Donald L. Snyder Workshop A workshop to honor Donald L. Snyder on the occasion of his 65th birthday, organized by friends and colleagues.

Brown Hall 100
Tentative Schedule of Talks

Marc Raichle

Washington University

Tom R. Miller

Nuclear medicine employs radioactive materials to image physiological and biochemical processes in the human body. The emitted gamma rays and X-rays form images, sometimes two-dimensional renditions of the radiation, but frequently in tomographic, cross-sectional form somewhat similar to computed tomography (CT). Don pioneered development of the tomographic reconstruction algorithms. In the mid-70s he wrote a paper, still widely quoted today, deriving the formulas for the sampling interval required to avoid aliasing in the filtered backprojection algorithm. He then worked with the inventors of positron tomography (PET), developing the algorithm for incorporating the time of flight of the photons into the reconstruction. Perhaps most significant was Don's work in iterative reconstruction applied to PET and single-photon tomography (SPECT). He co-authored one of the first two papers presenting the theory of maximum-likelihood reconstruction with the expectation-maximization algorithm (MLEM) in SPECT. In another series of papers, also widely referenced today, Don and colleagues explained the "noise breakup" problem in ML reconstruction, showing it was fundamental to the reconstruction and presenting ways to "regularize" the resulting images. More recently, Don and co-workers published papers quantitatively demonstrating the value of iterative reconstruction in nuclear medicine. Today, extensions of Don's work at Washington University and elsewhere are widely applied in commercial systems, leading to more accurate diagnosis of disease by nuclear techniques.

David G. Politte and Lewis J. Thomas, Jr.

Washington University

The two-decade span of the 1970s and 1980s was a period of intensive development of medical imaging instrumentation at the Washington University School of Medicine, particularly of the modality of positron-emission tomography (PET). Overall development of a series of tomographs was centered at the Mallinckrodt Institute of Radiology within the Division of Radiation Sciences. Many of the concomitant algorithms for processing data from these instruments were developed at the Biomedical Computer Laboratory (BCL) by Donald Snyder and his students. Early work included a novel derivation of the widely used filtered backprojection image-reconstruction algorithm and a fundamental sampling theorem that specifies a relationship between the angular and spatial sampling of the measured data. Subsequently developed tomographs augmented the usual data collected by PET by measuring the time-of-flight difference between a pair of photons that arose from a common positron- electron annihilation. A generalization of filtered backprojection to include the new information, called confidence weighting, was derived and was shown to possess a minimax property such that it minimized the worst case reconstruction error among a wide class of linear algorithms. A significant departure from the previous algorithmic work was the development, starting in 1982, of a maximum-likelihood method based on the iterative expectation- maximization algorithm. This method was first applied to real (not simulated) PET data at Washington University in 1982. Images produced by this method were found to have superior resolution and signal-to-noise ratio compared to images produced by confidence weighting, but images produced during the iterations were found to eventually suffer from two degradations, termed the noise and edge artifacts. These artifacts were shown to be fundamental, and regularization methods based on Grenander's method of sieves were developed to mitigate them. The maximum-likelihood method developed initially for application to data from positron-emission tomography has been adapted for use with data from a variety of applications, including optical sectioning microscopy and electron-microscopic autoradiography at BCL as well as for single- photon emission computed tomography and astronomical imaging.

The breadth of Dr. Snyder's contributions to BCL's program in quantitative biomedical imaging and supportive activities is reflected in his work in other areas including analysis of radioisotope-tracer data, vascular modeling, mass spectrometry data analysis, modeling the relation between ventricular dysrhythmias and sudden cardiac death, optical-communication- system theory, myocardial infarct-size estimation, modeling for cardiac-output estimation, ultrasonic tomography for tissue characterization, analysis of visual-field data in ocular hypertension, and ventricular boundary extraction from video- angiograms, to name a few.

by Pierre Moulin, Ph.D.

University of Illinois at Urbana-Champaign

This talk has two parts. The first is a brief discussion of significant advances that have ben made in the area of universal and adaptive image modeling during the last ten years; these advances have been largely motivated by the need to develop compression algorithms that perform well for a very broad class of images. The second part of the talk will describe the application of some of these sophisticated image models to statistical imaging.

