We address the problem of image formation in transmission tomography when metal objects of known composition and shape, but unknown pose, are present in the scan subject. Using an alternating minimization (AM) algorithm, derived from a model in which the detected data are viewed as Poisson-distributed photon counts, we seek to eliminate the streaking artifacts commonly seen in filtered back projection images containing high-contrast objects. We show that this algorithm, which minimizes the I-divergence (or equivalently, maximizes the log-likelihood) between the measured data and model-based estimates of the means of the data, converges much faster when knowledge of the high-density materials (such as brachytherapy applicators or prosthetic implants) is exploited. The algorithm incorporates a steepest descent-based method to find the position and orientation (collectively called the pose) of the known objects. This pose is then used to constrain the image pixels to their known attenuation values, or, for example, to form a mask on the "missing" projection data in the shadow of the objects. Results from two-dimensional simulations are shown in this paper. The extension of the model and methods used to three dimensions is outlined.

The reconstruction of tomographic images is often treated as a linear deblurring problem. When a high-density, man-made metal object is present somewhere in the image field, it is a deblurring problem in which the unknown function has a component that is known except for some location and orientation parameters. We first address general linear deblurring problems in which a known function having unknown parameters is present. We then show how the resulting iterative solution can be applied to tomographic imaging in the presence of man-made foreign objects, and we apply the result, in particular, to X-ray computed tomography imaging used in support of brachytherapy treatment of advanced cervical cancer.

The emergent role of information theory in image formation is surveyed. Unlike the subject of information-theoretic communication theory, information-theoretic imaging is far from a mature subject. The possible role of information theory in problems of image formation is to provide a rigorous framework for defining the imaging problem, for defining measures of optimality used to form estimates of images, for addressing issues associated with the development of algorithms based on these optimality criteria, and for quantifying the quality of the approximations. The definition of the imaging problem consists of an appropriate model for the data and an appropriate model for the reproduction space, which is the space within which image estimates take values. Each problem statement has an associated optimality criterion that measures the overall quality of an estimate. The optimality criteria include maximizing the likelihood function and minimizing mean squared error for stochastic problems, and minimizing squared error and discrimination for deterministic problems. The development of algorithms is closely tied to the definition of the imaging problem and the associated optimality criterion. Algorithms with a strong information-theoretic motivation are obtained by the method of expectation maximization. Related alternating minimization algorithms are discussed. In quantifying the quality of approximations, global and local measures are discussed. Global measures include the (mean) squared error and discrimination between an estimate and the truth, and probability of error for recognition or hypothesis testing problems. Local measures include Fisher information.

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.

Iterative deblurring methods using the expectation maximization (EM) formulation and the algebraic reconstruction technique (ART), respectively, are adapted for metal artifact reduction in medical computed tomography (CT). In experiments with synthetic noise-free and additive noisy projection data of dental phantoms, it is found that both simultaneous iterative algorithms produce superior image quality as compared to filtered backprojection after linearly fitting projection gaps. Furthermore, the EM-type algorithm converges faster than the ART-type algorithm in terms of either the I-divergence or Euclidean distance between ideal and reprojected data in the authors' simulation. Also, for a given iteration number, the EM-type deblurring method produces better image clarity but stronger noise than the ART-type reconstruction. The computational complexity of EM- and ART-based iterative deblurring is essentially the same, dominated by reprojection and backprojection. Relevant practical and theoretical issues are discussed.

A method of sieves using splines is proposed for regularizing maximum-likelihood estimates of power spectra. This method has several important properties, including the flexibility to be used at multiple resolution levels. The resolution level is defined in terms of the support of the polynomial B-splines used. An expression for the optimal rate of growth of the sieve is derived using a discrepancy measure derived from the Kullback-Leibler divergence of parameterized density functions. While the sieves may be defined on nonuniform grids, in the case of uniform grids the optimal sieve size corresponds to an optimal resolution. Iterative algorithms for obtaining the maximum-likelihood sieve estimates are derived. Applications to spectrum estimation and radar imaging are proposed.>

The expectation-maximization (EM) algorithm for maximizing likelihood functions, combined with the Viterbi algorithm, is applied to the problem of sequence detection when symbol timing information is not present. Although the EM algorithm is noncausal, results obtained using the algorithm on the problem of nonsynchronized sequence detection indicate that it converges most of the time in three iterations, making it both of theoretical and of practical interest. A practical algorithm based on the EM algorithm is introduced. It reduces the computational burden and improves performance by making use of timing estimates in previous observation windows.>

A novel approach is developed for forming high-resolution images of radar targets from delay Doppler spotlight-mode radar data. This approach is based on a model for the target's reflectivity in terms of wide-sense stationary, uncorrelated scatterers having complex-valued Gaussian statistics. The imaging problem is to estimate the target's scattering function in terms of radar-echo data acquired with a series of target illuminations. A method is developed for solving this multidimensional spectrum estimation problem through the use of maximum-likelihood estimation implemented using the expectation-maximization algorithm.>

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.>

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.

