Graph wavelet transform for image texture classiﬁcation

Graph is a data structure that can represent complex relationships among data. Graph signal processing, unlike traditional signal processing, explicitly considers the structure and relationship among the signal samples. Graph wavelet transform can provide a multiscale analysis for the graph signal. It is well known that texture is a region property in an image, which is characterized with the intensity and relationship among pixels. In this context of the graph signal processing framework, an image texture can be considered as the signal on the graph. Therefore, a texture classiﬁcation method based on graph wavelet transform is proposed. Speciﬁcally, image textures are decomposed into multiscale components by using two-channel graph wavelet ﬁlter banks. Then the local singular value decomposition is applied to each subband. In order to improve the noise-resistant ability, the maximum, mean and median values of the local singular values of graph-wavelet transformation coef-ﬁcients are extracted. Finally, the Weibull distributions are used to model those extracted values to describe the image textures. The experiments on the benchmark texture datasets are conducted to demonstrate the effectiveness of the proposed method.


INTRODUCTION
Texture is an important property of the visual scene and provides a power cue for identification and categorization of different objects and natural scenes in the human visual system [1]. In the field of image processing and computer vision, texture classification refers to as assigning an unknown texture sample to one of a set of predefined texture categories, which has attracted a lot of attention [2]. It has been applied in many fields, such as medical diagnosis [3,4], forest species recognition [5] and face recognition [6].
The core issue of texture classification is to describe the image texture with effective features. Tuceryan and Jain [1] classified the various texture feature extraction methods before 1993 into five categories: statistical methods, geometric methods, structural methods, mathematical model methods and signal processing methods [7].
The wavelet transform [8,9], as a classical signal processing technique, possesses better time-frequency localization and multiresolution analysis properties. Thus, various wavelet transform based methods have been proved for the image texture description. Mallat [8] firstly applied the discrete wavelet  [10] proposed the wavelet frame transform based texture classification. Qiao et al. [11] made use of the complex wavelet frame transform, and combined the magnitude and phase of the complex coefficients into a real measure to characterize the texture. Due to the approximately shift invariance and good directional selectivity of the dual-tree complex wavelet transform (DT-CWT), Celik and Tjahjadi [12] suggested that the statistical measures of magnitude and phase of the DT-CWT coefficients are better than the DWT based method. Ji et al. [13] applied the multi-fractal spectrum (MFS) analysis on wavelet coefficients to extract robust texture descriptors for texture classification. Wang et al. [14] extracted the texture feature based on the fractal dimensions and barycentric coordinates of the bit planes of the wavelet coefficients in both the detail wavelet subbands and approximate wavelet subbands. Yang et al. [15] obtained the image pyramid by applying dual-tree complex wavelet transform (DTCWT) on the original image, and then generated the local binary patterns (LBP) in DTCWT domain to extract features. In recent years, Rieze wavelets [16] and steerable circular harmonic wavelets [17] have shown great potential in texture classification.
Graph signal processing is one of the emerging fields in signal processing. In past years, Crovella and Kolaczyk [22] designed wavelet-like functions on graphs which are localized in space and time. In 2011, Hammond et al. [23] constructed a class of wavelet operators in the graph spectral domain, which leads to the research on applying the concepts of filter banks from classical signal processing to graph signals. Narang and Ortega [24] constructed critically sampled wavelet filter banks on graph for analyzing graph-signals defined on any arbitrary finite weighted graph. Then the biorthogonal wavelet filter banks on graph [25] have been introduced. Tanaka and Sakiyama [26] proposed Mchannel oversampled filter banks for graph signals. Jiang et al. [27] designed biorthogonal graph filter banks with the liftingbased method. Teke et al. [28] extended the classical multirate signal processing theory to graph signals. Nicolas and Pierre [29] leveraged the graph wavelet to complete the multiscale detection of communities in networks. In recent years, graph wavelets are combined with the convolutional neural network (CNN), and the graph wavelet neural network is proposed in the works [30,31].
Texture is a region property in an image, which is characterized with the intensities and relationships among pixels. Therefore, the texture features, that describe the relationships among the pixel intensities have been demonstrated power discriminative performance in texture classification, such as GLCM, Markov random field, LBP, some variants of the LBP [32], contourlet transform [33,34] and wavelet transforms. Although the traditional wavelet transforms have been successfully applied in texture feature extraction, it only implicitly considers the local structure of a texture by using the filter banks. Unlike traditional signal processing, graph signal processing explicitly considers the structure and relationship among the signal samples. In this context, an image texture can be considered to be the graph signal with very special structures. The graph wavelet transform (wavelet transform on graph) combines the advantages of both the wavelet transform and graph representation of the signal. It can be reasonably applied to describe the image texture. Therefore, in this paper, we extract the image texture feature in the graph wavelet transform domain.
It is well known that the singular value decomposition (SVD that is related with Principal Component Analysis) is a matrix decomposition method, in which larger singular values point to the major information of the matrix. The truncation of lower singular values also results in very good approximation of the original matrix, which can be considered as extracting the major information or denoising. Thus, the SVD (or PCA) based texture features are designed to be tolerance to image blur and noise corruption [35]. Ramakrishnan et al. [36] proposed a new approach based on multiwavelets transformation and singular value decomposition, in which the energy, entropy, local homogeneity and max-min ratio are extracted from the selected singular values as image texture features. Selvan et al. [37] modeled the singular value distribution of wavelet transform coefficients with an exponential function. The estimated six model parameters are taken as image texture features. However, Kim and So [38] demonstrated that it was hardly expected that a six-dimensional feature vector would yield a state-of-the-art performance, and then they proposed a modified texture feature based Haar wavelet transform and SVD.
For a texture image, the singular value decomposition of the whole coefficient matrix of a wavelet subband only captures the major information of that subband. It is difficult to characterize the local property of the texture. Meanwhile, the exponential distribution model parameters of those singular values hardly describe a texture. Therefore, in this paper, we divide the graph wavelet subband into overlapping regions and apply SVD in each region. Then the maximum values, mean values and median values of each subband are modeled with Weibull distributions, respectively. The model parameters are accumulated into the feature vector for texture classification.
The remaining structure of this paper is as follows. Section 2 mainly introduces the graph signal processing and graph wavelet filter banks. The scheme of this paper including graph wavelet transform for image textures, local singular value decomposition and Weibull distribution model of wavelet coefficients are proposed in Section 3. In Section 4, we conduct the texture classification experiments and demonstrate the performance of the proposed method. Finally, our work is concluded in Section 5.

