studying convergence of markov chain monte carlo algorithms using coupled sample paths abstract i describe a simple procedure for investigating the convergence properties of markov chain monte carlo sampling schemes the procedure uses coupled chains from the same sampler obtained by using the same sequence of random deviates for each run by examining the distribution of the iteration at which all sample paths couple convergence properties for the system can be established the procedure also provides a simple diagnostic for detecting modes in multimodal posteriors several examples of the procedure are provided in ising models the relation between the correlation parameter and the convergence rate of rudimentary gibbs samplers is investigated in another example the effects of multiple modes on the convergence of coupled paths are explored using mixtures of bivariate normal distributions the technique is also used to evaluate the convergence properties of a gibbs sampling scheme applied to a model for rat growth rates

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

hidden markov models and disease mapping we present new methodology to extend hidden markov models to the spatial domain and use this class of models to analyze spatial heterogeneity of count data on a rare phenomenon this situation occurs commonly in many domains of application particularly in disease mapping we assume that the counts follow a poisson model at the lowest level of the hierarchy and introduce a finite mixture model for the poisson rates at the next level the novelty lies in the model for allocation to the mixture components which follows a spatially correlated process the potts model and in treating the number of components of the spatial mixture as unknown inference is performed in a bayesian framework using reversible jump markov chain monte carlo the model introduced can be viewed as a bayesian semiparametric approach to specifying flexible spatial distribution in hierarchical models performance of the model and comparison with an alternative well known markov random field specification for the poisson rates are de

the stability of inversion formulas for toeplitz matrices abstract it is shown that under suitable assumptions the well known formulas for the inverse of toeplitz matrices that are due to gohberg and semencul and heinig are weakly stable i e they are numerically forward stable if the matrices that are by assumption nonsingular are actually well conditioned the same is true for another less known pair of inversion formulas that only involve the left biorthogonal szego polynomials

edge preserving tomographic reconstruction with nonlocal regularization tomographic image reconstruction using statistical methods can provide more accurate system modeling statistical models and physical constraints than the conventional filtered backprojection fbp method because of the ill posedness of the reconstruction problem a roughness penalty is often imposed on the solution to control noise to avoid smoothing of edges which are important image attributes various edge preserving regularization methods have been proposed most of these schemes rely on information from local neighborhoods to determine the presence of edges in this paper we propose a cost function that incorporates nonlocal boundary information into the regularization method we use an alternating minimization algorithm with deterministic annealing to minimize the proposed cost function jointly estimating region boundaries and object pixel values we apply variational techniques implemented using level sets methods to update the boundary estimates then using the most recent boundary estimate we minimize a space variant quadratic cost function to update the image estimate for the positron emission tomography transmission reconstruction application we compare the bias variance tradeoff of this method with that of a conventional penalized likelihood algorithm with local huber roughness penalty

recursive based pcg methods for toeplitz systems with nonnegative generating functions in this paper we consider the solutions of symmetric positive definite but ill conditioned toeplitz systems an x b here we propose to solve the system by the recursive based preconditioned conjugate gradient method the idea is to use the inverse of am the principal submatrix of an with the gohberg semencul formula as a preconditioner for an the inverse of am can be generated recursively by using the formula until m is small enough the construction of the preconditioners requires only the entries of an and does not require the explicit knowledge of the generating function f of an we show that if f is a nonnegative bounded and piecewise continuous even function with a finite number of zeros of even order the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers hence the conjugate gradient method when applied to solving the preconditioned system converges very quickly numerical results are included to illustrate the effectiveness of our approach

a new algorithm for solving toeplitz systems of equations abstract we present some recurrences that are the basis for an algorithm to invert an n n toeplitz system of linear equations with computational complexity o n log 2 n the recurrences used are closely related to the zohar trench and bareiss algorithms but do not have any obvious connection with other asymptotically fast algorithms for the inversion of toeplitz systems

