Stochastic relaxation, gibbs distributions, and the bayesian restoration of images.

Geman, Stuart; Geman, Donald

doi:10.1109/tpami.1984.4767596

Cited by 17,949Image and Signal Denoising MethodsAdvanced Image Processing TechniquesImage Enhancement Techniques

Stochastic relaxation, gibbs distributions, and the bayesian restoration of images.

Geman, Stuart, Geman, Donald·1984·IEEE Transactions on Pattern Analysis and Machine Intelligence

DOI 10.1109/tpami.1984.4767596

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Abstract

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 embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.

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Previous network models that implement sampling in neural networks are mostly based on Gibbs (or general Metropolis-Hastings) sampling.
“Previous network models that implement sampling in neural networks are mostly based on a special sampling algorithm called Gibbs (or general Metropolis-Hastings) sampling [9], [17], [18], [37].”
From Neural dynamics as sampling: a model for stochastic computation in recurrent networks of spiking neurons (2011) · cites this paper as [9]

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“We sample the posterior by Gibbs sampling (Geman & Geman, 1984) with Pólya–Gamma augmentation (Polson et al., 2013).”

Authors: Geman, S., & Geman, D.
Affiliations: Brown University (US)
University of Massachusetts Amherst (US)
Venue: IEEE Transactions on Pattern Analysis and Machine Intelligence
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Published: 1984-11
Volume: PAMI-6
Issue: 6
Pages: 721-741
ISSN: 0162-8828 (ISSN-L)
DOI: 10.1109/tpami.1984.4767596
OpenAlex: W2020999234
Cited by: 17,949
References: 45
Citations by year: 2026: 202 · 2025: 299 · 2024: 306 · 2023: 335 · 2022: 401 · 2021: 469 · 2020: 471 · 2019: 506 · 2018: 504 · 2017: 512 · 2016: 612 · 2015: 626 · 2014: 705 · 2013: 590 · 2012: 764
FWCI: 109.97
Type: journal article
Language: en
Open access: closed
Licence: ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html (vor)
Locations: IEEE Transactions on Pattern Analysis and Machine Intelligence (publishedVersion)
PubMed (publishedVersion)
Topics: Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Image and Signal Denoising Methods
Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Advanced Image Processing Techniques
Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Image Enhancement Techniques
Keywords: image restoration, Markov random fields, Boltzmann distribution, stochastic relaxation, annealing, maximum a posteriori estimation, image degradation, blurring, noise, image modeling, statistical mechanics analogy

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Stochastic relaxation, gibbs distributions, and the bayesian restoration of images.

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