Multiscale Directional Filter Bank with Applications to Structured And

Pattern Recognition 40 (2007) 1182–1194 www.elsevier.com/locate/pr

Multiscale directional ﬁlter bank with applications to structured and random texture retrieval

K.-O. Cheng∗, N.-F. Law, W.-C. Siu

Centre for Multimedia Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

Received 15 July 2005; received in revised form 16 June 2006; accepted 31 July 2006

Abstract In this paper, multiscale directional filter bank (MDFB) is investigated for texture characterization and retrieval. First, the problem of aliasing in decimated bandpass images on directional decomposition is addressed. MDFB is then designed to suppress the aliasing effect as well as to minimize the reduction in frequency resolution. Second, an entropy-based measure on energy signatures is proposed to classify structured and random textures. With the use of this measure for texture pre-classification, an optimized retrieval performance can be achieved by selecting the MDFB-based method for retrieving structured textures and a statistical or model-based method for retrieving random textures. In addition, a feature reduction scheme and a rotation-invariant conversion method are developed. The former is developed so as to find the most representative features while the latter is developed to provide a set of rotation-invariant features for texture characterization. Experimental works confirm that they are effective for texture retrieval. ᭧ 2006 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved.

Keywords: Texture characterization; Texture retrieval; Directional ﬁlter bank; Multiscale directional ﬁlter bank; Rotation-invariant features

1. Introduction filtering approaches including wavelet [10,11], Gabor filters [1,12], steerable pyramid [13] and directional filter bank As texture is one of the basic attributes of natural images, (DFB) [14] characterize textures in the frequency domain. texture analysis has attracted much attention on areas such Among the three categories, MPEG-7 has adopted Gabor- as computer vision, content-based image retrieval (CBIR), like filtering for the texture description [15]. The rationale remote sensing, medical imaging and quality inspection, behind is that visual cortex is sensitive to localized fre- etc. In particular, much research [1–4] has been done on quency components [16]. It has been shown that the direc- texture description for CBIR so as to manage the continu- tion together with scale information is important for texture ously growing multimedia data. The commonly used meth- perception. ods for texture characterization can be divided into three The filtering schemes, such as Gabor filters and steerable categories; statistical, model-based and filtering approaches pyramid, are developed for image analysis in a multiple [5]. Statistical methods such as co-occurrence features [6,7] scale and direction manner. Although Gabor filters and describe the tonal distribution in textures. Model-based steerable pyramid provide higher angular resolution than methods such as Markov random field (MRF) [8] and si- the wavelet transform, they are overcomplete in both scale multaneous autoregressive (SAR) models [9] provide a and directional decomposition. This in turn implies that they description of texture in terms of spatial interaction while are less computationally efficient than the wavelet approach [17]. For directional decomposition, DFB [18–20] has been ∗ Corresponding author. Tel.: +852 2766 6201; fax: +852 2362 8439. proposed as a highly computationally efficient tool. DFB E-mail addresses: [email protected] (K.-O. Cheng), is maximally decimated and so is not overcomplete. One ennfl[email protected] (N.-F. Law), [email protected] (W.-C. Siu). of the main disadvantages of DFB is the lack of multiscale

0031-3203/$30.00 ᭧ 2006 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2006.07.014 K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194 1183 property. Recently, pyramidal directional filter banks (PDFB) or contourlet transform [21,22] have been proposed to solve this problem by combining the DFB with Laplacian pyramid (LP) [23]. Although the LP is somehow redundant, the combined approach is still computationally efficient while providing a high angular resolution. In Ref. [24], the PDFB is modified as multiscale directional filter bank (MDFB) to have a fine high-frequency decomposition. In both the PDFB and MDFB, various lowpass filters can be used in the LP while still maintaining perfect reconstruction. Usually, the filters have the stopband edge greater than /2, e.g. quadrature mirror filter (QMF) and “9-7” biorthogonal filter. However, the use of these lowpass filters has a shortcoming that aliasing occurs in the passbands after downsampling, i.e. those at scales other than the first one. When the DFB is applied on the decimated lowpass image, the aliasing components will be decomposed at the same time. However, the orientations of the aliasing components can be very different from those of non-aliasing components for some directional subbands [25]. Therefore, further analysis should be performed to study and remove Fig. 1. Frequency partitioning in DFB. the aliasing effect so as to improve the use of MDFB for texture characterization. Besides the use of lowpass filters in the LP, there are still in Section 2. Then Section 3 addresses the aliasing problem many issues about the use of MDFB for texture character- associated with the directional decomposition of bandpass ization. In particular, the retrieval performance for random images in the LP. We will also describe a way to adapt the textures should be improved since most features commonly bandlimiting constraint in Refs. [17,25] to the MDFB so as extracted from filtering approaches lack statistical descrip- to alleviate the aliasing problem. In Section 4, a measure of tion. One way to enhance the retrieval performance of texture regularity based on the MDFB features is proposed. the MDFB is to unify it with statistical or model-based With the use of this measure, a hybrid system combining the approaches. It has been shown in Refs. [4,26] that the MDFB and the model-based algorithm is then developed. unified approaches can take the advantages of filtering A feature reduction method targeted for structured textures approaches for structured textures and statistical or model- is also developed so as to speed up the searching. Section 5 based approaches for random textures. Thus, we will first describes the development of the rotation-invariant MDFB study ways for characterizing structured and random tex- features. Finally, Section 6 concludes the paper. tures using MDFB. After this pre-classification, MDFB will be used for retrieval of structured textures while model- based approach will be utilized to retrieve random textures. 2. Background Furthermore, feature reduction and rotation-invariance are often concerned in practice. The retrieval time usually in- 2.1. Directional filter bank creases with the number of features so that features used for texture description should be as few as possible while The DFB [18–20] performs directional decomposition maintaining good retrieval accuracy. On the other hand, the with partitioning of a frequency plane in wedge-shaped query texture image provided by a user may have different regions as shown in Fig. 1. The DFB is computationally ef- orientations from the texture images stored in the databases ficient due to its tree structure. In the first two stages of the of the retrieval system. Therefore, the features representing tree structure, the DFB has a two-band filter bank structure the texture images should be insensitive to rotation. We stud- given in Fig. 2(a). The two-band filter bank consists of two ied ways to reduce the number of features as well as use of H 2-D() H 2-D() complementary fan-shaped filters 0 and 1 . rotation-invariant features for texture description. To split the input spectrum into two wedge-shaped regions, In this paper, we focus on various issues associated with the input signal is fed into the respective filters for de- the use of the MDFB for texture characterization and re- composition. Two directional subbands are obtained after trieval. This includes the aliasing problem, the combination decimation on a quincunx lattice using the downsampling of MDFB and model-based texture description algorithm, 1 −1 11 matrix, Q = or Q = . feature reduction for structured textures and the develop- 0 11 1 −11 ment of a rotation-invariant texture description. This paper For stages subsequent to the second stage, a two- is organized as follows. Background about MDFB is given band filter bank shown in Fig. 2(b) is used. This kind of 1184 K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194

