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A modified Gabor function for content based image retrieval
Pattern Recognition Letters 28 (2007) 293–300 www.elsevier.com/locate/patrec
A modified Gabor function for content based image retrieval Challa S. Sastry *, M. Ravindranath, Arun K. Pujari, B.L. Deekshatulu Artificial Intelligence Laboratory, Department of Computer and Information Sciences, University of Hyderabad, Hyderabad, AP 500 046, India Received 24 May 2005; received in revised form 10 July 2006 Available online 20 September 2006 Communicated by M.-J. Li
Abstract As the Gabor filters are direction dependent, the Gabor transform of an image is to be performed for all chosen directions. Thus the set of angles used in Gabor feature extraction does affect the results in applications such as Content Based Image Retrieval (CBIR). In the present work, we modify the Gabor filter suitably in such a way that the modified function besides being free from the choice of angles is as effective as the Gabor function itself. Additionally, our method of extraction of features is invariant to rotation in images. Our simulation results demonstrate that the modified Gabor based method being useful for CBIR shows better retrieval performance than the standard Gabor based method. 2006 Elsevier B.V. All rights reserved. Keywords: Gabor function; Modified Gabor function; Content based image retrieval; Rotational invariance
1. Introduction Content Based Image Retrieval (CBIR) from large image databases has been an active area of research for long due to its applications in various fields like satellite imaging, medicine, etc. CBIR systems extract features from the raw images and calculate an associative measure (similarity or dissimilarity) between a query image and database images based on these features. Hence the feature extraction is a very important step and the effectiveness of a CBIR system depends typically on the method of extraction of features from raw images. Several methods achieving effective feature extraction have been proposed in the literature (Li and Taylor, 2005; Kyrki et al., 2004; Manjunath and Ma, 1996; Manthalkar et al., 2003; Park and Yang, 2001), to name a few.
*
Corresponding author. Tel.: +91 9849362656; fax: +91 4023011683. E-mail addresses: [email protected] (C.S. Sastry), mrn_royal@ rediffmail.com (M. Ravindranath), [email protected] (A.K. Pujari), [email protected] (B.L. Deekshatulu). 0167-8655/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.patrec.2006.07.015
The Gabor filter is widely used to extract the texture features from images for image retrieval. This use is motivated by many factors (Castelli and Bergman, 2002; Manjunath and Ma, 1996). Some of the methods (Chan and Coghill, 2001; Li and Taylor, 2005; Kyrki et al., 2004; Manthalkar et al., 2003; Park and Yang, 2001) proposed for feature extraction use direction dependent Gabor filters, and consequently the feature vectors become direction dependent. In Gabor methods, a particular set of Gabor filters (corresponding to different angles) is chosen, which determines the quality of result in applications such as CBIR. To get rid of the angle dependence, some types of permutations on feature matrices are taken in (Kyrki et al., 2004). In CBIR images are retrieved from a data bank using features that best describe the orientation of objects in the query image. In the traditional application of Gabor filters the chosen directions may not correspond to the orientation of the contents in the query image. Therefore any method that extracts features independent of orientation in the image is desirable. Thus rotation invariance is particularly useful when one wants to retrieve images having same content but indifferent orientation.
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The basic objective in the present work is to modify the Gabor function suitably in such a way that the resulting function besides inheriting good properties of Gabor filters is a Radial Basis function (RBF), which is an angle independent function. Hence no specific set of angles is required for feature extraction. The main features of the present algorithm are: (1) it uses images in cartesian domain avoiding the nonlinear polar transformation, and certain approximations resulting therefrom, (2) it does not require, unlike standard Gabor method, direction dependent filters for the extraction of information pertaining to different directions, which minimizes the amount of computation. Additionally, our feature extraction procedure is independent of presence of rotation in images, and hence is useful for rotation independent CBIR. The paper is organized as follows: In Section 2, we discuss about the Gabor function, its properties, and standard way of feature extraction using Gabor function. Later on, pointing out the need, we go on modifying the Gabor function and present our method of feature extraction. In the last two sections, we present some comparisons and our simulation results as applicable to CBIR respectively. 2. Gabor transform
band-pass filter with certain optimal localization properties both in spatial and frequency domains. The Gabor function, which is the modulation by a sinusoid of Gaussian function, has the presence of undulations along x-direction, as shown in Fig. 1 as 3D profiles of real and imaginary components of (1). This oscillatory behaviour, an approximation property, captures the texture variations in an image along x-direction. A class of self-similar Gabor filters can be obtained by appropriate dilations and rotations of grx ;ry as follows: ðx cos hþy sin hÞ2 ðx sin hþy cos hÞ2 12 þ 1 2 r2 r x y grx ;ry ;h ðx; yÞ ¼ e 2prx ry e2piW ðx cos hþy sin hÞ ;
where h determines the orientation of the filter. As the axes for grx ;ry ;h are rotated by h, the function grx ;ry ;h has undulations along h-direction. Given an input image f, the Gabor transform of it is the convolution with f of a set of Gabor filters of different preferred orientations and scales, as follows: F rx ;ry ;h ðx; yÞ :¼ f Hgrx ;ry ;h ðx; yÞ Z 1 Z 1 ¼ f ðn; gÞ grx ;ry ;h ðx n; y gÞ dn dg; 1
A 2D Gabor function is defined as 2 2 12 x 2 þ y 2 1 grx ;ry ðx; yÞ ¼ e rx ry e2piWx ; 2prx ry
ð1Þ
where rx and ry are the scaling parameters of the filter, and W is central frequency. The function grx ;ry acts as a local
1
where g represents the complex conjugate of g. Normally, in Gabor feature extraction methods, F rx ;ry ;h is computed for all x and y, and for different triplets (rx, ry, h). Based on common observation (Chan and Coghill, 2001; Li and Taylor, 2005; Manthalkar et al., 2003; Park and Yang, 2001), the mean and variance, i.e.,
