Brushlet Features for Texture Image Retrieval
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1 DICTA00: Dgtal Image Computng Technques and Applcatons, 1 January 00, Melbourne, Australa 1 Brushlet Features for Texture Image Retreval Chbao Chen and Kap Luk Chan Informaton System Research Lab, School of EEE Nanyang Technologcal Unversty, Sngapore Emal: {P & eklchan}@ntu.edu.sg Ch-Fa Chen Dept. of Electrcal Engneerng I-Shou Unversty, Tawan Emal: cfchen@su.edu.tw Abstract In ths paper, we employ a new adaptve bass of functons brushlets for extractng texture propertes. Brushlets are functons whch are well localzed wth only one peak n the frequency doman. Hence, a representaton of texture n terms of spatal frequency dstrbutons can be constructed. The Brushlet features are used n texture mage retreval experments to assess ts effectveness by comparng wth retreval results obtaned usng other commonly used wavelet and Gabor based representatons. Experments usng the Brodatz texture database ndcates that the brushlet features acheve a very good retreval comparable to and slghtly better than that usng Gabor features and better than the wavelet features. The advantage of the brushlet features s that they requre far less computaton to extract than the Gabor features for achevng comparable performance. 1. Introducton Texture can be found on many types of objects, such as woods, fabrcs, and n natural scene. In a dgtal mage lbrary, mages contanng varous types of textures are also often found. Texture s also one of the key descrptors for representng mage content n the proposed MPEG-7 standards. The ablty to effcently descrbe and analyze textured pattern s then of fundamental mportance to mage analyss as well as to general mage retreval. Many tools can be used to descrbe textures. In mage retreval, wavelet bass and Gabor bass functons have been used to represent textures. Wavelets provde an octave based decomposton of the Fourer plane wth a poor angular resoluton. Wavelet packets make t possble to adaptvely construct an optmal tlng of the Fourer plane and have been used n texture classfcaton [5]. However the tensor product of two real valued wavelet packets s always assocated wth four symmetrc peaks n the frequency plane. It s therefore not possble to selectvely tune and localze a unque frequency. It s also not orentaton selectve. Gabor flters are very popular n mage analyss [][][7] and they show excellent texture analyss ablty. Comparng to wavelet bass, Gabor bass s both frequency tunable and orentaton selectve. However, a flter bank consstng of many Gabor flters are often constructed to cover the entre frequency spectrum and the orgnal mage must convolve wth all these flters to extract texture features. Obvously, the computaton load s hgh. In order to obtan a better angular resoluton than the standard wavelet packets we can expand the Fourer plane nto a set of bruslets. Brushlets were proposed by F. G. Meyer and R. R. Cofman and have been appled n mage compresson [8]. A brushlet s a functon reasonably well localzed wth only one peak n frequency. Furthermore, the brushlet s a complex valued functon wth a phase. The phase of the D brushlet provdes valuable nformaton about the orentaton. These propertes make the brushlets sutable for texture and drectonal mage analyss. The objectve of ths paper s to study the use of ths new adaptve bass of functons brushlets for extractng texture propertes. Ths paper s organzed as follows. In secton, we revew the constructon of brushlet bases. In secton we descrbe the mage retreval algorthm based on a brushlet decomposton of the mage. Results of experments are presented n secton. Conclusons are gven n secton 5.. The Brushlet Bases In sgnal processng, a local analyss of the Fourer spectrum of a sgnal s often nterested. In order to analyze the local frequency content of a sgnal, a smooth wndow functon s used to cut the support of the sgnal nto adjacent ntervals. Then a local Fourer analyss s performed nsde each nterval. Let G = {g(t m) exp(πnt) : m, n Z} be a collecton of functons n L (R), where g s some fxed squarentegrable bump. The Balan-Low theorem [1] states that f G s an orthogonal bass, then the Hesenberg product of g