Fig. 3. System block diagram for construction of each level in LP, where 2-D hL [n] is the 2-D lowpass filter implemented using the 1-D filter hL[n] through separable filtering.

Fig. 2. Two-channel fan-shaped filter bank (a) for the first two stages and (b) for the third and later stages. two bandpass images. The lowpass image is not considered for texture characterization. This is due to the non-ideal two-band filter bank still uses two complementary fan- frequency responses in real implementation which causes shaped filters for spectrum splitting. Since the next level uneven distribution of low-frequency components into dif- directional components have a parallelogram-shaped sup- ferent directional subbands [14]. Another reason for omitting port, resampling using some kind of unimodular matrix the lowpass image is that textures are often well character- Ri is required to change the support into fan-shaped for ized by the high- and mid-frequency components [28]. h [n] decomposition. There are four kinds of unimodular matri- In the MDFB, a 1-D lowpass filter l of cut-off fre- 11 1 −1 10 quency l is firstly applied on the input image on rows and ces, R = , R = , R = and 0 01 1 01 2 11 columns separately. In Ref. [24], the cut-off frequency l / 10 is 3 4. The first scale subimage is then obtained as the R = . Their uses depend on the shape of the 3 −11 difference of the input image and the lowpass image. Sub- support region of the required directional components. Af- sequent multiscale decomposition on this lowpass image is ter splitting, the signals are then decimated by matrix Qk performed by using the LP, as depicted in Fig. 3. Therefore, i(i ) to produce two subband signals. Further decomposition can the subimage 2 is the bandpass image of the level i − h [n] be achieved by feeding the subband signals into this type 1 in the LP. The 1-D prototype lowpass filter L used of two-channel filter bank. in the LP usually has stopband edge s which is greater / Apart from direct implementation, there is an efficient than 2 and gives rise to aliasing. Despite that, the LP still implementation of the two-band filter banks with a ladder allows perfect reconstruction by using the difference of the structure [27]. Under the ladder structure, signals are filtered original image and the approximation from the lowpass im- after downsampling. In addition, separable filtering can be age. After performing multiscale decomposition, the DFB is applied independently on each subimage for directional used. For the downsampling matrix Q1, the resulting two complementary fan-shaped filters are given as follows: analysis. For subsequent discussion, a multiscale directional decomposition is represented as (d1,d2,...,dn), where di −2N −1 −1 D z − z (−z0z )(−z0z1) is the number of directional decompositions at scale i and n H 2- (z ,z ) = 0 0 1 , (1) 0 0 1 2 is the total number of bandpass images (scales). An example of frequency partitioning is given in Fig. 4. H 2-D(z ,z ) =−z−4N+1 − (−z z−1)(−z z ) 1 0 1 0 0 1 0 1 × H0(z0,z1), (2) 3. Choice of hL[n] without aliasing effect where (z) is a 1-D lowpass filter, which has even length, N, and linear phase for the FIR design. The function of the 1-D lowpass filter hL[n] in LP is to de- compose an image iteratively into a set of bandpass images 2.2. Multiscale DFB i− i− of passbands [−, ]2\[−c, c]2, {[−c/2 1, c/2 1]2 i i S− As the DFB does not possess multiscale property, it has \[−c/2 , c/2 ]2|i = 1, 2,...,S − 1} and [−c/2 1, S− been combined with certain multiscale schemes for image c/2 1]2, where S is the number of decomposition lev- analysis [21,22,24]. For example, the LP is used in Refs. els and c is the cut-off frequency of hL[n]. In MDFB, c [21,22]. Unlike the critically sampled wavelet scheme, its is normally selected to be /2. However, non-ideal filters bandpass images do not suffer from frequency scrambling. hL[n] usually have non-zero values at frequency higher than The oversampled multiscale scheme has been improved for /2. According to Shannon sampling theorem, there would texture characterization [24] to provide one more bandpass be aliasing after downsampling by 2. The effect of alias- image by splitting the first bandpass image in the LP into ing on a multiscale decomposition is usually termed a shift K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194 1185

