Showing papers on "Maximum a posteriori estimation published in 1999"

Nonlinear spatial normalization using basis functions

Abstract: We describe a comprehensive framework for performing rapid and automatic nonlabel-based nonlinear spatial normalizations. The approach adopted minimizes the residual squared difference between an image and a template of the same modality. In order to reduce the number of parameters to be fitted, the nonlinear warps are described by a linear combination of low spatial frequency basis functions. The objective is to determine the optimum coefficients fur each of the bases by minimizing the sum of squared differences between the image and template, while simultaneously maximizing the smoothness of the transformation using a maximum a posteriori (MAP) approach. Most MAT approaches assume that the variance associated with each voxel is already known and that there is no covariance between neighboring voxels. The approach described here attempts to estimate this variance from the data, and also corrects fur the correlations between neighboring voxels. This makes the same approach suitable for the spatial normalization of both high-quality magnetic resonance images, and low-resolution noisy positron emission tomography images. A fast algorithm has been developed that utilizes Taylor's theorem and the separable nature of the basis functions, meaning that most of the nonlinear spatial variability between images can be automatically corrected within a few minutes. Hum. Brain Mapping 7:254-266, 1999. (C) 1999 Wiley-Liss, Inc.

Convergence of a stochastic approximation version of the EM algorithm

Abstract: The expectation-maximization (EM) algorithm is a powerful computational technique for locating maxima of functions. It is widely used in statistics for maximum likelihood or maximum a posteriori estimation in incomplete data models. In certain situations, however, this method is not applicable because the expectation step cannot be performed in closed form. To deal with these problems, a novel method is introduced, the stochastic approximation EM (SAEM), which replaces the expectation step of the EM algorithm by one iteration of a stochastic approximation procedure. The convergence of the SAEM algorithm is established under conditions that are applicable to many practical situations. Moreover, it is proved that, under mild additional conditions, the attractive stationary points of the SAEM algorithm correspond to the local maxima of the function. presented to support our findings.

Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors

Abstract: Research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near-ideal estimation performance in various asymptotic frameworks. However, image processing practice has shown that universal thresholding methods are outperformed by simple Bayesian estimators assuming independent wavelet coefficients and heavy-tailed priors such as generalized Gaussian distributions (GGDs). In this paper, we investigate various connections between shrinkage methods and maximum a posteriori (MAP) estimation using such priors. In particular, we state a simple condition under which MAP estimates are sparse. We also introduce a new family of complexity priors based upon Rissanen's universal prior on integers. One particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length (MDL) principle. We develop analytical expressions for the shrinkage rules implied by GGD and complexity priors. This allows us to show the equivalence between universal hard thresholding, MAP estimation using a very heavy-tailed GGD, and MDL estimation using one of the new complexity priors. Theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis-specifications of the prior-a basic concern in image processing applications.

Simultaneous maximum a posteriori reconstruction of attenuation and activity distributions from emission sinograms

Abstract: In order to perform attenuation correction in emission tomography an attenuation map is required. The authors propose a new method to compute this map directly from the emission sinogram, eliminating the transmission scan from the acquisition protocol. The problem is formulated as an optimization task where the objective function is a combination of the likelihood and an a priori probability. The latter uses a Gibbs prior distribution to encourage local smoothness and a multimodal distribution for the attenuation coefficients. Since the attenuation process is different in positron emission tomography (PET) and single photon emission tomography (SPECT), a separate algorithm for each case is derived. The method has been tested on mathematical phantoms and on a few clinical studies. For PET, good agreement was found between the images obtained with transmission measurements and those produced by the new algorithm in an abdominal study. For SPECT, promising simulation results have been obtained for nonhomogeneous attenuation due to the presence of the lungs.

Estimating Multiple Classification Latent Class Models.

Abstract: This paper presents a new class of models for persons-by-items data. The essential new feature of this class is the representation of the persons: every person is represented by its membership tomultiple latent classes, each of which belongs to onelatent classification. The models can be considered as a formalization of the hypothesis that the responses come about in a process that involves the application of a number ofmental operations. Two algorithms for maximum likelihood (ML) and maximum a posteriori (MAP) estimation are described. They both make use of the tractability of the complete data likelihood to maximize the observed data likelihood. Properties of the MAP estimators (i.e., uniqueness and goodness-of-recovery) and the existence of asymptotic standard errors were examined in a simulation study. Then, one of these models is applied to the responses to a set of fraction addition problems. Finally, the models are compared to some related models in the literature.