Imaging problems are typically ill-posed, and complexity measures of the type above can be used as flexible regularization functionals. Algorithmic issues for complexity-regularized estimation will be described, and nonasymptotic estimation bounds will be presented. The talk will conclude with a connection between complexity-regularized estimation and rate-distortion theory.

Gary E. Christensen, D.Sc.

A fundamental problem with a large class of image registration techniques is that the estimated transformation from image A to B does not equal the inverse of the estimated transform from B to A. This inconsistency is a result of the matching criteria's inability to uniquely describe the correspondences between two images. This research seeks to overcome this limitation by jointly estimating the transformation from A to B and from B to A while enforcing the consistency constraint that these transforms are inverses of one another. The transformations are further restricted to preserve topology by constraining them to obey the laws of continuum mechanics. A new parameterization of the transformation based on a Fourier series in the context of linear elasticity is presented. Results are presented using both Magnetic Resonance and X-ray Computed Tomography Imagery. It is shown that joint estimation of a consistent set of forward and reverse transformations constrained by linear-elasticity gives better registration results than using either constraint alone or none at all. Results will be presented for which we have used this new registration technique to compute and visualize average 3D anatomical image volumes.

Daniel R. Fuhrmann

Washington University

Restriction-fragment mapping is one tool used in large-scale efforts in genome sequencing. It involves the creation, in the laboratory, of hundreds or thousands of electrophoretic gel images (each roughly 1k x 1k) which must be carefully analyzed, either by human or by machine. One gel image consists of 121 vertical lanes, each 9 pixels wide, and each lane in turn contains a number of diffuse horizontal stripes, or bands. The primary signal processing task is the detection and localization of these bands. In addition, one out of every 5 lanes contains crucial calibration information which allows one to convert the band locations to molecular fragment size, the genetic quantity of interest.

In this talk we will describe a software package under development, called BANDLEADER, for this automated image analysis problem. The software is planned for production use at the WU Genome Sequencing Center and for sequencing the mouse genome at the Genome Sequence Centre of the British Columbia Cancer Research Agency. The forward model for the data generation and the resulting stochastic inverse problem will be presented. We will describe a number of interesting signal processing sub-problems which have been encountered, such as polynomial curve-fitting with linear inequality constraints and search algorithms on a lattice for model selection.

Prakash Narayan

University of Maryland
College Park, MD 20742
USA

Bixio Rimoldi

Ecole Polytechnique Federale de Lausanne
Switzerland

In tribute to Don Snyder's expository skills and commitment to excellence in teaching, we shall address three simple topics that we have found to be of pedagogical value in a first course on information theory.

The first topic concerns the output statistics of an additive white Gaussian noise channel when the input is used to transmit codewords selected uniformly from a capacity-achieving codebook. This is a special case of a more general result by Han and Verdu. The merit of the special Gaussian case presented here is in its simplicity and transparency.

The second topic is a converse for the Gaussian multiple-access coding theorem based on the converse for the point-to-point Gaussian channel. The standard way to prove that the operational capacity region is included in the information capacity region is by means of Fano's inequality or Gallager's error-exponent technique. An alternative and straightforward proof that we suggest has the pedagogical merit of emphasizing the tight connection that exists between the Gaussian multiple-access channel and the (point-to-point) Gaussian noise channel.

The third topic involves the penalty incurred in source coding when the code is not matched to the source. For variable-length lossless coding, it is well known that the excess average codeword length due to mismatch is given by the Kullback-Leibler divergence between the ``true" source and the ``wrong" source. In an analogous problem

for lossy source coding, we shall show that a notion of rate-distortion discrepancy between the two sources provides a characterization of the expected excess rate due to mismatch.

C.N. Georghiades and P. Spasojevic

Texas A&M University,

Since its introduction in the late seventies as a general iterative procedure for producing maximum-likelihood estimates in cases a direct approach is computationally or analytically intractable, the expectation-maximization (EM) algorithm has been used with increasing frequency in a wide variety of application areas. Perhaps not surprisingly, one of the areas that has seen a significant increase in the use of the algorithm over the last 10 years is telecommunications. In this manuscript, we describe some of the varied uses of the EM algorithm that appeared in the literature in the area of telecommunications since its introduction, and include some new results on the algorithm which have not appeared elsewhere in the literature.

Anekal B. Sripad
Technical Manager, Bell Laboratories
Optical Networking Group
Lucent Technologies, North Andover, MA.