Signal sets are identified that maximize the cutoff rate region for a multiple-access channel with an additive white Gaussian noise, in which the demodulator output alphabet is allowed to be infinite ("infinitely soft decisions"). The optimizing designs consist of a simplex signal set for each sender, such that each sender's set is orthogonal to those of the other senders. For "second moment" and for "fractional out-of-band-energy" bandwidth constraints on the signals of each sender, conditions are derived under which mutually orthogonal simplex sets are still optimal. For the second moment constraint, simplex sets derived from sinusoidal functions yield an optimal design and, for the out-of-band energy constraint, simplex Sets derived from prolate spheroidal wave functions are optimal. Choices of signal sets that maximize the cutoff rate region for an additive shot-noise limited multiple-access optical channel, subject to average energy and peak amplitude constraints, are also identified.

A model for the heterodyne optical channel that is consistent with experimental observations is introduced, and a general maximum-likelihood sequence estimation receiver that accounts for laser phase instabilities is proposed using the generalized log-likelihood functional. The performance of various suboptimal "estimator-correlator" receivers, obtained from optimal, nonimplementable receiver equations, is studied by simulation and compared to that of an incoherent receiver for practical parameter values. It is observed that, for hard decisions, the proposed receiver performs significantly better than an incoherent receiver for bit intervals larger than about half the coherence time of the laser oscillators, with larger gains to be expected when soft decisions on long sequences are made. A way of reducing the complexity of the soft-decisions receiver for long sequences is discussed.

Coherent heterodyne detection in optical communication is degraded by phase instabilities present when a semiconductor laser is used as a local oscillator. Postdetection processing that accounts for these instabilities is suggested, based on a diffusion-process model for oscillator instabilities and a maximum-likelihood sequence-estimator of information symbols modulating the received light.

The maximum-likelihood rule for locating a frame-synchronization pattern in a direct-detection optical communication system employing eitherQ-ary pulse-position modulation or on-off keying is identified. We show that equivalent performance for the simple correlation and postdctection-correlation rules used in practice can require substantially more than a 3 dB increase in signal power over that required using the maximum-likelihood rule.

We show thatQ-ary pulse-position modulation with raised cosine pulses minimizes the average power (photons/s) required to communicate at a specified throughput rate (nats/s) over a band-limited, noisy optical channel. The best choice ofQis identified, as are other encoder parameters.

The cutoff rate region for block coding of a two-sender one-receiver multiple-access channel is shown to be the same with and without frame synchronization between the two senders.

The cutoff rate is derived for a digital communication system employing an optical carrier and direct detection. The coordinated design of the encoder, optical modulator, and demodulator is then studied using the cutoff rate as a performance measure rather than the more commonly employed error probability. Modulator design is studied when transmitted optical signals are subject simultaneously to average-energy and peak-value constraints. Pulse-position modulation is shown to maximize the cutoff rate when the average-energy constraint predominates, and the best signals when the peak-value constraint predominates are identified in terms of Hadamard matrices. A time-sharing of these signals maximizes the cutoff rate when neither constraint dominates the other. Problems of efficient energy utilization, choice of input and output alphabet dimension, and the effect of random detector gain are addressed.

Receiver designs for maximum likelihood sequence estimation of digital data transmission through randomly dispersive channels are developed. The random channels are constrained to have finite memory of durationLin the sense that the casual minimum mean square error estimate of the channel output at any given time is a function of only the most recentLTseconds of observations. Examples of the channels included are the time varying Rayleigh, Rician and lognormal fading channels whose covariance functions have finite support, phase unsynchronized channels which satisfy the finite-memory constraint and point process channels. The information sequence may be either a stream of independent, equally-likely symbols of binary orM-ary alphabets or the output sequence of a trellis or convolutional encoder.

The evaluation of error-probability bounds for binary detection problems involving continuons-time stochastic processes as signals is considered. These bounds are of interest because, in even the simplest detection problems, the computation of the exact probabilities of error is usually mathematically intractable. The method used consists of applying some results from martingale theory to detection and estimation problems. Only discontinuous observations that contain the rate process associated with a counting process are considered. The problem addressed is to evaluate Chernoff bounds on error probabilities for the likelihood-ratio test. The solution procedure consists of a measure transformation technique that makes it possible to obtain an expression for the Chernoff bound in terms of an expectation of a multiplicative functional of the conditional mean signal (rate process) estimates. If the processes involved are Markov, it is then possible to represent the above expression as a solution to a partial differential equation that is derived from the backward equation of Kolmogorov. The above procedure is repeated when the optimal estimates are replaced by suboptimal estimates. Examples are given to illustrate the technique.

The development of an approach for obtaining statistical inferences about nonobservable processes that influence a processy(\cdot)which can be observed directly and which is assumed to be a mixture of continuous and discontinuous components is continued. The approach is based on probability-measure transformations and consists of finding the conditional probability of a nonobservable event in terms of the prior probability of that event and a functional of the observationsy(\cdot). The topics studied include optimal filtering, smoothing, and prediction estimates of the nonobservable process;M-ary hypothesis testing; performance lower-bounds; and stochastic control.