RNA secondary structure prediction using stochastic context-free grammars and evolutionary history.

Abstract: Motivation: Many computerized methods for RNA secondary structure prediction have been developed. Few of these methods, however, employ an evolutionary model, thus relevant information is often left out from the structure determination. This paper introduces a method which incorporates evolutionary history into RNA secondary structure prediction. The method reported here is based on stochastic context-free grammars (SCFGs) to give a prior probability distribution of structures. Results: The phylogenetic tree relating the sequences can be found by maximum likelihood (ML) estimation from the model introduced here. The tree is shown to reveal information about the structure, due to mutation patterns. The inclusion of a prior distribution of RNA structures ensures good structure predictions even for a small number of related sequences. Prediction is carried out using maximum a posteriori estimation (MAP) estimation in a Bayesian approach. For small sequence sets, the method performs very well compared to current automated methods. Contact: bk@imf.au.dk.

Bayesian multi-camera surveillance

Abstract: The task of multicamera surveillance is to reconstruct the paths taken by all moving objects that are temporally visible from multiple non-overlapping cameras. We present a Bayesian formalization of this task, where the optimal solution is the set of object paths with the highest posterior probability given the observed data. We show how to efficiently approximate the maximum a posteriori solution by linear programming and present initial experimental results.

Introduction to Optimal Estimation

Abstract: 1 Introduction.- 1.1 Signal Estimation.- 1.2 State Estimation.- 1.3 Least Squares Estimation.- Problems.- 2 Random Signals and Systems with Random Inputs.- 2.1 Random Variables.- 2.2 Random Discrete-Time Signals.- 2.3 Discrete-Time Systems with Random Inputs.- Problems.- 3 Optimal Estimation.- 3.1 Formulating the Problem.- 3.2 Maximum Likelihood and Maximum a posteriori Estimation.- 3.3 Minimum Mean-Square Error Estimation.- 3.4 Linear MMSE Estimation.- 3.5 Comparison of Estimation Methods.- Problems.- 4 The Wiener Filter.- 4.1 Linear Time-Invariant MMSE Filters.- 4.2 The FIR Wiener Filter.- 4.3 The Noncausal Wiener Filter.- 4.4 Toward the Causal Wiener Filter.- 4.5 Derivation of the Causal Wiener Filter.- 4.6 Summary of Wiener Filters.- Problems.- 5 Recursive Estimation and the Kaiman Filter.- 5.1 Estimation with Growing Memory.- 5.2 Estimation of a Constant Signal.- 5.3 The Recursive Estimation Problem.- 5.4 The Signal/Measurement Model.- 5.5 Derivation of the Kaiman Filter.- 5.6 Summary of Kaiman Filter Equations.- 5.7 Kaiman Filter Properties.- 5.8 The Steady-state Kaiman Filter.- 5.9 The SSKF as an Unbiased Estimator.- 5.10 Summary.- Problems.- 6 Further Development of the Kaiman Filter.- 6.1 The Innovations.- 6.2 Derivation of the Kaiman Filter from the Innovations.- 6.3 Time-varying State Model and Nonstationary Noises.- 6.4 Modeling Errors.- 6.5 Multistep Kaiman Prediction.- 6.6 Kaiman Smoothing.- Problems.- 7 Kaiman Filter Applications.- 7.1 Target Tracking.- 7.2 Colored Process Noise.- 7.3 Correlated Noises.- 7.4 Colored Measurement Noise.- 7.5 Target Tracking with Polar Measurements.- 7.6 System Identification.- Problems.- 8 Nonlinear Estimation.- 8.1 The Extended Kalman Filter.- 8.2 An Alternate Measurement Update.- 8.3 Nonlinear System Identification Using Neural Networks.- 8.4 Frequency Demodulation.- 8.5 Target Tracking Using the EKF.- 8.6 Multiple Target Tracking.- Problems.- A The State Representation.- A.1 Discrete-Time Case.- A.2 Construction of State Models.- A.3 Dynamical Properties.- A.4 Discretization of Noise Covariance Matrices.- B The z-transform.- B.1 Region of Convergence.- B.2 z-transform Pairs and Properties.- B.3 The Inverse z-transform.- C Stability of the Kaiman Filter.- C.1 Observability.- C.2 Controllability.- C.3 Types of Stability.- C.4 Positive-Definiteness of P(n).- C.5 An Upper Bound for P(n).- C.6 A Lower Bound for P(n).- C.7 A Useful Control Lemma.- C.8 A Kaiman Filter Stability Theorem.- C.9 Bounds for P(n).- D The Steady-State Kaiman Filter.- D.2 A Stabilizability Lemma.- D.3 Preservation of Ordering.- D.5 Existence and Stability.- E Modeling Errors.- E.1 Inaccurate Initial Conditions.- E.2 Nonlinearities and Neglected States.- References.