Fig. 4. Frequency partitioning (colored regions) of a frequency plane for Fig. 6. Illustration of aliasing in the frequency domain. (a) Ideal fre- H 2-D(z ,z ) multiscale directional decomposition (4, 8, 4). quency response of a non-decimated directional subband, D 0 1 . (b) Ideal frequency response of the directional subband for the decimated 2-D 2-D bandpass image at the second level of the LP, HB (z0,z1)HD (z0,z1). 2-D (c) Frequency response of the 2-D lowpass filter HL (z0,z1) commonly used in the LP. (d) Ideal frequency response of the non-decimated direc- H 2-D(z2,z2)H 2-D(z2,z2) tional subband in the second scale B 0 1 D 0 1 with dashed 2-D line indicating the filtering of the non-ideal lowpass filter HL (z0,z1).

bandpass response will not affect the following discussion. 2-D Furthermore, HB (z0,z1) can be approximated in terms of 2-D HL (z0,z1) as Fig. 5. Equivalent filter bank for one of the directional subbands in the D second scale (a) with upsampled filters in the DFB (H 2- (z ,z )) and 2-D 2-D 2 D 0 1 HB (z ,z ) = 1 − 0.25|HL ()| . (4) 2-D 0 1 (b) with upsampled filter (HSD (z0,z1)) in the MDFB. 2-D Hence, HB (z0,z1) is a bandpass filter with passband [−, ]2\[−/2, /2]2. The ideal frequency responses of variance problem in the time/spatial domain [17,29]. One the bandpass directional component before and after upsam- can also perform a frequency domain analysis on aliasing D D pling, i.e. represented by terms H 2- (z ,z )H 2- (z ,z ) [25]. Consider the PDFB that consists of the three levels LP B 0 1 D 0 1 and H 2-D(z2,z2)H 2-D(z2,z2), are illustrated in Fig. 6(b) and three levels DFB at the second scale. The filter bank for B 0 1 D 0 1 H 2-D(z ,z ) one of the directional subbands in the second scale can be and (d), respectively. In practice, the filter L 0 1 is realized as in Fig. 5(a). Using noble identity [30], the block non-ideal which means that there is non-zero frequency re- [−/ , / ]2 diagram of the filter bank can be redrawn in terms of one sponse outside the passband region 2 2 , especially non-decimated filter H 2-D(z ,z ) and one downsampler as near the transition band as shown in Fig. 6(c). This results SD 0 1 H 2-D(z2,z2)H 2-D(z2,z2) shown in Fig. 5(b). The non-decimated filter response is in passing undesired response of B 0 1 D 0 1 given by as indicated by the dashed line in Fig. 6(d). This undesired response comes from adjacent scale. However, its orienta- 2-D 2-D 2-D 2 2 2-D 2 2 HSD (z0,z1) = HL (z0,z1)HB (z ,z )HD (z ,z ), tion is very different from the desired response. There is 0 1 0 1 ◦ (3) about 73.7 deviation. In fact, this deviation in orientation is subband dependent. For subbands closer to direction 2-D ◦ where HL (z0,z1) is the 2-D lowpass filter in the LP, ±45 , the difference in orientation is larger. For example, 2-D HB (z0,z1) is the bandpass filter used to model the system with directional subband given in Fig. 7(a), less than half 2-D response of the bandpass image of the LP and HD (z0,z1) of deviation in the case of Fig. 6(d) is found as illustrated is the upsampled filter of one of the subbands in the DFB. by Fig. 7(b). Whatever thedirectional subbands involved, As the transition band of the bandpass response of LP will the aliasing must be avoided in order to have a precise 2-D not be concerned, the use of HB (z0,z1) to model the decomposition for directional analysis in each scale. 1186 K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194

drawback of using a lowpass filter with stopband edge at /2 is broadening of the bandwidth of the highest frequency subband. In these cases, the additional splitting of the highest frequency band in the MDFB becomes more important in order to maintain the radial frequency resolution for texture characterization. In this paper, the lowpass filter hl[n] for the additional splitting is designed so that the highest frequency band is split into two bands of nearly equal bandwidth. In particular, hl[n] is designed such that its frequency response Hl()=0.5 when =0.65 for non-aliasing filters, binom7 H ()= Fig. 7. Aliasing in the frequency domain for another directional subband. and ER13. For aliasing filters, QMF13 and Daub14, l (a) Ideal frequency response of a non-decimated directional subband, 0.5 when = 0.75. 2-D HD (z0,z1). (b) Ideal frequency response of the non-decimated direc- In summary, with the use of bandlimiting constraints for H 2-D(z2,z2)H 2-D(z2,z2) h [n] tional subband in the second scale B 0 1 D 0 1 with dashed L , precise directional decomposition can be achieved in 2-D line indicating the filtering of the non-ideal lowpass filter HL (z0,z1). downsampled bandpass images in LP. However, there would be a drawback of broadening the highest frequency passband. In order to maintain high radial frequency resolution, the highest frequency passband is split into two passbands of approximately equal bandwidth using a lowpass filter hl[n].

