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Multivariate Generalized Gaussian Mixture Model I.J. Image, Graphics and Signal Processing, 2016, 3, 45-54 Published Online March 2016 in MECS (http://www.mecs-press.org/) DOI: 10.5815/ijigsp.2016.03.06 Studies on Texture Segmentation Using D- Dimensional Generalized Gaussian Distribution integrated with Hierarchical Clustering K. Naveen Kumar Department of IT, GIT, GITAM University, Visakhapatnam Email: [email protected], [email protected] K. Srinivasa Rao Department of Statistics, Andhra University, Visakhapatnam Email: [email protected] Y.Srinivas Department of IT, GIT, GITAM University, Visakhapatnam Email: [email protected] Ch. Satyanarayana Department of CSE, JNTUK-Kakinada, Kakinada Email: [email protected] Abstract—Texture deals with the visual properties of an spatial inter relationships between the pixels in an image. image. Texture analysis plays a dominant role for image Texture usually refers the pattern in an image which segmentation. In texture segmentation, model based includes coarseness, complexity, fineness, shape, methods are superior to model free methods with respect directionality and strength [1,2]. Several segmentation to segmentation methods. This paper addresses the methods for texture segmentation have been developed application of multivariate generalized Gaussian mixture for analysing the images considering the texture surfaces probability model for segmenting the texture of an image [3-8]. Among these methods, model based methods using integrating with hierarchical clustering. Here the feature probability distribution gained lot of importance. These vector associated with the texture is derived through methods are often known as parametric texture DCT coefficients of the image blocks. The model classification methods. parameters are estimated using EM algorithm. The Several model based texture classification methods initialization of model parameters is done through have been developed using Gaussian distribution or hierarchical clustering algorithm and moment method of Gaussian mixture distribution [9-11]. The major estimation. The texture segmentation algorithm is drawback of the texture segmentation method based on developed using component maximum likelihood under Gaussian model or Gaussian mixture models is the Bayesian frame. The performance of the proposed segmentation quality metrics still remain inferior to the algorithm is carried through experimentation on five standard values such as PRI close to one, GCE close to image textures selected randomly from the Brodatz zero and VOI being low. This is due to the fact that the texture database. The texture segmentation performance feature vector associated with texture of the image measures such as GCE, PRI and VOI have revealed that regions may not be meso-kurtic. To improve the this method outperform over the existing methods of efficiency of the texture segmentation algorithm, one has texture segmentation using Gaussian mixture model. to consider the generalization of the Gaussian This is also supported by computing confusion matrix, distribution for characterising the feature vector accuracy, specificity, sensitivity and F-measure. associated with the texture of the image region. With this motivation, a texture segmentation algorithm is Index Terms—Multivariate generalised Gaussian developed and analysed using multivariate generalized mixture model, texture segmentation, EM-algorithm, Gaussian mixture model. The generalized Gaussian DCT coefficients, segmentation quality metrics. distribution includes several of the platy-kurtic, lepto- kurtic and meso-kurtic distributions. This also includes Gaussian distribution as a particular case [12]. The I. INTRODUCTION feature vector of the texture associated with the image is extracted through DCT coefficients using the heuristic The arrangement of constituent particles of a material arguments of Yu-Len Huang(2005) [13]. Assuming that is referred as Texture. The texture is influenced by the feature vector of the whole image is characterised by Copyright © 2016 MECS I.J. Image, Graphics and Signal Processing, 2016, 3, 45-54 46 Studies on Texture Segmentation Using D-Dimensional Generalized Gaussian Distribution integrated with Hierarchical Clustering the multivariate generalized Gaussian mixture model ij x ( A( ) ij ij with the feature vector, the segmentation algorithm by D ij K( ) ij integrating heuristic method of segmentation, g(x / )ij ij e(2) r 2 hierarchical clustering [14] is developed. j1 ij The rest of the paper is organised as follows. Section 2 deals with the generalized Gaussian mixture model and where, ,, are location, scale and shape its properties. Section 3 deals with extraction of the ij ij ij feature vector using DCT coefficients. Section 4 deals parameters. with extraction of model parameters using EM algorithm. 1/ 2 Section 5 is concerned with initialisation of parameters (3/ ) with hierarchical clustering and moment method of (1/ ) K( ) estimation. Section 6 deals with texture segmentation (1/ ) under Bayesian using component mixture model. Section /2 7 deals with performance evaluation of proposed (3/ ) A( ) algorithm through experimentation on five images taken (1/ ) from Brodatz texture dataset [15]. Section 8 deals with comparative study of proposed algorithm with that of other model based Gaussian segmentation algorithms. with ()denoting gamma function. Section 9 is to present the conclusions along with future Each parameter 0 controls the shape of GGD. scope for further research in this area. Expanding and rearranging the terms in (2), we get 1/ 2 /2 ij (3/ ) (3/ ) xij ij ULTIVARIATE ENERALIZED AUSSIAN IXTURE II. M G G M D (1/ ) (1/ ) ij MODEL g(xr / ) e (1/ )2 j In texture analysis, the entire image texture is j1 (3) considered as a union of several repetitive patterns. In this section, we briefly discuss the probability This implies, distribution (model) used for characterizing the feature D (4) vector of the texture. After extracting the feature vector 1 xij ij of each individual texture it can be modeled by a suitable exp 21A( , ) probability distribution such that the characteristics of j1 A( , ) (1 ) the feature vector should match the statistical theoretical characteristics of the distribution. The feature vector When =1, the corresponding generalized Gaussian characterizing the image is to follow M-component corresponds to a Laplacian or double exponential mixture distribution. Therefore we develop and analyze distribution. When =2, the corresponding generalized the textures in an image by considering that the feature vectors representing textures follow M-component Gaussian corresponds to Gaussian distribution. In limiting cases, equation (4) converges to a multivariate generalized Gaussian mixture distribution , (MGGMM) model. The joint probability density function uniform distribution in ( 3 , 3 ) and when of the feature vector associated with each individual , the distribution becomes degenerate are in texture is 0 x M (1) p(xr / ) w i g i x rr, i1 where, xr (x rij ), j=1,2,…..D, is D dimensional random vector represents the feature vector. i = 1, 2, .M representing the groups, r = 1, 2….T representing the samples. is a parametric set such that , w is the component weight such that (,,) i M and is the probability of ith class gxi rr, Fig.1. Generalized Gaussian pdf’s for different values of shape w1i i1 parameter . representing by the feature vectors of the image and the D-dimensional generalized Gaussian distribution is of The mean value of the generalized Gaussian the form[16]. distribution is Copyright © 2016 MECS I.J. Image, Graphics and Signal Processing, 2016, 3, 45-54 Studies on Texture Segmentation Using D-Dimensional Generalized Gaussian 47 Distribution integrated with Hierarchical Clustering xij ij these, the 2D discrete cosine transformation is robust and 1 A( , ) ij E(x ) xe dx simple for extracting the features. No serious attempt is ij 1 2 (1 )A( , ) made to utilise DCT coefficients for extracting the (5) feature of the texture of the image even though the DCT coefficients are capable of maintaining regularity, The GGD is symmetric with respect to , hence the complexity of the texture. This is possible since the DCT odd center moments are zero i.e., t , t = uses orthogonal transformation of the cosine function. E xij ij 0 The advantage is DCT can convert the texture of the 1,3,5,…. The even center moments can be obtained from image from time domain to frequency domain [19]. This absolute center moments and given by motivated us to consider DCT coefficients for extracting the feature vector associated with texture of the image. r / 2 To compute the feature vector associated with image 2 1 t 1 ij texture, we divide whole image into M x M blocks. t Exij ij Following the heuristic arguments given by Gonzalez 31 [20], 2D DCT coefficients in each block are computed. (6) These coefficients are selected in a zigzag pattern up to number of 16 coefficients in each block. The 16 The variance is coefficients are considered since in many of the face recognition algorithms, it is established that 16 var(x) = E(x x)2 E(x ) 2 2 coefficients provide sufficient information in the block. (7) These 16 coefficients in each block are considered as a sample feature vector and the total N=M x M blocks The model can have one covariance matrix for a provide Nx16 data matrix for the feature vector of the generalized Gaussian density of the class. The covariance whole image. matrix can be full or diagonal. In this
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