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2016 23rd international conference on pattern recognition icpr cancun center cancun mexico december 4 8 2016 texture classification with discrete fractional fourier transform liying zheng and yan chu school of ...

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          2016 23rd International Conference on Pattern Recognition (ICPR)
          Cancún Center, Cancún, México, December 4-8, 2016
                   Texture Classification with Discrete Fractional 
                                                        Fourier Transform 
                                                                                  
                                                               Liying Zheng* and Yan Chu* 
                                                        School of Computer Science and Technology 
                                                                Harbin Engineering University 
                                                                         Harbin, China 
                                                                  zhengliying@hrbeu.edu.cn 
                                                                    chuyan@hrbeu.edu.cn 
                                                                                  
                                                                                  
          Abstract—Based on the fact that the discrete fractional Fourier          proposed by Candon in 1937. The idea was then reintroduced 
          transform (DFrFT) of a signal consists of not only spatial but           and re-interpreted by Namias [12] and McBride and Kerr [13], 
          frequency characteristics, a novel texture classification algorithm      respectively. So far, great attention has been paid to the FrFT, 
          is presented in this paper. A 1-D local DFrFT is proposed and the        resulting in many applications of the FrFT on the areas such as 
          fractional frequency histogram is established to represent a             radar signal processing[14-16], medical ultrasound imaging 
          texture. Moreover, the conceptions such as the between-class             [17, 18], and image encryption[19-21]. Recently, some efforts 
          scatter and the within-class scatter, the coarse-to-fine searching,      have been done on applying the FrFT to the field of pattern 
                                  2-statistic distance are also adopted to 
          and the minimum χ                                                        recognition. Kong et al. employed the amplitudes of the 2-D 
          complete the classification. The simulation results on four groups       FrFT to represent the emotional state of a person [19]. Jing et 
          of benchmark texture images validate the performance of the              al. proposed a face recognition approach by combining the 
          proposed texture classification method.                                  FrFT with the discrimination analysis technique [20]. Based on 
              Keywords-  discrete fractional Fourier transform; texture            the 2-D discrete FrFT (DFrFT) and the principal component 
          classification; fractional frequency histogram                           analysis (PCA), Liu and Wang also proposed a method to 
                                                                                   address the problem of face recognition [21]. Wang and Wang 
                                  I.     INTRODUCTION                              applied the personal features derived from the FrFT, i.e. mean 
                                                                                   energy within critical bands, to identify speakers [22]. Singh et 
              Image texture which contains important information for               al. completed high-resolution SAR images categorization by 
          object recognition and scene interpretation is referred to as the        using the log cumulants of the FrFT coefficients[23].  
          arrangement of certain basic elements. The ability of                        Studies have shown that the FrFT is intimately related to 
          discriminating almost all types of textures of human visual              the time frequency representation of the Cohen class. 
          system attracts the researchers from various fields, especially          Specifically, the FrFT of a signal produces a rotation of its WD 
          from computer vision society, to pay much attention to the task          in phase space [24].It can be deduced that many important 
          of texture classification.                                               local characteristics of a signal can be revealed from its FrFT. 
              One problem in texture analysis is the description of texture        In this sense, the FrFT can potentially be used for analyzing 
          characteristics. Some popular texture descriptors are based on           textures that have obviously spatial/spatial frequency 
          statistical methods, such as gray level co-occurrence  characteristics. In this paper, an efficient texture classification 
          matrices[1], while others are based on modeling techniques,              method based on the DFrFT is presented. Several 1-D local 
          such as Markov random field [2, 3] and fractal based modeling            directional DFrFTs are firstly performed on a locally centered 
          [4]. Studies on human visual system show that the process of             texture image. The DFrFT frequencies are then used to 
          texture segmentation requires high resolution in both spatial            generate a histogram. Next, the scatter matrices of between-
          and frequency domains. Thus, to describe a texture, many                 class and within-class, as well as a coarse-to-fine searching 
          time-frequency analysis tools have been employed.  For                   technique are adopted to estimate the optimal DFrFT order. 
          example, Reed and Wechsler proposed a texture analysis                                                                                     2
                                                                                   Finally, the texture is classified by using the minimum χ -
          method by using the 2-D pseudo-Wigner distribution (PWD).                statistic distance classifier.  
