Bayesian Restoration Research Articles

Discussion on the Meeting on the Gibbs Sampler and Other Markov Chain Monte Carlo Methods

Discussion on the Meeting on the Gibbs Sampler and Other Markov Chain Monte Carlo Methods

Discussion of the Paper by Vardi and Lee

Discussion of the Paper by Vardi and Lee

Markov modeling for Bayesian restoration of two-dimensional layered structures

Bayesian estimation of two-dimensional stratified structures is described. The major point addressed is the derivation of a statistical prior model that adequately describes such layered media. This problem is of interest in applications in which the data are generally processed in one dimension only. In order to take local interactions into account, a Markovian description is used. The model is derived so as to fulfill a set of constraints that summarize physical and geometrical characteristics of the problem as well as practical requirements. The resulting class of Markov random fields presents a unilateral structure on a nonrectangular lattice and a hierarchical organization which involves a line process. In addition, it is shown to be an extension of one-dimensional models already in use. The properties of the model are investigated, and its practicality is demonstrated by an application to seismic deconvolution. Simulation results show significant improvements with respect to the usual one-dimensional methods. >

Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images*

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, non-linear 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.

Bayesian Restoration of Single-Channel Patch Clamp Recordings

The technique of patch clamp recording makes possible the measurement of current flowing through a single ion channel in a cell membrane. Examination of such recordings suggests that the current is quantal in nature, alternating in a seemingly random manner between "on" and "off," but the recordings are corrupted by noise from a variety of sources. In this paper we propose and illustrate methods for restoring the underlying quantal signal from such noisy measurements. The methods use a Markov chain prior distribution for the underlying quantal process and base the restoration on the resulting posterior distribution.

Discussion of the Paper by Hall and Johnstone

Discussion of the Paper by Hall and Johnstone

Discussion of the Paper by Thomas

Journal of the Royal Statistical Society: Series B (Methodological)Volume 53, Issue 1 p. 22-26 DiscussionFree Access Discussion of the Paper by Thomas First published: 1991 https://doi.org/10.1111/j.2517-6161.1991.tb01807.xAboutPDF ToolsExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat REFERENCES IN THE DISCUSSION Cannings, C. and Thompson, E. A. (1981) Genealogical and Genetic Structure. Cambridge: Cambridge University Press. Clayton, D. G. (1990) A Monte Carlo method for Bayesian inference in frailty models. Biometrics, to be published. Gelfand, A. E. and Smith, A. F. M. (1990) Sampling based approaches to calculating marginal densities. J. Am. Statist. Ass., to be published. Geman, S. and Geman, D. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattn Anal. Mach. Intell., 6, 721– 741. Lauritzen, S. L. and Spiegelhalter, D. J. (1988) Local computations with probabilities on graphical structures and their application to expert systems (with discussion). J. R. Statist. Soc. B, 50, 157– 224. Schagen, I. P. (1980) Stochastic interpolation applied to the optimization of expensive objective functions. In Compstat 1980, Proc. Computational Statistics (eds M. M. Barritt and D. Wishart), pp. 362– 367. Vienna: Physica. Sheehan, N. A. (1989) Image processing procedures applied to the estimation of genotypes on pedigrees. Technical Report 176. Department of Statistics, University of Washington, Seattle. Smith, H. F. (1936) A discriminant function for plant selection. Ann. Eugen., 7, 240– 250. Spiegelhalter, D. J. (1990) Fast algorithms for probabilistic reasoning in influence diagrams, with application in genetics and expert systems. In Influence Diagrams, Belief Nets and Decision Analysis (eds R. M. Oliver and J. Q. Smith), pp. 361– 384. Chichester: Wiley. Thomas, A. (1988) Discussion on Local computations with probabilities on graphical structures and their application to expert systems (by S. L. Lauritzen and D. J. Spiegelhalter). J. R. Statist. Soc. B, 50, 215. Thomas, A. (1990) Genotypic inference with the Gibbs sampler. to be published. Thompson, E. A. (1986) Genetic epidemiology: A review of the statistical basis. Statist. Med., 5, 291– 302. Volume53, Issue11991Pages 22-26 This article also appears in:Discussion Papers ReferencesRelatedInformation

Experiments with an adaptive Bayesian restoration method

This paper describes a Bayesian restoration method applied to two-dimensional measured images, whose detector response function is not completely known. The response function is assumed Gaussian with standard deviation depending on the estimate of the local density of the image. The convex hull of the K-nearest neighbours ( KNN) of each ‘on’ pixel is used to compute the local density. The method has been tested on ‘sparse’ images, with and without noise background.

Restoration of binary images using stochastic relaxation with annealing

This paper investigates the application of variations of Stochastic Relaxation with Annealing (SRA) as proposed by Geman and Geman [1] to the Bayesian restoration of binary images corrupted by white noise. After a general review we present some specific prior models and show examples of their application. It appears that a proper selection of the prior model is critical for the success of the method. We obtained better results on artificial images which fitted the model closely than on real images for which there was no precise model.

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

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.

Bayesian Restoration of Partially Coherent Imagery in Conventional and Scanning Microscopy

The use of bayesian techniques for image restoration under conditions of partial coherence in both CTEM and STEM is investigated. Amplitude or phase objects are considered, assuming only a single intensity measurement. The weak-phase approximation is not used. An iterative algorithm for determining the maximum a posteriori probability restored image is determined by using the steepest ascent method. The effect of coherence and noise on the object restorability is assessed.