auxiliary variable methods for markov chain monte carlo with applications abstract suppose that one wishes to sample from the density x using markov chain monte carlo mcmc an auxiliary variable u and its conditional distribution u x can be defined giving the joint distribution x u x u x a mcmc scheme that samples over this joint distribution can lead to substantial gains in efficiency compared to standard approaches the revolutionary algorithm of swendsen and wang is one such example besides reviewing the swendsen wang algorithm and its generalizations this article introduces a new auxiliary variable method called partial decoupling two applications in bayesian image analysis are considered a binary classification problem in which partial decoupling out performs swendsen wang and single site metropolis methods and a positron emission tomography pet reconstruction that uses the gray level prior of geman and mcclure a generalized swendsen wang algorithm is developed for this problem which reduces the computing time to the point where mcmc is a viab

bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

estimating optimal parameters for mrf stereo from a single image pair this paper presents a novel approach for estimating the parameters for mrf based stereo algorithms this approach is based on a new formulation of stereo as a maximum a posterior map problem in which both a disparity map and mrf parameters are estimated from the stereo pair itself we present an iterative algorithm for the map estimation that alternates between estimating the parameters while fixing the disparity map and estimating the disparity map while fixing the parameters the estimated parameters include robust truncation thresholds for both data and neighborhood terms as well as a regularization weight the regularization weight can be either a constant for the whole image or spatially varying depending on local intensity gradients in the latter case the weights for intensity gradients are also estimated our approach works as a wrapper for existing stereo algorithms based on graph cuts or belief propagation automatically tuning their parameters to improve performance without requiring the stereo code to be modified experiments demonstrate that our approach moves a baseline belief propagation stereo algorithm up six slots in the middlebury rankings

hyperparameter estimation for satellite image restoration using a mcmc maximum likelihood method abstract the satellite image deconvolution problem is ill posed and must be regularized herein we use an edge preserving regularization model using a function involving two hyperparameters our goal is to estimate the optimal parameters in order to automatically reconstruct images we propose to use the maximum likelihood estimator mle applied to the observed image we need sampling from prior and posterior distributions since the convolution prevents use of standard samplers we have developed a modified geman yang algorithm using an auxiliary variable and a cosine transform we present a markov chain monte carlo maximum likelihood mcmcml technique which is able to simultaneously achieve the estimation and the reconstruction

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

a generalized em algorithm for 3 d bayesian reconstruction from poisson data using gibbs priors a generalized expectation maximization gem algorithm is developed for bayesian reconstruction based on locally correlated markov random field priors in the form of gibbs functions and on the poisson data model for the m step of the algorithm a form of coordinate gradient ascent is derived the algorithm reduces to the em maximum likelihood algorithm as the markov random field prior tends towards a uniform distribution three different gibbs function priors are examined reconstructions of 3 d images obtained from the poisson model of single photon emission computed tomography are presented

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

inhomogeneous prior models for image reconstruction abstract over recent years the use of homogeneous gibbs prior models in image processing has become widely accepted there has been however much discussion over precisely which models are most appropriate for most applications the simplest gaussian model tends to oversmooth reconstructions so it has been rejected in favor of various edge preserving alternatives we claim that the problem is not with the gaussian family but rather with the assumption of homogeneity in this article we propose an inhomogeneous gaussian random field as a general prior model for many image processing applications the simplicity of the gaussian model allows rapid calculation and the flexibility of the spatially varying prior parameter allows varying degrees of spatial smoothing this approach is in the spirit of adaptive kernel density methods where only the choice of the variable window width is important the analysis of real single photon emission computed tomography data is used to illustrate the methods and simu

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

a theoretical study of the contrast recovery and variance of map reconstructions from pet data the authors examine the spatial resolution and variance properties of pet images reconstructed using maximum a posteriori map or penalized likelihood methods resolution is characterized by the contrast recovery coefficient crc of the local impulse response simplified approximate expressions are derived for the local impulse response crcs and variances for each voxel using these results the authors propose a practical scheme for selecting spatially variant smoothing parameters to optimize lesion detectability through maximization of the local crc to noise ratio in the reconstructed image

a class adaptive spatially variant mixture model for image segmentation we propose a new approach for image segmentation based on a hierarchical and spatially variant mixture model according to this model the pixel labels are random variables and a smoothness prior is imposed on them the main novelty of this work is a new family of smoothness priors for the label probabilities in spatially variant mixture models these gauss markov random field based priors allow all their parameters to be estimated in closed form via the maximum a posteriori map estimation using the expectation maximization methodology thus it is possible to introduce priors with multiple parameters that adapt to different aspects of the data numerical experiments are presented where the proposed map algorithms were tested in various image segmentation scenarios these experiments demonstrate that the proposed segmentation scheme compares favorably to both standard and previous spatially constrained mixture model based segmentation

a monte carlo method for high dimensional integration a new method for the numerical integration of very high dimensional functions is introduced and implemented based on the metropolis monte carlo algorithm the logarithm of the high dimensional integral is reduced to a 1 dimensional integration of a certain statistical function with respect to a scale parameter over the range of the unit interval the improvement in accuracy is found to be substantial comparing to the conventional crude monte carlo integration several numerical demonstrations are made and variability of the estimates are shown