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Fig. (a) Real part of g, (b) imaginary part of g, (c) real part of G and (d) imaginary part of G, (e) Gaussian function and (f) real part of R 2p1.2piW e ðx cos hþy sin hÞ dh, when rx = ry = W = 1; illustrating that pointwise multiplication of (e) and (f) gives (c), as shown in (5). 0
1 2p
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M rx ;ry ;h ¼ V rx ;ry ;h ¼
Z Z
1
Z
1
jF rx ;ry ;h ðx; yÞj dx dy
1 1 1 Z 1
1
ð3Þ 2
jF rx ;ry ;h ðx; yÞ M rx ;ry ;h j dx dy;
1
of the energy distribution of F(., ., rx, ry, h) are computed to identify a texture. The signature or compact representation of an image is, then, generated by considering M rx ;ry ;h and V rx ;ry ;h in vector form for different rx, ry and h. Since the Gabor filter grx ;ry ;h is angle dependent, the effective signature generation of an image requires the use of proper choice of angles over [0, 2p), and further the computation of convolution for all those angles is needed to be carried out. Besides, the signatures so generated, in general, are not invariant to the presence of rotation in images. Motivated by this, we propose to make the Gabor function angle independent by taking integration of it over all possible angles. Let us define the modified Gabor function Grx ;ry the following way: Grx ;ry ðx; yÞ ¼
¼
1 2p
Z
2p 0
1 4p2 rx ry
grx ;ry ;h ðx; yÞ dh Z 2p 1 ðx cos hþy sin hÞ2 þðx sin hþy cos hÞ2 2 r2 r2 x y e 0
e2piW ðx cos hþy sin hÞ dh 2 2 2pp 2pp 2pp x cos ð þy sin ð þy cos ð ðx sin ð2pp P Þ P ÞÞ P Þ P ÞÞ 1 ð P 1 þ X 2 1 r2 r2y x e 2pP rx ry p¼0 2pp
2pp
e2piW ðx cos ð P Þþy sin ð P ÞÞ ;
ð4Þ
where P is the number of angles used to discretize the integration in (4). It should be mentioned here that the integration of Gabor filters over all angles does not give Gaussian function. For example, in the simple case rx = ry, one can write (4) as 2 2 Z 2p 12 x 2 þ y 2 1 1 rx ry Grx ;ry ðx; yÞ ¼ e e2piW ðx cos hþy sin hÞ dh: 2prx ry 2p 0 ð5Þ 1 2p
R 2p
2piW ðx cos hþy sin hÞ
As the function e dh 6 1, from (5), the 0 function Grx ;ry is different from the Gaussian function, which is shown in Fig. 1, by setting P in (4) to 32. Since for every h, grx ;ry ;h is square integrable, so is Grx ;ry , the convolution f HGrx ;ry is well defined for a square integrable function f. Since grx ;ry ;h is the rotated copy of grx ;ry ;0 and Grx ;ry is the average of sum of grx ;ry ;h for all rotations, one may observe that the volumes enveloped by the surfaces Grx ;ry , grx ;ry ;h and grx ;ry ;0 over XY plane are same, that is Z 1 Z 1 Z 1 Z 1 Grx ;ry ðx; yÞ dx dy ¼ grx ;ry ;h ðx; yÞ dx dy 1 1 1 1 Z 1 Z 1 grx ;ry ;0 ðx; yÞ dx dy: ð6Þ ¼ 1
1
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Therefore, one may have Z 1 Z 1 b rx ;ry ð0; 0Þ ¼ Grx ;ry ðx; yÞ dx dy G 1 1 Z 1 Z 1 ¼ grx ;ry ;h ðx; yÞ dx dy ¼ g^rx ;ry ;h ð0; 0Þ 1 1 Z 1 Z 1 b rx ;ry ðn; gÞ dn dg Grx ;ry ð0; 0Þ ¼ G 1 1 Z 1 Z 1 ¼ g^rx ;ry ;h ðn; gÞ dn dg ¼ grx ;ry ;h ð0; 0Þ: 1
1
ð7Þ The hat operation used in (7) stands for the Fourier transform operation. As the low-pass or high-pass behaviour of a smooth filter depends on its values at the origin in frequency domain, the coincidence of Grx ;ry and grx ;ry ;h values at the origin in both space and frequency domains implies that Grx ;ry and grx ;ry ;h are filters of same bandpass. Therefore, since grx ;ry ;h has oscillatory behaviour along h-direction, from (4) and (7), the modified Gabor function possesses undulations along all directions, (shown in Fig. 1), and acts, like the Gabor function grx ;ry ;h , as a local band-pass filter. Although we do not know if Grx ;ry , like grx ;ry ;h , generates an unconditional (Riesz (Daubechies, 1992)) basis for the class of square integrable functions, due to 0 to 2p integration in the definition of Grx ;ry , when we use the choice of scales as used in (Manjunath and Ma, 1996), the frequency response of filters tiles the frequency plane into annular regions having no gaps, unlike the standard Gabor filters that tile the frequency plane into ellipses with some gaps (Fig. 1 in Manjunath and Ma, 1996). Hence the features extracted using the modified Gabor filter can be as effective. We define our feature vectors whose elements are the means M rx ;ry ;f ;i and the corresponding variances V rx ;ry ;g;i of f over the concentric circular regions bounded by circles of radii Ri and Ri+1 for different i, as Z Z jGrx ;ry Hf ðx; yÞj dx dy M rx ;ry ;f ;i ¼ D Z Z i ð8Þ 2 V rx ;ry ;f ;i ¼ ðjf HGrx ;ry ðx; yÞj M rx ;ry ;f ;i Þ dx dy: Di
where Di ¼ fðx; yÞjR2i 6 x2 þ y 2 < R2iþ1 g is the region over which the double integration is performed in the above equations. In addition to being a local bandpass function, Grx ;ry is a radial basis function (RBF), i.e., "(x, y) and h, Grx ;ry ðx cos h þ y sin h; x sin h þ y cos hÞ ¼ Grx ;ry ðx; yÞ: ð9Þ It is to be mentioned here that (9) holds irrespective of choices of rx and ry. As an additional feature, the above property and the definition of our mean and variance components ensure the rotation invariance in our feature extraction process as follows: Let h(x, y) = f(x cos h + y sin h, x sin h + y cos h), i.e., h is the rotation of f by an angle h. Then the means M rx ;ry ;f ;i of f and M rx ;ry ;h;i of h are equal, i.e.,
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M rx ;ry ;h;i ¼
Z Z
jGrx ;ry Hhðx; yÞj dx dy
Di
¼
Z Z
jGrx ;ry Hf ðx cos h þ y sin h; x sin h
Di
þ y cos hÞj dx dy Z Z ¼ jGrx ;ry Hf ðx; yÞj dx dy ¼ M rx ;ry ;f ;i ;
ð10Þ
Di
and the corresponding variances V rx ;ry ;g;i , g = f, h, are also equal, i.e., Z Z 2 V rx ;ry ;h;i :¼ ðjhHGrx ;ry ðx; yÞj M rx ;ry ;h;i Þ dx dy Di Z Z ðjf HGrx ;ry ðx; yÞj M rx ;ry ;f ;i Þ2 dx dy ¼ Di
¼ V rx ;ry ;h;i :
ð11Þ
