Bootstrap Approximation of Gibbs Measure for Finite-range Potential in Image Analysis

—This paper presents a Gibbs measure approximation method through the adjustment of the associated estimated potential. We use the information criterion to prove the accuracy of this approach and the bootstrap computation method to determine the explicit form. The Gibbs sampler is the tool of our simulations while taking advantage of the use of the only one MCMC inside of the multiple necessary MCMC in the classical approximation. We focus on the validity of our approach for the Gibbs measure of a Markov Random Field with an interaction potential function and the associated uniqueness condition. Some theoretical and numerical results are given.


INTRODUCTION
It is well known that in computer vision, one of the first aims is to determine, for generic prior Gibbs model, the correct form of the potential function.Certainly, this later will be better if the parametric estimation is good enough [5], but this requires both ideal conditions and expensive costs.Furthermore, the Gibbs Random Fields have become an efficient instrument in image analysis.So, the associated statistical inference has attracted a great deal of interest, because of its great adequacy in important applications related to image processing, computer vision, neural modeling and perceptual inference.Nevertheless, to estimate the true parameter for the Gibbs model required a high cost in terms of computation time and modeling conditions.For example, the use of the maximum likelihood estimation (MLE) was impossible to be calculated and was substituted by the pseudolikelihood estimation [5] [18].Fortunately, the intense developments in statistics accompanied by the evolution in computer systems allowed maximum likelihood estimation for Gibbs Random Fields to be constructed.For this, we propose the use of the resampling method [3] through Markov Chain Monte Carlo (MCMC).The bootstrap computation method and MCMC with Gibbs sampler is used to retrieve ever more the desired potential function for the parameter Gibbs distribution.Some technical changes make use of one MCMC instead of two chains, which is very promising to reduce the computational time.We prove that the KL-distance is minimized for the adjusted potential function.Moreover, the adjustment method proposed in this paper keeps the features of prior potential function for the associated Gibbs measure.This paper is organized as follows.In section 2, we present the necessary context of the Gibbs models that describes the validity of the proposed approximation.In section 3, we present the steps of the approximation proposed in this paper in particular the study of the information criterion for a Gibbs model from what we inspire the proper form of the adjusted potential and prove the accuracy of the associated approximation even though by the use of one MCMC .In section 4, we present some numerical results to explain the feasibility of the usefulness of the approximated expressions.

II. THE THEORITICAL APPROACH OF A GIBBS RANDOM FIELD WITH POTENTIAL INTERACTION FUNCTION
Before presenting the adjustment method for a parametric potential interaction function of a Gibbs distribution, it is necessary to review the concepts and results related to the Gibbs measure, which will clarify the necessity and importance of using the bootstrap approach and techniques that we introduce in this context.

A. Click and neighborhood system
It is quite obvious that a digital image is modeled by a matrix of data on a network instead of a linear data base.The shape of such model is a Random Field ( ) instead of an ARMA model for example.
In the network , a system of neighborhood * + is defined as follows:  is reduced to a single site  or it contains at least two elements and each pair (s, t) of elements is formed of neighboring sites (with regard to ).The boundary of a non-empty subset of the network S is a subset defined as: for there exists an element such that, Given a distance on S, the neighborhood of a site with respect to a finite-range is: .takes its values in the set of configurations ( ) is the set of the different levels of a pixel s.The first way to define a probability measure of the random field on is to give a Kolmogorov projective family of marginal distribution on finite subsets V of S. Nevertheless, the right way is to define a kernel family of conditional probabilities [11], which is more appropriate in case of image processing and analysis.In other words, given a probability measure on , the conditional probability kernel on a subset V of S is defined as follows: is the conditional expectation on the Borel space ( ) given a configuration on the outside ( ) of V, such that for all and ( ) , we have almost surely: , ( This can be written as the following: almost surely for all ( ).It is expressed as an operation between two kernels and : Using the operation above, we give the definition of a specification below.( ) denotes the set of finite subsets of the network S.

Definition 1.1
A specification is a family ( ) ( ) of probability kernels satisfying: c) For all and ( ) .Then, it follows that for a Markov Random Field given a Gibbs measure , the associated specification can be written as (Dobrushin-Lanford-Ruelle-equation): The sufficient condition of the existence of a Gibbs measure given a specification can be written as: ) and ( ) are the restriction configurations to ( ) and respectively.It should be noted that the condition ( 7) is true for a Markov Random Field with finiterange potential.

C. The interaction potential function of a GibbsRandom field
For a Markov Random field, the associated Gibbs measure " " is defined via the specification like: for all ( ) ( ).Then " " is a positive measure defined on and ( ) is the energy function given by the potential ( ) ( ) such that, for ( ) and ) is the concatenation of the configuration on D with boundary condition on (S-D).We can recall the different results in this context; however, we are interested in conditions of existence and uniqueness of the Gibbs measure given a potential function specification.It is used in measuring the accuracy of parametric model estimation for a Markov random field.Because of this, it is introducing a quantity, for a given site on S private the origin o: The sup is taken over all configurations and on * + identical everywhere except on s, and ‖ ‖ denotes the total variation norm of a measure defined by: measures the maximum influence of the modality at the site s on the conditional distribution at the origin network of S. So, the uniqueness condition of a Gibbs measure [11] is of the form:

∑
(11) This is rewritten in [7] for a specification with a potential invariant under translation as well; , , such that, The expression ( 12) is more significant and useful than (11).

D. Markov Chain Monte Carlo and paremeter estimation for Gibbs distribution
We can find some papers that dealt with the maximum likelihood estimation for a Gibbs Random Field [5] [19].The main problem in this topic is essentially the large computation time for a complete determination of the MLE based on maximizing: ) for an observed configuration of the Gibbs Random Field on a subset ( ) , where is a boundary configuration outside D. the expression (13) is the parametric version of (8) with respect to the parametric potential ( ( )) ( ) .The normalization constant is: Recently the construction of the MLE is allowed due to computational system evolution.It is to solve the derivative equation: , to examine the evolution of this function with regard to the difference between , using the MCMC simulated until different two states(N=600 and N=1000).The case indicated by NA means that the denominator of ( ) in (31) becomes a very high number.This gives the shape of the two graphs of this function as follows: We can notice that the approximate function keeps the theoretical properties of the initial function, such as The main advantage of this approximation is in its easy use while keeping its strong properties as information criterion and adjusted term for the estimated potential.This assures the validity of the approximation of the Gibbs measure given above.

Fig. 2 .
Fig. 2. The variation of the approximated function based on the difference between ; the dotted-line curve for N=600 stats of the MCMCm and the continuous-line curve for N=1000 the boundary of a subset of length is,