GRAPH WAVELET TRANSFORM
A graph represents a set of elements and a set of pairwise relationships between those elements, in which the element is called vertex, and the relationship is called edge. Formally, a graph G is defined as G = (V, E), where V represents the set of vertices, and the E represents the set of edges. In this paper, we only consider the undirected graph without self-loops. The connectivity of a graph can be represented by an adjacency matrix A, which is defined as w i, j is the weight that describes the relationship between vertices i and j. A graph signal f : V → R is defined on the graph, which is indexed by the vertices and may be represented by a vector. When the image is represented as a graph signal, the grey values of pixels and the relationships between different pixels can be considered. The degree matrix D is a diagonal matrix, whose the i th diagonal element is d i that represents the degree of vertex i, whose value is the sum of the weight of all edges connected to node i. The unnormalized graph Laplacian matrix is defined as L = D − A. The symmetric normalized graph Laplacian matrix is = I − D −1∕2 AD −1∕2 , where I is the identity matrix. It is a symmetric positive semi-definite matrix and has a set of real eigenvalues 0 ≤ 1 ≤ 2 ≤ ⋯ ⋯ ≤ N ≤ 2. Recently, researchers build a graph signal processing framework from the graph spectral theory, such as graph Fourier transform, graph filtering and graph wavelet filter banks, which provides potential transforms for image texture analysis.

