i withi cluster k. L is the total umber of pixels ad K=S R is the umber of filters i the bak correspodig to the dimesio of the feature space. Fisher criterio, J, is give as: J = η η 2 σ + σ (4) where η ad η 2 are the projectios of the meas µ ad µ 2, ad σ 2 ad σ 2 2 are the variaces of the distributios of the projected feature vectors. J is the distace betwee two clusters relative to their size. The larger the value of J, the better the discrimiatio betwee two textures performed by the filters. Figure 2. Boxplot represetatio of Fisher criterio values obtaied with the three filter baks (from left to right: G, G2, ad H). The Fisher criterio results show that better cluster separatio, i.e. better texture discrimiatio, is obtaied from Gabor (G) ad Gabor-like Hermite (G2) filter baks. Ideed, from fig. 2, oe ca see that their discrimiatio is similar. 4. TEXTURE SEGMENTATION RESULTS We have applied a supervised classificatio method, the k-meas algorithm, to the texture feature vectors (which form vectors i the multidimesioal feature space) obtaied as eergy outputs of each of the three filter baks G, G2 ad H. I this case, k is the umber of classes or textures i the images (k=5). The resultig classificatio, based o the texture discrimiatio performed by the extracted texture feature vectors, yields a segmeted image. Fig. 3 shows two textured images, with five textures each oe, ad their segmeted images. A classificatio similarity betwee G ad G2 is expressed by the percetage of correctly classified pixels as show i table. Figure. Set of texture images for computig Fisher criterio values. Similar to [], we have applied each of the three filter baks to each of the textures show i fig. We have made our experimets o a well-kow set of images by Brodatz [20]. We chose, for Gabor (G) ad Gabor-like Hermite (G2) filter baks, S=5 scales ad R=4 orietatios, which results i a bak of K=20 filters. By settig D=5, oe obtais the same umber of chaels for the cartesia Hermite (H) filter bak. This oe is, i such a way, a ormalizatio of the space dimesio, so we ca obtai vectors of the same dimesio K. For simplicity, we have oly take the first L=000 pixels from each image. We have computed, for M=2 textures (fig. ), all pair-wise combiatios of Fisher criterio. Thus, there are M(M-)/2=66 Fisher criterio values. The statistics ad distributios of these values, for each of the filter baks, are show i the boxplot represetatio of fig. 2. Figure 3. Two texture images ad segmetatio results with the k- meas classificatio. (a) Origial image with k=5 textures, (b) segmetatio with G, (c) segmetatio with G2, (d) segmetatio with H, ad (e) perfect segmetatio. Table. Percetage of correctly classified pixels usig the k- meas classificatio for the images i fig. 3. G G2 H Image Image

5 Texture segmetatio results cofirm that G ad G2 are equivalet models from a visual viewpoit ad they perform similar classificatios. 5. PARAMETRIC TEXTURE MODEL The Gabor-like Hermite MCF-bak preseted i the sectio 2 is applied to decompose a give textured image ito a set of filtered images that represet the image iformatio at differet frequecies ad at differet orietatios. Therefore, each of the chael outputs of the filter bak ca be cosidered as oe compoet of a texture feature vector of dimesio S R. Thus, there are as much feature vectors as pixels i the image. For our applicatio, we chose S = 4 scales ad R = 6 orietatios, which results i a bak of 24 filters. There is the a importat dimesioality icreasig which is S R = 24 times the origial image size. Dimesioality reductio is thus a importat goal i image idexig techiques, sice oe eeds to store such idexes. Parametric texture models use combiatios of parameters to characterize textures [2]. We oly keep parameters which describe well the essetial structure of texture. For this purpose, we have tested differet combiatios of parameters ad we have foud that the best results, for texture idexig, are obtaied for oly cosiderig the spatial auto-correlatio of coefficiets of each subbad. Sice auto-correlatios imply a spatial lag from the cetral pixel i both x- ad y-directios, we have the fixed it to M=7 as i [2]. It the represets, for each of the subbads, a dimesioality reductio which goes from the image size to (M 2 +)/2 parameters, sice the spatial correlatio is symmetric. It yields S R (M 2 +)/2 = 600 parameters, which is a sigificat dimesioality reductio. These parameters represet structures i images (e.g., edges, bars, corers). Because of the multi-scale approach, there is a differet aalysis widow size for each of the scales S. Lower scales are related to higher frequecies (fie levels) ad thus widow sizes are shorter. O the other had, higher scales are related to lower frequecies (coarse levels) ad thus widow sizes are larger. The more the scale icreases, the more the widow size does too. I order to compare spatial iteractios i coarser levels where smoothig is very high, we have subsampled the auto-correlatios by a factor of two, i both directios, from a fie scale level to the ext coarser level. 