The talk provides an overview on some of the advances in optical technology that is revolutionizing the optical communication network architectures. Over the past few years, the network traffic has grown exponentially and new the technologies have been introduced to address the demand. This has caused massive upgrades of the existing network infrastructures and deployment of new networks. The primary focus of this talk will be on the transport networking including optical layer protection, bandwidth management, network management and interoperability.

Jose' M. F. Moura

LIDS, EECS, MIT, on sabbatical leave from ECE, CMU

In this talk I will describe SPIRAL a signal processing library whose goal is to generate automatically implementations of classes of important signal processing algorithms that are optimized to a given computing platform. This project is a joint effort of a team led by CMU and including signal processing researchers, mathematicians, computer engineers, and computer scientists from CMU, Drexel University, University of Illinois at Urbana Champagne, University of Southern California, and a start-up company MathStar. The motivation for SPIRAL is two fold. The wide spread availability of new and old sensors collecting inordinate amounts of data creates the necessity of faster codes that extract the relevant information from the data, significantly reducing the storage and transmission bandwidth required. Secondly, to fully exploit the computing technology, there is the need to break the current vicious cycle where the development of fast codes for signal processing algorithms lags the rapid deployment of ever faster computing platforms with the net result that fine tuned codes end-up running on obsolete hardware. SPIRAL focus on transform algorithms and their applications in signal and image processing, e.g., SAR, hyperspectral, and MRI sensors. SPIRAL is decomposed into four major blocks. The first is a meta-level block that derives automatically fast versions for the signal processing algorithm of interest - think of the FFT and the DFT. The second is the formula generator block that uses this fast algorithm version as a rule that when applied recursively generates many different equivalent formulas implementing the same fast algorithm. The third is the translation block that compiles for each formula an optimized code implementation. The last is a feedback learning block that uses performance models and real performance data to search the formula and formula implementation spaces for the optimal formula and its optimal implementation. The talk will focus on the group representation theoretic infrastructure underlying the meta-level block and the tensor product mathematical framework underlying the formula generator block. The relations and the interplay between group representations, symmetry, signal extensions, Markov random fields, and fast signal transforms will be described.

Julius L. Goldstein, Ph.D.

Washington University

Rice's (1948) formulation of the energy statistics of duration-limited (T) and band-limited (W) Gaussian noise plus tone has played a key role (Green & Swets, 1966) in motivating and modeling psychophysical research on sound discrimination. It has become apparent that this corpus of experimental research is more consistent with auditory peak detection. For example, auditory discrimination of Gaussian noise intensity for TW>20 requires a mean intensity change of ~1dB, which is larger than the energy variability of the stimulus. Therefore, discrimination has been assumed to be limited by an internal noise equivalent to 1 dB. However, this is contradicted by other experiments that imply a much smaller internal noise. A more successful account of intensity discrimination as well as other experiments is provided by the statistics of the largest envelope peak within the observation window (T). Studies (Goldstein, 1996, 1999) of Rice's (1945) analytic formulations of the properties of envelope peaks, as well as Monte-Carlo simulations of noise show that peak detection adds ~1 dB from phase variability to the classical Chi-Squared energy variability. Comparisons of internal and external sources of detection variability (i.e., standard deviations) suggest a match between sensory biology and the environment for noise intensity discrimination. This research also suggests that Rice's (1945) results for noise-envelope peaks and its elaborations should receive more attention in the statistical signal-theory curriculum. (Supported by NSF Grant IBN-9728383).

Bob Morley, Ph.D.

Washington University

A survey of several applications for Washington University's Magneprint(TM) and the architecture of a custom VLSI circuit will be presented. Authentication applications include credit cards, checks and notes in the banking industry, as well as drivers' licenses and assundry products and memorabilia. The technology will also shown to be useful for internet commerce. General requirements for a VLSI implementation will be described.

Benjamin J. Slocumb, Ph.D.

Senior Research Engineer
Georgia Tech Research Institute

A number of signal processing applications exist where information must be extracted by processing the discrete event times in a pulse train. One example is the signal analysis and identification tasks performed by passive radar intercept receivers. The receiver operates by measuring each pulse time of arrival (TOA), and then performs pulse sorting to estimate the pulse repetition interval (PRI). PRI estimation is complicated by the presence of TOA jitter, and false and missing TOA data. The presence of simultaneous emitters forces the receiver to "deinterleave" the pulse trains before the PRI can be estimated.