A space-time point process is a stochastic process having as realizations points with random coordinates in both space and time. We define a general class of space-time point processes which we term {\em analytic}. These are point processes that have only finite numbers of points in finite time intervals, absolutely continuous joint-occurrence distributions, and for which points do not occur with certainty in finite time intervals. Analytic point processes possess an intensity determined by the past of the point process. As a class, analytic point processes remain closed under randomization by a parameter. The problem we consider is that of estimating a random parameter of an observed space-time point process. This parameter may be drawn from a function space and can, therefore, model a random variable, random process, or random field that influences the space-time point process. Feedback interactions between the point process and the randomizing parameter are included. The conditional probability measure of the parameter given the observed space-time point process is a sufficient statistic for forming estimates satisfying a wide variety of performance criteria. A general representation for this conditional measure is developed, and applications to filtering, smoothing, prediction, and hypothesis testing are given.

The observation processyconsidered is an additive composition of continuous and discontinuous components. The additive Gaussian, point, and jump process models, treated separately in the past, are all included here simultaneously. Representations foryin terms of its innovations and following a Girsanov-type measure transformation are derived. These are then used to develop a measure form of Bayes' rule that provides a convenient tool for the study of estimation and decision problems arising in a variety of applications including communication and control.

In this correspondence, we examine the problem of estimating the centroid of a normally shaped intensity function of a time-space point process in terms of an observed realization of the point process. The centroid is assumed to move randomly as a projection of a Gauss-Markov process. We find that the centroid is conditionally normal given the observations. Dynamical equations are given for the conditional mean and covariance of the centroid. The resulting filter is nonlinear, but closely related to the continuous-discrete Kalman-Bucy filter. An application to optical position sensing is given.

A model that is sufficiently general to describe the predominant statistical characteristics of the output of many real optical detectors is formulated. This model is used to study the optimum receiver processing for direct-detection optical communication systems. In particular, the structures of detectors and estimators for randomly filtered doubly stochastic Poisson processes observed in additive white Gaussian noise are considered. Representations for the posterior statistics of a vector-valued Markov process that modulates the intensity of the doubly stochastic Poisson process are obtained. Quasi-optimum estimators and detectors are specified in general terms and specialized for several important applications. These include a demodulator for subcarrier angle modulation, a detector structure for binary signaling with known intensities, and a detector structure for binary signaling in the turbulent atmosphere.

The performance in estimating a Wiener process that modulates the phase of a subcarrier in a direct-detection optical communication system is studied. An approximation is derived for the steady-state probability density of the estimation error. The meansquare estimation error predicted using this density is compared to theoretical lower bounds on the error and to the sample meansquare error obtained by simulation.

Lower bounds are given on the attainable mean-square performance in causally tracking the phase and frequency of a subcarrier that modulates an optical carrier in a direct-detection opticalcommunication system.

Equations are derived that describe the time evolution of the posterior statistics of a general Markov process that modulates the intensity function of an observed inhomogeneous Poisson counting process. The basic equation is a stochastic differential equation for the conditional characteristic function of the Markov process. A separation theorem is established for the detection of a Poisson process having a stochastic intensity function. Specifically, it is shown that the causal minimum-mean-square-error estimate of the stochastic intensity is incorporated in the optimum Reiffen-Sherman detector in the same way as if it were known. Specialized results are obtained when a set of random variables modulate the intensity. These include equations for maximum a posteriori probability estimates of the variables and some accuracy equations based on the Cramér-Rao inequality. Procedures for approximating exact estimates of the Markov process are given. A comparison by simulation of exact and approximate estimates indicates that the approximations suggested can work well even under low count rate conditions.

Some general equations are derived for the smoothing density of a Markov process that modulates the intensity of an observed doubly stochastic Poisson process. The equations are in terms of filtering densities that are specified by recursive equations.

Sufficient conditions for the realization of a causal filter-squarer receiver for the detection of a Gaussian signal in white Gaussian noise are established. Special consideration is given to a Gaussian signal that is the projection of a vector Gauss-Markov process. For this case, the causal filter-squarer receiver is shown to be related to the Kalman-Bucy filter for estimatng the underlying vector Gauss-Markov process.

A new approach is presented for estimating stochastic signals. The approach is based on the use of Markov processes and state-variable concepts. Equations are presented for approximate minimum mean-square error estimates of a Markovian state vector observed in a signal in which it is imbedded nonlinearly. A general model for analog communication via randomly time-varying channels is given and related to the state-vector estimation problem. The model includes as special cases such linear and nonlinear modulation schemes as AM, PM, FM, and PM_{n}/PM; and such continuous channels as Rayleigh and Rician channels, fixed channels with memory, and diversity channels. The state-variable approach leads automatically to physically realizable demodulators whose outputs are approximate MMSE estimates of the stochastic message and, if desired, the channel disturbances. Special consideration is given to angle modulation schemes.