Elsevier

Pattern Recognition Letters

Volume 18, Issue 14, December 1997, Pages 1479-1493
Pattern Recognition Letters

Cluster ZTP in the recovery of an image

https://doi.org/10.1016/S0167-8655(97)00146-3Get rights and content

Abstract

Cluster ZTP (zero-temperature process) is proposed as a method of image recovery and examined for 8-valued images. The cluster ZTP is an iterative algorithm for the search of an approximately optimal solution of an energy minimization problem.

Introduction

In statistical-mechanical techniques for image restoration, the analysis is reduced to an energy minimization problem using regularization theory (Poggio et al., 1985). The optimal solution is found by using the graduated non-convexity (GNC) algorithm as an iterative algorithm (Blake, 1989). In order to avoid local minima, a Markov random field (MRF) model is introduced and an annealing procedure is applied to the minimization (see, for example, (Chellappa and Jain, 1993)). Geman and Geman (1984) first introduced the MRF model with line processes for image restoration and segmentation. It is applicable to segmentation and was applied to texture analysis by Derin and Elliott (1987). Jeng and Woods (1991) extended the MRF model so as to be applicable to gray-level images. Their MRF model is based on the multi-valued Ising model with line processes and is equivalent to the weak membrane model except for the inclusion of interactions between line processes. The model is called the compound Gauss–Markov random field (CGMRF) model. In the MRF model, a deterministic algorithm for minimization can be constructed by adopting the mean-field approximation; some GNC-like mean-field annealing (MFA) algorithms were proposed in (Geiger and Girosi, 1991; Bilbro et al., 1992; Bedini et al., 1994, Bedini et al., 1995). The MFA algorithm is constructed from a deterministic equation for the marginal probability at a single site, and the optimal solution is determined from the obtained marginal probabilities. A simple algorithm called the iterated conditional mode (ICM) method was proposed by Besag (1986); see also (Geman et al., 1990). The present authors suggested the use of ICM at the last stage of the MFA in (Morita and Tanaka, 1997), which will be cited as I in the following. The ICM algorithm can erase noise when the noise is in an isolated site, but it cannot erase noise when two or more successive sites are affected by the noise. On the other hand, the MFA algorithm can deal with this problem at the cost of enormous computation time.

In statistical mechanics, many authors attempted to extend the mean-field approximation to a cluster-type version in order to take many-body correlation into account. The most familiar general technique is the cluster variation method (CVM) (see, for example, (Morita et al., 1994). In the CVM, the next approximation to the mean-field approximation is the pair approximation. Recently, the present authors (Tanaka and Morita, 1995) constructed deterministic equations for the marginal probability of a pair of nearest-neighbour lattice sites as well as that of a single lattice site, based on the pair approximation of the CVM, thus extending the MFA algorithm to the pair annealing (PA) algorithm (Tanaka and Morita, 1995), and showed that improved results are obtained. The ICM algorithm can be regarded as the zero-temperature limit of the MFA. It is interesting to extend the ICM algorithm to a cluster-type algorithm, so that noises where two or more successive sites are affected, might be erased. We shall refer to the cluster-type algorithm as the cluster zero-temperature process (ZTP) in the following discussion.

As mentioned above, the ICM is regarded as the limit at zero temperature of the mean-field approximation in statistical mechanics. Improved approximations over the mean-field approximation in the CVM are the pair approximation, the square approximation, etc. We shall consider the pair, the square, the Hi-no-ji and the Ta-no-ji approximations for the cluster ZTP in the present paper.

Computer programs for MFA and PA need storage for a lot of data in double precision and use a lot of main memory in the computer. On the other hand, in the iterative program for ICM, most of the storage needed is for integer data. Even when the simple ZTP, that is the ICM, algorithm is extended to cluster ZTP versions, the situation does not change. Moreover, since the cluster ZTP algorithm improves as we adopt larger clusters, we have only to consider the problem of constructing the energy function so as to obtain better recovered images, when we adopt a sufficiently large cluster.

In (Tanaka and Morita, 1995; Morita and Tanaka, 1996, Morita and Tanaka, 1997; Tanaka et al., 1996), the present authors studied the image restoration problem, using statistical-mechanical techniques. In the method, we estimate some properties of the original image to be recovered from a single given noisy image, with the aid of some a priori knowledge, and then seek an image which is nearest to the noisy image among the images satisfying the estimated properties (Morita and Tanaka, 1996). The problem is formulated as a variational calculation with constraints. Other authors, including Jeng and Woods (1991), applied the maximum likelihood approach to the determination of the a priori parameters. If they are determined by using the constraints, the meaning of the a priori parameters is clear. We consider that this point is an advantage of the method.