This talk is concerned with the application of adaptive Kalman filter and data association methods to the pulse train analysis and deinterleaving problems. The new techniques are presented that are based on a state-space model of the pulse train evolution process. A Kalman filter is implemented to estimate and track state parameters of a single pulse train. A Pulse Train Probabilistic Data Association Filter (PT-PDAF) is developed that handles significant false/missing TOA data. For pulse train deinterleaving, the problem is cast in switching system framework. The Interacting Multiple Models (IMM) approach is implemented to simultaneously estimate the pulse train PRIs. A new adaptive technique is developed to make the IMM robust to missing data, and a false-mode IMM is developed to handle false alarms.

M.I. Miller, Ph.D.

The Johns Hopkins University

Imaging systems to date have been largely unstructured. Pictures are most often generated without any knowledge of the structures and shapes being imaged. Imaging scientists find themselves in a similar position to those working in speech and language in the early 70's. The staggering implications of Chomsky's deeper structures on recognition systems had not yet been realized; all focus was on the sound.

Today, language recognition systems directly manipulate the deeper syntactic structures searching for consistent explanations of the acoustic sound or written text. So too in the 20th century for imaging science, most of the focus to date has been on the streams of pixels forming the image as the message.

The 21st Century looks very different. We should expect to see imaging systems of the future which manipulate complete CAD models of the structures being imaged.

Upon our entrance into the Radiology chamber, our digital Avitar will be loaded and our individualized CAT scan will be queried for consistent explanations supporting the recent weight gain that we have been denying to ourselves. That same Avitar will be beamed at highly compressed data rates across the planet as we teach geometry and calculus at the new Singapore University via our hand held Motorola video conferencing systems.

To accomplish all this, advances in imaging science will have to occur. In this talk we present, briefly, one of the most significant challenges: defining similarity between shapes and objects in the real world. The role of computational science in the construction of such similarity measures will be explored.

Acknowledgements:

This research was supported by NIH grants RO1-MH525158-01A1, RO1-NS35368-02, NSF BIR-9424264, and the ARO DAA-HO4-95-1-0494, ONR MURI N00014-98-1-0606.

Timothy J. Schulz

Michigan Technological University

Time-varying changes in the refractive index of Earth's atmosphere make surveillance and imaging from ground to space an extremely challenging activity. A major difficulty encountered when one attempts to restore resolution via post-detection processing of telescope imagery is that the precise form of the turbulence-induced distortions are unknown, and the imaging problem is one of blind deconvolution. In this talk, the imaging problem is posed in an estimation-theoretic framework whereby the object's incoherent scattering function is estimated through the simultaneous identification and correction of the distorting effects of atmospheric turbulence, and iterative processing methods are discussed and demonstrated.

Chrysanthe Preza

Washington University

An important application of quantitative DIC microscopy is mass determination of cells, based on a result shown in the early fifties: that the integrated optical-path-length (OPL) over a cell area is a measure of the total solid (DNA, protein) content of that cell. Recently we presented an iterative phase-estimation method developed for the computation of a specimen's phase function (or OPL distribution) from Differential-Interference-Contrast (DIC) microscopy images [1]. The method minimizes the least-squares discrepancy measure using the conjugate-gradient technique to estimate the OPL from multiple DIC images acquired at different specimen rotations. Regularization of the method is achieved using a quadratic smoothness penalty. An alternative discrepancy measure is the I-divergence which theoretically is a better choice when solving optimization problems that involve positive-valued functions.

In this talk we compare results obtained with the two different discrepancy measures, the least-squares measure and the I-divergence measure. For the comparison, rotational-diversity DIC images of a single bovine spermatozoa head acquired by rotating the cell were used. Comparison of the integrated optical path length (IOPL) computed from the

estimated phase images shows that the two discrepancy measures yield similar results.

Reference:

1) C. Preza, "Rotational-diverse phase estimation from differential-interfence-

contrast microscopy images", J. Opt. Soc. Amer. A, in press, 1999.