          They concluded that the WD-based method enjoys superior                      The remainder of this paper is organized as follows. 
          joint resolution [5]. Zhu et al described a framework for the            Section 2 briefly introduces the DFrFT. Section 3 describes the 
          WD-based texture analysis [6]. furthermore, other popular                1-D local directional DFrFT and the fractional frequency 
          techniques such as Gabor transform[7, 8], wavelet analysis[9,            histogram of a texture image in detail. Section 4 presents the 
          10], and windowed Fourier transform[11] have also been                   proposed DFrFT-based texture classification method. Section 5 
          employed to analyze textures.                                            gives the experimental results followed by some conclusions in 
              As a time-frequency analysis technique, the fractional               Section 6. 
          Fourier transform (FrFT) is a generalization of the 
          conventional Fourier transform, and its concept was initially 
              * Corresponding  author 
          978-1-5090-4846-5/16/$31.00 ©2016 IEEE                              2032
                                                                                                                     α
                                   II.     DISCRETE FRFT (DFRFT)                                            Since I     (x, y, m) is complex, one can use amplitude, phase 
                                                                                                                      d 
                  It is well known that a digital image is in discrete form.                           angle, or both, as the features to describe a texture. In this 
             Thus, in this paper, the discrete form of the FrFT has been                               paper, the amplitudes are employed for simplicity. Specifically, 
             employed. Let vector x = [ x(0), x(1), …., x(N-1) ] represent an                          the fractional Fourier frequencies corresponding to the sorted 
                                                               α         α             th              amplitudes are used. Firstly, |I(x,y,m)| are sorted in descending 
             N-length discrete time signal, and x  and F be the α  order                                                                        d
             DFrFT of x and the DFrFT matrix, respectively. Then, we have                              order. Then fractional Fourier frequencies corresponding to the 
                                                                                                       sorted |I(x,y,m)| are recoded. In accordance to the results shown 
                                                 x  FxT                                      (1)               d
                                                                                                       in fig.1, theoretically, the difference of the sorted fractional 
                  By now, several definitions of the DFrFT have been                                   Fourier frequencies of two signals will be more obvious than 
             proposed [25-28]. In this paper, the DFrFT proposed by                                    the traditional Fourier frequencies. 
             Ozaktas et al. [28] is employed for its excellent approximation 
             to the continuous FrFT. The DFrFT matrix of Ozaktas’                                      B.  Fractional Frequency Histogram and Its Similarity 
             definition is given by                                                                         To represent a texture, the fractional frequency histogram 
                                               F  DΛHlpΛJ                                   (2)      of the above 1-D local directional DFrFT is constructed. Let 
                                                                                                       I(x,y) be a texture image with size of W by H. The fractional 
             where D and J are matrices representing the decimation and                                frequency histogram of I(x,y) is given by 
             interpolation operations, respectively. Λ is a diagonal matrix                                               H(I,l,z) {H(I,l,z),H(I,l,z)...H(I,l,z)}       (5) 
             that corresponds to the chirp multiplication, and H                                                                           1            2             D
             corresponds to the convolution operation [28].                                    lp
                                                                                                            with                   H (I,l,z)          (zu(x,y,l))              (6) 
                  Unlike the discrete Fourier transform (DFT) which                                                                 d                          d
             represents a signal in pure frequency domain, the DFrFT of a                                                                         xy
             signal with 0 1is an intermediate status between the spatial                           where D is the number of directions. δ(.) is the Dirac delta 
                                                                                                       function.  z  0,1,2,...,| N         | with |N | being the length of 
             and the frequency domains.                                                                                                  d               d
                                                                                                       neighborhood sets at direction d and in this paper we 
                 III.    LOCAL DIRECTIONAL DFRFT AND THE FRACTIONAL                                    let| N0 || N1 || ND |. u(x, y,l)  is the fractional Fourier 
                                      FREQUENCY HISTOGRAM                                                                                     d
                                                                                                       frequency corresponding to the sorted| I(x, y,m)|. 
                                                                                                                                                            d
             A.  1-D Local Directional DFrFT                                                                                                                                   2
                                                                                                            The similarity of two textures is measured by the χ -statistic 
                  Let I (x, y, n) be the nth directional neighbor of an image                                                           ~
                        d                                                                              distance [29]. Let I and I be two texture images with the same 
             I(x,y) with direction d. To eliminate the effects due to                                                                         ~
             illumination, I (x, y, n) is firstly locally centered by using                            size, and H(I, l, z) and  H(I ,l,z)be the fractional frequency 
                                d                                                                                                   ~
                                         ˆ                                                       (3)   histograms of I and I , respectively. The similarity of I and 
                                         Id (x, y,n)  Id (x, y,n)  I(x, y)                            ~
             Define the 1-D local directional DFrFT as                                                  I is given by 
                                                                                                                                D |I| |Nd|                            ~        2
                                                                ˆ                              (4)              2     ~                    (Hd(I,l,z)Hd(I,l,z))
                                    Id (x, y,m)  F (m,n)Id(x,y,n)                                             (I,I )                                            ~                 (7) 
                                                   nNd                                                                       d1l 0 z0      Hd(I,l,z)Hd(I,l,z)
                                           th                      α 
             where Id (x,y,m) is the α  order DFrFT, F is the DFrFT matrix                             where | I |W H . 
             given in (2), and N  is the neighborhood indices in direction d. 
                                      d
             As shown in fig.1, for each pixel P, there are four directional                                    IV.     TEXTURE CLASSIFICATION WITH THE DFRFT 
                                                                   0      0      0             0
             neighborhood sets which are along 0 , 45 , 90  and 135 , 
                                                                                        α
             respectively. According to the property of the DFrFT, I d (x, y,                          A.  Summary of the Proposed Method 
             m) can reveal both spatial and frequency characteristics of the                                Fig.2 summarizes the major stages of the proposed texture 
             neighborhoods of I(x, y).                                                                 classification method. The neighbors along direction d of each 
                                                                                                       pixel of a texture image I are firstly obtained with the similar 
                                                                                                       template shown in Fig.1, resulting in a 1-D series {I(x,y, 
                                                                                                       n )|d=0, 1, 2, 3 and n N }. Then, the locally centered 
                                                                                                         d                             d       d
                                                                                                                ˆ
                                                                                                       series I(x, y,nd ) are obtained by using (3). Next, the optimal 
                                                                                                         th
                                                                                                       α  order 1-D local directional DFrFT is performed according 
                                                                                                       to (1) and (2), getting a 1-D series I(x, y,m) in the fractional 
                                                                                                                                                        d
                                                                                                       Fourier domain. Next, the fractional Fourier amplitudes 
                                                                                                       | I(x, y,m) | are sorted in descending order followed by 
                                                                                                          d
                                                                                                       recording the corresponding fractional frequencies 
                            Figure1. Illustration of directional neighborhoods.                        corresponding to the sorted amplitudes. Next, the fractional 
                                                                                                       frequency histogram is calculated with (5) and (6). Finally, a 
                                                                                                 2033
             type of classifier can be adopted for completing the  where S  and S  are the scatter matrices of between-class and 
                                                                                                               b         w
             classification. Here, since one of our targets is that of                               within-class, respectively. tr(.) means the trace of a matrix. 
             evaluating the performance of the proposed DFrFT texture                                Here,  S  and S  are computed according to the fractional 
                                                                                                               b          w
             descriptor, the minimum distance classifier is adopted.                                 frequency histogram obtained by (6) and (8). Please refer to 
                                                                                                     [20] For details.  
             B.  Representation of a Texture                                                             Here, to get the optimal transform order, a coarse-to-fine 
                 Assuming that there are C classes of textures in the training                       searching procedure which is similar to the algorithm in [30] is 
             set, the average fractional frequency histogram is employed to                          adopted. Let Δα0, λ, and ε be three positive numbers less than 
                               th
             represent the c  class, i.e.                                                            1, and αopt denote the optimal transform order .The coarse-to-
                                                     Nc                                              fine searching procedure consists of the following five steps. 
                                                 1             (c,i)                                     Step1, α        = 0, α     = 0.995, Δα = Δα ; 
                                H(c,l,z)         c H(I            ,l, z)                     (8)                  begin        end                      0
                                                N i1                                                    Step 2, Let α = {α          , Δα+α        , 2Δα+α        , …, α      }; 
                                                                                                                                begin         begin          begin        end
                                          c                                          th
             where c = 1,2,…C. N  is the training samples in the c  class.                               Step 3, Compute J with (9) for each α; 
              (c,i)               th                     th
             I    denotes the i  sample in the c  class. To classify a texture, 
                                       2                                                                 Step 4, Select the α which maximizes the J as α                 ; 
             we compute the χ -statistic distance between its spectral                                                                                                opt
             histogram and the average spectral histogram of each class by                               Setp5, α         = max{0, α  – 0.5Δα}, α                 = min{1, α  + 
             using (8). Then, the image is classified into the class with the                                       begin                 opt                 end                opt
                           2                                                                         0.5Δα}, Δα= Δα×λ, go to Step 2 until Δα≤ε.  
             minimum χ -distance.                                                                        Comparing to [30], the range of α is [0,0.995] rather than 
             C.  Optimal Transform Order Estimation                                                  [0,1.0], since we find that when α =1.0, J may achieve the 
                 The optimal DFrFT order α is a critical parameter for the                           extreme, but the performance of the  proposed classification 
             proposed texture classification algorithm. To estimate this                             algorithm is probably poor. 
             parameter, the between-class and the within-class scatter                                          V.      E
             matrices have been adopted [20]. An α which can maximize (9)                                                 XPERIMENTAL RESULTS AND ANALYSIS 
             is selected as the optimal transform order.                                             A.  Experimental Data 
                                     J  tr(Sb)/tr(Sw)                                     (9)           The texture classification method discussed in Section 3 
                                                                                                     and Section 4 has been implemented in Matlab R2007b. The 
                                                                                                     experimental dataset are consists of four groups of benchmark 
                                                                                                     texture images which have been shown in Fig.3, being named 
                                                                                                     as Group1, Group2, Group3, and Group4, respectively.  
                                                                                                     Group1 whose images are the same with those in fig.1 of [11] 
                                                                                                     consists of 16 textures with the size of 128×128. The first four 
                                                                                                     textures in Group1 were generated by Gaussian random fields, 
                                                                                                     and the other 12 textures were chosen from Brodatz album. 10 
                                                                                                     Brodatz textures in Group2 are D4, D9, D19, D21, D24, D28, 
                                                                                                     D29, D36, D37 and D38 with the size of 320×320. These 
                                                                                                     textures are the same with fig.11(h) of [31]. Group3 is from 
                                                                                                     MIT Vision Texture database consisting of Fabric.0009, 
                                                                                                     Fabric.0016, Fabric.0019, Flowers.0005, Food.0005, 
                                                                                                     Leaves.0003, Misc.0000, Misc.0002, Sand.0000, and 
                                                                                                     Stone.0004 with the size of 256×256. The textures in Group3 
                                                                                                     are the same with fig.11(i) of [31]. Group4 is from the fifth test 
                                                                                                     suite of Outex texture database, consisting of 
                                                                                                     Outex_TC_00005, whose 24 textures are canvas001, 002, 003, 
                                                                                                     005, 006, 009, 011, 021, 022, 023, 025, 026, 031, 032, 033, 
                                                                                                     035, 038, 039, carpet002, carpet004, carpet005, carpet009, 
                                                                                                     tile005 and tile006. The size of each texture in Group4 is 
                                                                                                     320×320. 
                                                                                                         To generate training and test data, each texture in 
                                                                                                     Group1~Group4 was divided into a number of patches. A total 
                                                                                                     of 21 patches of each texture in Group1 were randomly 
                                                                                                     selected as training data, while the other 28 were used for test 
                                                                                                     data. The randomly selected half patches of each texture in 
                                                                                                     Group2, Group3 and Group4 were used for training while the 
                      Figure2. Flowchart of the DFrFT-based texture classification.                  other half part for testing. All experiments were repeated 10 
                                                                                                     times and the average performance of the proposed method has 
                                                                                                     been evaluated. 
                                                                                               2034
                                                                                                  TABLE I.         CLASSIFICATION RATES WITH DIFFERENT METHODS (%) 
                                                                                                   Texture       DFrFT Liu Azencott DFT Singh
                                                                                                    Groups 
                                                                                                    Group1        98.21 99.78 95.1  87.1  71.6 
                                                                                                    Group2         100 83.1 100  90 100
                                                                                                    Group3        85.62 79.1 85.94  81.8 87.2 
                                                                                                    Group4 88.93 -- 55.86 83.2 51.3 
                                                                                                     
                                                                                                    The worst result of the proposed DFrFT method is on 
                                                                                               Group3. These textures which is the same with fig.11(i) in[31] 
                                                                                               is rather difficult to discriminate. In fact, all the heuristic 
                                                                                               methods selected in [31] yield the classification rate of about 
                                                                                               10% which is much lower than our method. 
                                                                                                    Moreover, table I illustrates that from the view of the 
                                                                                               uniformity of the results, the proposed DFrFT method is better 
                                                                                               than other methods. The mean and the standard deviation of the 
                                                                                               DFrFT are 0.932 and 0.07, while they are 0.843 and 0.198 for 
                Figure3. Textures. (a) Group1, (b) Group2, (c) Group3, and (d) group4.         Azencott’s method, 0.855 and 0.0373 for the DFT, and 0.775 
                                                                                               and 0.3 for Singh’s method. 
