Problems of Information Transmission, Vol. 40, No. 3, 2004, pp. 279–295. Translated from Problemy Peredachi Informatsii, No. 3, 2004, pp. 108–125.
Original Russian Text Copyright
2004 by Descombes, Zhizhina.
Gibbs Field Approaches in Image Processing Problems
INRIA, Sophia-Antipolis, France
Institute for Information Transmission Problems, RAS, Moscow
Received May 6, 2003; in ﬁnal form, May 6, 2004
Abstract—In this paper, we address the problem of image denoising using a stochastic diﬀer-
ential equation approach. Proposed stochastic dynamics schemes are based on the property of
diﬀusion dynamics to converge to a distribution on global minima of the energy function of the
model, under a special cooling schedule (the annealing procedure). To derive algorithms for
computer simulations, we consider discrete-time approximations of the stochastic diﬀerential
equation. We study convergence of the corresponding Markov chains to the diﬀusion process.
We give conditions for the ergodicity of the Euler approximation scheme. In the conclusion,
we compare results of computer simulations using the diﬀusion dynamics algorithms and the
standard Metropolis–Hasting algorithm. Results are shown on synthetic and real data.
1. INTRODUCTION. BAYESIAN APPROACH IN IMAGE RESTORATION PROBLEMS
One of the main problems of image processing is to create fast algorithms which can be applied to
image denoising and restoration problems. Image denoising is of fundamental importance for visual-
izing and interpreting images and also as a preprocessing step to improve the performance of image
processing tasks such as classiﬁcation, segmentation, or feature extraction. There exist numerous
approaches to image denoising based on ﬁltering. Indeed, noise is typically a high-frequency compo-
nent of an image. However, high frequencies contain also some important features of an image, such
as edge information. Simple low-pass ﬁltering results in blurring the image during the denoising pro-
cess. Therefore, the trade-oﬀ consists in preserving edges and informative features while denoising.
Besides ﬁltering, model-based approaches are traditionally used in image processing problems.
The most popular method of this type is the Bayesian approach based on the analysis of the
a posteriori distribution on the conﬁguration space of the model. The goal of the Bayesian image
restoration is to ﬁnd a conﬁguration which maximizes the a posteriori distribution. This conﬁgu-
ration is accepted as the denoised image. There exist diﬀerent methods in the framework of the
Bayesian approach. They depend on the form of the a posteriori distribution and on the way of
obtaining the denoised image (deterministic or stochastic). Here, we consider stochastic algorithms
under the a posteriori Gibbs distribution with an energy function (the Hamiltonian of the Gibbs
ﬁeld) depending on a given noisy image. Our restoration algorithm is a further application of the
Gibbs ﬁeld methods in image processing. It is based on the ergodic properties of diﬀusion dynamics
and subsequent use of approximation techniques.
The proposed algorithm has the structure of a stochastic iterative scheme, where the next conﬁg-
uration on each iteration step is found by a distribution depending on the current conﬁguration, the
Supported in part by the French–Russian A.M. Lyapunov Institute for Applied Mathematics and Com-
puter Science, Grant no. 98-02; INTAS, Grant no. 99-00559; NATO, NATO–Russia Grant CLG 980107;
Russian Foundation for Basic Research, project no. 02-01-00444; and the Scientiﬁc School Program, project
2004 MAIK “Nauka/Interperiodica”