3.1. Retrieval based on MDFB

Each subband in MDFB represents texture characteristics in a particular scale and direction. Thus, energy signatures of subbands in MDFB can be used as texture features. The commonly used energy signatures are L1 and L2 norms [3]. From our experiences, L1 norm gives slightly better retrieval performance for textures and is used throughout this paper. The L1 norm energy signature for the subband of scale s and direction d is deﬁned as N f = 1 |x | s,d N s,d,n , (5) n=1

Fig. 8. Filter response of the QMF filter of length 13 (QMF13) and where {xs,d,n} is the set of the subband coefficients and N is Daubechies filter of length 14 (Daub14), binomial filter of length 7 the total number of coefficients in that subband. The feature (binom7) and the designed equiripple filter of length 13 (ER13). vector is obtained by concatenating the features at different scales and directions as In order to reduce the undesired filtering, the aliasing in f = (f , ,f , ,...,fS,D − ,fS,D ), (6) the LP is minimized by imposing the bandlimiting constraint 1 1 1 2 S 1 S to ensure zero response above /2onhL[n] [17,25].For where S and Di are the total number of scales and the total example, a binomial filter of length 7 can be selected be- number of directions for the ith scale, respectively. For sim- cause its stopband energy is significantly small at frequency ilarity measure of two texture images, the weighted sum of larger than /2. Apart from binomial-type filters, the LP fil- absolute difference is used. The weighted sum of absolute ter can be designed using Parks–McClellan algorithm [31] difference between images u and v is computed by by requiring the stopband edge to be /2. Fig. 8 shows the S Ds u v magnitude responses of four types of filters, (1) binomial fs,d − fs,d d(u, v) = filter of length 7 (binom7), (2) an equiripple filter of length , (7) s= d= s,d 13 (ER13) designed using Parks–McClellan algorithm given 1 1 the bandlimiting constraint and two lowpass filters with cut- where the weight s,d is the standard deviation of features at off frequency /2 of similar length, (3) QMF of length 13 scale s and direction d over the entire database. In retrieval, (QMF13) and (4) Daubechies filter of length 14 (Daub14). K textures with the least distance from a query texture are As binom7 and ER13 have stopband edge at /2 so they are returned as the results, where K is set to be the size of the referred as non-aliasing filters whilst QMF13 and Daub14 query texture’s class in the database. The retrieval accuracy filters have non-zero values beyond /2 so they are re- is evaluated as the average percentage of correctly retrieved ferred as aliasing filters. From Fig. 8, it is obvious that the images for the queries. K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194 1187

Table 1 Table 2 Retrieval accuracy (%) of MDFB based on aliasing and non-aliasing ﬁlters Retrieval accuracy (%) for same number of directional decompositions in for different number of scales all the four scales

Decomposition QMF13 Daub14 Binom7 ER13 Decomposition Binom7 ER13

(8, 8, 8) 64.1 63.4 69.6 69.1 (4, 4, 4, 4) 69.3 69.7 (8, 8, 8, 8) 68.3 68.0 72.0 72.2 (8, 8, 8, 8) 72.0 72.2 (8, 8, 8, 8, 8) 70.0 68.9 71.5 72.1 (16, 16, 16, 16) 71.6 71.5 (32, 32, 32, 32) 70.7 70.5