a model for segmentation and analysis of noisy images abstract this article proposes a statistical model for image generation that provides automatic segmentation of images into intensity differentiated regions and facilitates the quantitative assessment of uncertainty associated with identified image features the model is specified hierarchically within the bayesian paradigm at the lowest level in the hierarchy a gibbs distribution is used to specify a probability distribution on the space of all possible partitions of the discretized image scene an important feature of this distribution is that the number of partitioning elements or image regions is not specified a priori at higher levels in the hierarchical specification random variables representing emission intensities are associated with regions and pixels observations are assumed to be generated from exponential family models centered about these values

instability of the hocking stewartson pulse and its implications for three dimensional poiseuille flow the linear stability problem for the hocking stewartson pulse obtained by linearizing the complex ginzburg landau cgl equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to stability exponents a numerical algorithm based on the compound matrix method is developed for computing the evans function using values in the cgl equation associated with spanwise modulation of plane poiseuille flow we show that the hocking stewartson pulse associated with points along the neutral curve is always linearly unstable due to a real positive eigenvalue implications for the spanwise structure of nonlinear poiseuille problem between parallel plates are also discussed

ml parameter estimation for markov random fields with applications to bayesian tomography markov random fields mrfs have been widely used to model images in bayesian frameworks for image reconstruction and restoration typically these mrf models have parameters that allow the prior model to be adjusted for best performance however optimal estimation of these parameters sometimes referred to as hyperparameters is difficult in practice for two reasons i direct parameter estimation for mrfs is known to be mathematically and numerically challenging ii parameters can not be directly estimated because the true image cross section is unavailable we propose a computationally efficient scheme to address both these difficulties for a general class of mrf models and we derive specific methods of parameter estimation for the mrf model known as generalized gaussian mrf ggmrf we derive methods of direct estimation of scale and shape parameters for a general continuously valued mrf for the ggmrf case we show that the ml estimate of the scale parameter spl sigma has a simple closed form solution and we present an efficient scheme for computing the ml estimate of the shape parameter p by an off line numerical computation of the dependence of the partition function on p we present a fast algorithm for computing ml parameter estimates when the true image is unavailable to do this we use the expectation maximization em algorithm we develop a fast simulation method to replace the e step and a method to improve the parameter estimates when the simulations are terminated prior to convergence experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

image restoration using gibbs priors boundary modeling treatment of blurring and selection of hyperparameter the authors propose a bayesian model for the restoration of images based on counts of emitted photons the model treats blurring within the context of an incomplete data problem and utilizes a gibbs prior to model the spatial correlation of neighboring regions the gibbs prior includes line sites to account for boundaries between regions and the line sites are assigned continuous values to permit efficient estimation using a method called iterative conditional averages in addition the effect of blurring in masking differences between images and the effects of misspecifying the amount of blurring are discussed

deterministic homogenization on problems of minimal hypersurface type abstract we study here the deterministic homogenization for a particular family of convex integral functionals namely problems of minimal hypersurfaces by the sigma convergence method relooked especially in appropriate representation of non periodic functions and in the extension of sigma convergence on the l 1 space we obtain a general homogenization result and thanks to the particularity of minimal hypersurfaces we derive a macroscopic integral functional we then provide a number of physical applications including the periodic case of this result

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

adaptive langevin sampler for separation of t distribution modelled astrophysical maps we propose to model the image differentials of astrophysical source maps by student s t distribution and to use them in the bayesian source separation method as priors we introduce an efficient markov chain monte carlo mcmc sampling scheme to unmix the astrophysical sources and describe the derivation details in this scheme we use the langevin stochastic equation for transitions which enables parallel drawing of random samples from the posterior and reduces the computation time significantly by two orders of magnitude in addition student s t distribution parameters are updated throughout the iterations the results on astrophysical source separation are assessed with two performance criteria defined in the pixel and the frequency domains

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

a maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography the expectation maximization method for maximum likelihood image reconstruction in emission tomography based on the poisson distribution of the statistically independent components of the image and measurement vectors is extended to a maximum aposteriori image reconstruction using a multivariate gaussian a priori probability distribution of the image vector the approach is equivalent to a penalized maximum likelihood estimation with a special choice of the penalty function the expectation maximization method is applied to find the a posteriori probability maximizer a simple iterative formula is derived for a penalty function that is a weighted sum of the squared deviations of image vector components from their a priori mean values the method is demonstrated to be superior to pure likelihood maximization in that the penalty function prevents the occurrence of irregular high amplitude patterns in the image with a large number of iterations the so called checkerboard effect or noise artifact

fully bayesian reconstructions from single photon emission computed tomography data abstract with suitably chosen priors bayesian models are useful in image reconstruction i consider reconstructions from single photon emission computerized tomography data using a gibbs pairwise difference prior a fully bayesian approach is presented where the prior parameters are considered drawn from hyperpriors the approach is problematical because the normalization constant in the prior is an intractable function of its parameters here the constant is estimated off line by reverse logistic regression markov chain monte carlo methods are used on simulated and real data to gain estimates of the posterior mean and pixelwise credibility bounds the resulting reconstructions are compared to those obtained by filtered back projection maximum likelihood em and the one step late solution to the fixed parameter posterior mode

