86 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 1, JANUARY 1997 Express Letters Markov Random Field Image Segmentation Using Cellular Neural Network Tam´ as Szir´ anyi and Josiane Zerubia Abstract— Markovian approaches to early vision processes need a huge amount of computing power. These algorithms can usually be implemented on parallel computing structures. With the Cellular Neural Networks (CNN), a new image processing tool is coming into consid- eration. Its VLSI implementation takes place on a single analog chip containing several thousands of cells. Herein we use the CNN UM architecture for statistical image segmentation. The Modified Metropolis Dynamics (MMD) method can be implemented into the raw analog architecture of the CNN. We are able to implement a (pseudo) ran- dom field generator using one layer (one memory/cell) of the CNN. We can introduce the whole pseudostochastic segmentation process in the CNN architecture using 8 memories/cell. We use simple arithmetic functions (addition, multiplication), equality-test between neighboring pixels and very simple nonlinear output functions (step, jigsaw). With this architecture, a real VLSI CNN chip can execute a pseudostochastic relaxation algorithm of about 100 iterations in about 1 ms. In the proposed solution the segmentation is unsupervised. We have developed a pixel-level statistical estimation model. The CNN turns the original image into a smooth one. Then we have two gray-level values for every pixel: the original and the smoothed one. These two values are used for estimating the probability distribution of region label at a given pixel. Using the conventional first-order Markov Random Field (MRF) model, some misclassification errors remained at the region boundaries, because of the estimation difficulties in case of low SNR. By using a greater neighborhood, this problem has been avoided. In our CNN experiments, we used a simulation system with a fixed-point integer precision of 16 bits. Our results show that even in the case of the very constrained conditions of value-representations (the interval is (-64, +64), the accuracy is 0.002) can result in an effective and acceptable segmentation. I. IMAGE SEGMENTATION AND MARKOV RANDOM FIELDS For all early vision processes modeled by MRF, the problem is posed as one of minimizing a cost function which is constructed from the observed data, a priori information on the world and constraints. The cost function obtained is usually nonconvex and several relaxation techniques have been proposed to reach an op- timum. The first group of methods deals with stochastic relaxation and is based on Simulated Annealing (SA) [4]. These algorithms converge asymptotically toward the global minimum but require a great deal of computation. The second group of methods is related to deterministic relaxation. These techniques are suboptimal but require less computational time than the previous ones. This is why so many deterministic relaxation algorithms have been recently investigated (Graduated Non Convexity (GNC), Iterated Conditional Mode (ICM) [2], Mean Field Annealing (MFA) [10], Modified Metropolis Dynamics (MMD) [5]). Manuscript received September 9, 1996. This work was supported in part by the Balaton program of the National Committee for Technological Development of Hungary and the French Ministry of Foreign Affairs. This paper was recommended by Associate Editor J. Pineda de Gyvez. T. Szir´ any is with the Analogical and Neural Computing Systems Research Laboratory, Computer and Automation Institute, Hungarian Academy of Sciences, (MTA SZTAKI), H-1518 Budapest, Hungary. J. Zerubia is with the INRIA Sophia-Antipolis, 06902 Sophia-Antipolis Cedex, France. Publisher Item Identifier S 1057-7122(97)00216-X. A. Markovian Image Model for Segmentation Herein, we are interested in the following problem: we are given a set of sites each of which may belong to any one of classes (or equivalently take any label from ). We are also given a MRF [4] on these sites, defined by the so-called clique potentials and by a neighborhood-system . Let denote a clique of and the set of all cliques. Also . A global discrete labeling assigns one label ( ) to each site in . The restriction of to the sites of a given clique is denoted by . The definition of the MRF is completed by the knowledge of the clique potentials for every in and every in , where is the set of all possible discrete labelings. Given a set of image data where stands for the gray-level at pixel . A very general problem is to find the labeling which maximizes . In order to be able to implement the proposed method on CNN, we had to introduce important modifications with respect to the original model proposed in [5]. • first, we worked with a larger neighborhood (third-order MRF (12 neighbors), using only the cliques of order 1 ( ) and 2( ), i.e., the singletons and the doubletons). • second, we have made the assumption that we could work at the pixel level (because computation needs to be very local for the CNN). The observation at pixel ( ) is supposed to come from an original gray level value ( ), disturbed by an additive Gaussian white noise ( ), . Furthermore, we assume that a crude approximation of could be obtained by the following mean and standard deviation : and (1) where is a smoothed value of obtained through anisotropic diffusion for instance. Then, following the probability formula in [5], we use an ad hoc criterion to get a final segmentation : where corresponds to the mean gray level value associated to the class at site . It is automatically estimated considering the main peaks in the histogram of input (or smoothed input) image. Using the above equation, it is easy to define the local energy of any labeling at site : (2) where is equal to ( ) if and to otherwise. A large value of will favor homogeneous regions. The estimation of is done through the energy minimization using a MMD. B. The MMD algorithm Basically, the MMD algorithm [5] is just a modified version of the Metropolis Dynamics which turns the algorithm into a pseudos- tochastic relaxation. The difference between the original Metropolis method and the MMD is the choice of the threshold used in the 1057–7122/97$10.00  1997 IEEE