Graph filtering
The spectral representation of the graph signal is related to the graph Laplacian matrix L or the normalized version , which is a symmetric positive semi-definite matrix. We follow the work [25,26], the matrix L (or ) can be decomposed with the eigenvalue decomposition as follows, where u 1 , u 2 , ⋯ u N −1 , and u N are the eigenvectors corresponding to the eigenvalues 1 , 2 , ⋯ N −1 , and N , respectively. Similar to the traditional Fourier transform, the eigenvalues are considered as the graph frequencies and eigenvectors as the graph Fourier transform basis. Therefore, for a graph signal f ∈ R N , the graph Fourier transform is defined as By using the eigenspace projection matrix The graph filter for graph signal in the spectral domain is defined as follows, where (L) is the set of eigenvalues of the graph Laplacian matrix L (or ). The filter H acts on a given graph signal f by modulating each Fourier mode as Due to the high complexity of the eigenvalue decomposition, the Chebyshev polynomial approximation is used to implement the graph signal filtering [23].

Graph wavelet filter banks
The critically sampled two-channel wavelet filter banks for the bipartite graph signal is shown in Figure 1. It is assumed that the signal f is defined on the bipartite graph G = (L, H, E). The graph filters H 0 and H 1 are analysis filters, and G 0 and G 1 are synthesis filters, which can be defined as Equation (3) 0 Two-channel graph wavelet filter bank with different graph filter kernels h 0 ( ), h 1 ( ), g 0 ( ) and g 1 ( ). The operators ↓ J L and ↑ J L are downsampling and upsampling functions in the first channel, respectively. In order to design critically sampled wavelet filter banks, Narang [24] defines the diagonal downsampling matrix J L = diag{J L (n)}, in which The downsampling and then upsampling operator can be represented as where I is the identity matrix. In the second channel, the diagonal downsampling matrix is J H = −J L . Based on the perfect reconstruction condition, Narang et al. [24] gives the necessary and sufficient condition for the perfect reconstruction condition in bipartite graphs filter banks as follows, They also extend the well-known quadrature mirror filter (QMF) solution to the case of bipartite graphs, such as the ideal kernel and Meyer kernel functions.

2D graph wavelet transform
A digital image is a regular 2D signal, which can be considered as a graph signal by connecting each pixel in the image to its adjacent pixels (vertices) and defining the pixel value as the value of the graph signal. Formally, we define an image texture as a signal on the undirected graph G = (V, E), in which V is the set of pixels (vertices), and E is the set of links (edges) of pixels. Three typical graphs [24] for an image are shown in Figure 2. The graph in Figure 2a is 8-connected image graph that is 4partite, in which each pixel is a vertex on the graph and each vertex is connected to its eight adjacent vertices. The graph wavelet filter banks are implemented on bipartite graphs. Therefore 2D graph wavelet transform on an image can be performed by decomposing an arbitrary graph into bipartite subgraphs. For example, the 8-connected graph can be  Figure 2b,c [24]. Similar to the traditional 2D separable wavelet transform, twochannel graph wavelet filter banks are applied in a cascaded manner, by filtering along a series of bipartite subgraphs of the original graph. Specifically, the graph analysis filters are firstly applied on the first bipartite graph B 1 (shown in Figure 2b). After downsampling the filtering signals, the graph analysis filtering and downsampling are operated on the signals on two disjoint graphs B 2 (L 1 ) and B 2 (H 1 ), separately, which is shown in Figure 3a. Narang [24] has also proven that the graph wavelet decomposition on bipartite graphs in Figure 3a is the same as the implementation in Figure 3b. In Figure 3, the graph filter kernels h 0 and h 1 are used to construct the graph filters with Equation (3) on the corresponding bipartite subgraphs. The downsampling operator ↓ J i j (i = L 1 , H 1 and j = L 2 , H 2 ) is to preserve those vertices corresponding to J i (⋅) = 1 and J j (⋅) = 1.

FEATURE EXTRACTION AND CLASSIFICATION
It is well known that the traditional wavelet transform possesses better time-frequency localization and multiscale analysis properties. The wavelet transform based texture feature extraction method explores those advantages and provides excellent texture descriptors. However, the wavelet transform does not explicitly consider the local relationship of a texture. Therefore, in this paper, an image texture is represented as a graph signal. Graph wavelet transform combines the advantages of wavelet transform and graph signal processing. It provides a multiscale analysis way for the graph signal. This new transform is used to obtain the multiscale representation of an image texture. The second key component of our method is the local singular value decomposition, because the truncation of lower singular values can be considered as extracting the major information or denoising. Unlike the general truncation method, we take the median, maximum and mean values of the local singular values in each local region. Finally, we use the Weibull distributions to characterize their statistical properties. The model parameters serve as the texture feature. The image texture feature extraction method proposed in this paper is summarized in Figure 5, which includes the following aspects: 1) graph wavelet transform of texture images; 2) local singular value decomposition; 3) Weibull distribution modeling and parameter estimation.

Graph wavelet transform
One of the advantages of the critically sampled two-channel graph-based wavelet transform is that the number of transform coefficients is the same as the number of image pixels. Thus this critically sampled transform has been used in image compression [39]. However, as mentioned in the work [10], the overcomplete wavelet decomposition is much better than the complete wavelet transformation for texture analysis. Therefore, we combine the overcomplete transform with critically sampled transform as shown in Figure 4. Compared with the critically sampled graph wavelet transform shown in Figure 3b, our graph wavelet transform omits the downsampling operation in the LH, HL and HH channels  The proposed image texture feature extraction method and only preserves the downsampling operator in the LL channel. That is to say, after one level decomposition, the sizes of graph wavelet coefficients of the LH, HL and HH channels equal to the size of the original image texture, and the size of the LL channel is one fourth of the size of the original image. Then the next level decomposition is performed on the LL channel.
In the traditional wavelet frame transform [10], the LL channel is not downsampled. But the analysis filters must be upsampled by inserting a zero between two filter taps. For graph filters, they are defined in the graph spectral domain. We cannot simply unsampled the graph filters for the next level decomposition in the vertex domain. Therefore, in order to obtain the multiscale overcomplete graph wavelet decomposition, we apply the downsampling operator to the LL channel.

Local singular value decomposition
Singular value decomposition is a very powerful tool, which is successfully applied in the dimensionality reduction, solving linear equations and noise reduction [36]. Let Y be a matrix of size m × n, its SVD is defined as where Σ is a diagonal m × n matrix with non-negative real diagonal elements. The diagonal entries of Σ are singular values. The left singular vectors and right singular vectors form the orthogonal matrices U and V, respectively. The superscript T denotes the matrix transpose. Selvan et al. [37] showed that the singular value decomposition of wavelet transform coefficients can be used for texture classification. Kim and So [38] modified the SVD based texture feature in the Haar wavelet domain, and demonstrated the improved classification performance. Nevertheless, it is well known that texture is a region property. The singular value decomposition of the whole coefficient matrix of a wavelet subband only capture the major information of that subband. It is difficult to characterize the local property of the texture. Meanwhile, the number of non-zero singular values equals to the rank FIGURE 6 Local singular value decomposition. We choose 4 × 4 window to calculate local singular values of a coefficient matrix. When the size of the texture sample is small, it is hard to obtain a good probability distribution model for fitting the singular values, especially in the critically sampled wavelet domain, in which the size of each subband at next level decomposition is one quarter of the size of the subband at the current level. Therefore, in this paper, we make use of local SVD of graph wavelet subbands to extract the texture feature.
As for a graph wavelet subband, the local SVD is shown in the Figure 6. We perform the singular value decomposition on the region within the window. Then the window is moved to the next coefficient to perform local singular value decomposition on the windowed region. That is to say, we use the SVD to extract the major information in the local region. Selecting the size of the window plays an important role in texture feature extraction. It can directly affect the performance of texture classification. At present, there is no general way to determine the optimal size of the window without supervision. Considering the computational complexity and ensuring a good texture classification, we empirically choose a window with the size of 4 × 4.
It is well known that low singular value truncation is considered as a noise reduction method. This manipulation can reduce the noise influence in texture classification when the image texture is corrupted by noise. There are many methods for low singular value truncation in the literature. For example, it can define a measure called index of separability [36] as to select the significant singular values. This truncation method works well when number of singular values is large. However, in the local SVD with 4 × 4 window, there are 4 singular values. It seems that the general truncation does not suitable for texture description.
In order to extract the texture features and make the proposed method to be noise-resistant to some extent, we take the median, maximum and mean values of the local singular values in each window, and form three sets respectively, which will be modeled with Weibull distributions later. It is assumed that there are N local regions in a subband, and the singular values of the i th region form a vector S i . We can obtain three sets F 1 , F 2 and F 3 , which are composed of the median, maximum and mean values of every vector S i , respectively.

Weibull distribution model
The Weibull distribution has been widely used in various applications, such as reliability analysis and weather forecasting. In general, there are one-parameter Weibull distribution, twoparameter Weibull distribution, three-parameter Weibull distribution and hybrid Weibull distribution. From the perspective of probability and statistics, the probability density function of the two-parameter Weibull distribution [40] is defined as follows, where x is a random variable, > 0 is the proportional parameter, k > 0 is the shape parameter. Obviously, the Weibull distribution function becomes an exponential distribution, if the shape parameter k equals to 1. When k = 2, it is a Rayleigh distribution. If k = 3.458, it is most similar to a normal distribution. Thus the Weibull distribution possesses great potential in the data modeling. When we model the extracted sets F1, F2 and F3 in a subband, the key issue is to estimate the distribution parameters. Without loss of generality, let the elements of the set F1 to be 0 ≤ x 1 ≤ x 2 ≤ … ≤ x N , it can be determined the likelihood function from the Weibull probability density function as, Taking the partial derivative of the logarithm of the likelihood function to be zero, we obtain the following equations, from which the parameters can be estimated. In order to verify the fitting performance of Weibull distribution, we choose a Brodatz image texture, and model the maximum, mean and median values of local singular values in graph wavelet subbands. The fitting results on LL, LH channels of the first level graph wavelet decomposition are shown in Figure 7. It can be seen that the Weibull distribution works well in the local singular value fitting experiments.

Feature extraction and classification
For an image texture, the feature extraction procedures are summarized as follows, 1) Decompose the image texture with L-level graph wavelet filter banks as shown in Figure 5, and obtain graph wavelet subbands WS l,c (l = 1, … , L and c = LL, LH, HL, HH). 2) In each subband WS l,c , implement SVD on the local region and form the sets F l,c, j , in which j = max, mean, median. 3) Model F l,c, j with Weibull distribution, and estimate the parameters k l,c, j and l,c, j . 4) Accumulate all parameters k l,c, j and l,c, j into a feature vector to describe the texture.
During the texture classification, we need measure the similarity of feature vectors. As for Weibull distributions, the Kullback-Leibler divergence between two distributions f (x| 1 , k 1 ) and g(x| 2 , k 2 ) is given by Bauckhage [41] as follows, However, Kullback-Leibler divergence is not symmetric, the symmetrized Kullback-Leibler divergence,

EXPERIMENTAL RESULTS AND DISCUSSIONS
In our experiment, five different experimental datasets constructed by four benchmark texture datasets are used to evaluate the proposed method.
The first experimental dataset contains 30 texture images (listed in Table 1) which are selected from the Brodatz texture database as shown in Figure 8 [10]. There are 112 textures in dataset, and the size of each image is 512 × 512. The second experimental dataset contains the whole Brodatz dataset. In our experiments, each texture image is subdivided into 64  Table 2 [10]. Each texture image is divided into 64 nonoverlapping sub-images of size 64 × 64. Therefore, the training dataset and testing dataset all consist of 1600 texture images and 32 images per class.
We choose Outex_TC_00010 database as the fourth dataset in our experiment which contains 24 types of texture images as shown in Figure 9. For each class, there are 20 non-overlapping 128 × 128 texture samples in each rotation angle under a given illumination condition. The sample of illumination "inca" and angle 0 • in each class are employed to train the classifier [28]. In order to highlight the advantages of the method proposed in this paper, we extract the texture with the size of 64 × 64 in the centre of the image for the experiment. Therefore, there are 24 × 20 training textures and 24 × 8 × 20 testing samples.
The fifth experimental dataset is constructed from UIUC texture database as shown in Figure 10, which involves 25 texture classes and 40 texture images for each class. The resolution of each texture image is 640 × 480 pixels [42]. Similar to the previous database, we reduce the image size to one quarter of the original size. In the experiment, we select the first 20 texture samples to test the method, and the last 20 images for training the classifier.
Furthermore, in order to eliminate the different dynamic ranges in the gray levels of subimages derived from the same original image, each subimage is individually normalized to zero-mean and unit variance. In our experiments, we make use of Nearest Neighbour classifier with the similarity measure Equation (20). The average classification correct rate, ACCR = number o f correctly classi fied textures total number o f test textures × 100%, (20) are utilized to evaluate the classification performance. We will conduct the classification experiments by considering the following aspects.

1) Number of graph wavelet decomposition levels. It aims
to determine whether the number of decomposition levels influences the discrimination power for texture classification and evaluate the computational complexity of the feature extraction. 2) Different wavelet kernel functions. We will use different graph wavelet kernel functions to test the classification performance of the proposed texture feature. 3) Feature selection. The sequential forward floating selection (SFFS) method is used to further verify the discrimination power of the introduced feature. 4) Comparison with the existing methods. We will compare our experimental results with the state-of-art works to test the performance of the proposed method. 5) Noise resistance performance. In our experiments, the zero mean additive white Gaussian noises are added into the texture samples to test the noise resistance performance of the introduced method.

Number of graph wavelet decomposition levels
In order to determine whether the number of decomposition levels influence the discrimination power for texture classification, and choose a reasonable decomposition level for the texture feature extraction, we conduct the texture classification test on five datasets with different levels of graph wavelet decomposition. For better comparison, we list the average time consuming of the feature extraction for each texture image. The ACCRs with the Meyer graph wavelet kernel function and corresponding extraction time are shown in Table 3.
It can be seen that the ACCRs of the texture features extracted from the two-level decomposition is about 1.6% better than that of the one-level decomposition, and the extraction time is about 0.2-0.3s longer than that with the one level decomposition. Meanwhile, except the UIUC database, with the increase of the decomposition level after two levels, the ACCRs will decrease, and the computational complexity is increasing. Especially for the four level decomposition, the ACCRs are much lower (around 4.0-10.2%) than those of the two level decomposition. The reason is that for a texture image with the size of 64 × 64, the size of each subband at the fourth level will be 8 × 8, which will result in 25 values in the median, maximum and mean sets, respectively. It is difficult to effectively estimate the parameters of Weibull distribution with those random samples. Therefore, in the following experiments, we set the decomposition level L to be 2.

4.2
Different graph wavelet kernel functions Narang [24] proposed the graph-QMF that is to design a single spectral kernel function h 0 ( ) and select other spectral kernel functions as This solution extends the well-known quadrature mirror filter (QMF) solution to the case of bipartite graphs. Therefore, as the traditional wavelet kernel, the ideal spectral kernel function h 0 ( ) is defined as The Meyer spectral kernel is where We can also construct the spectral kernel function with the traditional Daubechies wavelet filters.
For example, the low-pass Daubechies wavelet filter is h 0 = {h 0 (0), h 0 (1), … , h 0 (N − 1)} , then the corresponding spectral kernel function for the graph wavelet can be defined as, where j = √ −1 is used to represent the complex value. This spectral kernel can be considered as the scaled amplitude spectrum.
In this section, we decompose the image texture with ideal spectral kernel (Ideal), Meyer spectral kernel (Meyer) and Daubechies wavelet (the length of filter is 6) kernel (DB6) based graph wavelet filters to extract the texture feature, and texture classification experiments are also conducted with feature selection. The results are listed in Table 4. It can be found that three wavelet kernels bring competitive classification performance. For the first experimental dataset (30 Brodatz tex-ture), the results of graph filters based on Daubechies wavelet with filter length 6 are slightly worse than that of Meyer kernel based graph wavelet filters. On the second, third, fourth and fifth experimental datasets, the ACCRs of Daubechies wavelet with filter length 6 based graph wavelet filters are the best one. In order to fairly evaluate our method and compare it with other existing methods, we choose the Meyer spectral kernel for the graph wavelet transform in the following experiments.