3.2. Retrieval experiment using non-aliasing LP decomposition Table 3 Retrieval accuracy (%) for various combinations of 4 and 8 directional To investigate texture retrieval based on the MDFB with decompositions in the four scales h [n] non-aliasing filters L , a texture retrieval experiment on Decomposition Binom7 ER13 a database derived from 111 images in the Brodatz texture album [32] is performed. Each of the 111 album im- (4, 4, 4, 8) 70.0 70.8 ages is a 512 × 512 8-bit gray level image. To construct the (4, 4, 8, 4) 71.2 71.3 database, nine 128 × 128 non-overlapping subimages are (4, 8, 4, 4) 70.9 71.2 extracted at the center of each album image. This results (8, 4, 4, 4) 69.9 70.0 in 999 images of 111 texture classes. Every subimage ex- (4, 4, 8, 8) 70.9 71.5 tracted from the same album image is considered to be in (4, 8, 4, 8) 71.1 71.9 the same texture class. To prevent the bias due to similar in- (8, 4, 4, 8) 70.8 71.1 tensity in images of the same class, all images are subjected (4, 8, 8, 4) 71.7 71.8 to histogram equalization. In order to study the influence of (8, 4, 8, 4) 71.6 71.2 aliasing, we vary the number of scales but fix the number of (8, 8, 4, 4) 71.2 71.1 directional decomposition to be 8 for each scale in this exper- (4, 8, 8, 8) 71.7 72.1 iment. Filters used for hL[n] include those discussed in the (8, 4, 8, 8) 71.5 71.8 previous section, i.e. QMF13, Daub14, binom7 and ER13. (8, 8, 4, 8) 71.8 71.7 A lowpass filter hl[n] of half amplitude at frequency 0.75 (8, 8, 8, 4) 72.1 71.7 is associated with aliasing filters hL[n] for further decom- The highest retrieval accuracy among the combinations of same set of position in the highest frequency band. For non-aliasing fil- directional decompositions is highlighted. ters hL[n], hl[n] with half amplitude at frequency 0.65 is used. Table 1 summarizes retrieval accuracies for the MDFB the scales higher than the fifth one, should have no or very based on different scale decompositions. For the same little improvement in the cases of aliasing filters as well. number of scales, the retrieval accuracy obtained using non- aliasing filters is always higher than that using aliasing fil- 3.3. Retrieval with different directional decomposition in ters. With non-aliasing filters, the highest retrieval accuracy scales is achieved when four scales are used. The retrieval accuracy is about 72%, which is higher than those with aliasing As shown in Table 1, four scales are important for texture filters by approximately 4% for the same number of scales retrieval in the MDFB with the non-aliasing filters hL[n]. and 2% for the best results. From the results of non-aliasing In this section, we investigate the retrieval performance of filters, it can also be found that the retrieval accuracy in- different directional decomposition in the scales through increases up to four scales and has no improvement when tense simulations. First, let us fix the number of directional the fifth scale, which corresponds to a scale of frequency decompositions to be 4, 8, 16 and 32 in all the four scales. below /16, is added. This confirms our discussion in The results are given in Table 2. It can be seen that when Section 2.2 that the mid- to high-frequency texture informa- the number of directions is 8, the retrieval accuracy is the tion is more important than the low-frequency information. highest for both the two non-aliasing LP filters. Second, In the cases of aliasing filters, the retrieval performance is we perform further analysis using different combinations improved up to five scales. However, it should be noted that of 4 and 8 directional decompositions in the four scales. the aliasing filters have bandwidth about double of that of Note that, increasing the number of directional components the non-aliasing ones (as illustrated in Fig. 8) so their fifth increases the angular resolution, while at the same time scale corresponds to the frequency range of the fourth scale increases the number of features used in texture characteri- in the cases of the non-aliasing filters. In addition, the re- zation and retrieval. The retrieval accuracies for these combi- trieval accuracy also tends to be stable when the number of nations are summarized in Table 3. With two or more scales scales increases. The low-frequency components, probably of 8 directional decompositions, the retrieval accuracy can 1188 K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194 be maintained very close to the decomposition (8, 8, 8, 8). as given in Table 4. The complexity of steerable pyramid is For ER13 filter, decomposition (4, 8, 8, 4) has retrieval accu- higher and proportional to number of directions D because racy about 0.4% less than that of decomposition (8, 8, 8, 8) of its overcomplete directional decomposition. For Gabor but with 8 features less. The retrieval accuracy of decompo- filters, if the filtering is performed in frequency domain, the O(DSN2 N) sition (4, 8, 8, 8) is 0.1% less while the number of features complexity is log2 . The complexity of Gabor used is 4 less. From these results, retrieval performance is filters is the highest as the decomposition is overcomplete in higher in most cases when higher angular resolution is used both scales and directions. In summary, the MDFB approach in mid-frequency range. For example, decomposition (4, 8, has a low computational complexity while gives comparable 8, 8) has retrieval accuracy which is 0.4% higher than that retrieval accuracy as Gabor filters and steerable pyramid. of decomposition (8, 8, 8, 4). This can be explained by the brief that mid-frequency components are more important for characterizing textures [10]. Similar experiments have been 4. Use of MDFB in classification of structured textures conducted using different combinations of 8 and 16 directional decompositions in the four scales. However, the high- A way to classify textures is to specify its perceptual char- est improvement was found to be about 0.1% despite that acteristics such as regularity. Regularity is closely related more features were used. From these experiments, we can to directionality [33]. We have seen that the MDFB can be see that the optimal number of orientations is about 8. For used to capture texture characteristics at various scales and the first and fourth scales, the number of orientations can be orientations. Hence, the MDFB can be used to character- reduced to be 4 without significant drop in retrieval accuracy ize texture regularity. Regular or structured textures usually because the mid-frequency components can well describe consist of dominant periodic patterns. Therefore, orienta- textural features. tions of the patterns are consistent throughout. For random or unstructured textures, there is no well-defined direction- 3.4. Performance comparison ality so the distribution of signal energy should be roughly uniform over all orientations. In our classification algorithm, The performance of the MDFB for texture retrieval has the directionality is measured based on the similarity of en- been compared with the Gabor filters in polar form [12] ergy signatures. Shannon entropy, which gives a measure of and steerable pyramid [17] in terms of retrieval accuracy “concentration” for a sequence [34], is used. For a directional and computational complexity. The comparison results are energy sequence, {ed }, Shannon entropy is calculated as summarized in Table 4. For texture retrieval, the database 1 e 2 e 2 used is the same as before. The L norm and sum of abso- E({e }) =− d d d e ln e , lute difference are used as subband features and similarity d measure, respectively. The number of scales of Gabor filters 1/2 and steerable pyramid is 4 as it is frequently used in liter- 2 where e = ed . (8) ature. In fact, our results of the MDFB in Section 3.2 that d the frequency components below /16 are insignificant for texture characterization agree with the use of the first four A large value of E is obtained if energy values are nearly the scales. For Gabor filters, different numbers of orientations same, and vice versa. In MDFB, each scale will give one have been tested and the results are provided in Table 5. value of E. To determine whether a texture is structured or The highest retrieval accuracy of 72.5% is achieved for six random, the minimum of the entropy over all scales is used orientations in the four scales. For the steerable pyramid, as the overall regularity measure of textures. The reason is six orientations and four scales decomposition is performed. that a structured texture should have dominant directional The retrieval accuracy is found to be 69.6%. In Table 4, features in at least one scale. The regularity of a texture is these results are compared with the MDFB approach using thus given by decomposition (4, 8, 8, 8) for ER13 filter. It can be seen that R = min Es, (9) the retrieval performance of the MDFB is comparable to the s Gabor filters while higher than steerable pyramid by 2.5%. In terms of the number of features, the MDFB approach where Es is the entropy of the normalized directional en- uses 28 features, which are more than the 24 features in both ergy sequence at scale s. If the regularity is below a pre-set the Gabor filters and steerable pyramid cases. However, in threshold, the texture is classified to be structured. Oth- Section 4.3, a feature reduction scheme for the MDFB ap- erwise, the texture is classified as a random texture. The proach will be proposed which can significantly lower the threshold can be determined to have the value for the best computational complexity. For the complexity in feature ex- retrieval performance on a given training set of textures. It traction, the MDFB is the best. Unlike these two schemes, should be noted that random textures may be confused with the directional decomposition in MDFB is not overcomplete. textures with circular patterns such as tree–ring patterns. For decomposition with S scales and D directions of an im- The textures with circular patterns would give low value of N ×N O(N2 D) age of size , MDFB has complexity of log2 R even if the patterns are arranged regularly. K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194 1189

