Texture Image Retrieval based on Log-Gabor Features

Texture Image Retrieval Based on Log-Gabor Features Rodrigo Nava1 , Boris Escalante-Ramı́rez2, and Gabriel Cristóbal3 1 3 Posgrado en Ciencia e Ingenierı́a de la Computación, Universidad Nacional Autónoma de México, Mexico City, Mexico [email protected] 2 Departamento de Procesamiento de Señales, Facultad de Ingenierı́a, Universidad Nacional Autónoma de México, Mexico City, Mexico [email protected] Instituto de Óptica, Spanish National Research Council (CSIC), Serrano 121, Madrid 28006, Spain [email protected] Abstract. Since Daugman found out that the properties of Gabor filters match the early psychophysical features of simple receptive fields of the Human Visual System (HVS), they have been widely used to extract texture information from images for retrieval of image data. However, Gabor filters have not zero mean, which produces a non-uniform coverage of the Fourier domain. This distortion causes fairly poor pattern retrieval accuracy. To address this issue, we propose a simple yet efficient image retrieval approach based on a novel log-Gabor filter scheme. We make emphasis on the filter design to preserve the relationship with receptive fields and take advantage of their strong orientation selectivity. We provide an experimental evaluation of both Gabor and log-Gabor features using two metrics, the Kullback-Leibler (DKL ) and the Jensen-Shannon divergence (DJ S ). The experiments with the USC-SIPI database confirm that our proposal shows better retrieval performance than the classic Gabor features. Keywords: Gabor filters, Image retrieval, Jensen-Shannon divergence, Log-Gabor filters, Texture analysis. 1 Introduction Due to the massive amount of digital image collections, visual information retrieval has become an active research area. The content-based image retrieval approach (CBIR) is based on extracting the content of visual information such as color [1] or textures [2] and its goal is to retrieve images from a data bank using features that best describe objects in a query image [3]. Image characterization by feature extraction is used to catch similarities among images. Hence, it is a crucial stage in CBIR. Theoretically, having more features implies a greater ability to discriminate images. However, this is not always true, because not all features are important for understanding or representing a visual scene [4]. L. Alvarez et al. (Eds.): CIARP 2012, LNCS 7441, pp. 414–421, 2012. c Springer-Verlag Berlin Heidelberg 2012  Texture Image Retrieval Based on Log-Gabor Features 415 Texture is one of the most important features in image retrieval [5], [6]. It provides a robust mathematical description of the spatial distribution of gray levels within a bounded neighborhood and refers to visual patterns that have properties of homogeneity [7]. However, texture characterization is not an easy problem because some spatial patterns can be quite simple as stripes while others can exhibit complex behavior like those in natural images. From a mathematical point of view, it is usual to analyze the spatial distributions as intensity variations from deterministic –where textures contain periodic patterns– to randomness – where textures look like unstructured noise. Since texture is a fundamental image property that describes a perceptually homogeneous region, the HVS requires that textures can be extracted and processed in an optimal way. Spectral methods for characterizing textures have proven to be powerful tools [8]. These methods collect a distribution of filter responses and extract features from the first and second order statistics [9]. Especially, the use of Gabor filters in texture analysis was motivated due to the studies of Daugman on visual modeling of simple cells. He found out that the experimental findings on orientation selectivity of visual cortical neurons were previously observed by Hubel and Wiesel in human beings