Bayesian estimation and testing of structural equation models

Abstract: The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters of underidentified models, as we illustrate on a simple errors-in-variables model.

Shadow puppetry

Abstract: The mapping between 3D body poses and 2D shadows is fundamentally many-to-many and defeats regression methods, even with windowed context. We show how to learn a function between paths in the two systems, resolving ambiguities by integrating information over the entire length of a sequence. The basis of this function is a configural and dynamical manifold that summarizes the target system's behaviour. This manifold can be modeled from data with a hidden Markov model having special topological properties that we obtain via entropy minimization. Inference is then a matter of solving for the geodesic on the manifold that best explains the evidence in the cue sequence. We give a closed-form maximum a posteriori solution for geodesics through the learned density space, thereby obtaining optimal paths over the dynamical manifold. These methods give a completely general way to perform inference over time-series; in vision they support analysis, recognition, classification and synthesis of behaviours in linear time. We demonstrate with a prototype that infers 3D from monocular monochromatic sequences (e.g., back-subtractions), without using any articulatory body model. The framework readily accommodates multiple cameras and other sources of evidence such as optical flow or feature tracking.

Bayesian and regularization methods for hyperparameter estimation in image restoration

Abstract: In this paper, we propose the application of the hierarchical Bayesian paradigm to the image restoration problem. We derive expressions for the iterative evaluation of the two hyperparameters applying the evidence and maximum a posteriori (MAP) analysis within the hierarchical Bayesian paradigm. We show analytically that the analysis provided by the evidence approach is more realistic and appropriate than the MAP approach for the image restoration problem. We furthermore study the relationship between the evidence and an iterative approach resulting from the set theoretic regularization approach for estimating the two hyperparameters, or their ratio, defined as the regularization parameter. Finally the proposed algorithms are tested experimentally.

On the Bernstein-von Mises Theorem with Infinite Dimensional Parameters

Abstract: If there are many independent, identically distributed observations governed by a smooth, finite-dimensional statistical model, the Bayes estimate and the maximum likelihood estimate will be close. Furthermore, the posterior distribution of the parameter vector around the posterior mean will be close to the distribution of the maximum likelihood estimate around truth. Thus, Bayesian confidence sets have good frequentist coverage properties, and conversely. However, even for the simplest infinite-dimensional models, such results do not hold. The object here is to give some examples.

A theoretical study of the contrast recovery and variance of MAP reconstructions from PET data

Abstract: 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.

Structure learning in conditional probability models via an entropic prior and parameter extinction

Abstract: We introduce an entropic prior for multinomial parameter estimation problems and solve for its maximum a posteriori (MAP) estimator. The prior is a bias for maximally structured and minimally ambig...

Comparative Assessment of Independent Component Analysis (ICA) for Face Recognition

Abstract: This paper addresses the relative usefulness of Independent Component Analysis (ICA) for Face Recognition. Comparative assessments are made regarding (i) ICA sensitivity to the dimension of the space where it is carried out, and (ii) ICA discriminant performance alone or when combined with other discriminant criteria such as Bayesian framework or Fisher’s Linear Discriminant (FLD). Sensitivity analysis suggests that for enhanced performance ICA should be carried out in a compressed and whitened Principal Component Analysis (PCA) space where the small trailing eigenvalues are discarded. The reason for this finding is that during whitening the eigenvalues of the covariance matrix appear in the denominator and that the small trailing eigenvalues mostly encode noise. As a consequence the whitening component, if used in an uncompressed image space, would fit for misleading variations and thus generalize poorly to new data. Discriminant analysis shows that the ICA criterion, when carried out in the properly compressed and whitened space, performs better than the eigenfaces and Fisherfaces methods for face recognition, but its performance deteriorates when augmented by additional criteria such as the Maximum A Posteriori (MAP) rule of the Bayes classifier or the FLD. The reason for the last finding is that the Mahalanobis distance embedded in the MAP classifier duplicates to some extent the whitening component, while using FLD is counter to the independence criterion intrinsic to ICA.