Table 4 Summary of comparison results

Approach Gabor ﬁlters Steerable pyramid MDFB

Retrieval accuracy (%) 72.5 69.6 72.1 Number of features 24 24 28 O(DSN2 N) O(DN2)O(N2 D) Complexity in feature extraction log2 log2

Table 5 Retrieval accuracy of four scales Gabor wavelet with various number of orientations

Number of orientations 4 6 8 10 12 14 16 18 Retrieval accuracy (%) 71.6 72.5 72.2 71.7 71.6 71.3 71.0 70.8 Number of orientations 20 22 24 26 28 30 32 Retrieval accuracy (%) 70.4 70.4 69.8 70.1 69.7 69.9 69.3

Table 6 Entropy-based regularity of six texture samples

Texture D1 D17 D74 D86 D91 D102

(8,8,8,8) 1.553 1.876 1.897 1.840 1.692 1.838 (16,16,16,16) 1.952 2.336 2.559 2.521 2.384 2.112

The dynamic range of entropy is different for different number of directions.

4.1. Finding structured and random content as well as random textures, a hybrid system which com- bines features from MDFB and a model-based approach In this part, classification performance of structured called multiresolution simultaneous autoregressive (MR- and random textures using the MDFB-based features is SAR) model [35] is investigated and implemented. First, a examined. Table 6 gives the regularity value R based on query texture is classified to be either structured or random decomposition (8, 8, 8, 8) and (16, 16, 16, 16) of six texture using the method described in Section 4.1. If the query samples from the Brodatz texture album as illustrated in texture is classified to be structured, features extracted Fig. 9. Images D1, D17 and D102 are regarded as structured from MDFB are selected for retrieval. Otherwise, features while D74, D86 and D91 are considered to be random. The extracted from MRSAR models are used. As shown in Sec- regularity value of structured texture D1 is significantly tion 4.1, the multiscale directional features can distinguish smaller than that of random texture D74 in both decompo- structured and random textures. Therefore, the multiscale sitions. However, the more complicated structured texture directional features can be used for pre-classification of D17 has higher regularity value than the random textures textures in structure as well as in the retrieval of any query D86 and D91 when decomposition (8, 8, 8, 8) is used. For texture classified to be structured. decomposition (16, 16, 16, 16), the situation is improved Experiments have been performed to investigate the re- so that the regularity value of D17 is smaller than that of trieval performance of the hybrid system. To avoid the bias D86 and D91. If the threshold for classification is set to be due to unequal number of structured and random textures, 2.36, the structured and random textures can be classified a database composed of 20 classes for each structured and successfully. This implies that having higher directional random textures is selected. The images are derived from decomposition such as (16, 16, 16, 16) is necessary for the Brodatz album images, D1, D4, D6, D8, D15, D21, D22 classifying structured textures. D23, D27, D30, D34, D49, D50, D51, D52, D53, D54, D55, D56, D58, D62, D63, D66, D67, D68, D73, D75, D77, D79, 4.2. Pre-classification of textures before retrieval D83, D85, D90, D92, D95, D99, D100, D106, D108, D111 and D112. For the hybrid retrieval system, the MDFB ap- As a filtering approach, the MDFB features can charac- proach with decomposition (16, 16, 16, 16) is used. The terize periodic structured textures effectively. For random lowpass filter in LP is ER13. For MRSAR approach, Gaus- textures, the MDFB approach may not work as effective sian filter is chosen for multiscale decomposition and the as statistical approaches and model-based approaches due number of scales is four. From each scale, four SAR coef- to ambiguous directionality in their patterns. In order to ficients and standard deviation of estimation error are cal- optimize the performance of the MDFB method for a more culated as the features. This gives 20 features in total. The general texture database, i.e. the one with structured textures weighted sum of absolute difference similar to Eq. (7) is 1190 K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194

Table 7 Comparison for the hybrid retrieval system, the retrieval system based on MDFB only and that based on MRSAR only

Database Retrieval accuracy (%) for hybrid system Retrieval accuracy (%) for MDFB Retrieval accuracy (%) for MRSAR Quantity of texture images

All 81.5 79.3 79.2 360 Structured 96.8 96.8 92.1 175 Random 67.0 62.7 67.0 185

lower retrieval accuracy but is compensated in the case of the hybrid system. The experimental results justify the use of pre-classiﬁcation based on entropy for optimizing the retrieval performance of MDFB.

4.3. Feature reduction for structured textures

Besides the retrieval accuracy, another main concern in texture retrieval is the searching time. As the number of computations in similarity measurement increases with the number of features, an efficient retrieval scheme should use features as few as possible to speed up the searching. As discussed in Section 4.2, structured textures are extracted and processed using the MDFB approach in the hybrid system. Feature reduction method for MDFB can then be focused for structured textures only. For the wavelet packet energy features used in Ref. [36], a few most dominant energy signatures were selected for texture representation whilst the performance could still be maintained at a high level. A similar feature reduction scheme is used here because structured textures usually have significant energy in the dominant orientations. The m highest energy signatures of a query texture and the corresponding subband energy signatures of textures in the database are selected for similarity measurement. However, somewhat unlike in the scheme of wavelet packet energy features, the corresponding features of a texture in the database, which do not belong to its m most dominant energy signatures, are set to zero. The reason is that the non-dominant energy signatures usually correspond to subbands of little directional information and can be greatly affected by noises. By setting them to zero, Fig. 9. Six texture samples used in regularity measurement in Table 6. the retrieval error can be reduced. Images from left to right and top to bottom: D1, D17, D74, D86, D91 Fig. 10 shows an example of reconstruction of a structured and D102. texture using the lowpass subband and different numbers of dominant subbands. Here, dominant subbands refer to the used as their similarity measure. In the pre-classification, the subbands with large energy signature. With only the low- threshold for regularity is chosen so as to achieve the high- pass subband, most of textural characteristics are lost. By est retrieval accuracy. The experimental results are summa- increasing the number of dominant subbands, the structure rized in Table 7. The overall retrieval accuracy of the hybrid of the texture in the reconstructed images becomes clearer. system is 81.5% which is 2.2% and 2.3% higher than those The reconstructed image looks similar to the original one of MDFB and MRSAR approaches, respectively. In the re- when first 28 dominant subbands are included in the recon- trieval, 175 samples are classified to be structured while 185 struction. This shows the importance of dominant subbands samples are regarded as random. The pre-classification re- for structured texture description. sults closely match the texture distribution in the databases. Further experiments were performed to evaluate the fea- As shown in the table, the retrieval accuracy of MDFB for ture reduction scheme for the MDFB features in texture structured textures is 96.8% and higher than that of MRSAR retrieval. The setup of the experiment was the same as in by 4.7%. For random textures, the MDFB approach gives Section 4.2 except that the proposed feature reduction K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194 1191

structured textures versus the number of features is plot- ted in Fig. 11. From the results, high retrieval accuracy can be found even for small feature sets. The highest retrieval rate is achieved when 37 features are used. The value is 97.8%, which results in 82.0% overall retrieval accuracy in the hybrid system. The minimum number of features for performance comparable to the complete set of features is 19, which is less than one-third of the complete feature set. The results show that the proposed feature reduction method based on the MDFB features is effective for structured textures.

5. Rotation-invariant features

Each band of the DFB corresponds to a frequency region at a particular orientation. If the input image is rotated, the sequence of subband energies is circularly shifted. Similar to the case of Gabor filtering [37], the magnitudes of discrete Fourier transform (DFT) coefficients of a directional energy sequence can be used as rotation-invariant features. To ac- commodate this idea to features extracted from the MDFB, the DFT is applied on the features weighted by global standard deviation in each scale, i.e. {ês,d}=DFT {es,d/s,d} for an energy sequence {es,d} in scale s. A rotation-invariant feature vector can then be formed as ˆ fs = (|ês,1|, |ês,2|,...,|ês,Ds |), (10)

where Ds is the number of directions in scale s. Due to symmetry of magnitude spectrum, almost half of the size of the feature vector can be reduced. The feature vector can Fig. 10. Reconstruction of structured texture D1. Images from left to right then be formed as and top to bottom: original image, reconstructed images using the lowpass subband and the first m dominant subbands, m = 0, 14, 28, 42 and 56. ˆ fs = (|ês,1|,a|ês,2|,...,a|ês,Ds /2|, |ês,Ds /2+1|), (11) where a is the scaling factor for coefficients except those at frequency 0 and , which is used to adjust the similarity ˆ ˆ measure based on f√s to be the same as that based on fs. For example, a = 2 for Euclidean distance and a = 2 for sum of absolute difference, respectively. The rotation- invariant multiscale directional feature vector is given by concatenating the feature vector in each scale as fˆ = (fˆ, fˆ,...,fˆ ) 1 2 S , (12) where S is the total number of scales. Since the feature vectors have changed, some modifications are required in the retrieval algorithms as presented in Sections 3 and 4. First, sum of absolute difference without weighting is used as the similarity measure. Second, if pre-classification for texture structure is desired, it should be performed before Fig. 11. Retrieval accuracy (%) versus number of features used. applying the DFT. Besides these two modifications, other steps remain unchanged. The Brodatz texture album is used to test the retrieval per- method was applied to the MDFB features. The 175 query formance using the proposed rotation-invariant multiscale textures, which were pre-classified to be structured, were directional features. Each Brodatz texture image is divided investigated. A graph showing the retrieval accuracy of the into four non-overlapped regions of size 256 × 256. After 1192 K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194 that, each divided region is rotated from 0◦ to 165◦ 6. Conclusions with a step size of 15◦ to generate 12 rotated subimages. The center part of each rotated subimage is then cropped to The use of multiscale directional filter bank (MDFB) is have a size of 128 × 128. The database is thus formed with analyzed for texture characterization and retrieval in this pa- 5328 images and each class consists of 48 texture images. per. First, we study the influence of aliasing in Laplacian The performance for two decompositions (8, 8, 8, 8) and pyramid (LP) on the directional decomposition of bandpass (16, 16, 16, 16) based on LP with filter ER13 are examined. images. It is found that the aliasing mixes frequency com- The retrieval accuracy and number of features are given in ponents in different orientations which results in inaccurate Table 8. The results for features without rotation-invariant directional decomposition. Hence, the multiscale decompo- conversion are included for reference only. Obviously, the sition in MDFB is designed by considering a bandlimiting rotation-invariant features give much higher retrieval accu- constraint on the lowpass filter used in LP so as to eliminate racy than the original ones. It can also be found that unlike the aliasing. Experimental results show that the features ex- the case without rotation, when the number of angular de- tracted based on MDFB without aliasing in the LP decompo- composition increases from 8 to 16, the retrieval accuracy sition give higher retrieval accuracy. On the other hand, it is is improved from 59.2% to 60.3%. This is due to the fact found that finer directional decomposition in mid-frequency that high angular resolution is essential to distinguish finely range usually results in better texture characterization as rotated texture images. most texture patterns are quasi-periodic in nature. Our com- Comparative studies about rotation-invariant texture parative study shows that the MDFB approach is comparable classification were performed with one existing approach to Gabor filters in polar form but better than steerable pyra- using rotation-invariant polar-wavelet texture features [2]. mid in retrieval accuracy. However, directional decomposi- Retrieval experiment for our proposed algorithm was per- tion in MDFB is maximally decimated. At the same time, formed on the same database used in the reference paper the filtering in MDFB is separable in nature. These imply [2]. The database is a subset of the database used previ- that the computational complexity of the MDFB is much ously with 25 texture classes only. The retrieval accuracies smaller than that of the Gabor filters and steerable pyramid. of our proposed algorithm and the algorithm using polar- The MDFB has been utilized for classifying structured wavelet features are provided in Table 9. Our proposed and random textures. The classification is based upon the algorithm uses a significantly smaller number of features similarity measurement of directional subband energy sig- than the polar-wavelet features. Moreover, the retrieval natures in each scale using the Shannon entropy. With a performance of the case with decomposition (16, 16, 16, pre-classification of structured textures, a hybrid retrieval 16) provides about 2% increase in retrieval accuracy while system is developed in which the MDFB features are se- the number of features used is only 36, in comparison to lected for characterizing structured textures, while a model- 96 in the polar-wavelet approach. This shows the effec- based method multiresolution simultaneous autoregressive tiveness of our proposed rotation-invariant features based (MRSAR) model is used for characterizing random tex- on MDFB. tures. Experimental results show that for a database of 40 classes with equal number of structured and random textures, the overall retrieval accuracy of the proposed hybrid Table 8 system can be improved by about 2.2% as compared with Comparison in retrieval accuracy (%) between using the proposed rotation- the approach based only on MDFB. In addition, a feature invariant multiscale directional features and the proposed features in reduction scheme has been proposed to select the most Section 3 for the rotated texture database characteristics MDFB features for texture representation. Decomposition Rotation-invariant Non-rotation-invariant Finally, MDFB-based rotation invariant features have been proposed. They are computed as the magnitudes of (8, 8, 8, 8) 59.2 25.7 (16, 16, 16, 16) 60.3 23.0 the discrete Fourier transform (DFT) of the directional energy signatures in each scale. Compared with the existing