6. GLOBAL TEXTURE FEATURE EXTRACTION AND INDEXING As we have already metioed i the itroductio of this work, our approach addresses both global ad local texture feature extractio ad idexig. I this sectio, idexes characterize globally the texture i the whole image ad each of the images i the database represet oly oe texture. 6. Texture Database ad Similarity Measure I this study, texture uder cosideratio are either gray-scale or lumiace-based. We have made our experimets o a wellkow set of images by Brodatz [20]. The texture database cosists of 2 differet texture images where each oe has a image with size of 640x640 pixels. Each of the images is divided ito 9 256x256 overlappig subimages, thus creatig a database of 008 images. We have the applied the texture feature extractio approach described i the previous sectio to every subimage i the database. The resultig feature vectors were saved as texture image idexes. Thus, oe has a texture-based retrieval system i which distaces betwee the query patter ad patters i the database are computed. We have used as distace - ρ, where ρ is a ormalized correlatio coefficiet i [-;]. It is computed as a ormalized dot product. Let x ad x 2 be two colum vectors of autocorrelatio parameters (with legth of 600) of two textures. The distace betwee them is the computed as: ( ) d = x x x x x x (5) T T T I the ideal case all the top 9 retrievals, which have the lowest distace to the query, are from the same large image. We have measured the performace of the idexig system i terms of the average retrieval rate, which is defied as the average percetage umber of patters belogig to the same image as the query patter i the top 9 matches (average recall). This oe is show i fig. 5, where the horizotal axis represets the umber of retrieved images ad the vertical axis represets the percetage retrieval performace. However, this way of assessig our idexig system is ot quite adapted to the groud-truth. Subimages comig from the same large image ca have a texture visually differet. 6.2 Experimetal Results ad Discussio The average retrieval rate (retrieval efficiecy) accordig to the umber of top matches cosidered is 8.29% for the top 9 retrievals. It meas that, i average, about less tha 2 textures amog the 9 most relevat oes are apparetly ot well classified. However, i geeral, they have a visual similarity from the texture viewpoit (fig. 4: D88-6 ad D89-9). The visual similarity (groud-truth) is cosidered by our texture idexig method whereas the retrieval efficiecy, which is used as a system assessmet, does ot. Ideed, the latter oly takes ito accout the correct matches which come from the same large image. Hece, the misclassificatios due to the assessmet are compesated by the visual similarity accordig to the followig reasos: () Some texture subimages which result from two differet large images have a similar visual aspect (fig 4.: D88-6 ad D89-9). The method is right i cosiderig them as similar textures (9 top matches) whereas the system assessmet is wrog i cosiderig them as two differet textures (because of two large images). (2) Some texture subimages which result from the same large image have differet visual aspects (fig 4.: D59-3 ad D59-7). This is the case of images i the database which have large structures ad their extracted subimages have o loger the same visual structures. The method is right i ot matchig them (differet textures). However, the system assessmet supposes that they have to be associated. O the other had, the method performs well sice it matches D59-3 with D88-6 i the 9 top (see fig 4). (3) Some texture images are icluded twice i the database: the image itself ad its egative (images above D00). The method cosiders them as idetical textures, because of the correlatiobased distace. This oe is right sice texture is more related to 5