and cats [10], [11], [12]. Gabor filters represent timevarying signals in terms of functions that are localized in both time and frequency domains. These functions described by the product of a Gaussian function and a sinusoid constitute a unique family of linear filters that behave optimally in the sense that their simultaneous resolution in both domains is maximal [13]. Manjunath and Ma in [14] proposed a method for texture analysis. The input images are filtered using a set of Gabor filters and the mean and standard deviation are taken to build a feature vector. Their method is generally accepted as a benchmark method for texture retrieval. However, Gabor filters have not zero mean, which produces a non-uniform coverage of the Fourier domain. This distortion may cause fairly poor pattern retrieval accuracy [15]. In this paper, we propose a simple yet efficient image retrieval approach based on a novel log-Gabor filter scheme. In Section 2, the classic Gabor filter and the log-Gabor model proposal are presented. In Section 3, the DKL and DJS are described. In Section 4, we compare retrieval accuracy of both Gabor and log-Gabor filter banks over the USC-SIPI database [16]. Finally, our work is summarized in Section 5. 2 Bio-Inspired Models for Texture Feature Extraction Daugman [11] proposed a 2D extension of the Gabor filters –receptive fields are deployed in two dimensions– and showed that they occupy an irreducible volume in the four-dimensional (4D) hyperspace where the four orthogonal axes correspond to spatial (x, y) and frequency (u, v) variables. The joint 2D resolution achieves the lower bound of the 2D uncertainty principle as follows: 1 (∆x) (∆y) (∆u) (∆v) ≥ 16π 2. The canonical 2D Gabor filter in spatial domain is defined as: g (x, y) = e − 21  (x−x0 )2 +γ 2 (y−y0 )2 α2  +i(2π[u0 (x−x0 )+v0 (y−y0 )]+φ) (1) 416 R. Nava, B. Escalante-Ramı́rez, and G. Cristóbal where (x0 , y0 ) are the center of the filter, (u0 , v0 ) and φ represent the radial frequency and the phase of the sinusoidal signal respectively. (α, γ) are the space constants of the Gaussian envelope along x and y axes respectively and they control the filter bandwidth. Here, we assume the use of real Gabor filters (just the even part) centered at the origin. Therefore, we obtain the next expression that provides a suitable symmetric filter for detecting salient edges [17] as follows: g (x, y) = e − 12  x2 +γ 2 y2 α2  cos (2πu0 x) (2) Using the rotation matrix, Rθ = [cos θ, − sin θ; sin θ, cos θ] and applying in Eq. 2 yields the 2D polar Gabor representation as follows: g (x, y) = e with − 21  x̃2 +γ 2 ỹ2 α2  cos (2πu0 x̃) x̃ = x cos θ − y sin θ ỹ = x sin θ + y cos θ (3) (4) The frequency and orientation selectivity properties of Gabor filters can be more explicit in Fourier domain. The Fourier transform of g (x, y) is given by: Ĝ (u, v) = e +e   −2π 2 α2 (ũ−u0 cos θ)2 + γ12 (ṽ+u0 sin θ)2   −2π 2 α2 (ũ+u0 cos θ)2 + γ12 (ṽ−u0 sin θ)2 (5) where (ũ, ṽ) = (u cos θ + v sin θ, −u sin θ + v cos θ). Ĝ (u, v) represents a rotated Gaussian function by an angle θ with u0 frequency units shifted along the axes. Psychophysical experiments showed that frequency bandwidths of simple cells are about one octave apart [11], [18], [19]. The half-amplitude bandwidth of the frequency response, Bu , satisfies this condition and is linked to central frequency u0 as follows:    log (2) 2Bu + 1 α= √ (6) 2πu (2Bu − 1) In order to determine the optimum angular bandwidth Bθ we considered an isotropic Gabor filter. Hence, we forced γ = 1.  log (2) α   (7) = √ γ 2πu tan B2θ in this way, Bθ ≈ 36◦ is obtained, but for computational efficiency Bθ = π6 was chosen. Although Gabor filters possess a number of interesting mathematical properties (they have a smooth and indefinitely differentiable shape and they do not have side lobes neither in space nor frequency domain) they present a main drawback, the filter averaging is not null and therefore the DC component influences intermediate bands. They overlap more at lower frequencies than in higher ones yielding a non-uniform coverage of the Fourier domain, (see Fig. 1(a)). Texture Image Retrieval Based on Log-Gabor Features (a) 417 (b) Fig. 1. Profiles of the frequency response of (a) Gabor and (b) log-Gabor filters. Note that the DC component is minimized by introducing the ln function. 