Maximum a posteriori linear regression for hidden Markov model adaptation.

Abstract: In the past few years, transformation-based model adaptation techniques have been widely used to help reducing acoustic mismatch between training and testing conditions of automatic speech recognizers. The estimation of the transformation parameters is usually carried out using estimation paradigms based on classical statistics such as maximum likelihood, mainly because of their conceptual and computational simplicity. However, it appears necessary to introduce some constraints on the possible values of the transformation parameters to avoid getting unreasonable estimates that might perturb the underlying structure of the acoustic space. In this paper, we propose to introduce such constraints using Bayesian statistics, where a prior distribution of the transformation parameters is used. A Bayesian counterpart of the well known maximum likelihood linear regression (MLLR) adaption is formulated based on maximum a posteriori (MAP) estimation. Supervised, unsupervised and incremental non-native speaker adaptation experiments are carried out to compare the proposed MAPLR approach to MLLR. Experimental results show that MAPLR outperforms MLLR.

Linear and Nonlinear Image Deblurring: A Documented Study

Abstract: Nonlinear image deblurring procedures based on probabilistic considerations are widely believed to outperform conventional linear methods. This paper is exclusively concerned with nonsmooth images such as those that occur in biomedical imaging, where reconstruction of high frequency detail is of prime interest, and where avoidance of a priori smoothness constraints is a major concern. The theoretical basis behind each of the following nonlinear procedures is examined: the Lucy--Richardson method, the maximum likelihood E-M algorithm, the Poisson maximum a posteriori method, and the Nunez--Llacer version of the maximum entropy method. A linear iterative method, VanCittert's iteration, is also studied. It is shown that each of the first three methods, as well as VanCittert's method, lack a necessary ingredient for successful solution of the ill-posed deblurring problem, while in the maximum entropy method, the enforced smoothness may have adverse consequences in medical imaging. A direct linear method, the slow evolution from the continuation boundary (SECB) method, designed specifically for nonsmooth images, is also considered. That method is stabilized by constraining the blurring operator as well as the solution and does not require smoothness constraints. It is shown that useful error estimates can be obtained in the SECB method while this is impossible in Tikhonov's method without a priori bounds on derivatives of the unknown solution. Reconstruction experiments on low noise synthetic MRI data show that thousands of iterations are necessary to achieve sufficient resolution in the iterative procedures. However, the SECB method provides higher resolution at considerable savings in computer time. At high noise levels, the iterative algorithms are shown to diverge. At these same noise levels, the SECB method produces reconstructions comparable in quality to those that would be obtained in the iterative methods, were one able to terminate the divergent algorithm at that iteration which best approximates the true solution in the L1 norm.

Optical diffusion tomography by iterative- coordinate-descent optimization in a Bayesian framework

Abstract: Frequency-domain diffusion imaging uses the magnitude and phase of modulated light propagating through a highly scattering medium to reconstruct an image of the spatially dependent scattering or absorption coefficients in the medium. An inversion algorithm is formulated in a Bayesian framework and an efficient optimization technique is presented for calculating the maximum a posteriori image. In this framework the data are modeled as a complex Gaussian random vector with shot-noise statistics, and the unknown image is modeled as a generalized Gaussian Markov random field. The shot-noise statistics provide correct weighting for the measurement, and the generalized Gaussian Markov random field prior enhances the reconstruction quality and retains edges in the reconstruction. A localized relaxation algorithm, the iterative-coordinate-descent algorithm, is employed as a computationally efficient optimization technique. Numerical results for two-dimensional images show that the Bayesian framework with the new optimization scheme outperforms conventional approaches in both speed and reconstruction quality.