2 must be nfnte! In order to crcumvent ths obstacle rased by the Balan-Low theorem, varous Wlson bases [][6] have been constructed that use snes and cosnes rather than exponentals. Such a mechansm results n bass functons that contan equal energy at both postve and negatve frequences. However, usng exponentals s much preferred because the phase of the exponentals wll provde nformaton about the drecton of the pattern when descrbng mages n two dmensons. Wth the smooth local perodzaton technque ntroduced n [9], we can avod the Balan- Low phenomenon whle usng exponental functons that have all but an arbtrarly small amount of energy localzed n just the postve part of the frequency spectrum..1. Brushlet Fundamentals Consder a cover R = n=+ n= [α n, α n+1 ). Let = α n+1 α n, and c n = α n+α n+1. Around each α n we defne a neghborhood of radus ɛ. Let r be a ramp functon such that { 0 f t 1 r(t) =, 1 f t 1 and r (t) + r ( t) = 1, t R. Let b be the bump functon supported on [ ɛ, ɛ], b(t) = r ( ) ( t ɛ r t ɛ). Let wn be the wndowng functon supported on [ ɛ, + ɛ] ( ) r t+ ɛ f t [ ɛ, + ɛ) w n (t) = 1 f t [ + ɛ, ( ) ɛ). r t+ ɛ f t [ ɛ, + ɛ] ( ) Let e m,n (t) = 1 exp πm t α n the collecton of exponental functons. By usng the FFT and smooth local perodzaton technque [9], the 1D brushlet bass {v m,n } can be constructed [8] v m,n = exp(πα n t) exp(π t)[( 1) m ŵ σ ( t m) sn(π t)ˆbσ ( t + m)], (1) where σ = ɛ s defned as a steeess factor of the wndow w n and w σ (t) = w n ( t) s supported on [ 1 σ, 1 + σ]. Bump functon b σ (t) = b n ( t) s supported on [ σ, σ]. ŵ σ and ˆb σ are the nverse FFT of w σ and b σ, repectvely. Note that, n (1), appears as a scalng factor of the analyss, and m s the translaton ndex of the brushlet. v m,n has an expresson smlar to a wavelet. However, as opposed to a real valued wavelet, v m,n s a complex valued functon wth a phase. The phase encodes the orentaton of the brushlet pattern n the two-dmensonal case. The functon v m,n s composed of two terms, localzed around m and around m, that are oscllatng wth the frequency c n = αn+αn+1. The frst term s an exponental multpled by the wndow ŵ σ. Snce ˆb σ σ, the second term can be made as small as possble... The D Brushlets In the two dmensonal case we defne two parttons of R, j=+ j= [α j, α j+1 ) and k=+ k= [β k, β k+1 ). Let p j = α j+1 α j, q k = β k+1 β k, c j = α j+α j+1, and d k = β k +β k+1. Consder the tlng obtaned by the lattce rectangles [α j, α j+1 ) [β k, β k+1 ). The tensor product of bases v m,j and v n,k sequence, v m,j v n,k, s an orthonormal bass for L (R ). The tensor product v m,j (x) v n,k (y) s an orented pattern oscllatng wth the frequency (c j, d k ) and localzed at ( m p j, n q k ). To llustrate the orentaton selectve property of the D brushlet, we have calculated the brushlet expanson of the Brodatz mage D07 (see Fgure 1). A frst expanson was performed wth a parttonng of the Fourer plane nto four quadrants. The four sets of brushlets have the orentaton π + k π, k = 0, 1,,. A second expanson has been performed usng a fner grd. Each quadrant was further dvded nto four sub-quadrants. The sxteen set of brushlets have twelve dfferent orentatons. The orentatons π + k π are assocated wth two dfferent frequences. The four lattce squares around the orgn characterze the DC terms of the expanson. The other squares correspond to hgher frequency textures.. Image Retreval wth Brushlet Features In ths secton we ntroduce the feature defntonsbased on the brushlet decomposton of textured mages, and ther use n mage retreval experments..1. Texture Feature Extracton and Representaton For a gven retreval mage sample I, we extract ts texture feature usng a level brushlet decomposton. The decomposton dvdes the spectrum nto 16 blocks wth 1 dstnct drectons as shown n the bottom rght part of Fgure 1. Denote the brushlet coeffcents n each block I (j, k) ( = 1,..., 16, j = 1,..., R, and k = 1,..., C, where R and C are the number of rows and columns n each block, respectvely.), then we use the mean µ and the standard devaton σ ( = 1,..., 16) of the brushlet coeffcents to represent the blocks for retreval purpose: µ = 1 RC R j C I (j, k), () k
3 where d (m, n) = µ (m) µ (n) α(µ ) + σ (m) σ (n) α(σ ). (6) Orgnal D07 α(µ ) and α(σ ) are the standard devatons of the respectve features over the entre mage database, and are used to normalze the ndvdual feature components... Texture Retreval Algorthm Upon presentaton of a query pattern, the pattern s processed to compute the feature vector as n Equaton. The dstance d(m, n), where m s the query pattern and n s a pattern from the database for all mages n the database, s computed. The dstances are then sorted n ascendng order and the closest set of N patterns are then retreved Expermental Results and Level 1 decomposton Level decomposton Fgure 1. Brushlet decompostons (magnal parts) of Brodatz texture D07 and ther assocated drectons σ = 1 R C ( I (j, k) µ ). () RC 1 j A feature vector s now constructed usng µ and σ as feature components.. Dstance Measure k F = [µ 1 σ 1 µ σ... µ 16 σ 16 ]. () Consder two mage patterns m and n, and let F (m) and F (n) represent the correspondng feature vectors. Then the dstance between the two patterns n the feature space s defned to be d(m, n) = 16 d (m, n), (5) In ths secton, we present our expermental results. The texture database used n the experments conssts of the 11 texture classes from the Brodatz album. Each of the mages s dvded nto 16 non-overlapg sub-mages of sze 18 18, thus creatng a database of 179 texture mages. All 179 mages are used as query. When an mage s used as a query, t s removed from the database. The performance