2.1 Log-Gabor Filters Log-Gabor filters, firstly proposed by D. Field [20], are defined in the frequency domain as Gaussian functions shifted from the origin due to the singularity of the log function. They always have a null DC component and can be optimized to produce filters with minimal spatial extent in an octave scale multiresolution scheme, (see Fig. 1(b)). Log-Gabor filters can be splited into two components: radial and angular filters, Ĝ (ρ, θ) = Ĝρ Ĝθ , as follows: Ĝ (ρ, θ) = e − 12  ( uρ0 ) αρ log( u0 ) log 2 e − 21  (θ−θ0 ) αθ 2 (8) where (ρ, θ) represent the polar coordinates, u0 is the central frequency, θ0 is the orientation angle. αρ and αθ determine the scale and the angular bandwidth respectively. We set αρ = 0.75 that results in minimal overlap among scales one octave apart and αtheta = pi 6 as it was mentioned before. In order to better cover the Fourier plane even scales are rotated by a constant factor consisting of the half a distance between filter centers, (see Fig. 2(c)), [21]. (a) (b) (c) Fig. 2. Half-amplitude bandwidth of the frequency response of (a) an ensemble of Gabor filters. (b) Contour comparison between Gabor and log-Gabor filters before rotating the log-Gabor even bands. (c) Log-Gabor filters (1 octave bandwidth). 418 3 R. Nava, B. Escalante-Ramı́rez, and G. Cristóbal Texture Retrieval Based on Entropy Information As in [14], any image coefficient, C(s,θ) , defined as C(s,θ) = I (x, y) ⋆ g (x, y)(s,θ) where I (x, y) is the given image, g (x, y)(s,θ) is the filter at the scale s and orientation θ, and ⋆ indicates the convolution, represents texture characteristics in a particular scale and orientation. Thus, energy signatures such as the mean 2 µ(s,θ) and the variance σ(s,θ) can be used as texture features for constructing a feature vector as follows:   2 2 (9) , . . . , µ(s−1,θ−1) , σ(s−1,θ−1) t = µ(0,0) , σ(0,0) Although the Kullback-Leibler divergence –a generalization of Shannon’s entropy– is not a true metric rather it is a relative entropy, it can be used as a suitable descriptor for measuring distances between histograms or feature vectors. Then, the distance between two texture images A and B with tA and tB as the corresponding feature vectors is defined as:  b−1  tB (i) DKL (A, B) = (10) tB (i) log tA (i) i=0 where b is the length of the feature vectors tA and tB . In addition, the Jensen-Shannon divergence [22] denoted by ψ can be used for evaluating distance between two textures as follows:  ψ = 2DJS (A, B) (11) where DJS (A, B) = 4  1 A+B DKL A, 2 2  A+B 1 + DKL B, 2 2 (12) Experimental Results We used the USC-SIPI texture database [16], to measure retrieval accuracy (RA) of both Gabor and log-Gabor filters. USC-SIPI consists of twenty gray-scale textures of 512 × 512 pixels. Each image was divided into sixteen 128 × 128 non-overlapping patches, thus creating a database of 320 texture images. The resulted patches were processed with a filter bank (4 scales and 6 orientations) in order to form 320 feature vectors of 48 bins-length each. Each feature vector is a query pattern and was used to calculate distances among the 320 textures. The distances were sorted in increasing order and the closest sixteen patches were retrieved. We must note that in [14] the mean and the standard deviation were used to form a query image. Here we use the mean and the variance because they improve the retrieval performance. The average retrieval rate (ARR) is the standard metric for evaluating CBIR systems and is listed in Table 1 for the different texture images