            B.  Optimal Transform Order for Each Group                                              Finally, we compared the computational cost of the 
                The coarse-to-fine searching shown in section 4.3 is                           proposed method to three of the above mentioned methods. 
            adopted to estimate the optimal DFrFT order for each group of                      The average computational cost for classifying a texture  with 
            textures. For Group1 and Group3, Δα  = 0.1, λ = 0.1, and ε                         the proposed DFrFT, Azencott’s method, the DFT, and Singh’s 
                                                            0                                  method are 155.2 milliseconds (ms), 6.2ms, 21.3ms, and 
            =0.01. And for Group2 and Group4, α  = 0.1, λ = 0.1, and ε 
                                                             0                                 61.5ms, respectively. Clearly, the proposed method is much 
                                        | N   || N ||N |8
            =0.1. Here, we let              0      1             3       .The optimal          slower than other methods, since both calculating the 1-D local 
            DFrFT orders are 0.84, 0.3, 0.91 and 0.4 for Group1, Group2,                       DFrFT and establishing the fractional frequency histogram are 
            Group3, and Group4, respectively.                                                  time consuming.  At this stage, therefore, our method is not fit 
            C.  Comparison to Other Methods                                                    for real time applications where the speed of an algorithm is 
                                                                                               crucial. However, its time cost for processing a texture with 
                We compared the proposed DFrFT-based method with                               size of 320 by 320 is about 155 ms, meaning 39 textures per 
            Liu’s spectral histogram method [29], Azencott's windowed                          second, which may still satisfy most applications that don’t 
            Fourier transform [11], traditional DFT, and Singh’s FrFT[23].                     have a strict requirement on the speed. 
            Here, | N0 || N1 || N3 |8 , the DFrFT orders are 0.84, 
                                                                                                                          VI.     C
            0.3, 0.91, and 0.4 for Group1, Group2, Group3, and Group4,                                                              ONCLUSIONS 
            respectively. Here, we let the DFrFT order be equal to 1 to get                         Based on the fact that the DFrFT of a signal reveals both 
            the results of the traditional DFT. TableI lists the classification                spatial and frequency characteristics, a novel texture 
            rates. Note that the result of Liu’s method for Group4 is missed                   classification method has been proposed in this paper. A local 
            since all results of this method are from [29]. Furthermore, the                   directional DFrFT is proposed and the fractional frequency 
            result of Azencott for Group1 is from [11].                                        histogram is generated from the discrete fractional Fourier 
                For the textures in Group1, the methods in [11] and [29]                       frequencies corresponding to the sorted amplitudes of the local 
            give 22 and 2 misclassified patches, i.e., the classification rates                directional DFrFTs of a texture. Furthermore, by using the 
            of 95.1% and 99.78%. The proposed method gives 8  matrices of the between-class scatter and the within-class 
            misclassified patches with classification rate of 98.21%. The                      scatter, as well as the coarse-to-fine searching strategy, the 
            classification rates of other two methods on Group1 are lower                      optimal DFrFT order has been estimated. The comparison to 
            than 90%, especially for Singh’s method whose rate is only                         the four existing methods validates the proposed DFrFT-based 
            71.6%. The main reason for the poor performance of Singh’s                         texture classification method.  One problem with the proposed 
            method is that “it is dependent upon the application”[23]. For                     method is that of its time cost. For the future work, we need to 
            the textures in Group2 and in Group3, the proposed method                          study how to optimize the method to improve its efficiency. 
            outperforms the DFT and Liu’s method, and is competitive to 
                                                                                                                              R
            the other two methods, i.e. Azencott’s method and Singh’s                                                           EFERENCES 
            method. For Group4, the performance of the proposed method                         [1]   R. M. Haralick, "Statistical and structural approaches to texture," in 
            is much better than the DFT, Singh’s or Azencott’s methods. It                           Proceedings of the IEEE, 1979, pp. 786-804. 
            is worth to mention that the best result of Group4 submitting to                   [2]   H. Derin and W. S. Cole, "Segmentation of textured images using Gibbs 
            the Outex website is 86% [32], which is still lower than the                             random fields," Computer Vision, Graphics, and Image Processing vol. 
            result of the proposed DFrFT method.                                                     35, pp. 72-98, 1985. 
                                                                                          2035
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...Rd international conference on pattern recognition icpr cancun center mexico december texture classification with discrete fractional fourier transform liying zheng and yan chu school of computer science technology harbin engineering university china zhengliying hrbeu edu cn chuyan abstract based the fact that proposed by candon in idea was then reintroduced dfrft a signal consists not only spatial but re interpreted namias mcbride kerr frequency characteristics novel algorithm respectively so far great attention has been paid to frft is presented this paper d local resulting many applications areas such as histogram established represent radar processing medical ultrasound imaging moreover conceptions between class image encryption recently some efforts scatter within coarse fine searching have done applying field statistic distance are also adopted minimum kong et al employed amplitudes complete simulation results four groups emotional state person jing benchmark images validate perf...

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