Parameter Estimation of Markov Random Field Model of Image Textures

Transcription

1 Michał Strzelecki Andrzej Materka Institute of Electronics Technical University of Łódź ul. Stefanowskiego 18/ Łódź, POLAND Parameter Estimation of Markov Random Field Model of Image Textures ABSTRACT The issue of image feature selection for textured image segmentation is addressed in this paper. It is pointed out that the popular coocurrence matrix statistical method of feature extraction may not be the most efficient one for this task. Usefulness of Random Markov Field parameters as image features is postulated and investigated by means of a numerical analysis. It is found out that MRF parameters can be useful for image segmentation provided they are estimated in an image window of a properly large size. Moreover, the number of features necessary for successful texture classification can be reduced by using the MRFs rather then statistical features, for a wide class of textured images. 1. INTRODUCTION Segmentation of visual textures is one of the most important problems of image processing. Texture may be defined as spatially random distributed structural pattern. It is an important characteristic of many types of visual images, such as aerial photographs or microscopic biomedical images. Analytical description and automatic recognition of textured images poses a complex image recognition problem. Segmentation and classification of

2 textured images is very important in medical diagnosis, where cells or fragments of tissues, represented in majority of images, can be considered as textures. Recently, there were some works presented on segmentation of textured images [3]. Among them, the Coocurrence Matrix Method for obtaining a set of features for texture identification was used. Feedforward multilayer neural networks were used as texture classifiers. The design of VLSI analogue realisation of such networks was also presented [4]. The set of features used for texture classification is calculated based on the so called coocurrence matrix (CM). This matrix is evaluated for textured image. These features do not describe an analysed image directly; they describe texture properties formally by using greylevel transformation contained in the coocurrence matrix. One can suppose, that some other set of features derived directly from the analysed image could be more adequate for texture classification. The second important problem is the choice of number and type of features used for texture description. This problem has to be solved individually for each class of analysed textures. In this paper an alternative approach to texture representation is presented. The texture is assumed as a realisation of a Markov Random Field (MRF) [1,2]. MRFs have proved to be good models for different classes of textures [1]. Parameters of MRF uniquely describe a modelled texture and can be derived directly from the analysed image, provided MRF is an adequate image model. 2. ESTIMATION OF MRF PARAMETERS Let X[i,j] be an image array, with size NxN, x[i,j] denote the integer grey level value in range of [0, L-1] at location (i,j). X[i,j] is an MRF realisation if p( x[ i, j] x[ k, l], k 1,..., i 1, i 1,..., N, l 1,..., j 1, j 1,..., N) p( x[ i, j] x[ k, l], ( k, l) N v ) (1) where p(. ) denotes conditional probability of x[i,j], given field realisation X[i,j] and Nv denotes a neighbour set. The meaning of Nv is illustrated by Fig. 1. As indicated there, a firstorder MRF is one whose neighbour set contains elements marked with "1", the second-order MRF is that for which the neighbour set contains elements marked both with "1" and "2", and so on. In general, the neighbour set of pixel x[i,j] is defined as

3 N {( k, l) ( k, l) - ( i, j) K, ( k, l) ( i, j)} (2) v where. is the Euclidean distance, v is the order of the process and Kv is a square of Euclidean distance of x[i,j] to its farthest neighbour. Kv takes values 1, 2, 4, 5, 8 for v=1, 2, 3, 4, 5, respectively. For simplicity, we will refer to binary MRFs, i.e. those with L=2. The conditional probability for a second-order MRF is given by v x[ i, j] T e p( x[ i, j] x[ k, l]) 1 e T where T a b ( x[ i 1, j] x[ i 1, j]) b ( x[ i, j 1] x[ i, j 1]) b ( x[ i 1, j 1] x[ i 1, j 1]) b ( x[ i 1, j 1] x[ i 1, j 1]) (3) where a, b1 1, b1 2, b2 1, b2 2 are field parameters. The set of these parameters fully define a texture modelled by a second-order MRF. Some examples of first-order MRFs (b2 1 =b2 2 =0) are shown on Fig. 2. Image segmentation can be performed based on estimated MRF parameters. The technique used to estimate parameters of a given MRF realisation (i.e. given image texture) is based on numerical maximisation of the conditional likelihood L [2]: N N L = p( x[ i, j ] x[ k, l ], { k, l} N v ) i= 1 j 1 (4) The factors in (4) are not independent. Evaluation of L is performed for disjoint sets of points in matrix X[i,j], called codings [2]. These codings are chosen so that their points are independent. This means that the points x[i,j] and x[i',j'], (i,j) (i',j') in any coding are not neighbours in the MRF sense. For example, first-order MRF requires two codings as shown on Fig 3. The independent points are marked by and, respectively. The estimation of parameter vector is performed by maximising (4) separately for the two codings, and the final estimate is the average of values obtained.