Computation of the traditional Gabor function requires less computational overheads than the proposed modification. However, we achieve rotation invariance at the expense of an extra computation. On the other hand, if we attempt to achieve rotation invariance through traditional approach we have to compute the Gabor function in several directions and take the average. This requires far more computational overhead. That is, using Gabor function oriented along different directions and considering extracting features from images as shown in (3), and then taking their average appropriately over all those angles, i.e., M 0rx ;ry ¼ Meanh fM rx ;ry ;h g and V 0rx ;ry ¼ Meanh fV rx ;ry ;h g as features, one can achieve rotation invariance in feature extraction. But in this process, one has to first compute f w gh for every h, which is computationally more involved than our method, because our method being free from the choice of angles does not involve computation of convolution for every sampled value of h. 3. Some comparisons In the introduction part, we have already discussed as to how the method differs from the ones using standard Gabor function. In recent literature, wavelet transform has been widely used for texture classification/CBIR (Jafari-Khouzani and Soltanian-Zadeh, 2005; Li and Taylor, 2005; Manthalkar et al., 2003; Sastry et al., 2004). In (Li and Taylor, 2005), the authors have compared in detail the performances of various texture classification methods, namely dyadic wavelet, wavelet frame, Gabor wavelet and steerable pyramids, and have observed that the Gabor based method gives superior performance than the rest on texture images. Standard wavelet transform captures variations along specific (namely, horizontal, diagonal and vertical) directions due to the use of separable wavelets. In (Jafari-Khouzani and Soltanian-Zadeh, 2005), the authors have employed Radon transform to detect the principal direction of texture. Then, rotating principal direction in texture to 0,
they have applied wavelet transform to achieve rotation invariance in texture classification. As they have stated, when dealing with complex images such as those having straight lines along several directions, finding principal direction through their method is ambiguous. Additionally, the method is computationally a bit more intense due to computation of Radon transform along different directions and spacings. Recently, some of us have proposed (Sastry et al., 2004) a wavelet like representation formula using orthonormal wavelets for rotation invariant feature extraction. One distinct advantage of the present method is that it can be implemented easily. This is due to the fact that the Gabor function enjoys closed form representation and in absolute terms it decays very fast. On the other hand, in (Sastry et al., 2004) the radially extended functions, Ur, Wr (especially Ur) are relatively poorly localized in spatial domain [As one may see in (3.10) in Sastry et al., 2004, the modulus function introduces discontinuity at origin in the frequency domain of Ur, Wr, which results in the poor localization in spatial domain of Ur, Wr (especially Ur due to its unit value at origin in frequency domain).]. In our simulation work, we observe that the present modified Gabor approach and the wavelet based method proposed in (Sastry et al., 2004) give comparable results. 4. Simulation results In order to compare our method with the standard Gabor feature extraction method, we generate feature vectors of size 32 in both cases, using four choices of (rx, ry) pairs, viz rx = ry = 1, 2, 3, 4. Here, it is to be mentioned that there is no specific reason for taking same values for both rx and ry. One may as well take distinct rx and ry. In our feature extraction method involving the use of modified Gabor function, we use four concentric circular regions over which we compute our features (4 means + 4 variances for each of four (rx, ry) pairs) to generate a feature vector of size 32. While, in the standard method of feature extraction involving the use of standard Gabor function, to generate a feature vector of size 32 (1 mean + 1 variance for each of four (rx, ry) pairs and each of four angles), we use four angles f0; p2 ; p; 3p ; 2pg and the same 2 choice of (rx, ry) pairs. In our simulation work, we choose the numbers Ri in (3) in such a way that the computation of features over concentric circular regions involves nearly equal number of pixels. In discrete setting, if the matrix f HGrx ;ry is of size m · n for a given pair (rx, ry), the regions, namely D1 ¼ fðp; qÞ : p2 þ q2 < mn g D2 ¼ fðp; qÞ : 4p mn 2 2 mn mn 2 6 p þ q < g, D ¼ fðp; qÞ : 6 p þ q2 < 3mn g and 3 4p 2p 2p 4p 3mn 2 2 D4 ¼ fðp; qÞ : 4p 6 p þ q g, involve nearly equal number of pixels, and using (10) and (11) we compute our features to generate a feature vector of size 32. After extracting the features, to show the usefulness of them, we consider the application of CBIR. The use of the features in CBIR requires the use of an appropriate similarity measure for comparing the query image with
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Fig. 2. Comparison of performances of results obtained using standard (dashed line) and modified (continuous line) Gabor functions for all 116 classes of images. The horizontal lines represent the total average performance of both methods; which demonstrates that the modified Gabor method shows better retrieval performance than the standard Gabor method in almost all cases. In this figure, x- and y-axes, respectively represent the classes of images and the average retrieval performance of both methods.
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2003). Since the objective of this paper is to modify Gabor function for easier implementation and better retrieval, as our similarity measure, we use the simple Euclidean distance, which is defined for two feature vectors f = (f1, f2, . . . , fn) and g = (g1, g2, . . . , gn) as !12 n X 2 jfi gi j : dðf ; gÞ ¼
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Fig. 3. Precision–recall graph for Brodatz database.
those in the database. Several similarity measures based on common distance functions such as Euclidean, Mahalonibis, etc. are defined in literature (Kamarainen et al.,
We have carried our simulation work on Brodatz (Brodatz, 1996) texture image database of size 1855 consisting of 116 classes of images of size 128 · 128. Considering each image of every class as query image, we have counted the number of images of the same class, appearing in the first 15 matches, that have been retrieved by both the modified and standard Gabor based methods. We have then computed the average retrieval performance of both methods for each class, which we have plotted in Fig. 2. It may be noted in Fig. 2 that for almost all classes, the modified Gabor method has shown better retrieval performance than the standard Gabor method. This may be attributed to the better filling of frequency plane by modified Gabor
Fig. 4. Comparison of retrieval performances of standard and modified Gabor methods. First two rows are of standard method with different sets of four angles. Third row is of modified method. The image on the first column represents query image. The first two rows justify that the retrieval performance of standard Gabor method gets affected by the choice of angles.