6 spatial structures tha their lumiace variatios. O the other had, the system assessmet cosiders them as differet textures. Figure 4. Some examples of retrieved textures. From left to right: D88-6, D89-9, D59-3, ad D59-7. Therefore, oe should take as much the visual similarity as the retrieval efficiecy ito cosideratio, sice both of them are ivolved i which oe cosiders as a relevat retrieval. Fig. 8 shows some example retrievals where the first top image is cosidered as the query for the system assessmet ad the followig images i the colum are i the top most similar oes. Figure 5. Percetage retrieval performace (average recall). 7. LOCAL TEXTURE FEATURE EXTRACTION AND INDEXING I this sectio, we address the case of idexig ad retrievig a image composed of several texture regios. Sice texture iformatio varies at distict spatial locatios, we apply the texture feature extractio method of the previous sectio to spatial eighborhoods localized at positios of a sample grid of the image. The sample grid is computed adaptively so that dimesioality reductio is achieved. As we have already metioed, this is a importat costrait i image idexig techiques. Aother importat issue i texture image browsig ad retrieval is ot oly retrievig images that match similar texture regios with respect to a texture query but locatig i such a way these regios that help a user visualize their respective positios withi the retrieved images. I image searchig applicatios oe is iterested i havig a geeral isight of such texture regios matchig a query patter. This characteristic simplifies the task of texture localizatio sice oly eighbored poits belogig to the matched texture are useful. This way of texture localizatio differs from texture segmetatio ad texture edge detectio. The former obtai a segmeted image as a set of regios which are homogeeous i texture whereas the latter locate cotours delimitig each texture regio i the image. However, i both cases, the boudary betwee texture regios is fiely detected. 7. Implemetatio Details 7.. Sample grid Local texture feature extractio cosists of applyig the parametric texture model of sectio 5 to spatial eighborhoods localized at positios of a sample grid of the image. The maximum size of the sample grid is the image size, where each positio correspod to each pixel i the image. I this case, there is a vector of 600 parameters associated to each pixel i the image. Hece, the set of texture features icreases dramatically ad thereby the costrait of dimesioality reductio is ot achieved for idexig purposes. Ideed, oe should store 600 times the image size, which is ot suitable as a idex at all. O the other had, the miimum size of the sample grid is oe, the cetral pixel i the image, which correspod to the global feature extractio scheme preseted i sectio 6. Nevertheless, this oe supposes a uique uiform texture i the image, which is ot the case. Hece, it is ot suitable either. Therefore, there is a tradeoff betwee accuracy localizatio ad idex dimesio, which is related to the size of the sample grid. Thus, a represetative idex should be of a dimesioality lower tha or equal to the image size. I order to achieve accurate localizatio while avoidig dimesioality icreasig with respect to the image size, the sample grid is computed adaptively i such a way the resultig idex has the same size as the image size. Two subsamplig steps, T x ad T y, i x- ad y-directios, respectively, are thus computed for idicatig the positios i the image of the sample grid. The square size of the eighborhoods is equal to the maximum of subsamplig steps, so that there is o overlappig betwee them at two adjacet positios, at least i oe directio, whereas their tile cover all the image. Let us defie the followig parameters: N c = S R (M 2 +)/2, the total umber of parameters of local autocorrelatio at each positio of the sample grid;. X s = X r X c, the image size of X r rows ad X c colums; K, the estimated umber of samples i the grid; N, the eighborhood size; ad N p, the resultig umber of samples i the grid whe subsamplig steps, T x ad T y, are used. K = X s / Nc, T / x = X c K, Ty = X r / K N = max( Tx, Ty ), N p = X c / Tx X r / T y (6) The total umber of values, i.e. the idex size, which is close to the image size, is obtaied as N c N p. Fig. 6 shows a iput image composed of four differet square-shaped texture regios ad its sample grid. 6

