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

@article{Geman1984StochasticRG, 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}, year={1984}, volume={PAMI-6}, pages={721-741} }

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…

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## References

SHOWING 1-10 OF 49 REFERENCES

### Markov Random Field Texture Models

- MathematicsIEEE Transactions on Pattern Analysis and Machine Intelligence
- 1983

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.

- Computer ScienceJournal 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.

- Mathematics
- 1972

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

- MathematicsIEEE Trans. Inf. Theory
- 1983

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

- Mathematics, Computer ScienceIEEE Trans. Inf. Theory
- 1965

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

- Computer Science, MathematicsICASSP
- 1984

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

- Computer ScienceIEEE 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

- MathematicsIEEE Trans. Commun.
- 1976

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…

### OPTIMAL PERCEPTUAL INFERENCE

- Computer Science
- 1983

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.