Markovian reconstruction using a GNC approach

Abstract: This paper is concerned with the reconstruction of images (or signals) from incomplete, noisy data, obtained at the output of an observation system. The solution is defined in maximum a posteriori (MAP) sense and it appears as the global minimum of an energy function joining a convex data-fidelity term and a Markovian prior energy. The sought images are composed of nearly homogeneous zones separated by edges and the prior term accounts for this knowledge. This term combines general nonconvex potential functions (PFs) which are applied to the differences between neighboring pixels. The resultant MAP energy generally exhibits numerous local minima. Calculating its local minimum, placed in the vicinity of the maximum likelihood estimate, is inexpensive but the resultant estimate is usually disappointing. Optimization using simulated annealing is practical only in restricted situations. Several deterministic suboptimal techniques approach the global minimum of special MAP energies, employed in the field of image denoising, at a reasonable numerical cost. The latter techniques are not directly applicable to general observation systems, nor to general Markovian prior energies. This work is devoted to the generalization of one of them, the graduated nonconvexity (GNC) algorithm, in order to calculate nearly-optimal MAP solutions in a wide range of situations. In fact, GNC provides a solution by tracking a set of minima along a sequence of approximate energies, starting from a convex energy and progressing toward the original energy. In this paper, we develop a common method to derive efficient GNC-algorithms for the minimization of MAP energies which arise in the context of any observation system giving rise to a convex data-fidelity term and of Markov random field (MRF) energies involving any nonconvex and/or nonsmooth PFs. As a side-result, we propose how to construct pertinent initializations which allow us to obtain meaningful solutions using local minimization of these MAP energies. Two numerical experiments-an image deblurring and an emission tomography reconstruction-illustrate the performance of the proposed technique.

Multitarget detection/tracking for monostatic ground penetrating radar: application to pavement profiling

Abstract: Monostatic ground penetrating radar (GPR) has proven to be a useful technique in pavement profiling. In road and highway pavements, layer thickness and permittivity of asphalt and concrete can be estimated by using an inverse scattering approach. Layer-stripping inversion refers to the iterative estimation of layer properties from amplitude and time of delay (TOD) of echoes after their detection. This method is attractive for real-time implementation, in that accuracy is improved by reducing false alarms. To make layer stripping useful, a multitarget detection/tracking (D/T) algorithm is proposed. It exploits the lateral continuity of echoes arising from a multilayered medium. Interface D/T means that both detection and tracking are employed simultaneously (not sequentially). For each scan, both detection of the target and tracking of the corresponding TOD of the backscattered echoes are based on the evaluated a posteriori probability density. The TOD is then estimated by using the maximum a posteriori (MAP) or the minimum mean square error (MMSE) criterion. The statistical properties of a scan are related to those of the neighboring ones by assuming, for the interface, a first-order Markov model.

Bayesian estimation of precipitating cloud parameters from combined measurements of spaceborne microwave radiometer and radar

Abstract: The objective of this paper is to evaluate the potential of a Bayesian inversion algorithm using microwave multisensor data for the retrieval of surface rainfall rate and cloud parameters. The retrieval scheme is based on the maximum a posteriori probability (MAP) method, extended for the use of both spaceborne passive and active microwave data. The MAP technique for precipitation profiling is also proposed to approach the problem of the radar-swath synthetic broadening; that is, the capability to exploit the combined information also where only radiometric data are available. In order to show an application to airborne data, two case studies are selected within the Tropical Ocean-Global Atmosphere Coupled Ocean-Atmosphere Response Experiment (TOGA-COARE). They refer to a stratiform storm region and an intense squall line of two mesoscale convective systems, which occurred over the ocean on February 20 and 22, 1993, respectively. The estimated rainfall rates and columnar hydrometeor contents derived from the proposed algorithms are compared to each other and to radar estimates based on reflectivity-rainrate (Z-R) relationships. Results in terms of reflectivity profiles and upwelling brightness temperatures, reconstructed from the estimated cloud structures, are also discussed. A database of combined measurements acquired at nadir during various TOGA-COARE flights, is used for applying the radar-swath synthetic broadening technique in the case of an along-track radar-failure countermeasure. A simulated test of the latter technique is performed using the case studies of February 20 and 22, 1993.

Principal manifolds and Bayesian subspaces for visual recognition

Abstract: We investigate the use of linear and nonlinear principal manifolds for learning low dimensional representations for visual recognition. Three techniques: principal component analysis (PCA), independent component analysis (ICA) and nonlinear PCA (NLPCA) are examined and tested in a visual recognition experiment using a large gallery of facial images from the "FERET" database. We compare the recognition performance of a nearest neighbour matching rule with each principal manifold representation to that of a maximum a posteriori (MAP) matching rule using a Bayesian similarity measure derived from probabilistic subspaces, and demonstrate the superiority of the latter.

Histogram clustering for unsupervised image segmentation

Abstract: This paper introduces a novel statistical mixture model for probabilistic grouping of distributional (histogram) data. Adopting the Bayesian framework, we propose to perform annealed maximum a posteriori estimation to compute optimal clustering solutions. In order to accelerate the optimization process, an efficient multiscale formulation is developed. We present a prototypical application of this method for the unsupervised segmentation of textured images based on local distributions of Gabor coefficients. Benchmark results indicate superior performance compared to K-means clustering and proximity-based algorithms.

Probabilistic Methods for Support Vector Machines

Abstract: I describe a framework for interpreting Support Vector Machines (SVMs) as maximum a posteriori (MAP) solutions to inference problems with Gaussian Process priors. This can provide intuitive guidelines for choosing a 'good' SVM kernel. It can also assign (by evidence maximization) optimal values to parameters such as the noise level C which cannot be determined unambiguously from properties of the MAP solution alone (such as cross-validation error). I illustrate this using a simple approximate expression for the SVM evidence. Once C has been determined, error bars on SVM predictions can also be obtained.

Pattern discovery via entropy minimization

Abstract: We propose a framework for learning hidden-variable models by optimizing entropies, in which entropy minimization, posterior maximization, and free energy minimization are all equivalent. Solutions for the maximum a posteriori (MAP) estimator yield powerful learning algorithms that combine all the charms of expectation-maximization and deterministic annealing. Contained as special cases are the methods of maximum entropy, maximum likelihood, and a new method, maximum structure. We focus on the maximum structure case, in which entropy minimization maximizes the amount of evidence supporting each parameter while minimizing uncertainty in the sufficient statistics and cross-entropy between the model and the data. In iterative estimation, the MAP estimator gradually extinguishes excess parameters, sculpting a model structure that reflects hidden structures in the data. These models are highly resistant to over-fitting and have the particular virtue of being easy to interpret, often yielding insights into the hidden causes that generate the data.

Finding people by sampling

Abstract: We show how to use a sampling method to find sparsely clad people in static images. People are modeled as an assembly of nine cylindrical segments. Segments are found using an EM algorithm and then assembled into hypotheses incrementally, using a learned likelihood model. Each assembly step passes on a set of samples of its likelihood to the next; this yields effective pruning of the space of hypotheses. The collection of available nine-segment hypotheses is then represented by a set of equivalence classes, which yield an efficient pruning process. The posterior for the number of people is obtained from the class representatives. People are counted quite accurately in images of real scenes using a MAP estimate. We show the method allows top-down as well as bottom up reasoning. While the method can be overwhelmed by very large numbers of segments, we show that this problem can be avoided by quite simple pruning steps.

Improvements of the Maximum Pseudo-Likelihood Estimators in Various Spatial Statistical Models

Abstract: Maximum pseudo-likelihood estimation has hitherto been viewed as a practical but flawed alternative to maximum likelihood estimation, necessary because the maximum likelihood estimator is too hard to compute, but flawed because of its inefficiency when the spatial interactions are strong. We demonstrate that a single Newton-Raphson step starting from the maximum pseudo-likelihood estimator produces an estimator which is close to the maximum likelihood estimator in terms of its actual value, attained likelihood, and efficiency, even in the presence of strong interactions. This hybrid technique greatly increases the practical applicability of pseudo-likelihood-based estimation. Additionally, in the case of the spatial point processes, we propose a proper maximum pseudo-likelihood estimator which is different from the conventional one. The proper maximum pseudo-likelihood estimator clearly shows better performance than the conventional one does when the spatial interactions are strong.

Covariance approximation for fast and accurate computation of channelized Hotelling observer statistics

Abstract: We describe a method for computing linear observer statistics for maximum a posteriori (MAP) reconstructions of PET images. The method is based on a theoretical approximation for the mean and covariance of MAP reconstructions. In particular, we derive here a closed form for the channelized Hotelling observer (CHO) statistic applied to 2D MAP images. We show reasonably good correspondence between these theoretical results and Monte Carlo studies. The accuracy and low computational cost of the approximation allow us to analyze the observer performance over a wide range of operating conditions and parameter settings for the MAP reconstruction algorithm.