7 Figure 6. Sample grid of a image with four texture regios. Top row: D06 ad D49. Bottom row: D8 ad D Local Autocorrelatio The local autocorrelatio is computed for each eighborhood at positios of the sample grid. This process requires to compute 600 parameters at each positio, which ivolves the 24 filters of the bak ad the 25 lags for the autocorrelatio values. However, i order to optimize computatios, we have performed a filterigbased local autocorrelatio. For each of the bads, 25 matrices with size of the image have bee obtaied. Their values are the resultig products betwee the bad ad its shifted versios. The, for each of the matrices, a covolutio with a 2Drectagular widow (size of NxN) is computed. This is equivalet to take summatios at each pixel positio. Fially, with the subsamplig steps, T x ad T y, i x- ad y-directios, respectively, the local autocorrelatio values are take at the sample grid positios. At the ed, oe has 600 subsampled matrices, whose elemets correspod to a vector of 600 parameters at the located positio i the sample grid. 7.2 Experimetal Results ad Discussio Retrieval is based o a similarity measure. We apply the same distace criterio preseted i the previous sectio betwee each of the texture feature vectors at each positio of the sample grid ad the texture feature vector of a give query patter. Notice that the query patter is idexed by the methodology preseted i sectio 6, sice it is supposed to be a homogeeous texture i the whole image. The eighborhoods with a small distace value idicate zoes that match the searchig query. Thus, we ca use a distace map, i.e. a grayscale image associated to distace values at each positio of the sample grid, where each tile represets a idexed eighborhood, to visualize the locatio of the retrieved texture. This way of localizig texture regios that match the query patter ca be see as a texture regio locatio without correspodece, sice texture regio idetificatio is based o a matchig criterio ad there is ot a accurate boudary detectio. Besides, a aalyzed texture regio by a sample grid implies that the local texture ca match a global oe sice there might be local similarities from the structure viewpoit. Figure 7. Distace map of two query texture resposes. Left colum: texture query. Middle colum: gray distace map. Right colum: biary map. Top row: D0. Bottom row: D75. Fig. 7 shows the distace map of two query patters. Oe correspods to a regio of the aalyzed image preseted i fig. 6 whereas the other is ot preset at all. The distace values are rescaled ito 256 gray levels (from 0 to 255) for display purposes. Each tile of the distace map correspods to a tile at a specified positio of the sample grid of the origial image. From fig. 7, it ca be show that a preset texture i the aalyzed image is detected (the image is thus retrieved) ad the associated regio is localized visually by the values of lower distaces. A biarizatio of distace map ehaces localizatio (where black zoes are the detected texture regio with respect to the query). I our experimets, we have chose for biarizatio a percetage threshold of 20% (p = 0.2). The threshold distace, d thr, is easily computed as d thr = p d max +( p) d mi, where d max ad d mi are the maximum ad miimum distace values, respectively. However, dark iformatio represetig the smaller distaces are ot the same from oe map to aother. Further post-processig might improve regio localizatio. For istace, oe could validate a subregio texture if it has a miimum umber of adjacet tiles. Of course, this does ot allow to detect regios smaller tha a miimal size. From fig. 7, oe ca see that a query texture which is ot preset i the aalyzed image leads i geeral to a (biary) distace map that does ot localize ay texture. However, retrieval could occur sice some tiles are ayway matched. It could be iterpreted as a error, but it depeds o the observer sice perhaps there is locally a similarity, although image bouds could cause iexact detectio due to discotiuity effects. 8. CONCLUSION We have implemeted a efficiet texture feature extractor for texture discrimiatio based o a Gabor-like Hermite filter bak, which is a multi-scale decompositio of steered discrete Hermite filters. The Fisher criterio results show that better cluster separatio, i.e. better texture discrimiatio, is obtaied from Gabor (G) ad Gabor-like Hermite (G2) filter baks. Ideed, from fig. 2, oe ca see that their discrimiatio is similar. O the other had, the cartesia Hermite (H) filter bak perform less texture discrimiatio. Moreover, texture segmetatio results cofirm that G ad G2 are equivalet models from a visual viewpoit ad they perform similar classificatios. Thus, texture discrimiatio is improved by the Gabor-like Hermite filter bak over the origial cartesia Hermite filter bak. 7