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function than standard Gabor function. The total average retrieval (i.e., the average of average retrieval over 116 classes) performances of modified and standard methods are 8.3440 and 4.5522, respectively, which are shown in
Fig. 2 through horizontal lines. The motive behind using small set of parameters (angles and scales) in our simulation work is to show that, even with feature vectors of smaller sizes, we can have better retrieval performance
Fig. 5. Images retrieved by both methods for the query images shown on the first column. For each query image, the images shown on first and second rows are respectively of standard and modified methods.
Fig. 6. For the query images on the first column of two classes of four rotated images, the images on the other than columns are those retrieved by the modified Gabor method.
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Fig. 7. Precision–recall graph for texture database obtained from University of Oulu.
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Fig. 8. Precision–recall graph for satellite image database.
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using modified Gabor function. One may, however, improve the performance of standard (and also modified) method by using better choices of parameters (for example, as used in (Manjunath and Ma, 1996)). As it is known, the precision–recall curve is the most popular way to evaluate the performance of an image retrieval algorithm. Recall is the percentage of total relevant images retrieved, and is defined (Castelli and Bergman, 2002) as Recall ¼
Number of relevant images retrieved : Total number of relevant images
Precision refers to the capability of the system to retrieve only relevant images. Precision can be expressed as
Precision ¼
299
Number of relevant images retrieved : Total number of images retrieved
We have computed precision–recall values for 50 queries belonging to different classes in Brodatz image database. We believe that our results are unbiased because Brodatz image database is well known for texture image retrieval/ classification. In terms of precision–recall, we have shown the performances of both standard, modified Gabor based methods and our wavelet method (Sastry et al., 2004) in Fig. 3 with x-axis denoting recall and y-axis denoting average precision. Fig. 3 indicates that the performance of modified Gabor method is better than its standard counterpart, while the modified Gabor and wavelet based methods have shown nearly similar performance.
Fig. 9. Retrieval performance of the modified Gabor based method on the University of Oulu texture database.
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For the query image on the first column of Fig. 4, we have implemented both the standard and modified methods. In the implementation of standard Gabor feature extraction method, we have taken two sets of angles, viz f0; p2 ; p; 3p ; 2pg and fp4 ; 3p ; 5p ; 7pg. It may be seen in Fig. 4 2 4 4 4 that the retrieval performance of standard Gabor feature extraction method gets affected by the choice of angles. Fig. 5 displays the performance of both methods for some query images shown therein. For the first three query images in Fig. 5, retrieval performance of proposed method is very high. For the fourth query, although the proposed method shows average retrieval performance, the images so retrieved are visually similar to query image. As for the fifth query image, despite the content being very less in it, the proposed method has retrieved those images that have similar patterns. The Eqs. (10) and (11) prove the rotation invariance in feature vectors that one can obtain using the modified Gabor function. To justify this experimentally, we have used two classes of rotated images shown on the first four columns in Fig. 6. For the two query images on the first column of Fig. 6, the first five images retrieved by the modified Gabor algorithm, other than query images, are shown on the last 5 columns of Fig. 6. We have also carried out our experimental work on larger texture database of 4320 images obtained from University of Oulu1 and another database downloaded from the internet.2 The University of Oulu texture database consists of 24 texture classes with each class consisting of 180 images of size 128 · 128 with different orientations. The second database consists of 10 classes and 200 high resolution satellite images of size 128 · 128 in each class. As before, for each image of both databases, we have evaluated the retrieval performances of both standard and modified Gabor methods. The precision–recall plots reported in Figs. 7 and 8 indicate that the modified Gabor based method and the wavelet based method in (Sastry et al., 2004) show better performance than the standard Gabor based method. We have created Graphical User Interface to make the user more comfortable, and Fig. 9 shows first 10 images retrieved using modified Gabor method. 5. Conclusion In this paper, we modify the Gabor function so suitably that it can be used for applications like rotation invariant CBIR without a specific set of angles, which makes the
1 2
http://www.outex.oulu.fi. http://www-db.stanford.edu/wangz/image.vary.jpg.tar.
method relatively less intense computationally. Our computational results and comparison of performances of the standard and modified Gabor based methods on different databases justify that the modified Gabor method shows superior performance in the retrieval of images of same class. Acknowledgements The authors are thankful to the reviewers for their suggestions. The authors acknowledge valuable suggestions made by Dr. C. Bhagvati, Department of Computer and Information Sciences, University of Hyderabad, Hyderabad, India, during the preparation of the present work. References Brodatz, P., 1996. Textures: Aphotographic Album for Artists and Designers. Dover Publication, New York. Castelli, V., Bergman, L., 2002. Image Databases – Search and Retrieval of Digital Imagery. John Wiley & Sons, Inc. Chan, Woei, Coghill, G., 2001. Text analysis using local energy. Pattern Recognition 34 (12), 2523–2532. Daubechies, I., 1992. Ten lectures on wavelets. CBMS-NSF Series in Appl. Math, No.61. SIAM Philadelphia. Jafari-Khouzani, Kourosh, Soltanian-Zadeh, Hamid, 2005. Radon transform orientation estimation for rotation invariant texture analysis. IEEE Trans. PAMI 27 (6). Kamarainen, J.K., Kyrki, V., Ilonen, J., Kalviainen, H., 2003. Improving similarity measures of histograms using smoothing projections. Pattern Recognition Lett. 24, 2009–2019. Kyrki, V., Kamarainen, J.K., Kalviainen, H., 2004. Simple Gabor feature space for invariant object recognition. Pattern Recognition Lett. 25, 11–318. Li, Shutao, Taylor, J.S., 2005. Comparison and fusion of multiresolution features for texture classification. Pattern Recognition Lett. 26, 633– 638. Manjunath, B.S., Ma, Y.S., 1996. Texture features for browsing and retrieval of image data. IEEE Trans. Pattern Anal. Machine Intell. 18 (8), 837–842. Manthalkar, R., Biswas, P.K., Chatterji, B.N., 2003. Rotation and scale invariant texture features using discrete wavelet packet transform. Pattern Recognition Lett. 24 (14), 2455–2462. Manthalkar, R., Biswas, P.K., Chatterji, B.N., 2003. Rotation invariant texture classification using even symmetric Gabor filters. Pattern Recognition Lett. 24 (12), 2061–2068. Park, H.J., Yang, H.S., 2001. Invariant object detection based on evidence accumulation and Gabor features. Pattern Recognition Lett. 22, 869– 882. Sastry, Ch.S., Pujari, A.K., Deekshatulu, B.L., Bhagvati, C., 2004. A wavelet based multiresolution algorithm for rotation invariant feature extraction. Pattern Recognition Lett. 25 (16), 1845–1855, December.
