Showing papers on "Maximum a posteriori estimation published in 1987"

Modeling and Segmentation of Noisy and Textured Images Using Gibbs Random Fields

Abstract: This paper presents a new approach to the use of Gibbs distributions (GD) for modeling and segmentation of noisy and textured images. Specifically, the paper presents random field models for noisy and textured image data based upon a hierarchy of GD. It then presents dynamic programming based segmentation algorithms for noisy and textured images, considering a statistical maximum a posteriori (MAP) criterion. Due to computational concerns, however, sub-optimal versions of the algorithms are devised through simplifying approximations in the model. Since model parameters are needed for the segmentation algorithms, a new parameter estimation technique is developed for estimating the parameters in a GD. Finally, a number of examples are presented which show the usefulness of the Gibbsian model and the effectiveness of the segmentation algorithms and the parameter estimation procedures.

A Comparison of Maximum Likelihood and Bayesian Estimators for the Three-Parameter Weibull Distribution

Abstract: Maximum likelihood and Bayesian estimators are developed and compared for the three‐parameter Weibull distribution. For the data analysed in the paper, the two sets of estimators are found to be very different. The reasons for this are explored, and ways of reducing the discrepancy, including reparametrization, are investigated. Our overall conclusion is that there are practical advantages to the Bayesian approach.

A Maximum a Posteriori Probability Expectation Maximization Algorithm for Image Reconstruction in Emission Tomography

Abstract: 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").

Scene Segmentation from Visual Motion Using Global Optimization

Abstract: This paper presents results from computer experiments with an algorithm to perform scene disposition and motion segmentation from visual motion or optic flow. The maximum a posteriori (MAP) criterion is used to formulate what the best segmentation or interpretation of the scene should be, where the scene is assumed to be made up of some fixed number of moving planar surface patches. The Bayesian approach requires, first, specification of prior expectations for the optic flow field, which here is modeled as spatial and temporal Markov random fields; and, secondly, a way of measuring how well the segmentation predicts the measured flow field. The Markov random fields incorporate the physical constraints that objects and their images are probably spatially continuous, and that their images are likely to move quite smoothly across the image plane. To compute the flow predicted by the segmentation, a recent method for reconstructing the motion and orientation of planar surface facets is used. The search for the globally optimal segmentation is performed using simulated annealing.

Simple Parallel Hierarchical and Relaxation Algorithms for Segmenting Noncausal Markovian Random Fields

Abstract: The modeling and segmentation of images by MRF's (Markov random fields) is treated. These are two-dimensional noncausal Markovian stochastic processes. Two conceptually new algorithms are presented for segmenting textured images into regions in each of which the data are modeled as one of C MRF's. The algorithms are designed to operate in real time when implemented on new parallel computer architectures that can be built with present technology. A doubly stochastic representation is used in image modeling. Here, a Gaussian MRF is used to model textures in visible light and infrared images, and an autobinary (or autoternary, etc.) MRF to model a priori information about the local geometry of textured image regions. For image segmentation, the true texture class regions are treated either as a priori completely unknown or as a realization of a binary (or ternary, etc.) MRF. In the former case, image segmentation is realized as true maximum likelihood estimation. In the latter case, it is realized as true maximum a posteriori likelihood segmentation. In addition to providing a mathematically correct means for introducing geometric structure, the autobinary (or ternary, etc.) MRF can be used in a generative mode to generate image geometries and artificial images, and such simulations constitute a very powerful tool for studying the effects of these models and the appropriate choice of model parameters. The first segmentation algorithm is hierarchical and uses a pyramid-like structure in new ways that exploit the mutual dependencies among disjoint pieces of a textured region.

Approximating the Distribution of the Maximum Likelihood Estimate of the Change-Point in a Sequence of Independent Random Variables

Abstract: The problem of estimating the change-point in a sequence of independent random variables is considered. As the sample sizes before and after the change-point tend to infinity, Hinkley (1970) showed that the maximum likelihood estimate of the change-point converges in distribution to that of the change-point based on an infinite sample. Letting the amount of change in distribution approach 0, it is shown that the distribution, suitably normalized, of the maximum likelihood estimate based on an infinite sample converges to a simple one which is related to the location of the maximum for a two-sided Wiener process. Numerical results show that this simple distribution provides a good approximation to the exact distribution (with an infinite sample) in the normal case. However, it is unclear whether the approximation is good for general nonnormal cases.

Stuart Geman and Donald Geman, “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images”;

Abstract: An original image has been blurred by some point spread function, and this blurred image has been corrupted by noise. The goal of processing is to recover the original image. The processing is based on a user-provided model that specifies the likelihood of a pixel having an intensity value similar to those of its neighbors (i.e., the spatial coherence in images), and on the data provided by the observed degraded image. The Geman and Geman paper is more general than this, showing how to reconstruct an original data set from the degraded observation of that data set using a model of the interaction between neighboring elements of the data set. The Geman and Geman paper can be divided into three parts: (1) The first part draws together ideas and theorems from the literature on Markov Random Fields (MRF) that allow one to specify the maximum a posteriori estimate of the state of the MRF given degraded observations of that MRF. The MRF is used as the formalism for describing images, and established theorems provide a means for specifying the probability of a particular original image given the observed degraded image. (2) The second part introduces the technique of simulated annealing as a mechanism for finding the image that maximizes the probability of it being a replica of the original image given the observations, and establishes convergence properties for these annealing procedures. (3) The third part introduces a model of spatial coherence in images. This model explicitly permits the placement of image boundaries that terminate this coherence.

Exact Bayesian analysis of two proportions

Abstract: Altham (1969) derived a relation between the cumulative posterior probability for association and the exact p-value in a 2 x 2 table. But she found that, in general, the exact posterior distribution of the chosen measure of association (odds ratio) was not easy to deal with. This paper covers generalizations of the Bayesian analysis in two directions. First, the posterior probability is formulated for a general comparative parameter, which implies that the analysis is not limited in application to problems involving odds ratio but can be parameterized in terms of a difference and a ratio of two proportions as well. Second, the formal analysis is extended to concern the non-null values of the three usual parameters of association. Under the model of a general beta (or a particular rectangular) prior distribution, the parameter-specific posterior functions are express- ible as finite sums of the beta-binomial type. The posterior distributions are immediately intelli- gible and provide for a uniform basis for the Bayesian analogues of interval estimation, point estimation and significance testing.

A maximum a posteriori estimator for trajectories of diffusion processes

Abstract: Let x t be a diffusion process observed via a noisy sensor, whose output is yt We consider the problem of evaluating the maximum a posteriori trajectory {xs0≤ s ≤ t Based on results of Stratonovich [1] and Ikeda-Watanabe [2], we show that this estimator is given by the solution of an appropriate variational problem which is a slight modification of the "minimum energy" estimator. We compare our results to the non-linear filtering theory and show that for problems which possess a finite dimensional solution, our approach yields also explicit filters. For linear diffusions observed via linear sensors, these filters are identical to the Kalman-filter

A multinomial Bayesian approach to the estimation of population and vocabulary size

Abstract: We approach estimation of the size of a population or a vocabulary through a Bayesian analysis of the multinomial distribution. We view the sample as being generated from such a distribution with an unknown number of cells and unknown cell probabilities, and develop a Bayesian procedure to estimate the number of cells and the coverage of the sample. The prior distribution of the number of cells is arbitrary. Given that number, the cell probabilities are assumed to follow a symmetric Dirichlet prior. A two-stage approach is developed for use when the flattening constant of the latter prior cannot be specified in advance. Our procedures are applied to samples of butterflies, insect species and alleles, to the works of Shakespeare and Joyce, and to Eldridge's sample of English words.

A parallel image segmentation algorithm using relaxation with varying neighborhoods and its mapping

Abstract: This paper presents a segmentation algorithm based on deterministic relaxation with varying neighborhood structures for the segmentation of noisy images. The image is modeled as a discrete-valued Markov random field (MRF), or equivalently a Gibbs random field, corrupted by additive, independent, Gaussian noise; although, additivity and Gaussian assumptions are not necessary for the algorithm. The algorithm seeks to determine the maximum a posteriori (MAP) estimate of the noiseless scene. Using varying neighborhoods during relaxation helps pick up certain directional features in the image which are otherwise smoothed out. The parallelism of the algorithm is underscored by providing its mapping to mesh-connected and systolic array processors suitable for VLSI implementation. Segmentation results are given for 2- and 4-level Gibbs distributed and geometric images corrupted by noise of different levels. A comparative study of this segmentation algorithm with other relaxation algorithms and a single-sweep dynamic programming algorithm, all seeking the MAP estimate, is also presented.

Dynamic speaker adaptation for feature-based isolated word recognition

Abstract: In this paper, we describe efforts to improve the performance of FEATURE, the Carnegie-Mellon University speaker-independent speech recognition system that classifies isolated letters of the English alphabet by enabling the system to learn the acoustical characteristics of individual speakers. Even when features are designed to be speaker-independent, it is frequently observed that feature values may vary more from speaker to speaker for a single letter than they vary from letter to letter. In these cases, it is necessary to adjust the system's statistical description of the features of individual speakers to obtain improved recognition performance. This paper describes a set of dynamic adaptation procedures for updating expected feature values during recognition. The algorithm uses maximum a posteriori probability (MAP) estimation techniques to update the mean vectors of sets of feature values on a speaker-by-speaker basis. The MAP estimation algorithm makes use of both knowledge of the observations input to the system from an individual speaker and the relative variability of the features' means within and across all speakers. In addition, knowledge of the covariance of the features' mean vectors across the various letters enables the system to adapt its representation of similar-sounding letters after any one of them is presented to the classifier. The use of dynamic speaker adaptation improves classification performance of FEATURE by 49 percent after four presentations of the alphabet, when the system is provided with supervised training indicating which specific utterance had been presented to the classifier from a particular user. Performance can be improved by as much as 31 percent when the system is allowed to adapt passively in an unsupervised learning mode. without any information from individual users.

A Bayesian approach to the estimation of variance components for the unbalanced one-way random model

Abstract: The estimation of variance components in the one-way random model with unequal sample sizes is studied. A simulation study that indicates that modes of posterior distributions have good sampling properties compared with other estimators is presented. The posterior distributions are calculated using a noninformative prior distribution that is uniform on the intraclass correlation. A simulation study for the estimation of the ratio of the variance components is also presented, together with a study of the sampling properties of highest posterior density regions for this ratio, Bayesian estimators appear to be viable competitors to the many classical alternatives in a sampling framework.

The two-component extreme value distribution for flood frequency analysis: Derivation of a new estimation method

Abstract: The two component extreme value (TCEV) distribution has recently been shown to account for most of the characteristics of the real flood experience. A new method of parameter estimation for this distribution is derived using the principle of maximum entropy (POME). This method of parameter estimation is suitable for application in both the site-specific and regional cases and appears simpler than the maximum likelihood estimation method. Statistical properties of the regionalized estimation were evaluated using a Monte Carlo approach and compared with those of the maximum likelihood regional estimators.

On Relaxation Algorithms Based on Markov Random Fields

Abstract: : Many computer vision problems can be formulated as computing the minimum energy states of thermal dynamic systems. However, due to the complexity of the energy functions, the solutions to the minimization problem are very difficult to acquire in practice. Stochastic and deterministic methods exist to approximate the solutions, but they fail to be both efficient and robust. This paper describes a new deterministic method--the Highest Confidence First algorithm--to approximate the minimum energy solution to the image labeling problem under the Maximum A Posteriori (MAP) criterion. This method uses Markov Random Fields to model spatial prior knowledge of images and likelihood probabilities to represent external observations regarding hypotheses of image entities. Following an order decided by a dynamic stability measure, the image entities make make local estimates based on the combined knowledge of priors and observations. The solutions so constructed compare favorably to the ones produced by existing methods and that the computation is more predictable and less expensive. Keywords: Image segmentation; Bayesian approach.

Statistics of the three-parameter weibull distribution

Abstract: We consider the estimation of the parameters of the three-parameter Weibull distribution, with particular emphasis on the unknown endpoint of the distribution. We summarize recent results on the asymptotic behaviour of maximum likelihood estimators. We continue with an example in which maximum likelihood and Bayesian estimators arc compared. We conclude that there are practical advantages to the Bayesian approach, but the study also suggests ways in which the maximum likelihood analysis may be improved.

Segmentation of noisy textured images using simulated annealing

Abstract: This paper presents a segmentation algorithm for noisy textured images. To represent noisy textured images, we propose a hierarchical stochastic model that consists of three levels of random fields: the region process, the texture processes and the noise. The hierarchical model also includes local blurring and nonlinear image transformation as results of the image corrupting effects. Having adopted a statistical model, the maximum a posteriori (MAP) estimation is used to find the segmented regions through the restored(noise-free) textured image data. Since the joint a posteriori distribution at hand is a Gibbs distribution, we use simulated annealing as a maximization technique. The simulated annealing based segmentation algorithm presented in this paper can also be viewed as a two-step iterative algorithm in the spirit of the EM algorithm [10].

Maximum Likelihood Estimation of the Distribution of Length at Age

Abstract: SUMMARY This paper exploits the pattern of missing values in a multivariate otolith-fish length data set to obtain the maximum likelihood estimates of the age-length distribution. Methodological tools include maximum likelihood estimation, multiple regression, distance measures, and diagnostic graphical procedures. Data from a sample of cutthroat trout are used to illustrate the method and a simulation study is presented to compare alternative methods of estimation.