4 3. RESULTS The estimation method described in section 2 was used to obtain a set of parameters of MRFs shown on Fig. 2. The sample textures are binary first-order MRFs with a size of 128x128 pixels each. The estimation should be performed in a square window of a small size rather than over the whole image. The window size is a compromise between segmentation accuracy and possibility of exploiting statistical information it contains. To find a proper size of the window, estimation with various window sizes was performed. For a given window size, a set of 32 samples was calculated, each sample containing a vector of MRF parameters estimated for random window location over the whole image. Such a procedure was repeated for 64 different window sizes. The results of estimation are shown on Fig. 4. It represents a distribution of mean values and standard deviations calculated for a set of 32 samples. As can be seen on Fig 4, the estimated parameter values stabilise close to values for which the sample MRF was generated, even for small window size, except for the sample from Fig. 2a, where a larger window size was required. The values of the standard deviation decrease for growing size of the analysis window. Fig. 4 demonstrates that, the window size of 55x55 pixels is appropriate for providing proper data for texture classification based on estimated parameters of sample MRFs. 4. DISCUSSION As shown in section 3, estimated MRF parameters can be used for texture segmentation, provided appropriate window size is used. For comparison, for the sample images from Fig. 2, features of other kind were also calculated by applying the Coocurrence Matrix Method [1]. Two features with maximum discrimination measure were chosen. They were evaluated for each sample image using the similar window size as for MRF parameters. Distributions of the two kinds of features are shown on Fig. 5. Analysing the distributions on Fig. 5, it is clear that based on statistical CM features one is not able to classify correctly the sample textures. This is because clusters corresponding to different textures overlap. On the other hand, the MRF parameters form the well separated clusters, good for classification of analysed class of textures. It proves the superiority of MRF parameters over statistical features. The drawback of MRF model approach is long time of estimation using a procedure described in section 3.

5 This time grows rapidly with increasing MRF order. Therefore the problem under investigation is to find faster estimation algorithms. REFERENCES [1] Cohen S. F., Cooper B. D., Simple Parallel Hierarchical and Relaxation Algorithms for Segmenting Noncausal Markovian Random Fields, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. PAMI-9, pp , [2] Hassner M., Sklansky J., The Use of Markov random fields as Models of Texture, Image Modelling, Academic Press, pp , [3] Strzelecki M., Statystyczne metody segmentacji obrazów zawierających tekstury z wykorzystaniem sieci neuronowych, Archiwum Informatyki Teoretycznej i Stosowanej, tom 5, pp , [4] Strzelecki M., The design aspects of analogue VLSI realisations of neural network classifier of textures, Proceedings of XVII KKTOiUE Conference, vol. 1, pp , Polanica Zdrój 1994.

6 x[i,j] Fig. 1 The neighbour set of x[i,j] a) b) c) Fig 2 Examples of textures modelled as 1st-order binary MRF generated for parameters a) a= -6, b=3 b) a= -4, b=2 c) a= -2, b=1 Fig. 3 The example of coding for 1st order MRF

7 Fig 4 Mean values and standard deviations of MRF parameters for textures from Fig. 2 as function of window size

8 Feature values for sample textures: - Fig. 2a - Fig. 2b - Fig. 2 Fig. 5 Normalised feature values for textures from Fig 2: a) statistical CM features [3] b) estimated MRF