A modified Gabor function for content based image retrieval Challa S. Sastry *, M. Ravindranath, Arun K. Pujari, B.L. Deekshatulu Artificial Intelligence Laboratory, Department of Computer and Information Sciences, University of Hyderabad, Hyderabad, AP 500 046, India Received 24 May 2005; received in revised form 10 July 2006 Available online 20 September 2006 Communicated by M.-J. Li
Abstract As the Gabor filters are direction dependent, the Gabor transform of an image is to be performed for all chosen directions. Thus the set of angles used in Gabor feature extraction does affect the results in applications such as Content Based Image Retrieval (CBIR). In the present work, we modify the Gabor filter suitably in such a way that the modified function besides being free from the choice of angles is as effective as the Gabor function itself. Additionally, our method of extraction of features is invariant to rotation in images. Our simulation results demonstrate that the modified Gabor based method being useful for CBIR shows better retrieval performance than the standard Gabor based method. 2006 Elsevier B.V. All rights reserved. Keywords: Gabor function; Modified Gabor function; Content based image retrieval; Rotational invariance
1. Introduction Content Based Image Retrieval (CBIR) from large image databases has been an active area of research for long due to its applications in various fields like satellite imaging, medicine, etc. CBIR systems extract features from the raw images and calculate an associative measure (similarity or dissimilarity) between a query image and database images based on these features. Hence the feature extraction is a very important step and the effectiveness of a CBIR system depends typically on the method of extraction of features from raw images. Several methods achieving effective feature extraction have been proposed in the literature (Li and Taylor, 2005; Kyrki et al., 2004; Manjunath and Ma, 1996; Manthalkar et al., 2003; Park and Yang, 2001), to name a few.
*
Corresponding author. Tel.: +91 9849362656; fax: +91 4023011683. E-mail addresses: [email protected] (C.S. Sastry), mrn_royal@ rediffmail.com (M. Ravindranath), [email protected] (A.K. Pujari), [email protected] (B.L. Deekshatulu). 0167-8655/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.patrec.2006.07.015
The Gabor filter is widely used to extract the texture features from images for image retrieval. This use is motivated by many factors (Castelli and Bergman, 2002; Manjunath and Ma, 1996). Some of the methods (Chan and Coghill, 2001; Li and Taylor, 2005; Kyrki et al., 2004; Manthalkar et al., 2003; Park and Yang, 2001) proposed for feature extraction use direction dependent Gabor filters, and consequently the feature vectors become direction dependent. In Gabor methods, a particular set of Gabor filters (corresponding to different angles) is chosen, which determines the quality of result in applications such as CBIR. To get rid of the angle dependence, some types of permutations on feature matrices are taken in (Kyrki et al., 2004). In CBIR images are retrieved from a data bank using features that best describe the orientation of objects in the query image. In the traditional application of Gabor filters the chosen directions may not correspond to the orientation of the contents in the query image. Therefore any method that extracts features independent of orientation in the image is desirable. Thus rotation invariance is particularly useful when one wants to retrieve images having same content but indifferent orientation.
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The basic objective in the present work is to modify the Gabor function suitably in such a way that the resulting function besides inheriting good properties of Gabor filters is a Radial Basis function (RBF), which is an angle independent function. Hence no specific set of angles is required for feature extraction. The main features of the present algorithm are: (1) it uses images in cartesian domain avoiding the nonlinear polar transformation, and certain approximations resulting therefrom, (2) it does not require, unlike standard Gabor method, direction dependent filters for the extraction of information pertaining to different directions, which minimizes the amount of computation. Additionally, our feature extraction procedure is independent of presence of rotation in images, and hence is useful for rotation independent CBIR. The paper is organized as follows: In Section 2, we discuss about the Gabor function, its properties, and standard way of feature extraction using Gabor function. Later on, pointing out the need, we go on modifying the Gabor function and present our method of feature extraction. In the last two sections, we present some comparisons and our simulation results as applicable to CBIR respectively. 2. Gabor transform
band-pass filter with certain optimal localization properties both in spatial and frequency domains. The Gabor function, which is the modulation by a sinusoid of Gaussian function, has the presence of undulations along x-direction, as shown in Fig. 1 as 3D profiles of real and imaginary components of (1). This oscillatory behaviour, an approximation property, captures the texture variations in an image along x-direction. A class of self-similar Gabor filters can be obtained by appropriate dilations and rotations of grx ;ry as follows: ðx cos hþy sin hÞ2 ðx sin hþy cos hÞ2 12 þ 1 2 r2 r x y grx ;ry ;h ðx; yÞ ¼ e 2prx ry e2piW ðx cos hþy sin hÞ ;
where h determines the orientation of the filter. As the axes for grx ;ry ;h are rotated by h, the function grx ;ry ;h has undulations along h-direction. Given an input image f, the Gabor transform of it is the convolution with f of a set of Gabor filters of different preferred orientations and scales, as follows: F rx ;ry ;h ðx; yÞ :¼ f Hgrx ;ry ;h ðx; yÞ Z 1 Z 1 ¼ f ðn; gÞ grx ;ry ;h ðx n; y gÞ dn dg; 1
A 2D Gabor function is defined as 2 2 12 x 2 þ y 2 1 grx ;ry ðx; yÞ ¼ e rx ry e2piWx ; 2prx ry
ð1Þ
where rx and ry are the scaling parameters of the filter, and W is central frequency. The function grx ;ry acts as a local
1
where g represents the complex conjugate of g. Normally, in Gabor feature extraction methods, F rx ;ry ;h is computed for all x and y, and for different triplets (rx, ry, h). Based on common observation (Chan and Coghill, 2001; Li and Taylor, 2005; Manthalkar et al., 2003; Park and Yang, 2001), the mean and variance, i.e.,