Comparison with existing methods
We will evaluate the proposed methods by comparing it with the existing techniques. The following approaches are included in our experiments.
1) GWT+Energy. The energy features, the mean values and standard deviations of amplitude of wavelet coefficients are always the baseline for texture classification. Therefore, we also extract these energy features for comparison. 2) DWT+GLCM. Gray level co-occurrence matrix (GLCM) is another baseline method for texture classification. The texture feature based on GLCM in the traditional wavelet domain is also used to be compared in the experiments. 3) GWT+GLCM. It is obtained by extracting GLCM based feature in the graph wavelet transform domain. 4) MDSV. Ramakrishnan et al. [36] applied the SVD decomposition in multiwavelet subbands. They extracted the energy, entropy, local homogeneity and max-min ratio of the singular values to characterize the texture. 5) CLBC. Zhao et al. [43] proposed the local binary count (LBC) based texture feature. In this framework, each pixel in the local region is turned to binary values by comparing it with the central pixel, and then the number of value 1′s is counted to form the feature. The completed LBC is also introduced to enhance the performance. The features CLBC_S/M, CLBC_S_M/C and CLBC_S/M/C are compared with our method. 6) CLDP. Hu et al. [44] proposed a new texture descriptor, completed local derivative pattern (CLDP). The new component in CLDP, with regarded as the directional derivative pattern, reflects the directional smoothness of local textures without increasing computation complexity. 7) CWT+GGD. Qiao et al. [11] combined the phase information and amplitude information of the complex wavelet coefficients into a real quantity, and then used the real generalized Gaussian distribution (GGD) to extract the feature for texture classification. 8) MDCM. Li et al. [45] developed the marginal distribution covariance model (MDCM) in the multiple wavelet domain. The texture features were extracted by using the orthogonal wavelet transform, Gabor wavelet transform and dual-tree complex wavelet transform for texture classification. 9) Jet Texton Learning. Roy et al. [46] used a new class of jet texton learning to classifying textures, in which a Jet space representation of the image is derived from several filter responses, the jet textons dictionary is learned using Kmeans clustering algorithm. The feature distribution of jet texton is considered as a model for texture classification.

Noise resistance performance
As we know, the low singular value truncation has the function of noise reduction. The paper extracts the mean values, the maximum values and the median values of the local singular values, and applies Weibull distributions. In order to test the noise resistance performance of the introduced method, we add the zero mean additive white Gaussian noise (AWGN) to the image textures, and apply the original classification method. The experimental results with different signal-to-noise ratios (SNR) are shown in Table 6. It can be seen from the Table 6 that when the SNR decrease, the ACCRs on those datasets become lower. If the SNR is greater than 30db, the noise has little effect on the accuracy of texture classification. When SNR is 20db, ACCRs are worse than those without adding AGWN by around 2.48%, 1.81%, 7.69%, 1.95% and 6.40% on the four datasets, respectively. Once SNR reaches to 10 dB, the performance of the pro-posed method becomes worse severely. Therefore, the proposed method has a noise resistance capability to some extent.

CONCLUSION
This paper introduces an image texture classification method based on graph wavelet filter banks. The image texture is considered as a graph signal. By using graph wavelet transform, the texture image is decomposed into the multi-scale representation. Meanwhile we make use of the local singular value decomposition to extract the local information of the texture image in the graph wavelet domain, and then model them with Weibull distributions to extract the texture feature. The experiments are conducted on four benchmark texture datasets, which includes graph wavelet decomposition levels, different graph spectral kernel functions, comparison with other existing methods and noise resistance performance evaluation. Experimental results show that the proposed texture features have better texture classification performance.