Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images

  title={Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images},
  author={Stuart Geman and Donald Geman},
  journal={IEEE Transactions on Pattern Analysis and Machine Intelligence},
  • S. Geman, D. Geman
  • Published 1 November 1984
  • Physics
  • IEEE Transactions on Pattern Analysis and Machine Intelligence
We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for… 

Figures from this paper

Unsupervised image restoration and edge location using compound Gauss-Markov random fields and the MDL principle

A new unsupervised discontinuity-preserving image restoration criterion is proposed, carried out by a continuation-type iterative algorithm which provides estimates of the number of discontinuities, their locations, the noise variance, the original images variance, and the original image itself (restored image).

From Particle Mechanics to Pixel Dynamics: Utilizing Stochastic Resonance Principle for Biomedical Image Enhancement

There is a noteworthy analogy between the statistical mechanical systems and the digital image processing systems. We can make pixel gray levels of an image correspondence to a discrete particles

Image-modeling Gibbs distributions for Bayesian restoration

Gibbs distributions have been widely used in the Bayesian approach to many image processing problems. However, little attention has been paid to whether or not the Gibbs distribution indeed models

Temperature and Gibbs Image Modeling

The Gibbs random eld (GRF) has become a popular image model with applications in restoration, segmen-tation, reconstruction, edge detection, compression, and motion estimation. Its synthesis of

Mean field approximation for PDE-Markov random field models in image analysis

Markov random fields (M.R.F.) on a lattice system and Gibbs distribution provide a wide area of models for interacting particle systems in image analysis, mechanical physics and statistical

Image Sequence Restoration Using Gibbs Distributions

This thesis addresses a number of issues concerned with the restoration of one type of image sequence, namely archived black and white motion pictures, by exploring the use of a general class of model known as Markov Random Fields, based on Gibbs distributions, by analogy with models from statistical physics.

Statistical-mechanical approach to image processing

The basic frameworks and techniques of the Bayesian approach to image restoration are reviewed from the statistical-mechanical point of view. First, a few basic notions in digital image processing

Statistical mechanics of image restoration

We develop the statistical mechanics formulation of the image restoration problem, pioneered by Geman and Geman (1984). Using Bayesian methods we establish the posterior probability distribution for

A Study on Application of Markov Random Fields in Image Restoration and its Efficiency

Bayesian statistical techniques and Markov random field (MRF) theory were used to restore a black and white binary image corrupted by additive Gaussian noise with zero mean and constant variance and high quality images were restored.

Some Extensions of the Spring Model for Image Processing

This paper shows that by allowing the state of the particles to take values on a space of discrete probability measures, one can use a generalized spring model for the empirical posterior marginal distributions of discrete MRF’s, such as the Ising model.



Markov Random Field Texture Models

The power of the binomial model to produce blurry, sharp, line-like, and blob-like textures is demonstrated and the synthetic microtextures closely resembled their real counterparts, while the regular and inhomogeneous textures did not.

Restoring with maximum likelihood and maximum entropy.

  • B. Frieden
  • Computer Science
    Journal of the Optical Society of America
  • 1972
A communication-theory model for the process of image formation is used and it is found that the most likely object has a maximum entropy and is represented by a restoring formula that is positive and not band limited.

Bayesian recursive image estimation.

A procedure for recursively estimating images that are characterized statistically by the mean and correlation functions associated with the random process representing the brightness level is

Estimation and choice of neighbors in spatial-interaction models of images

Some aspects of statistical inference for a class of spatial-interaction models for finite images are presented: primarily the simultaneous autoregressive (SAR) models and conditional Markov (CM)

Classification of binary random patterns

A novel extension of Markov chain methods into two dimensions leads to the Markov mesh which economically takes care of a much larger class of spatial dependencies.

Bayes smoothing algorithms for segmentation of images modeled by Markov random fields

The Bayes smoothing algorithm presented is an extension of a 1-D algorithm to 2-D and it yields the a posteriori distribution and the optimum Bayes estimate of the scene value at each pixel, using the total noisy image data.

Bayesian Methods in Nonlinear Digital Image Restoration

  • B. Hunt
  • Computer Science
    IEEE Transactions on Computers
  • 1977
A model is used which explicitly includes nonlinear relations between intensity and film density, by use of the D-log E curve, and a maximum a posteriori (Bayes) estimate of the restored image is derived.

Two-Dimensional Markov Representations of Sampled Images

This paper shows that, under certain weak restrictions, a two-dimensional discrete Markov process can be represented either "causally" by a one-sided difference equation, or "noncausally" by a


A particular nondeterministic operator is given, based on statistical mechanics, for updating the truth values of hypothcses, and a learning rule is described which allows a parallel system to converge on a set ofweights that optimizes its perccptt~al inferences.