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Fig. (a) Real part of g, (b) imaginary part of g, (c) real part of G and (d) imaginary part of G, (e) Gaussian function and (f) real part of R 2p1.2piW e ðx cos hþy sin hÞ dh, when rx = ry = W = 1; illustrating that pointwise multiplication of (e) and (f) gives (c), as shown in (5). 0
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M rx ;ry ;h ¼ V rx ;ry ;h ¼
Z Z
1
Z
1
jF rx ;ry ;h ðx; yÞj dx dy
1 1 1 Z 1
1
ð3Þ 2
jF rx ;ry ;h ðx; yÞ M rx ;ry ;h j dx dy;
1
of the energy distribution of F(., ., rx, ry, h) are computed to identify a texture. The signature or compact representation of an image is, then, generated by considering M rx ;ry ;h and V rx ;ry ;h in vector form for different rx, ry and h. Since the Gabor filter grx ;ry ;h is angle dependent, the effective signature generation of an image requires the use of proper choice of angles over [0, 2p), and further the computation of convolution for all those angles is needed to be carried out. Besides, the signatures so generated, in general, are not invariant to the presence of rotation in images. Motivated by this, we propose to make the Gabor function angle independent by taking integration of it over all possible angles. Let us define the modified Gabor function Grx ;ry the following way: Grx ;ry ðx; yÞ ¼
¼
1 2p
Z
2p 0
1 4p2 rx ry
grx ;ry ;h ðx; yÞ dh Z 2p 1 ðx cos hþy sin hÞ2 þðx sin hþy cos hÞ2 2 r2 r2 x y e 0
e2piW ðx cos hþy sin hÞ dh 2 2 2pp 2pp 2pp x cos ð þy sin ð þy cos ð ðx sin ð2pp P Þ P ÞÞ P Þ P ÞÞ 1 ð P 1 þ X 2 1 r2 r2y x e 2pP rx ry p¼0 2pp
2pp
e2piW ðx cos ð P Þþy sin ð P ÞÞ ;
ð4Þ
where P is the number of angles used to discretize the integration in (4). It should be mentioned here that the integration of Gabor filters over all angles does not give Gaussian function. For example, in the simple case rx = ry, one can write (4) as 2 2 Z 2p 12 x 2 þ y 2 1 1 rx ry Grx ;ry ðx; yÞ ¼ e e2piW ðx cos hþy sin hÞ dh: 2prx ry 2p 0 ð5Þ 1 2p
R 2p
2piW ðx cos hþy sin hÞ
As the function e dh 6 1, from (5), the 0 function Grx ;ry is different from the Gaussian function, which is shown in Fig. 1, by setting P in (4) to 32. Since for every h, grx ;ry ;h is square integrable, so is Grx ;ry , the convolution f HGrx ;ry is well defined for a square integrable function f. Since grx ;ry ;h is the rotated copy of grx ;ry ;0 and Grx ;ry is the average of sum of grx ;ry ;h for all rotations, one may observe that the volumes enveloped by the surfaces Grx ;ry , grx ;ry ;h and grx ;ry ;0 over XY plane are same, that is Z 1 Z 1 Z 1 Z 1 Grx ;ry ðx; yÞ dx dy ¼ grx ;ry ;h ðx; yÞ dx dy 1 1 1 1 Z 1 Z 1 grx ;ry ;0 ðx; yÞ dx dy: ð6Þ ¼ 1
1
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Therefore, one may have Z 1 Z 1 b rx ;ry ð0; 0Þ ¼ Grx ;ry ðx; yÞ dx dy G 1 1 Z 1 Z 1 ¼ grx ;ry ;h ðx; yÞ dx dy ¼ g^rx ;ry ;h ð0; 0Þ 1 1 Z 1 Z 1 b rx ;ry ðn; gÞ dn dg Grx ;ry ð0; 0Þ ¼ G 1 1 Z 1 Z 1 ¼ g^rx ;ry ;h ðn; gÞ dn dg ¼ grx ;ry ;h ð0; 0Þ: 1
1
ð7Þ The hat operation used in (7) stands for the Fourier transform operation. As the low-pass or high-pass behaviour of a smooth filter depends on its values at the origin in frequency domain, the coincidence of Grx ;ry and grx ;ry ;h values at the origin in both space and frequency domains implies that Grx ;ry and grx ;ry ;h are filters of same bandpass. Therefore, since grx ;ry ;h has oscillatory behaviour along h-direction, from (4) and (7), the modified Gabor function possesses undulations along all directions, (shown in Fig. 1), and acts, like the Gabor function grx ;ry ;h , as a local band-pass filter. Although we do not know if Grx ;ry , like grx ;ry ;h , generates an unconditional (Riesz (Daubechies, 1992)) basis for the class of square integrable functions, due to 0 to 2p integration in the definition of Grx ;ry , when we use the choice of scales as used in (Manjunath and Ma, 1996), the frequency response of filters tiles the frequency plane into annular regions having no gaps, unlike the standard Gabor filters that tile the frequency plane into ellipses with some gaps (Fig. 1 in Manjunath and Ma, 1996). Hence the features extracted using the modified Gabor filter can be as effective. We define our feature vectors whose elements are the means M rx ;ry ;f ;i and the corresponding variances V rx ;ry ;g;i of f over the concentric circular regions bounded by circles of radii Ri and Ri+1 for different i, as Z Z jGrx ;ry Hf ðx; yÞj dx dy M rx ;ry ;f ;i ¼ D Z Z i ð8Þ 2 V rx ;ry ;f ;i ¼ ðjf HGrx ;ry ðx; yÞj M rx ;ry ;f ;i Þ dx dy: Di
where Di ¼ fðx; yÞjR2i 6 x2 þ y 2 < R2iþ1 g is the region over which the double integration is performed in the above equations. In addition to being a local bandpass function, Grx ;ry is a radial basis function (RBF), i.e., "(x, y) and h, Grx ;ry ðx cos h þ y sin h; x sin h þ y cos hÞ ¼ Grx ;ry ðx; yÞ: ð9Þ It is to be mentioned here that (9) holds irrespective of choices of rx and ry. As an additional feature, the above property and the definition of our mean and variance components ensure the rotation invariance in our feature extraction process as follows: Let h(x, y) = f(x cos h + y sin h, x sin h + y cos h), i.e., h is the rotation of f by an angle h. Then the means M rx ;ry ;f ;i of f and M rx ;ry ;h;i of h are equal, i.e.,
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M rx ;ry ;h;i ¼
Z Z
jGrx ;ry Hhðx; yÞj dx dy
Di
¼
Z Z
jGrx ;ry Hf ðx cos h þ y sin h; x sin h
Di
þ y cos hÞj dx dy Z Z ¼ jGrx ;ry Hf ðx; yÞj dx dy ¼ M rx ;ry ;f ;i ;
ð10Þ
Di
and the corresponding variances V rx ;ry ;g;i , g = f, h, are also equal, i.e., Z Z 2 V rx ;ry ;h;i :¼ ðjhHGrx ;ry ðx; yÞj M rx ;ry ;h;i Þ dx dy Di Z Z ðjf HGrx ;ry ðx; yÞj M rx ;ry ;f ;i Þ2 dx dy ¼ Di
¼ V rx ;ry ;h;i :
ð11Þ