s measured n terms of the average retreval rate whch s defned as the average percentage number of patterns belongng to the same mage as the query pattern n the top N = 5 most smlar mages. To assess the usefulness of brushlets, we compare the retreval results obtaned by brushlet features wth those obtaned by Gabor and wavelet packet features. For the sake of farness, the decomposton levels performed n brushlet and wavelet packet expansons are set to and number of scales n the Gabor flterng s also set to. In our texture retreval applcaton, a par of two dstnct drectons, whch have a dfference of π, have the same effect on the expermental results. Therefore, the number of drectons of the Gabor flters are set to 6. The wavelet packets used n the experments are -tag Daubeches wavelet functons. The texture feature extracton and representaton, and dstance measure used n wavelet packet and Gabor based retreval algorthms are same as those used n brushlet based retreval algorthm. The average retreval rates for the 11 texture mage classes n the database are shown n Table 1. The overall retreval rates and standard devatons for the entre mage database are shown n Table. From Table, we can see the overall retreval of brushlet based algorthm s the best because that the dstrbuton of the retreval rates of brushlet based algorthm s also the most compact one as well as the margnally hgher averages than the Gabor based
4 algorthm. The computaton tmes for extractng features of one texture on a PENTIUM 5 CPU usng MATLAB 5..1 are shown n Table. From Table, we can see the brushlet feature extracton s sgnfcantly faster than the Gabor features. Hence, the brushlet features can be consdered better than the compared features for ths texture mage retreval. However we also note that there are several ndvdual mage classes bear very low retreval rate. For example, the rate of D0 s 0.70% and the rate of D0 s.%. Ths s due the bg varaton n the orgnal mages,.e the texture s non-homogenous. We dvded a bg mage nto 16 small mages and consder these mages formed an mage class n the mage database. Therefore, the mages n one of these mage class do not resemble each other. [6] H. Malvar. Lapped transform for effcent transform/subband codng. IEEE Trans. Acoust. Sgn. Speech Process. Machne, 8: , [7] B. S. Manjunath and W. Y. Ma. Texture features for browsng and retreval of mage data. IEEE Trans. Pattern Anal. Machne Intell., 18(8):87 8, [8] F. G. Meyer and R. R. Cofman. Brushlet: A tool for drectonal mage analyss and mage compresson. Appled and Computatonal Harmonc Analyss, :17 187, [9] M. V. Wckerhauser. Adapted Wavelet Analyss from Theory to Software. A K Peters, Ltd., A K Peters, Ltd., 89 Lnden Street, Wellesley, MA 0181, USA, Concluson In ths paper, a new texture analyss tool, the brushlet, s used to decompose a texture and represent the texture. Expermental results show that these brushlet features are qute effectve. Comparsons show the brushlets based features perform better than Gabor based features and much better than wavelet packets features n the terms of retreval rates and ther dstrbutons. Ths can be explaned as follows: The Gabor flters used n our experments are complex functons. They also provde excellent frequency localzaton ablty. Besdes, n a Gabor based system, arbtrary number of drectons and scales can be used. Ths ndcates that the analyss ablty of Gabor can be nfnte. However, all these come at the expense of computaton burden; wavelet packets of real functons cannot localze a unque frequency n the Fourer spectrum. The angular resoluton of brushlets s better than that of wavelet packets too. The man advantage of brushlets s less computaton requred for achevng comparable retreval performance as usng Gabor features. References [1] I. Daubeches. Ten lectures on wavelets. CBMS-NSF Conference Seres n Appled Mathematcs, SIAM Ed [] I. Daubeches, S. Jaffard, and J. L. Journé. A smple Wlson orthnormal bass wth exponental decay. SIAM J. Math. Anal., :55 57, [] J. G. Daugman. Complete dcrete D gabor transforms by neural networks for mage analyss and compresson. IEEE Trans. Acoust., Speech, Sgnal Processng, 6: , [] A. K. Jan and F. Farrokhna. Unsupervsed texture segmentaton usng Gabor flters. Pattern Recognton, (1): , [5] A. Lane and J. Fan. Texture classfcaton by wavelet packet sgnatures. IEEE Trans. Pattern Anal. Machne Intell., 15: , 199.
5 Table 1. Average retreval rate for the 11 texture mage classes n the database (B: Brushlet features, G: Gabor features, and W: Wavelet packet features) Class Rate (%) Class Rate (%) B G W B G W D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D Table. Overall retreval rates and standard devatons for the entre database Brushlet Gabor WP Overall Retreval Rate (%) Standard Devaton (%) Table. Average computaton tme for extractng features of one texture Brushlet Gabor WP Computaton Tme (Sec) Rato
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