In order to satisfy the constraints, we introduce Lagrange multipliers. In terms of statistical mechanics, the problem is that of obtaining the ground state which gives the minimum value of a Hamiltonian. In the present paper, the energy function and the minimum-energy configuration play the roles of the Hamiltonian and the ground state, respectively. The energy function involves a number of parameters which are the Lagrange multipliers. In this method, we calculate the minimum value of the energy for various sets of values of the parameters, and check how the constraints are satisfied in the configuration giving the minimum energy. We choose the set of parameters for which the constraints are best satisfied, and adopt it as the best set of the parameters. In the papers cited above, the present authors used an annealing method in obtaining the minimum-energy configuration. Since the annealing method required much CPU time, we proposed to use ICM in estimating rough values of the parameters, in I. We now develop cluster ZTP, which enables us to achieve the minimum-energy configuration in reasonable CPU time. The present method is closely related to the zero-temperature limit of the CVM (Morita, 1985).

In Section 2, we give a sketch of our variational method for the recovery of an image. In Section 3, we propose pair, square, Hi-no-ji and Ta-no-ji ZTP, which are collectively called cluster ZTP. ICM is sometimes referred to as simple ZTP. Numerical examples for 8-valued images are studied in Section 4, where a comparison of the present method with the MFA and PA algorithms is also given. Section 5contains concluding remarks, along with some preliminary results for 256-valued images.

In I, the problem of determining the best values of the Lagrange multipliers was discussed in detail. In (Tanaka et al., 1996), we discussed the problem of dealing with line processes with the aid of some Lagrange multipliers. In the present paper, we do not take up these problems but focus on showing how cluster ZTP gives the minimum-energy configuration in a short time. Numerical calculation is given for an energy function with fixed values of Lagrange multipliers. We now consider triplet interaction terms in addition to those considered in I, in order to improve the resulting image. The energy function we take up may be said to be one for the spin-S Ising model with 2S+1=8 states.

Section snippets

Image restoration problem

We consider a q-valued image on a square lattice, meaning that a site on the lattice can take one of q states. The lattice is assumed to consist of L sites and to satisfy the periodic boundary conditions, so that the lattice is on a torus. An image is represented by a set of L integers as z={zii=1,2,…,L}, where zi is the state of the ith site and is assumed to take a value from 0,1,2,…,q−1. The configurations of the original, damaged and recovered images, are represented by x={xii=1,2,…,L}, y

Pair ZTP, Square ZTP, Hi-no-ji ZTP and Ta-no-ji ZTP

In ICM, we choose a starting image z(0), choose a site i, generate a new zi for site i and change the value of zi when the change reduces the value of E(z|y) until no further change zi for any i reduces E(z|y). In cluster ZTP, we start with z=z(0) and change the configuration of a cluster when the change reduces the value of E(z|y) until no further reduction in the objective energy function is possible. In pair ZTP, the clusters are restricted to the pair-site clusters, each of which consists

Numerical examination

We report here the result of applying cluster ZTP to the example of the image given in I. The images we consider are 8-valued images on a 64×64 lattice. We consider the original image shown in Fig. 2(a), and the damaged images shown in Fig. 2(b) and Fig. 2(c). These damaged images are produced from the original image by choosing 30% and 50% of the sites and changing their states to values chosen from a uniform distribution of the 8 states. Hence the probability p(yi|xi) of yi is given byp(yi|xi

Concluding remarks

We have studied the problem of recovery of an 8-valued image. The problem was reduced to that of obtaining the minimum-energy configuration of an energy function. We have considered two problems in this study: (i) how to construct an energy function whose minimum-energy configuration is the original image, from knowledge only of a single damaged image; (ii) how to find the minimum-energy configuration of the energy function.

The first problem was investigated by using the Bayesian approach,

Acknowledgements

The first author (K.T.) is a visiting researcher, visiting Professor D.M. Titterington of the Department of Statistics, University of Glasgow, as an abroad-researcher of the Ministry of Education, Science, Sports and Culture of Japan, during the period from March, 1997 to February, 1998. K.T. is grateful to Professor Titterington, for hospitality and valuable comments on this work. The authors are thankful to Professor Titterington and Dr A.P. Dunmur for help in improving the manuscript.

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