Sequential designs for estimating the product of paraeers

Abstract: Suppose that from each of I populations an independent random variable with distribution depending on parameter can be observed The goal is estimation of the product of population parameters, using a Bayesian approach and allowing sequential allocation and sampling The Bayes estimator is used to estimate θ and thus, the problem involves two choices: the allocation procedure (where to take the observations) and the stopping rule (when to stop the experiment) Such a pair is called a policy, and policies are evaluated in terms of their expected total cost, here equal to the total Bayes risk For scaled squared error estimation loss sufficient conditions for asymptotic optimality (ao) of a policy are derived, and a simple aopolicy is construdted For restricted population and prior distributions,the allocation proportions of aopolicies are shown to be proportional to the population coefficient of variation and to not depend on g in the lossFinally,the theory is applied to the reliability problem i

Stochastic relaxation for MAP restoration of gray level images with multiplicative noise

Abstract: This paper is concerned with developing alagorithms for maximum a posteriori (MAP) restoration of gray level images degraded by multiplicative noise. The MAP algorithm requires the probability density function of the original undegraded image which is rarely available and the probability density function of the corrupting noise. By assuming that the original image is represented by a 2-D noncausal Gaussian Markov random field (GMRF) model, the MAP algorithm is written in terms of GMRF model parameters. The computer implementation of the MAP estimator equations is realized by a stochastic relaxation (SR) algorithm. The SR algorithm generates a sequence of images which converges in probability to the global MAP estimate. Several examples of restoration of the gray level image degraded by multiplicative noise are included.

A Bayesian-adaptive decision method for the V/UV/S classification of segments of a speech signal

Abstract: In this correspondence, a method for voiced (V), unvoiced (UV), or silence (S) classification of speech segments, based on the maximum a posteriori probability criterion, is presented. The a posteriori probabilities of the three classes are determined using a vector x = ( f 1 ,... , f L ) of measurements on the segment under consideration. It is assumed that the vector x has an L-dimensional Gaussian distribution with an expected random value also characterized by an L-dimensional Gaussian distribution. In addition, it is assumed that the sequence of the classes constitutes a first-order stationary Markov chain. The initial parameters are estimated in a training phase. During the application phase, the decision method is adapted by using the previous classifications in order to update the probability density function (pdf) of the expected random values.

A review of different estimation procedures in the Rasch model

Abstract: A short review of the different estimation procedures that have been used in association with the Rasch model is provided. These procedures include joint, conditional, and marginal maximum likelihood methods; Bayesian methods; minimum chi-square methods; and paired comparison estimation. A comparison of the marginal maximum likelihood estimation with all other estimation procedures is then provided. Marginal maximum likelihood estimation is defended as the best procedure, but serious numerical problems exist even when applying this method. These problems are especially evident for distribution-free marginal maximum likelihood estimation.

1 - Détection-estimation récursive rapide de séquences Bernoulli-gaussiennes

Abstract: This article deals with the problem of deconvolution of Bernoulli-Gaussian random processes observed through linear systems . This corresponds to situations that occur frequently in areas like geophysics, ultrasonic imaging or nondestructive inspection . Deconvolution of such signais is a detection-estimation problem which does not allow a purely linear data processing, and the nature of the difficulties greatly depends on the type of representation chosen for the linear system . ARMA representations yield a non-standard state driving noise detection-estimation problem whose resolution is complex and requires great computational efforts . AR representations and the use of multi-pulse coding techniques cannot account for nonminimal phase systems and exhibit the disadvantages of output-error type methods . None of these approaches provide any on-line processing ability . In the method proposed here, a degenerate state-space representation is used, and a maximum a posteriori detection step is inserted in an estimation loop by Kalman filtering . This allows deconvolution of Bernoulli-Gaussian processes in a globally recursive manner . Furthermore, fast modified Chandrasekhar equations can be used for the implementation of this procedure and produce significant savings in computational requirements . Simulation results are satisfactory, and are obtained with less computations than other existing methods .

Maximum likelihood using an array processor

Abstract: Maximum likelihood estimation (MLE) is a method used to calculate the parameters of a dynamic system. It can be applied to a large class of problems and has good statistical properties. The main disadvantage of the MLE method is the amount of computation required. This paper describes how the computation time can be reduced significantly by using an array processor. The estimation of the parameters of a dynamic model of the space station is used as an example to evaluate the method.

Parameter estimation using the autocorrelation of the discrete Fourier transform

Abstract: Optimal parameter estimation algorithms are developed using the maximum likelihood technique, when no statistics are available for the parameter. Sub-optimal parameter estimates, using one sample of the autocorrelation of the DFT, have been developed previously. In this paper, maximum likelihood estimates are derived, given the auto-correlation function of the received signal's DFT. These estimates sometimes require less computation time than conventional estimates, and frequently have a closed form or simple iterative implementation.