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

This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

The theory of phase-ordering dynamics that is the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a broken-symmetry phase, is reviewed, with the emphasis on recent developments. Interest will focus on the scaling regime that develops at long times after the quench. How can one determine the growth laws that describe the time dependence of characteristic length scales, and what can be said about the form of the associated scaling functions? Particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered, for example vector and tensor fields. The latter are needed, for example, to describe phase ordering in nematic liquid crystals, on which there have been a number of recent experiments. The study of topological defects (domain walls, vortices, strings and monopoles) provides a unifying framework for discussing coarsening in these different systems.

Theory of phase-ordering kinetics

One of Feynman's early applications of path integrals was to superfluid $^{4}\mathrm{He}$. He showed that the thermodynamic properties of Bose systems are exactly equivalent to those of a peculiar type of interacting classical "ring polymer." Using this mapping, one can generalize Monte Carlo simulation techniques commonly used for classical systems to simulate boson systems. In this review, the author introduces this picture of a boson superfluid and shows how superfluidity and Bose condensation manifest themselves. He shows the excellent agreement between simulations and experimental measurements on liquid and solid helium for such quantities as pair correlations, the superfluid density, the energy, and the momentum distribution. Major aspects of computational techniques developed for a boson superfluid are discussed: the construction of more accurate approximate density matrices to reduce the number of points on the path integral, sampling techniques to move through the space of exchanges and paths quickly, and the construction of estimators for various properties such as the energy, the momentum distribution, the superfluid density, and the exchange frequency in a quantum crystal. Finally the path-integral Monte Carlo method is compared to other quantum Monte Carlo methods.

Path integrals in the theory of condensed helium

Scaling theory of percolation clusters

http://dsec.pku.edu.cn/~tieli/notes/stoch_topics2015/BKL1975.pdf

A new algorithm for Monte Carlo simulation of Ising spin systems

https://ergodic.ugr.es/jmarro/papers/82%20lmk%20am30.pdf

Dynamical scaling of structure function in quenched binary alloys

The time evolution of the structure function and of the cluster (or grain) distribution following quenching in a model binary alloy with a small concentration of minority atoms is obtained from computer simulations. The structure function $\overline{S}(k,t)$ obeys a simple scaling relation, $\overline{S}(k,t)={K}^{\ensuremath{-}3}F(\frac{k}{K})$ with $K(t)\ensuremath{\propto}{t}^{\ensuremath{-}a}$, $a\ensuremath{\cong}0.25$, during the latter and larger part of the evolution. During the same period, the mean cluster size grows approximately linearly with time.

Computer Simulation of the Time Evolution of a Quenched Model Alloy in the Nucleation Region

The Green's-function Monte Carlo method has been used to calculate the properties of the ground-state $^{4}\mathrm{He}$ with several modern potentials. One potential, dubbed HFDHE2, by Aziz et al., gave very good agreement with the experimental equation of state in both the liquid and crystal phases. Other properties such as the structure factor and the momentum distribution also compare well with the experiment. In both phases, perturbative estimates of the three-body Axilrod-Teller potential were computed. When they are added to the two-body energies, the total energy is no longer in agreement with experiment. Further investigation on the nature of three-body interactions in the $^{4}\mathrm{He}$ may be necessary to elucidate this problem.

Modern potentials and the properties of condensed /sup 4/He

A new thermodynamic analysis is given for the equilibrium between a liquid cluster and the surrounding supersaturated gas phase in afinite constant volume. It is shown that for constant total density and intermediate volume this equilibrium is stable, although it is unstable for very large volume. We show that observation of the critical cluster sizel* then yields information on the surface free energy of the liquid cluster. The accuracy of previous approximate prescriptions for obtaining the free energy of physical clusters is investigated. As an application, the theory is used to analyze Monte Carlo simulations of the two-dimensional lattice gas model at low temperatures. We obtain cluster surface area, diffusivity, and free energy for clusters with 26≥l≥500. It is found that the capillarity approximation is inaccurate forl≥100, but the free energy of small clusters ishigher than the result of classical nucleation theory, in contrast to what one expects from Tolman-like corrections. We interpret these results, deriving low-temperature series expansions for very small clusters, thus showing that the capillarity approximation both underestimates the surface energy and overestimates the surface entropy of very small clusters. Finally, we use our results to give a speculative explanation of recent nucleation experiments. The dependence of the cluster diffusivity on cluster size is tentatively explained in terms of a crossover between two mechanisms yielding different power laws.

Malvin H. Kalos

Papers

A new algorithm for Monte Carlo simulation of Ising spin systems

Dynamical scaling of structure function in quenched binary alloys

Computer Simulation of the Time Evolution of a Quenched Model Alloy in the Nucleation Region

Modern potentials and the properties of condensed /sup 4/He

``Critical clusters'' in a supersaturated vapor: Theory and Monte Carlo simulation