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

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

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

bayes smoothing algorithms for segmentation of binary images modeled by markov random fields a new image segmentation algorithm is presented based on recursive bayes smoothing of images modeled by markov random fields and corrupted by independent additive noise the bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel given the total noisy image in a recursive way the a posteriori distribution together with a criterion of optimality then determine a bayes estimate of the scene the algorithm presented is an extension of a 1 d bayes smoothing algorithm to 2 d and it gives the optimum bayes estimate for the scene value at each pixel computational concerns in 2 d however necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm in particular the scene noiseless image is modeled as a markov mesh random field a special class of markov random fields and the bayes smoothing algorithm is applied on overlapping strips horizontal vertical of the image consisting of several rows columns it is assumed that the signal scene values vector sequence along the strip is a vector markov chain since signal correlation in one of the dimensions is not fully used along the edges of the strip estimates are generated only along the middle sections of the strips the overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image the bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel discrete or continuous random variables

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

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

asymptotics of toeplitz determinants and the emptiness formation probability for the xy spin chain we study an asymptotic behaviour of a special correlator known as the emptiness formation probability efp for the one dimensional anisotropic xy spin 1 2 chain in a transverse magnetic field this correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models for the xy spin chain the correlator can be expressed as the determinant of a toeplitz matrix and its asymptotical behaviours for n throughout the phase diagram are obtained using known theorems and conjectures on toeplitz determinants we find that the decay is exponential everywhere in the phase diagram of the xy model except on the critical lines i e where the spectrum is gapless in these cases a power law prefactor with a universal exponent arises in addition to an exponential or gaussian decay the latter gaussian behaviour holds on the critical line corresponding to the isotropic xy model while at the critical value of the magnetic field the efp decays exponentially at small anisotropy one has a crossover from the gaussian to the exponential behaviour we study this crossover using the bosonization approach

approximate inverse free preconditioners for toeplitz matrices abstract in this paper we propose approximate inverse free preconditioners for solving toeplitz systems the preconditioners are constructed based on the famous gohberg semencul formula we show that if a toeplitz matrix is generated by a positive bounded function and its entries enjoys the off diagonal decay property then the eigenvalues of the preconditioned matrix are clustered around one experimental results show that the proposed preconditioners are superior to other existing preconditioners in the literature

segmentation of macular fluorescein angiographies a statistical approach this paper is concerned with the use of bayesian methods in the segmentation of macular uorescein angiographies fluorescein angiography is used in ophthalmic practice to evaluate vascular retinopathies and choroidopathies sodium uorescein is injected in the arm s cubital vein of the patient and its distribution is observed along retinal vessels at certain times in this task a previous and essential step is the segmentation of the image into its relevant components in order to obtain this segmentation bayesian methods can be used because a previous knowledge about the spatial structure of the scene to be segmented is available in this kind of images the stochastic model assumed for the observed intensities is a simple model with a gaussian noise process which is statiscally independent between pixels the process of labels x is modelled as a markov random eld with a space dependent external eld expressing the anatomy of the ocular fundus and higher order interactions encouraging blood vessels to be thin and large this procedure is applied to di erent cases of diabetic retinophaty and vein occlusions two algorithms have been used to estimate x simulated annealing and iterated conditional modes in order to evaluate the accuracy of the estimation several error measures have been calculated 2001 pattern recognition society published by elsevier science ltd all rights reserved

contextual clustering for analysis of functional mri data presents a contextual clustering procedure for statistical parametric maps spm calculated from time varying three dimensional images the algorithm can be used for the detection of neural activations from functional magnetic resonance images fmri an important characteristic of spm is that the intensity distribution of background nonactive area is known whereas the distributions of activation areas are not the developed contextual clustering algorithm divides an spm into background and activation areas so that the probability of detecting false activations by chance is controlled i e hypothesis testing is performed unlike the much used voxel by voxel testing neighborhood information is utilized an important difference this is achieved by using a markov random field prior and iterated conditional modes icm algorithm however unlike in the conventional use of icm algorithm the classification is based only on the distribution of background the results from the authors simulations and human fmri experiments using visual stimulation demonstrate that a better sensitivity is achieved with a given specificity in comparison to the voxel by voxel thresholding technique the algorithm is computationally efficient and can be used to detect and delineate objects from a noisy background in other applications