Table 9 Comparisons of the proposed rotation-invariant multiscale directional features and rotation-invariant polar-wavelet features

Features used in retrieval Proposed approach with de- Proposed approach with de- Rotation-invariant polar- composition (8, 8, 8, 8) composition (16, 16, 16, 16) wavelet featuresa

Retrieval accuracy (%) 82.3 85.2 ∼ 83 Number of features 20 36 96

aThe results of rotation-invariant polar-wavelet features are cited from Ref. [2]. K.-O. Cheng et al. / Pattern Recognition 40 (2007) 1182–1194 1193

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About the Author—KIN-ON CHENG received the B.Eng. degree (ﬁrst-class honors) and M.Phil. degree in electrical and electronic engineering from The Hong Kong University of Science and Technology, 2001 and 2003, respectively. He is now working towards a Ph.D. degree from The Hong Kong Polytechnic University. His research interests include signal processing and texture classiﬁcation.

About the Author—NGAI-FONG LAW (M’98) received the B.Eng. degree with ﬁrst-class honors from the University of Auckland, Auckland, New Zealand, in 1993 and the Ph.D. degree from the University of Tasmania, Hobart, Australia, in 1997, both in electrical and electronic engineering. She is currently with the Electronic and Information Engineering Department, Hong Kong Polytechnic University. Her research interests include signal and image processing, wavelet transforms, image enhancement and compression. Recently, she has also been working on Web-based system design and video searching for Internet applications.

About the Author—WAN-CHI SIU (SM’90) received the Associateship degree from The Hong Kong Polytechnic University (formerly called the Hong Kong Polytechnic), the M.Phil. degree from The Chinese University of Hong Kong and the Ph.D. degree from Imperial College of Science, Technology, and Medicine, London, UK, in 1975, 1977 and 1984, respectively. He was with The Chinese University of Hong Kong between 1975 and 1980. He then joined The Hong Kong Polytechnic University as a Lecturer in 1980 and has became Chair Professor in 1992. He was Head of Department of Electronic and Information Engineering and subsequently Dean of the Engineering Faculty between 1994 and 2002. He is now Director of Centre for Multimedia Signal Processing of the same university. He has published over 200 research papers. His research interests include DSP, fast algorithms, transforms, wavelets, image and video coding and computational aspects of pattern recognition and neural networks. He is a Member of the Editorial Board of the Journal of VLSI Signal Processing Systems for signal, image and video technology and the EURASIP Journal on Applied Signal Processing. Dr. Siu was a Guest Editor of a special issue of the IEEE Transactions on Circuits and Systems II, published in May 1998, and was an Associate Editor of the same journal from 1995 to 1997. He was the General Chair or the Technical Program Chair of a number of international conferences. In particular, he was the Technical Program Chair of the IEEE International Symposium on Circuits and Systems (ISCAS’97) and the General Chair of the 2001 International Symposium on Intelligent Multimedia, Video and Speech Processing (ISIMP’2001), which were held in Hong Kong in June 1997 and May 2001, respectively. He was the General Chair of the 2003 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP’2003), which was held in Hong Kong. Between 1991 and 1995, he was a member of the Physical Sciences and Engineering Panel of the Research Grants Council (RGC), Hong Kong Government, and in 1994, he chaired the ﬁrst Engineering and Information Technology Panel to assess the research quality of 19 cost centers (departments) from all universities in Hong Kong. He is a Chartered Engineer and a Fellow of both the IEE and the HKIE.