Computation of the traditional Gabor function requires less computational overheads than the proposed modification. However, we achieve rotation invariance at the expense of an extra computation. On the other hand, if we attempt to achieve rotation invariance through traditional approach we have to compute the Gabor function in several directions and take the average. This requires far more computational overhead. That is, using Gabor function oriented along different directions and considering extracting features from images as shown in (3), and then taking their average appropriately over all those angles, i.e., M 0rx ;ry ¼ Meanh fM rx ;ry ;h g and V 0rx ;ry ¼ Meanh fV rx ;ry ;h g as features, one can achieve rotation invariance in feature extraction. But in this process, one has to first compute f w gh for every h, which is computationally more involved than our method, because our method being free from the choice of angles does not involve computation of convolution for every sampled value of h. 3. Some comparisons In the introduction part, we have already discussed as to how the method differs from the ones using standard Gabor function. In recent literature, wavelet transform has been widely used for texture classification/CBIR (Jafari-Khouzani and Soltanian-Zadeh, 2005; Li and Taylor, 2005; Manthalkar et al., 2003; Sastry et al., 2004). In (Li and Taylor, 2005), the authors have compared in detail the performances of various texture classification methods, namely dyadic wavelet, wavelet frame, Gabor wavelet and steerable pyramids, and have observed that the Gabor based method gives superior performance than the rest on texture images. Standard wavelet transform captures variations along specific (namely, horizontal, diagonal and vertical) directions due to the use of separable wavelets. In (Jafari-Khouzani and Soltanian-Zadeh, 2005), the authors have employed Radon transform to detect the principal direction of texture. Then, rotating principal direction in texture to 0,
they have applied wavelet transform to achieve rotation invariance in texture classification. As they have stated, when dealing with complex images such as those having straight lines along several directions, finding principal direction through their method is ambiguous. Additionally, the method is computationally a bit more intense due to computation of Radon transform along different directions and spacings. Recently, some of us have proposed (Sastry et al., 2004) a wavelet like representation formula using orthonormal wavelets for rotation invariant feature extraction. One distinct advantage of the present method is that it can be implemented easily. This is due to the fact that the Gabor function enjoys closed form representation and in absolute terms it decays very fast. On the other hand, in (Sastry et al., 2004) the radially extended functions, Ur, Wr (especially Ur) are relatively poorly localized in spatial domain [As one may see in (3.10) in Sastry et al., 2004, the modulus function introduces discontinuity at origin in the frequency domain of Ur, Wr, which results in the poor localization in spatial domain of Ur, Wr (especially Ur due to its unit value at origin in frequency domain).]. In our simulation work, we observe that the present modified Gabor approach and the wavelet based method proposed in (Sastry et al., 2004) give comparable results. 4. Simulation results In order to compare our method with the standard Gabor feature extraction method, we generate feature vectors of size 32 in both cases, using four choices of (rx, ry) pairs, viz rx = ry = 1, 2, 3, 4. Here, it is to be mentioned that there is no specific reason for taking same values for both rx and ry. One may as well take distinct rx and ry. In our feature extraction method involving the use of modified Gabor function, we use four concentric circular regions over which we compute our features (4 means + 4 variances for each of four (rx, ry) pairs) to generate a feature vector of size 32. While, in the standard method of feature extraction involving the use of standard Gabor function, to generate a feature vector of size 32 (1 mean + 1 variance for each of four (rx, ry) pairs and each of four angles), we use four angles f0; p2 ; p; 3p ; 2pg and the same 2 choice of (rx, ry) pairs. In our simulation work, we choose the numbers Ri in (3) in such a way that the computation of features over concentric circular regions involves nearly equal number of pixels. In discrete setting, if the matrix f HGrx ;ry is of size m · n for a given pair (rx, ry), the regions, namely D1 ¼ fðp; qÞ : p2 þ q2 < mn g D2 ¼ fðp; qÞ : 4p mn 2 2 mn mn 2 6 p þ q < g, D ¼ fðp; qÞ : 6 p þ q2 < 3mn g and 3 4p 2p 2p 4p 3mn 2 2 D4 ¼ fðp; qÞ : 4p 6 p þ q g, involve nearly equal number of pixels, and using (10) and (11) we compute our features to generate a feature vector of size 32. After extracting the features, to show the usefulness of them, we consider the application of CBIR. The use of the features in CBIR requires the use of an appropriate similarity measure for comparing the query image with
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Fig. 2. Comparison of performances of results obtained using standard (dashed line) and modified (continuous line) Gabor functions for all 116 classes of images. The horizontal lines represent the total average performance of both methods; which demonstrates that the modified Gabor method shows better retrieval performance than the standard Gabor method in almost all cases. In this figure, x- and y-axes, respectively represent the classes of images and the average retrieval performance of both methods.
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2003). Since the objective of this paper is to modify Gabor function for easier implementation and better retrieval, as our similarity measure, we use the simple Euclidean distance, which is defined for two feature vectors f = (f1, f2, . . . , fn) and g = (g1, g2, . . . , gn) as !12 n X 2 jfi gi j : dðf ; gÞ ¼
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Fig. 3. Precision–recall graph for Brodatz database.
those in the database. Several similarity measures based on common distance functions such as Euclidean, Mahalonibis, etc. are defined in literature (Kamarainen et al.,
We have carried our simulation work on Brodatz (Brodatz, 1996) texture image database of size 1855 consisting of 116 classes of images of size 128 · 128. Considering each image of every class as query image, we have counted the number of images of the same class, appearing in the first 15 matches, that have been retrieved by both the modified and standard Gabor based methods. We have then computed the average retrieval performance of both methods for each class, which we have plotted in Fig. 2. It may be noted in Fig. 2 that for almost all classes, the modified Gabor method has shown better retrieval performance than the standard Gabor method. This may be attributed to the better filling of frequency plane by modified Gabor
Fig. 4. Comparison of retrieval performances of standard and modified Gabor methods. First two rows are of standard method with different sets of four angles. Third row is of modified method. The image on the first column represents query image. The first two rows justify that the retrieval performance of standard Gabor method gets affected by the choice of angles.