8 Efficiet global idexig ad retrieval has bee achieved by a powerful dimesioality reductio, which is based o spatial autocorrelatios of all multi-chael outputs. Spatial local autocorrelatios at positios of a sample grid allow to idex ad localize without correspodece texture regios withi a image composed of several textures. Idexig uses the same texture feature extractio scheme ad texture regio localizatio is based o both gray-scale ad biary distace maps. The system assessmet for global idexig is ot suitable eough to defie a real retrieval efficiecy as a fuctio of the true similarity sice it is oly based o cosiderig similar subimages whe they result from the same large image. Despite this drawback, we have about 82% of good matches which icreases up to 88% (see fig. 5) if some images with large structures ad those with egatives are removed from the database. This proves that our method is suitable for texture idexig purposes, sice the oly available iformatio to perform texture discrimiatio ad idexig is that resultig from the steered filters of the Gaborlike Hermite filter bak. Thus, these filters characterize well eough texture. 9. ACKNOWLEDGMENTS This work was supported by the Natioal Coucil of Sciece ad Techology (CONACyT) of Mexico, grat 539, ad by the SEP of Mexico. 0. REFERENCES [] Zhag, J. ad Ta, T., Brief review of ivariat texture aalysis methods. Patter Recogitio, vol. 35, pp , [2] Majuath, B.S. ad Ma, W.Y., Texture features for browsig ad retrieval of image data, IEEE Tras. Patter Aalysis Mach. Itell., vol. 8, pp , 996. [3] Wu, P., Majuath, B.S., Newsam, S., ad Shi, H.D., A texture descriptor for browsig ad similarity retrieval. Sigal Processig: Image Commuicatio, vol. 6, o.,2, pp , [4] Huag, P.W. ad Dai, S.K., Image retrieval by texture similarity, Patter Recogitio, vol. 36, pp , [5] Puzicha, J., Hofma, T., ad Buhma, J.M., Noparametric similarity measures for usupervised texture segmetatio ad image retrieval, I Proc. IEEE Cof. Computer Visio ad Patter Recogitio, pp , 997. [6] Rade, T. ad Husøy, J.H., Filterig for texture classificatio: A comparative study, IEEE Tras. Patter Aalysis Mach. Itell., vol. 2, pp , 999. [7] Bovik, A.C., Clark, M., ad Geisler, W.S., Multichael texture aalysis usig localized spatial filters, IEEE Tras. Patter Aalysis Mach. Itell., vol. 2, pp , 990. [8] Che, C.C. ad Che, C.C., Filterig methods for texture discrimiatio, Patter Recogitio Letters, vol. 20, pp , 999. [9] Rade, T. ad Husøy, J.H., Multichael filterig for image texture segmetatio, Optical Egieerig, vol. 33, pp , 994. [0] Porat, M. ad Zeevi, Y.Y., The geeralized Gabor scheme of image represetatio i biological ad machie visio, IEEE Tras. Patter Aalysis Mach. Itell., vol. 0, pp , 988. [] Grigorescu, S.E., Petkov, N., ad Kruiziga, P., Compariso of texture features based o Gabor filters, IEEE Tras. Image Processig, vol., pp , [2] Martes, J.-B., The Hermite trasform Theory, IEEE Tras. Acoust., Speech, Sigal Processig, vol. 38, o. 9, pp , 990. [3] Youg, R.A., Lesperace, R.M., ad Meyer, W.W., The Gaussia derivative model for spatial-temporal visio: I. Cortical model. Spatial Visio, vol. 4, o. 3,4, pp , 200. [4] Rivero-Moreo, C.J. ad Bres, S., Les filtres de Hermite et de Gabor doet-ils des modèles équivalets du système visuel humai?, I Proc. ORASIS 03, pp , [5] Rivero-Moreo C.J. ad Bres, S., Coditios of similarity betwee Hermite ad Gabor filters as models of the huma visual system, I Petkov, N. ad Westeberg, M.A. (eds.): Computer Aalysis of Images ad Patters, Lectures Notes i Computer Sciece, vol. 2756, Spriger-Verlag, Berli Heidelberg, pp , [6] va Dijk, A.M. ad Martes, J.-B., Image represetatio ad compressio with steered Hermite trasforms, Sigal Processig, vol. 56, pp. -6, 997. [7] Fukuaga K., Itroductio to Statistical Patter Recogitio, 2d ed. New York: Academic Press, 99. [8] Koekoek, R. ad Swarttouw, R.F., The Askey-Scheme of Hypergeometric Orthogoal Polyomials ad its q- Aalogue. Techical Report 98-7, Delft Uiversity of Techology, Faculty of Iformatio Techology ad Systems, Departmet of Techical Mathematics ad Iformatics, The Netherlads, 998. [9] Freema, W.T. ad Adelso, E.H., The desig ad use of steerable filters, IEEE Tras. Patter Aalysis Mach. Itell., vol. 3, pp , 99. [20] Brodatz, P., Textures: A Photographic Album for Artists ad Desigers. New York: Dover, 966. [2] Portilla, J. ad Simocelli, E.P., A parametric texture model based o joit statistics of complex wavelet coefficiets, It. Joural Computer Visio, vol. 40, pp. 49-7,

9 Figure 8. Example of retrieval images where the first top image is cosidered as the query. 9