efficient variational bayesian approximation method based on subspace optimization variational bayesian approximations have been widely used in fully bayesian inference for approximating an intractable posterior distribution by a separable one nevertheless the classical variational bayesian approximation vba method suffers from slow convergence to the approximate solution when tackling large dimensional problems to address this problem we propose in this paper a more efficient vba method actually variational bayesian issue can be seen as a functional optimization problem the proposed method is based on the adaptation of subspace optimization methods in hilbert spaces to the involved function space in order to solve this optimization problem in an iterative way the aim is to determine an optimal direction at each iteration in order to get a more efficient method we highlight the efficiency of our new vba method and demonstrate its application to image processing by considering an ill posed linear inverse problem using a total variation prior comparisons with state of the art variational bayesian methods through a numerical example show a notable improvement in computation time

bayesian methods applied to survey data from archeological magnetometry a model is presented for the interpretation of magnetometer data in terms of archeological features beneath the ground it describes the detector s response to the assemblage of buried features incorporating both a spread function and a statistical error process as well as appropriate prior beliefs about the nature of archeological features the problem is to estimate the magnetic susceptibility of the buried features at each horizontal location a monte carlo markov chain approach is used to estimate magnetic susceptibilities and all prior parameters this requires estimation of the normalization constant of the gibbs prior distribution the approach is illustrated with both simulated data and measurements from an archeological site in the latter case the reconstruction of the buried features corresponds well with the archeologist s observations during subsequent excavation

markov random field models for high dimensional parameters in simulations of fluid flow in porous media we give an approach for using flow information from a system of wells to characterize hydrologic properties of an aquifer in particular we consider experiments where an impulse of tracer fluid is injected along with the water at the input wells and its concentration is recorded over time at the uptake wells we focus on characterizing the spatially varying permeability field which is a key attribute of the aquifer for determining flow paths and rates for a given flow experiment as is standard for estimation from such flow data we use complicated subsurface flow code that simulates the fluid flow through the aquifer for a particular well configuration and aquifer specification in particular the permeability field over a grid the solution to this ill posed problem requires that some regularity conditions be imposed on the permeability field typically this regularity is accomplished by specifying a stationary gaussian process model for the permeability field here we use an intrinsically stationary

bifurcation of homoclinic orbits to a saddle focus in reversible systems with so 2 symmetry we study reversible so 2 invariant vector fields in r4 depending on a real parameter e which possess for e 0 a primary family of homoclinic orbits t h0 s1 under a transversality condition with respect to e the existence of homoclinic n pulse solutions is demonstrated for a sequence of parameter values e n k 0 for k the existence of cascades of 2l3m pulse solutions follows by showing their transversality and then using induction the method relies on the construction of an so 2 equivariant poincare map which after factorization is a composition of two involutions a logarithmic twist map and a smooth global map reversible periodic orbits of this map corresponds to reversible periodic or homoclinic solutions of the original problem as an application we treat the steady complex ginzburg landau equation for which a primary homoclinic solution is known explicitly

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

fully bayesian estimation of gibbs hyperparameters for emission computed tomography data in recent years many investigators have proposed gibbs prior models to regularize images reconstructed from emission computed tomography data unfortunately hyperparameters used to specify gibbs priors can greatly influence the degree of regularity imposed by such priors and as a result numerous procedures have been proposed to estimate hyperparameter values from observed image data many of these procedures attempt to maximize the joint posterior distribution on the image scene to implement these methods approximations to the joint posterior densities are required because the dependence of the gibbs partition function on the hyperparameter values is unknown here the authors use recent results in markov chain monte carlo mcmc sampling to estimate the relative values of gibbs partition functions and using these values sample from joint posterior distributions on image scenes this allows for a fully bayesian procedure which does not fix the hyperparameters at some estimated or specified value but enables uncertainty about these values to be propagated through to the estimated intensities the authors utilize realizations from the posterior distribution for determining credible regions for the intensity of the emission source the authors consider two different markov random field mrf models the power model and a line site model as applications they estimate the posterior distribution of source intensities from computer simulated data as well as data collected from a physical single photon emission computed tomography spect phantom

a bayesian approach to characterizing uncertainty in inverse problems using coarse and fine scale information the bayesian approach allows one to easily quantify uncertainty at least in theory in practice however the markov chain monte carlo mcmc method can be computationally expensive particularly in complicated inverse problems we present a methodology for improving the speed and efficiency of an mcmc analysis by combining runs on different scales by using a coarser scale the chain can run faster particularly when there is an external forward simulator involved in the likelihood evaluation and better explore the posterior being less likely to become stuck in local maxima we discuss methods for linking the coarse chain back to the original fine scale chain of interest the resulting coupled chain can thus be run more efficiently without sacrificing the accuracy achieved at the finer scale