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

segmenting images corrupted by correlated noise image segmentation is fundamental to many image analysis problems it aims to partition a digital image into a set of nonoverlapping homogeneous regions the main contribution of this paper is the development of a new segmentation procedure which is designed to segment images corrupted by correlated noise this new segmentation procedure is based on rissanen s minimum description length mdl principle and consists of two components 1 an mdl based criterion in which the best segmentation is defined as its minimizer and 2 a merging algorithm which attempts to locate this minimizer the performance of this procedure is illustrated via a simulation study with promising results

minimax emission computed tomography using high resolution anatomical side information and b spline models in this paper a minimax methodology is presented for combining information from two imaging modalities having different intrinsic spatial resolutions the focus application is emission computed tomography ect a low resolution modality for reconstruction of radionuclide tracer density when supplemented by high resolution anatomical boundary information extracted from a magnetic resonance image mri of the same imaging volume the mri boundary within the two dimensional 2 d slice of interest is parameterized by a closed planar curve the cramer rao cr lower bound is used to analyze estimation errors for different boundary shapes under a spatially inhomogeneous gibbs field model for the tracer density a representation for the minimax mri enhanced tracer density estimator is obtained it is shown that the estimator is asymptotically equivalent to a penalized maximum likelihood pml estimator with resolution selective gibbs penalty quantitative comparisons are presented using the iterative space alternating generalized expectation maximization sage fm algorithm to implement the pml estimator with and without minimax weight averaging

on inversion of toeplitz matrices in 1992 labahn and shalom showed that the inverse of a toeplitz matrix can be reconstructed by two of its columns and by some entries of the original toeplitz matrix we here present a modification of this result and another shorter proof

optimal quasi optimal and superlinear band toeplitz preconditioners for asymptotically ill conditioned positive definite toeplitz systems in this paper we are concerned with the solution of n x n hermitian toeplitz systems with nonnegative generating functions f the preconditioned conjugate gradient pcg method with the well known circulant preconditioners fails in the case where f has zeros in this paper we consider as preconditioners band toeplitz matrices generated by trigonometric polynomials g of fixed degree l we use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n for each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy we obtain different estimates of l for which the total cost of the proposed pcg methods is optimal and the related rates of convergence are superlinear finally for the most economical strategy we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems

a minimum description length based image segmentation procedure and its comparison with a cross validation based segmentation procedure abstract image segmentation is a very important problem in image analysis as quite often it is a key component of a good practical solution to a real life imaging problem it aims to partition a digital image into a set of nonoverlapping homogeneous regions one approach to segmenting an image is to fit a piecewise constant function to the image and define the segmentation by the discontinuity points of the fitted function the article s first contribution is to present a new and automatic segmentation procedure which follows this piecewise constant function fitting approach this procedure is based on rissanen s minimum description length mdl principle and consists of two components a an mdl based criterion in which the best segmentation i e the best fitted piecewise constant function is defined as its minimizer and b a fast merging algorithm that attempts to locate this minimizer as a second contribution the new mdl based procedure is compared with a cross validation based segmentation

four short stories about toeplitz matrix calculations the stories told in this paper are dealing with the solution of finite infinite and biinfinite toeplitz type systems a crucial role plays the off diagonal decay behavior of toeplitz matrices and their inverses classical results of gelfand et al on commutative banach algebras yield a general characterization of this decay behavior we then derive estimates for the approximate solution of bi infinite toeplitz systems by the finite section method showing that the approximation rate depends only on the decay of the entries of the toeplitz matrix and its condition number furthermore we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems finally some quantitative results on the construction of preconditioners via circulant embedding are derived which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems

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

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

exact distribution of edge preserving map estimators for linear signal models with gaussian measurement noise we derive the exact statistical distribution of maximum a posteriori map estimators having edge preserving nongaussian priors such estimators have been widely advocated for image restoration and reconstruction problems previous investigations of these image recovery methods have been primarily empirical the distribution we derive enables theoretical analysis the signal model is linear with gaussian measurement noise we assume that the energy function of the prior distribution is chosen to ensure a unimodal posterior distribution for which convexity of the energy function is sufficient and that the energy function satisfies a uniform lipschitz regularity condition the regularity conditions are sufficiently general to encompass popular priors such as the generalized gaussian markov random field prior and the huber prior even though those priors are not everywhere twice continuously differentiable

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea