This paper presents a variational framework for dense diffeomorphic atlas-mapping onto high-throughput histology stacks at the 20 μm meso-scale. The observed sections are modelled as Gaussian random fields conditioned on a sequence of unknown section by section rigid motions and unknown diffeomorphic transformation of a three-dimensional atlas. To regularize over the high-dimensionality of our parameter space (which is a product space of the rigid motion dimensions and the diffeomorphism dimensions), the histology stacks are modelled as arising from a first order Sobolev space smoothness prior. We show that the joint maximum a-posteriori, penalized-likelihood estimator of our high dimensional parameter space emerges as a joint optimization interleaving rigid motion estimation for histology restacking and large deformation diffeomorphic metric mapping to atlas coordinates. We show that joint optimization in this parameter space solves the classical curvature non-identifiability of the histology stacking problem. The algorithms are demonstrated on a collection of whole-brain histological image stacks from the Mouse Brain Architecture Project.

In the field of Computational Anatomy, biological form (including our focus, neuroanatomy) is studied quantitatively through the action of the diffeomorphism group on example anatomies - a technique called diffeomorphometry. Here we design an algorithm within this framework to pass from dense objects common in neuromaging studies (binary segmentations, structural images) to a sparse representation defined on the surface boundaries of anatomical structures, and embedded into the low dimensional coordinates of a parametric model. Our main new contribution is to introduce an expanded group action to simultaneously deform surfaces through direct mapping of points, as well as images through functional composition with the inverse. This allows us to index the diffeomorphisms with respect to two-dimensional surface geometries like subcortical gray matter structures, but explicitly map onto cost functions determined by noisy 3-dimensional measurements. We consider models generated from empirical covariance of training data, as well as bandlimited (Laplace-Beltrami eigenfunction) models when no such data is available. We show applications to noisy or anomalous segmentations, and other typical problems in neuroimaging studies. We reproduce statistical results detecting changes in Alzheimer's disease, despite dimensionality reduction. Lastly we apply our algorithm to the common problem of segmenting subcortical structures from T1 MR images.

Diffeomorphic registration using optimal control on the diffeomorphism group and on shape spaces has become widely used since the development of the large deformation diffeomorphic metric mapping (LDDMM) algorithm. More recently, a series of algorithms involving sub-Riemannian constraints have been introduced in which the velocity fields that control the shapes in the LDDMM framework are constrained in accordance with a specific deformation model. Here, we extend this setting by considering, for the first time, inequality constraints in order to estimate surface deformations that only allow for atrophy, introducing for this purpose an algorithm that uses the augmented Lagrangian method. We prove the existence of solutions of the associated optimal control problem and the consistency of our approximation scheme. These developments are illustrated by numerical experiments on simulated and real data.

A method for automated location of shape differences in diseased anatomical structures via high resolution biomedical atlases annotated with labels from formal ontologies is described. In particular, a high resolution magnetic resonance image of the myocardium of the human left ventricle was segmented and annotated with structural terms from an extracted subset of the Foundational Model of Anatomy ontology. The atlas was registered to the end systole template of a previous study of left ventricular remodeling in cardiomyopathy using a diffeomorphic registration algorithm. The previous study used thresholding and visual inspection to locate a region of statistical significance which distinguished patients with ischemic cardiomyopathy from those with nonischemic cardiomyopathy. Using semantic technologies and the deformed annotated atlas, this location was more precisely found. Although this study used only a cardiac atlas, it provides a proof-of-concept that ontologically labeled biomedical atlases of any anatomical structure can be used to automate location-based inferences.

We present a new diffeomorphic surface mapping algorithm under the framework of large deformation diffeomorphic metric mapping (LDDMM). Unlike existing LDDMM approaches, this new algorithm reduces the complexity of the estimation of diffeomorphic transformations by incorporating a shape prior in which a nonlinear diffeomorphic shape space is represented by a linear space of initial momenta of diffeomorphic geodesic flows from a fixed template. In addition, for the first time, the diffeomorphic mapping is formulated within a decision-theoretic scheme based on Bayesian modeling in which an empirical shape prior is characterized by a low dimensional Gaussian distribution on initial momentum. This is achieved using principal component analysis (PCA) to construct the eigenspace of the initial momentum. A likelihood function is formulated as the conditional probability of observing surfaces given any particular value of the initial momentum, which is modeled as a random field of vector-valued measures characterizing the geometry of surfaces. We define the diffeomorphic mapping as a problem that maximizes a posterior distribution of the initial momentum given observable surfaces over the eigenspace of the initial momentum. We demonstrate the stability of the initial momentum eigenspace when altering training samples using a bootstrapping method. We then validate the mapping accuracy and show robustness to outliers whose shape variation is not incorporated into the shape prior.

Technological limitations pose a major challenge to acquisition of myocardial fiber orientations for patient-specific modeling of cardiac (dys)function and assessment of therapy. The objective of this project was to develop a methodology to estimate cardiac fiber orientations from in vivo images of patient heart geometries. An accurate representation of ventricular geometry and fiber orientations was reconstructed, respectively, from high-resolution ex vivo structural magnetic resonance (MR) and diffusion tensor (DT) MR images of a normal human heart, referred to as the atlas. Ventricular geometry of a patient heart was extracted, via semiautomatic segmentation, from an in vivo computed tomography (CT) image. Using image transformation algorithms, the atlas ventricular geometry was deformed to match that of the patient. Finally, the deformation field was applied to the atlas fiber orientations to obtain an estimate of patient fiber orientations. The accuracy of the fiber estimates was assessed using six normal and three failing canine hearts. The mean absolute difference between inclination angles of acquired and estimated fiber orientations was 15.4°. Computational simulations of ventricular activation maps and pseudo-ECGs in sinus rhythm and ventricular tachycardia indicated that there are no significant differences between estimated and acquired fiber orientations at a clinically observable level.

Atlas-driven morphometric analysis has received great attention for studying anatomical shape variation across clinical populations in neuroimaging research as it provides a local coordinate representation for understanding the family of anatomic observations. We present a procedure for generating atlas of subcortical and ventricular structures, including amygdala, hippocampus, caudate, putamen, globus pallidus, thalamus, and lateral ventricles, using the large deformation diffeomorphic metric atlas generation algorithm. The atlas was built based on manually labeled volumes of 41 subjects randomly selected from the database of Open Access Series of Imaging Studies (OASIS, 10 young adults, 10 middle-age adults, 10 healthy elders, and 11 patients with dementia). We show that the estimated atlas is representative of the population in terms of its metric distance to each individual subject in the population. In the application of detecting shape variations, using the estimated atlas may potentially increase statistical power in identifying group shape difference when comparing with using a single subject atlas. In shape-based classification, the metric distances between subjects and each of within-class estimated atlases construct a shape feature space, which allows for performing a variety of classification algorithms to distinguish anatomies.

In large-deformation diffeomorphic metric mapping (LDDMM), the diffeomorphic matching of images are modeled as evolution in time, or a flow, of an associated smooth velocity vector field v controlling the evolution. The initial momentum parameterizes the whole geodesic and encodes the shape and form of the target image. Thus, methods such as principal component analysis (PCA) of the initial momentum leads to analysis of anatomical shape and form in target images without being restricted to small-deformation assumption in the analysis of linear displacements. We apply this approach to a study of dementia of the Alzheimer type (DAT). The left hippocampus in the DAT group shows significant shape abnormality while the right hippocampus shows similar pattern of abnormality. Further, PCA of the initial momentum leads to correct classification of 12 out of 18 DAT subjects and 22 out of 26 control subjects

Functional and structural maps, such as a curvature, cortical thickness, and functional magnetic resonance imaging (MRI) maps, indexed over the local coordinates of the cortical manifold play an important role in neuropsychiatric studies. Due to the highly convoluted nature of the cerebral cortex and image quality, these functions are generally uninterpretable without proper methods of association and smoothness onto the local coordinate system. In this paper, we generalized the spline smoothing problem (Wahba, 1990) from a sphere to any arbitrary two-dimensional (2-D) manifold with boundaries. We first seek a numerical solution to orthonormal basis functions of the Laplace-Beltrami (LB) operator with Neumann boundary conditions for a 2-D manifold M then solve the spline smoothing problem in a reproducing kernel Hilbert space (r.k.h.s.) of real-valued functions on manifold M with kernel constructed from the basis functions. The explicit discrete LB representation is derived using the finite element method calculated directly on the manifold coordinates so that finding discrete LB orthonormal basis functions is equivalent to solving an algebraic eigenvalue problem. And then smoothed functions in r.k.h.s can be represented as a linear combination of the basis functions. We demonstrate numerical solutions of spherical harmonics on a unit sphere and brain orthonormal basis functions on a planum temporale manifold. Then synthetic data is used to quantify the goodness of the smoothness compared with the ground truth and discuss how many basis functions should be incorporated in the smoothing. We present applications of our approach to smoothing sulcal mean curvature, cortical thickness, and functional statistical maps on submanifolds of the neocortex.

The Biomedical Informatics Research Network (BIRN) is a National Institutes of Health (USA) initiative that fosters distributed collaborations in biomedical science by utilizing information technology innovations. Morphometry BIRN is one of its testbeds and has the goal to develop the ability to conduct clinical imaging studies across multiple sites, to analyze structural imaging data with the most powerful software regardless of development site, and to test new hypotheses on large collections of subjects with well-characterized image and clinical data. Through large-scale analyses of patient population data acquired and pooled across sites, we are investigating neuroanatomic correlates of Alzheimer's Disease Depression and Mild Cognitive Impairment subjects. This paper describes progress in multi-site image calibration and in software integration for multi-site image processing.

This paper proposes a method to match diffusion tensor magnetic resonance images (DT-MRI) through the large deformation diffeomorphic metric mapping of vector fields, focusing on the fiber orientations, considered as unit vector fields on the image volume. We study a suitable action of diffeomorphisms on such vector fields, and provide an extension of the large deformation diffeomorphic metric-mapping framework to this type of dataset, resulting in optimizing for geodesies on the space of diffeomorphisms connecting two images. Two different distance function of vector fields are considered. Existence of the minimizers under smoothness assumptions on the compared vector fields is proved, and coarse to fine hierarchical strategies are detailed, to reduce both ambiguities and computation load. This is illustrated by numerical experiments on DT-MRI heart and brain images

One of the central problems in Automated Target Recognition is to accommodate the infinite variety of clutter in real military environments. The principle focus of our paper is on the construction of metric spaces where the metric measures the distance between objects of interest invariant to the infinite variety of clutter. Such metrics are formulated using second-order random field models. Our results indicate that this approach significantly improves detection/classification rates of targets in clutter.

The human cerebral cortex is a laminar structure about 3 mm thick, and is easily visualized with current magnetic resonance (MR) technology. The thickness of the cortex varies locally by region, and is likely to be influenced by such factors as development, disease and aging. Thus, accurate measurements of local cortical thickness are likely to be of interest to other researchers. We develop a parametric stochastic model relating the laminar structure of local regions of the cerebral cortex to MR image data. Parameters of the model include local thickness, and statistics describing white, gray and cerebrospinal fluid (CSF) image intensity values as a function of the normal distance from the center of a voxel to a local coordinate system anchored at the gray/white matter interface. Our fundamental data object, the intensity-distance histogram (IDH), is a two-dimensional (2-D) generalization of the conventional 1-D image intensity histogram, which indexes voxels not only by their intensity value, but also by their normal distance to the gray/white interface. We model the IDH empirically as a marked Poisson process with marking process a Gaussian random field model of image intensity indexed against normal distance. In this paper, we relate the parameters of the IDH model to the local geometry of the cortex. A maximum-likelihood framework estimates the parameters of the model from the data. Here, we show estimates of these parameters for 10 volumes in the posterior cingulate, and 6 volumes in the anterior and posterior banks of the central sulcus. The accuracy of the estimates is quantified via Cramer-Rao bounds. We believe that this relatively crude model can be extended in a straightforward fashion to other biologically and theoretically interesting problems such as segmentation, surface area estimation, and estimating the thickness distribution in a variety of biologically relevant contexts.

In large-deformation diffeomorphic metric mapping (LDDMM), the diffeomorphic matching of images are modeled as evolution in time, or a flow, of an associated smooth velocity vector field v controlling the evolution. The initial momentum parameterizes the whole geodesic and encodes the shape and form of the target image. Thus, methods such as principal component analysis (PCA) of the initial momentum leads to analysis of anatomical shape and form in target images without being restricted to small-deformation assumption in the analysis of linear displacements. We apply this approach to a study of dementia of the Alzheimer type (DAT). The left hippocampus in the DAT group shows significant shape abnormality while the right hippocampus shows similar pattern of abnormality. Further, PCA of the initial momentum leads to correct classification of 12 out of 18 DAT subjects and 22 out of 26 control subjects.

Over the past several years our group has been studying biological shape in the emerging new discipline of computational anatomy (CA). CA consists of several components: (i) the construction of coordinatized anatomical manifolds, (ii) comparison of anatomical manifolds, and (iii) inference of morphometric change on anatomical manifolds. In this paper we focus on (ii) the comparison of anatomical shapes and structures in imagery via metric mapping. The purpose of this paper is to examine the generation of the geodesics associated with the metric from several points of view, the first the Euler equation describing the geodesic diffeomorphic flow, and the second the variational formulation of the geodesic in terms of the minimizing flow of vector fields which generate them.

This paper derives bounds on the performance of statistical object recognition systems, wherein an image of a target is observed by a remote sensor. Detection and recognition problems are modeled as composite hypothesis testing problems involving nuisance parameters. We develop information-theoretic performance bounds on target recognition based on statistical models for sensors and data, and examine conditions under which these bounds are tight. In particular, we examine the validity of asymptotic approximations to probability of error in such imaging problems. Problems involving Gaussian, Poisson, and multiplicative noise, and random pixel deletions are considered, as well as least-favorable Gaussian clutter. A sixth application involving compressed sensor image data is considered in some detail. This study provides a systematic and computationally attractive framework for analytically characterizing target recognition performance under complicated, non-Gaussian models and optimizing system parameters.

Asymptotic approximations to the partition function of Gaussian random fields are derived. Textures are characterized via Gaussian random fields induced by stochastic difference equations determined by finitely supported, stationary, linear difference operators, adjusted to be nonstationary at the boundaries. It is shown that as the scale of the underlying shape increases, the log-normalizer converges to the integral of the log-spectrum of the operator inducing the random field. Fitting the covariance of the fields amounts to fitting the parameters of the spectrum of the differential operator-induced random field model. Matrix analysis techniques are proposed for handling textures with variable orientation. Examples of texture parameters estimated from training data via asymptotic maximum-likelihood are shown. Isotropic models involving powers of the Laplacian and directional models involving partial derivative mixtures are explored. Parameters are estimated for mitochondria and actin-myocin complexes in electron micrographs and clutter in forward-looking infrared images. Deformable template models are used to infer the shape of mitochondria in electron micrographs, with the asymptotic approximation allowing easy recomputation of the partition function as inference proceeds.

This paper describes the generation of large deformation diffeomorphisms /spl phi/:/spl Omega/=[0,1]/sup 3//spl rlhar2//spl Omega/ for landmark matching generated as solutions to the transport equation d/spl phi/(x,t)/dt=/spl nu/(/spl phi/(x,t),t),t/spl isin/[0,1] and /spl phi/(x,0)=x, with the image map defined as /spl phi/(/spl middot/,1) and therefore controlled via the velocity field /spl nu/(/spl middot/,t),t/spl isin/[0,1]. Imagery are assumed characterized via sets of landmarks {x/sub n/, y/sub n/, n=1, 2, ..., N}. The optimal diffeomorphic match is constructed to minimize a running smoothness cost /spl par/L/spl nu//spl par//sup 2/ associated with a linear differential operator L on the velocity field generating the diffeomorphism while simultaneously minimizing the matching end point condition of the landmarks. Both inexact and exact landmark matching is studied here. Given noisy landmarks x/sub n/ matched to y/sub n/ measured with error covariances /spl Sigma//sub n/, then the matching problem is solved generating the optimal diffeomorphism /spl phi//spl circ/(x,1)=/spl int//sub 0//sup 1//spl nu//spl circ/(/spl phi//spl circ/(x,t),t)dt+x where /spl nu//spl circ/(/spl middot/)argmin/sub /spl nu/(/spl middot/)//spl int//sub 1//sup 1//spl int//sub /spl Omega///spl par/L/spl nu/(x,t)/spl par//sup 2/dxdt +/spl Sigma//sub n=1//sup N/[y/sub n/-/spl phi/(x/sub n/,1)]/sup T//spl Sigma//sub n//sup -1/[y/sub n/-/spl phi/(x/sub n/,1)]. Conditions for the existence of solutions in the space of diffeomorphisms are established, with a gradient algorithm provided for generating the optimal flow solving the minimum problem. Results on matching two-dimensional (2-D) and three-dimensional (3-D) imagery are presented in the macaque monkey.

This paper presents measures characterizing the information content of remote observations of ground scenes imaged via optical and infrared sensors. Object recognition is posed in the context of deformable templates; the special Euclidean group is used to accommodate geometric variation of object pose. Principal component analysis of object signatures is used to represent and efficiently accommodate variation in object signature due to changes in the thermal state of the object surface. Mutual information measures, which are independent of the recognition system, are calculated quantifying both the information gain due to remote observations of the scene and the information loss due to signature variability. Signature model mismatch is quantitatively examined using the Kullback-Leibler divergence. Expressions are derived quadratically approximating the posterior conditional entropy on the orthogonal group for high signal-to-noise ratio. It is demonstrated that quadratic modules accurately characterize the posterior entropy for broad ranges of signal-to-mise ratio. Information gain in multiple-sensor scenarios is quantified, and it is demonstrated that the cost of signature uncertainty for the Comanche series of FLIR images collected by the US Army Night Vision Electronic Sensors Directorate is approximately 0.8 bits with an associated near doubling of mean-squared error uncertainty in pose.

This article investigates the asymptotic performance of Bayesian target recognition algorithms using deformable-template representations. Rigid computer-aided design (CAD) models represent the underlying targets; low-dimensional matrix Lie-groups (rotation and translation) extend them to particular instances. Remote sensors observing the targets are modeled as projective transformations, converting three-dimensional scenes into random images. Bayesian target recognition corresponds to hypothesis selection in the presence of nuisance parameters; its performance is quantified as the Bayes' error. Analytical expressions for this error probability in small noise situations are derived, yielding asymptotic error rates for exponential error probability decay.

We consider the problem of recognizing CAD models at arbitrary orientations observed via the projective transformation on an imaging sensor with noise. Bounds on codebook size are established through the rate-distortion curve for a distortion measure derived from the Hilbert-Schmidt norm for elements of the orthogonal group.

Deformable template representations of observed imagery model the variability of target pose via the actions of the matrix Lie groups on rigid templates. In this paper, we study the construction of minimum mean squared error estimators on the special orthogonal group, SO(n), for pose estimation. Due to the nonflat geometry of SO(n), the standard Bayesian formulation of optimal estimators and their characteristics requires modifications. By utilizing Hilbert-Schmidt metric defined on GL(n), a larger group containing SO(n), a mean squared criterion is defined on SO(n). The Hilbert-Schmidt estimate (HSE) is defined to be a minimum mean squared error estimator, restricted to SO(n). The expected error associated with the HSE is shown to be a lower bound, called the Hilbert-Schmidt bound (HSB), on the error incurred by any other estimator. Analysis and algorithms are presented for evaluating the HSE and the HSB in cases of both ground-based and airborne targets.

Dynamic programming algorithms are presented for automated generation of length minimizing geodesics and curves of extremal curvature on the neocortex of the macaque and the visible human. Probabilistic models of curve variation are constructed in terms of the variability in speed, curvature, and torsion in the Frenet representation.

This paper develops mathematical representations for neuro-anatomically significant substructures of the brain and their variability in a population. The focus of the paper is on the neuro-anatomical variation of the geometry and the "shape" of two-dimensional surfaces in the brain. As examples, we focus on the cortical and hippocampal surfaces in an ensemble of Macaque monkeys and human MRI brains. The "shapes" of the substructures are quantified via the construction of templates; the variations are represented by defining probabilistic deformations of the template. Methods for empirically estimating probability measures on these deformations are developed by representing the deformations as Gaussian random vector fields on the embedded sub-manifolds. The Gaussian random vector fields are constructed as quadratic mean limits using complete orthonormal bases on the sub-manifolds. The complete orthonormal bases are generated using modes of vibrations of the geometries of the brain sub-manifolds. The covariances are empirically estimated from an ensemble of brain data. Principal component analysis is presented for characterizing the "eigen-shape" of the hippocampus in an ensemble of MRI-MPRAGE whole brain images. Clustering based on eigen-shape is presented for two sub-populations of normal and schizophrenic.

Proposes a framework for simultaneous detection, tracking, and recognition of objects via data fused from multiple sensors. Complex dynamic scenes are represented via the concatenation of simple rigid templates. The variability of the infinity of pose is accommodated via the actions of matrix Lie groups extending the templates to individual instances. The variability of target number and target identity is accommodated via the representation of scenes as unions of templates of varying types, with the associated group transformations of varying dimension. We focus on recognition in the air-to-ground and ground-to-air scenarios. The remote sensing data is organized around both the coarse scale associated with detection as provided by tracking and range radars, along with the fine scale associated with pose and identity supported by high-resolution optical, forward looking infrared and delay-Doppler radar imagers. A Bayesian approach is adopted in which prior distributions on target scenarios are constructed via dynamical models of the targets of interest. These are combined with physics-based sensor models which define conditional likelihoods for the coarse/fine scale sensor data given the underlying scene. Inference via the Bayes posterior is organized around a random sampling algorithm based on jump-diffusion processes. New objects are detected and object identities are recognized through discrete jump moves through parameter space, the algorithm exploring scenes of varying complexity as it proceeds. Between jumps, the scale and rotation group transformations are generated via continuous diffusions in order to smoothly deform templates into individual instances of objects.

This paper presents diffeomorphic transformations of three-dimensional (3-D) anatomical image data of the macaque occipital lobe and whole brain cryosection imagery and of deep brain structures in human brains as imaged via magnetic resonance imagery. These transformations are generated in a hierarchical manner, accommodating both global and local anatomical detail. The initial low-dimensional registration is accomplished by constraining the transformation to be in a low-dimensional basis. The basis is defined by the Green's function of the elasticity operator placed at predefined locations in the anatomy and the eigenfunctions of the elasticity operator. The high-dimensional large deformations are vector fields generated via the mismatch between the template and target-image volumes constrained to be the solution of a Navier-Stokes fluid model. As part of this procedure, the Jacobian of the transformation is tracked, insuring the generation of diffeomorphisms. It is shown that transformations constrained by quadratic regularization methods such as the Laplacian, biharmonic, and linear elasticity models, do not ensure that the transformation maintains topology and, therefore, must only be used for coarse global registration.

A general automatic approach is presented for accommodating local shape variation when mapping a two-dimensional (2-D) or three-dimensional (3-D) template image into alignment with a topologically similar target image. Local shape variability is accommodated by applying a vector-field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth (globally positive definite Jacobian). Smoothness is guaranteed without specifically penalizing large-magnitude deformations of small subvolumes by constraining the transformation on the basis of a Stokesian limit of the fluid-dynamical Navier-Stokes equations. This differs fundamentally from quadratic penalty methods, such as those based on linearized elasticity or thin-plate splines, in that stress restraining the motion relaxes over time allowing large-magnitude deformations. Kinematic nonlinearities are inherently necessary to maintain continuity of structures during large-magnitude deformations, and are included in all results. After initial global registration, final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem. Automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular. Application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability.

A maximum a posteriori (MAP) algorithm is presented for the estimation of spin-density and spin-spin decay distributions from frequency and phase-encoded magnetic resonance imaging data. Linear spatial localization gradients are assumed: the y-encode gradient applied during the phase preparation time of duration tau before measurement collection, and the x-encode gradient applied during the full data collection time t>/=0. The MRI signal model developed in M.I. Miller et al., J. Magn. Reson., ser. B (Apr. 1995) is used in which a signal resulting from M phase encodes (rows) and N frequency encode dimensions (columns) is modeled as a superposition of MN sinc-modulated exponentially decaying sinusoids with unknown spin-density and spin-spin decay parameters. The nonlinear least-squares MAP estimate of the spin density and spin-spin decay distributions solves for the 2MN spin-density and decay parameters minimizing the squared-error between the measured data and the sine-modulated exponentially decay signal model using an iterative expectation-maximization algorithm. A covariance diagonalizing transformation is derived which decouples the joint estimation of MN sinusoids into M separate N sinusoid optimizations, yielding an order of magnitude speed up in convergence. The MAP solutions are demonstrated to deliver a decrease in standard deviation of image parameter estimates on brain phantom data of greater than a factor of two over Fourier-based estimators of the spin density and spin-spin decay distributions. A parallel processor implementation is demonstrated which maps the N sinusoid coupled minimization to separate individual simple minimizations, one for each processor.

Fixed block coding schemes for Gibbs random fields are proposed in which the empirical expectations of the local interactions of the Gibbs measure is compared to its expectation with respect to all Gibbs measures having those interactions. The exponential decay of the error probabilities is proven and it is shown that the code rates equal the entropy of the random field. In addition, it is shown that any coding scheme based on regarding the field as a 1-D sequence of symbols has rate greater than the entropy of the field. The theory of fixed length coding is approached from the point of view of large deviations, both for the calculation of the error exponents or error probabilities and for the calculation of the encoding rates or the asymptotic combinatorics of the coding schemes. This approach is also applied to fixed length coding schemes of Markov sources for which estimates on the error exponents and on the rates are derived.>

Single photon emission computed tomography (SPECT) reconstructions performed using maximum a posteriori (penalized likelihood) estimation with the expectation maximization algorithm are discussed. Due to the large number of computations, the algorithms were performed on a massively parallel single-instruction multiple-data computer. Computation times for 200 iterations, using I.J. Good and R.A. Gaskins's (1971) roughness as a rotationally invariant roughness penalty, are shown to be on the order of 5 min for a 64x64 image with 96 view angles on an AMT-DAP 4096 processor machine and 1 min on a MasPar 4096 processor machine. Computer simulations performed using parameters for the Siemens gamma camera and clinical brain scan parameters are presented to compare two regularization techniques-regularization by kernel sieves and penalized likelihood with Good's rotationally invariant roughness measure-to filtered backprojection. Twenty-five independent sets of data are reconstructed for the pie and Hoffman brain phantoms. The average variance and average deviation are examined in various areas of the brain phantom. It is shown that while the geometry of the area examined greatly affects the observed results, in all cases the reconstructions using Good's roughness give superior variance and bias results to the two alternative methods.

A fully three-dimensional (3-D) implementation of the maximum a posteriori (MAP) method for single photon emission computed tomography (SPECT) is demonstrated. The 3-D reconstruction exhibits a major increase in resolution when compared to the generation of the series of separate 2-D slice reconstructions. As has been noted, the iterative EM algorithm for 2-D reconstruction is highly computational; the 3-D algorithm is far worse. To accommodate the computational complexity, previous work in the 2-D arena is extended, and an implementation on the class of massively parallel processors of the 3-D algorithm is demonstrated. Using a 16000- (4000-) processor MasPar/DECmpp-Sx machine, the algorithm is demonstrated to execute at 2.5 (7.8) s/EM-iteration for the entire 64x64x64 cube of 96 planar measurements obtained from the Siemens Orbiter rotating camera operating in the high-resolution mode.

The theory of large deviations for Gibbs random fields is used to show that the asymptotic number of bits per symbol for Ziv-Lempel codes in two dimensions is given by the maximal entropy of all Gibbs fields with the same interaction. The error-probability is shown to converge exponentially fast to zero. In addition, the stronger version of the Shannon-McMillan theorem proved by D.S. Ornstein and B. Weiss (1990) is formulated and proved in terms of the exponential decay of the probability of the nontypical sequences.>

The entropies and combinatorics of trees that branch according to fixed but finite numbers of rules are studied. Context-free grammars are used to categorize the ways in which nodes branch to yield daughter nodes, thus providing an organized setting to examine the entropies for random branching processes whose realizations are trees and whose probabilities are determined by probabilities associated with the substitution rules of the grammar. Normalized entropy rates H are derived for the critical branching rate and supercritical branching rate processes. An equipartition theorem is proved for the supercritical processes. A strong departure from classical theorems for Markov sources occurs for supercritical branching processes as the typical sets have supergeometric growth rates. The combinatorics of the set of all trees that can be generated from the context-free substitution rules is also studied.>

A general method is proposed for incorporating rule-based constraints corresponding to regular languages into stochastic inference problems, thereby allowing for a unified representation of stochastic and syntactic pattern constraints. The authors' approach establishes the formal connection of rules to Chomsky grammars and generalizes the original work of Shannon on the encoding of rule-based channel sequences to Markov chains of maximum entropy. This maximum entropy probabilistic view leads to Gibbs representations with potentials which have their number of minima growing at precisely the exponential rate that the language of deterministically constrained sequences grow. These representations are coupled to stochastic diffusion algorithms, which sample the language-constrained sequences by visiting the energy minima according to the underlying Gibbs probability law. This coupling yields the result that fully parallel stochastic cellular automata can be derived to generate samples from the rule-based constraint sets. The production rules and neighborhood state structure of the language of sequences directly determine the necessary connection structures of the required parallel computing surface. Representations of this type have been mapped to the DAP-510 massively parallel processor consisting of 1024 mesh-connected bit-serial processing elements for performing automated segmentation of electron-micrograph images.

Earlier work on parallel computation of generalized Bayesian hypothesis tests for hierarchical image reconstruction on massively parallel processor arrays is extended to incorporate pattern constraints specified with Boolean functions defined on symbolic imaging variables. This is based on a stochastic representation for memoryless Boolean functions following U. Grenander's work (1984) on metric pattern theory. Its application is presented to the segmentation of low-contrast textured images through an extension of J. Besag's (1986) ICM segmentation algorithm, and to image reconstruction with large point spread. Hierarchical image reconstruction in time-of-flight positron emission tomography at low count levels is described.>

Nonparametric Markov random field (MRF) texture modeling for the purpose of segmenting electron-microscope autoradiography (EMA) images is discussed. A Bayesian approach is assumed for addressing the basic problem of learning which model among a number of nonparametric MRF models best represents an observed texture. Nonparametric MRF models are inherently quite complex, prompting inclusion of a complexity measure within the Bayesian framework. The measure adopted is the Rissanen complexity, which quite naturally incorporates into the Bayesian analysis. The new Bayesian measure referred to as the minimum description length (MDL) then allows learning the conditional probabilities for the nonparametric MRF texture models of the mitochondria and background regions of the EMA image. Experiments show the results of segmenting an EMA image using these models.>

The authors present a single formulation for constrained imaging that fuses the problem of joint estimation of the continuous parameters using MAP (maximum a posteriori) and conditional-mean estimators with that of performing generalized Bayes hypothesis testing for the symbolic imaging variables. Coupling this with recent results on representing regular grammars via Gibbs' distributions makes it possible to incorporate into a single hierarchical framework the stochastic constraints relevant to continuous-valued parameters as well as language-theoretic constraints on the symbolic variables. The authors also present a method for performing the required computations on a massively parallel architecture, which makes it possible to update every variable at every level in the hierarchy in parallel. The conclusions obtained are supported with results for a Poisson imaging problem computed on a DAP-500 massively parallel processor with 1024 processing elements.>

The authors present results on learning regular grammars as well as developing extensions to learning multidimensional random fields. In learning a regular grammar, they use recent results on the stochastic representation of strongly connected regular grammars in order to derive an algorithm based on mutual information for learning the minimal state set as well as the production rules of the grammar. These learning results are then extended to multiple dimensions by extending the state structure of the regular grammar to the neighborhood structure of multidimensional random fields. This allows the authors to learn textures for image segmentation and reconstruction. The implementation of the learning algorithms on connection architectures is described.>

An approach to high-resolution imaging that starts with a model of the radar echo signal derived from the physics governing radar reflections is presented. The model has been used in the past to describe radar targets that are rough compared to the wavelength of the transmitted radiation. Without specifying precisely what the transmitted signal is, a general estimation-based procedure is derived for obtaining images. After discretizing the model, the radar imaging problem reduces to the task of estimating discretized second-order statistics of the reflectance process of the target. Maximum-likelihood estimates of these statics are obtained as the limit point of an expectation-maximization algorithm.>

A constrained maximum-likelihood estimator is derived by incorporating a rotationally invariant roughness penalty proposed by I.J. Good (1981) into the likelihood functional. This leads to a set of nonlinear differential equations the solution of which is a spline-smoothing of the data. The nonlinear partial differential equations are mapped onto a grid via finite differences, and it is shown that the resulting computations possess a high degree of parallelism as well as locality in the data-passage, which allows an efficient implementation on a 48-by-48 mesh-connected array of NCR GAPP processors. The smooth reconstruction of the intensity functions of Poisson point processes is demonstrated in two dimensions.>

It is shown that a sufficient condition for the likelihood function of a zero-mean Gaussian random vector with covariance R from some class of covariances R to be unbounded above over the set of positive-definite matrices in R is that some singular R/sub o/ exists in R whose range space contains the data. The results obtained imply that, for the spectrum estimation problem in which R is the class of Toeplitz covariances and only one long observation vector is available, by constraining the maximum-likelihood estimation problem to the class of Toeplitz matrices with nonnegative definite circulant extensions, a positive-definite solution is guaranteed to exist.>

The principle of maximum entropy has played an important role in the solution of problems in which the measurements correspond to moment constraints on some many-to-one mapping h(x). In this paper we explore its role in estimation problems in which the measured data are statistical observations and moment constraints on the observation function h(x) do not exist. We conclude that: 1) For the class of likelihood problems arising in a complete-incomplete data context in which the complete data x are nonuniquely determined by the measured incomplete data y via the many-to-one mapping y = h(x), the density maximizing entropy is identical to the conditional density of the complete data given the incomplete data. This equivalence results by viewing the measurements as specifying the domain over which the density is defined, rather than as a moment constraint on h(x). 2) The identity between the maximum entropy and the conditional density results in the fact that maximum-likelihood estimates may be obtained via a joint maximization (minimization) of the entropy function (Kullback-Liebler divergence). This provides the basis for the iterative algorithm of Dempster, Laird, and Rubin [1] for the maximization of likelihood functions. 3) This iterative method is used for maximum-likelihood estimation of image parameters in emission tomography and gammaray astronomy. We demonstrate that unconstrained likelihood estimation of image intensities from finite data sets yields unstable estimates. We show how Grenander's method of sieves can be used with the iterative algorithm to remove the instability. A bandwidth sieve is introduced resulting in an estimator which is smoothed via exponential splines. 4) We also derive a recursive algorithm for the generation of Toeplitz constrained maximum-likelihood estimators which at each iteration evaluates conditional mean estimates of the lag products based on the previous estimate of the covariance, from which the updated Toeplitz covariance is generated. We prove that the sequence of Toeplitz estimators has the property that they increase in likelihood, remain in the set of positive-definite Toeplitz covariances, and has all of its limit points stable and satisfying the necessary conditions for maximizing the likelihood.