used in this study. ARR is calculated by the following procedure: First, each texture (D*) is Texture Image Retrieval Based on Log-Gabor Features 419 Table 1. ARR for the 20 texture images, D* indicates the Brodatz texture. ARR is computed using Gabor and log-Gabor filter banks and both DKL and DJ S metrics. Gabor filters texture D1 D3 D4 D5 D6 D9 D10 D11 D15 D20 D24 D26 D56 D66 D93 D104 D105 D106 D109 D112 DKL # patches 256 186 256 220 256 231 191 256 183 256 240 256 256 178 231 256 136 134 163 200 (%) 100 72.65 100 85.93 100 90.23 74.60 100 71.48 100 93.75 100 100 69.53 90.23 100 53.12 52.34 63.67 78.12 DJS # patches (%) 256 100 181 70.70 256 100 224 87.5 256 100 233 91.01 194 75.78 256 100 179 69.92 256 100 238 92.96 256 100 256 100 178 69.53 234 91.40 256 100 136 53.12 136 53.12 163 53.67 198 77.34 log-Gabor filters DKL # patches 256 177 256 227 256 254 217 256 194 256 243 256 253 200 242 256 205 173 227 190 (%) 100 69.14 100 88.67 100 99.21 84.76 100 75.78 100 92.92 100 98.82 78.12 94.53 100 80.07 67.57 88.67 74.21 DJS # patches (%) 256 100 175 68.35 256 100 229 89.45 256 100 254 99.21 221 86.32 256 100 187 73.04 256 100 242 94.53 256 100 256 100 202 78.90 244 95.31 256 100 203 79.29 177 69.14 227 88.67 191 74.60 Table 2. FRR for Gabor and log-Gabor filters. Given a single query (patch), all the sixteen patches that belong to the same texture are retrieved. distance DKL DJS Gabor filters # patches (%) 146 45.62 147 45.93 log-Gabor filters # patches (%) 165 51.56 167 52.18 divided into 16 patches, from which each patch is used as a query. In the best case, one single query returns the 16 patches belonging to the same texture, and evaluating all the 16 queries return up to 256 patches from the same texture. Note that for the Gabor scheme, the lower rate achieved was with the D106 texture, the ARR was 52.34% and 53.12% using DKL and DJS respectively. On the contrary, the log-Gabor scheme achieved 67.57% and 69.14% of accuracy respectively, which represents 39 and 41 more patches classified correctly with DKL and DJS metrics respectively. In the ideal case, given a single query, all the sixteen patches that belong to the same texture should be retrieved. An important metric that assesses this specific case is called full retrieval rate (FRR) which measures the number of query patterns fully retrieved correctly. Our proposal achieves 52.18% of query patterns fully retrieved, it means a 6.56% higher rate compare to the Gabor scheme with 45.62%, (see Table 2). An overall retrieval rate (ORR) is presented in Table 3. The Gabor scheme achieves 84.78% and 84.80% of patches retrieved correctly with DKL and DJS respectively. On the other hand, the proposal here presented achieves 89.72% 420 R. Nava, B. Escalante-Ramı́rez, and G. Cristóbal Table 3. ORR for Gabor and log-Gabor schemes distance DKL DJS Gabor filters (%) 84.78 84.80 log-Gabor filters (%) 89.72 89.84 and 89.84% of patches retrieved correctly with DKL and DJS respectively. This represents an increase in the classification rate up to 4.94% using DKL and 5.04% using DJS . 5 Conclusions Here we presented the classic Gabor scheme for texture analysis and summarized its properties and drawbacks. Further, a novel scheme for CBIR was presented. This proposal based on log-Gabor filters has a strong correlation with the HVS. It may say that the proposal is a bio-inspired model where the parameters agreed with simple cells in the visual cortex. In addition, we evaluate the texture distances using two metrics, the well-known DKL and the Jensen-Shannon divergence, which boosts the retrieval process. The log-Gabor filtering approach outperforms the retrieval performance for the analyzed textures in comparison with the Gabor filters. Acknowledgments. This work has been sponsored by the grant UNAM PAPIIT IN113611 and TEC2010-20307 from the Spanish Ministry of Science and Innovation. R. 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