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function than standard Gabor function. The total average retrieval (i.e., the average of average retrieval over 116 classes) performances of modified and standard methods are 8.3440 and 4.5522, respectively, which are shown in
Fig. 2 through horizontal lines. The motive behind using small set of parameters (angles and scales) in our simulation work is to show that, even with feature vectors of smaller sizes, we can have better retrieval performance
Fig. 5. Images retrieved by both methods for the query images shown on the first column. For each query image, the images shown on first and second rows are respectively of standard and modified methods.
Fig. 6. For the query images on the first column of two classes of four rotated images, the images on the other than columns are those retrieved by the modified Gabor method.
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Fig. 7. Precision–recall graph for texture database obtained from University of Oulu.
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Fig. 8. Precision–recall graph for satellite image database.
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using modified Gabor function. One may, however, improve the performance of standard (and also modified) method by using better choices of parameters (for example, as used in (Manjunath and Ma, 1996)). As it is known, the precision–recall curve is the most popular way to evaluate the performance of an image retrieval algorithm. Recall is the percentage of total relevant images retrieved, and is defined (Castelli and Bergman, 2002) as Recall ¼
Number of relevant images retrieved : Total number of relevant images
Precision refers to the capability of the system to retrieve only relevant images. Precision can be expressed as
Precision ¼
299
Number of relevant images retrieved : Total number of images retrieved
We have computed precision–recall values for 50 queries belonging to different classes in Brodatz image database. We believe that our results are unbiased because Brodatz image database is well known for texture image retrieval/ classification. In terms of precision–recall, we have shown the performances of both standard, modified Gabor based methods and our wavelet method (Sastry et al., 2004) in Fig. 3 with x-axis denoting recall and y-axis denoting average precision. Fig. 3 indicates that the performance of modified Gabor method is better than its standard counterpart, while the modified Gabor and wavelet based methods have shown nearly similar performance.
Fig. 9. Retrieval performance of the modified Gabor based method on the University of Oulu texture database.
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For the query image on the first column of Fig. 4, we have implemented both the standard and modified methods. In the implementation of standard Gabor feature extraction method, we have taken two sets of angles, viz f0; p2 ; p; 3p ; 2pg and fp4 ; 3p ; 5p ; 7pg. It may be seen in Fig. 4 2 4 4 4 that the retrieval performance of standard Gabor feature extraction method gets affected by the choice of angles. Fig. 5 displays the performance of both methods for some query images shown therein. For the first three query images in Fig. 5, retrieval performance of proposed method is very high. For the fourth query, although the proposed method shows average retrieval performance, the images so retrieved are visually similar to query image. As for the fifth query image, despite the content being very less in it, the proposed method has retrieved those images that have similar patterns. The Eqs. (10) and (11) prove the rotation invariance in feature vectors that one can obtain using the modified Gabor function. To justify this experimentally, we have used two classes of rotated images shown on the first four columns in Fig. 6. For the two query images on the first column of Fig. 6, the first five images retrieved by the modified Gabor algorithm, other than query images, are shown on the last 5 columns of Fig. 6. We have also carried out our experimental work on larger texture database of 4320 images obtained from University of Oulu1 and another database downloaded from the internet.2 The University of Oulu texture database consists of 24 texture classes with each class consisting of 180 images of size 128 · 128 with different orientations. The second database consists of 10 classes and 200 high resolution satellite images of size 128 · 128 in each class. As before, for each image of both databases, we have evaluated the retrieval performances of both standard and modified Gabor methods. The precision–recall plots reported in Figs. 7 and 8 indicate that the modified Gabor based method and the wavelet based method in (Sastry et al., 2004) show better performance than the standard Gabor based method. We have created Graphical User Interface to make the user more comfortable, and Fig. 9 shows first 10 images retrieved using modified Gabor method. 5. Conclusion In this paper, we modify the Gabor function so suitably that it can be used for applications like rotation invariant CBIR without a specific set of angles, which makes the
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http://www.outex.oulu.fi. http://www-db.stanford.edu/wangz/image.vary.jpg.tar.
method relatively less intense computationally. Our computational results and comparison of performances of the standard and modified Gabor based methods on different databases justify that the modified Gabor method shows superior performance in the retrieval of images of same class. Acknowledgements The authors are thankful to the reviewers for their suggestions. The authors acknowledge valuable suggestions made by Dr. C. Bhagvati, Department of Computer and Information Sciences, University of Hyderabad, Hyderabad, India, during the preparation of the present work. References Brodatz, P., 1996. Textures: Aphotographic Album for Artists and Designers. Dover Publication, New York. Castelli, V., Bergman, L., 2002. Image Databases – Search and Retrieval of Digital Imagery. John Wiley & Sons, Inc. Chan, Woei, Coghill, G., 2001. Text analysis using local energy. Pattern Recognition 34 (12), 2523–2532. Daubechies, I., 1992. Ten lectures on wavelets. CBMS-NSF Series in Appl. Math, No.61. SIAM Philadelphia. Jafari-Khouzani, Kourosh, Soltanian-Zadeh, Hamid, 2005. Radon transform orientation estimation for rotation invariant texture analysis. IEEE Trans. PAMI 27 (6). Kamarainen, J.K., Kyrki, V., Ilonen, J., Kalviainen, H., 2003. Improving similarity measures of histograms using smoothing projections. Pattern Recognition Lett. 24, 2009–2019. Kyrki, V., Kamarainen, J.K., Kalviainen, H., 2004. Simple Gabor feature space for invariant object recognition. Pattern Recognition Lett. 25, 11–318. Li, Shutao, Taylor, J.S., 2005. Comparison and fusion of multiresolution features for texture classification. Pattern Recognition Lett. 26, 633– 638. Manjunath, B.S., Ma, Y.S., 1996. Texture features for browsing and retrieval of image data. IEEE Trans. Pattern Anal. Machine Intell. 18 (8), 837–842. Manthalkar, R., Biswas, P.K., Chatterji, B.N., 2003. Rotation and scale invariant texture features using discrete wavelet packet transform. Pattern Recognition Lett. 24 (14), 2455–2462. Manthalkar, R., Biswas, P.K., Chatterji, B.N., 2003. Rotation invariant texture classification using even symmetric Gabor filters. Pattern Recognition Lett. 24 (12), 2061–2068. Park, H.J., Yang, H.S., 2001. Invariant object detection based on evidence accumulation and Gabor features. Pattern Recognition Lett. 22, 869– 882. Sastry, Ch.S., Pujari, A.K., Deekshatulu, B.L., Bhagvati, C., 2004. A wavelet based multiresolution algorithm for rotation invariant feature extraction. Pattern Recognition Lett. 25 (16), 1845–1855, December.