Kurt Smith

Colliding with a DLS Being is a very rare superatomic event that evokes a wide tear in a Beings knowledge fabric - a tear which enables a shower of thinking, intuition, and adventure-seeking. In this talk I will provide evidence that the DLS Effect is an actual phenomenon and demonstrate with a sample set of One that it continues to affect.

Lyn Hibbard, PhD

Computerized Medical Systems, Inc.

Modern 3-D radiation treatment planning (RTP) aims to deliver prescribed dose over the tumor volume while minimally affecting surrounding tissues. RTP requires accurate models of the patient anatomies, laboriously obtained by manually tracing organ contours seen in CT imagery, using interactive graphics. We describe here an automated contouring program designed to accelerate plan preparation. In its semi-automated mode, the program computes a contour for an object given a user-input portion of the object's interior. In fully automatic operation, the program steps from section to section, combining previous- and current-section information to generate contours.

The contours are maximum a posteriori (MAP) estimates of soft tissue boundaries formed by combining region segmentation and gradient edge detection with prior contours' shapes. Region growth and segmentation is conducted by supervised classification of voxel textures. Initialized by the user input for some initial section, the classifier finds like-textured voxels in neighboring sections to propagate the region segmentation in 3-D. Contours correspond to maxima in an objective function for which the contour maximally (and simultaneously) agrees with the segmented-region edge, local gray level gradient maxima, and with prior contour shapes. Region- and gradient-edge agreements are posed as MAP problems, and maximization is carried out over the space of Fourier elliptic parameters defining the closed contour. The method successfully contours test images with signal-to-noise ratios < 2.0. Up to 80% of the sections of some abdominal organs can be computed without requiring subsequent editing. Results on several abdominal and pelvic cases will be demonstrated.

R. Martin Arthur

Electronic Signals and System Research Laboratory
Department of Electrical Engineering
Washington University, St. Louis, MO 63130

Conventional pulse-echo image generation, in which scattering potential is visualized, requires knowing the time of flight (TOF) for insonification in tissue. TOF depends on propagation distance and the speed of sound (SOS). Conventional imaging systems assume tissue is like water and SOS is constant. They also couple echo acquisition and image generation to enable real-time operation. This coupling limits the number of focal points and precludes adaptive image formation from the same echoes. On the other hand, synthetic-focus systems separate echo acquisition from image generation to allow both point-focus and adaptive image generation. Adaptive focusing under varying assumptions about SOS can improve pulse-echo images and elicit local values for SOS. To implement such a scheme, description of the variation of either SOS or TOF in tissue must be compatible with adaptive-focus image generation. TOF was used to characterize tissue rather than SOS because TOF is needed for image formation and because conversion of TOF to SOS is much easier than the inverse. Approximation of TOF with integer-degree polynomials in range and azimuth accurately represents TOF and reduces image generation time for constant SOS. Consequently, the utility of these polynomials to describe varying SOS was explored. Regions (64x64 mm) with step and tissue-like changes in SOS were simulated. Contrast of images of a point-like scatterer at 50 mm with a 32-element, linear array of 1.1 x 13 mm transducers, with center frequencies of 3.5 MHz, improved by 8-10 dB when actual SOS was taken into account. Assumed SOS maps were matched to within a few percent from integer-degree polynomials. If the coefficients of these polynomials can be estimated from measured backscattered signals, then their use with adaptive-focus imaging may lead to improved image quality along with SOS characterization in tissue.

Jeffrey H. Shapiro

Julius A. Stratton Professor
of Electrical Engineering
Massachusetts Institute of Technology

In semiclassical theory, light is a classical electromagnetic wave and the fundamental source of photodetection noise is the shot effect arising from the discreteness of the electron charge. In quantum theory, light is a quantum-mechanical entity and the fundamental source of photodetection noise comes from measuring the photon-flux operator. The Glauber coherent states are Gaussian quantum states which represent classical electromagnetic radiation. Quantum photodetection of these states yields statistics that are indistinguishable from the corresponding Poisson point-process results of semiclassical photodetection. Optical parametric interactions, however, can be used to produce other Gaussian quantum states, states whose photodetection behavior cannot be characterized semiclassically. We shall describe a unified analytical framework for Gaussian-state photodetection that includes the full panoply of non-classical effects that have been produced via parametric interactions.

Alfred Hero, Ph.D.

University of Michigan-Ann Arbor

Filtered Poison processes are useful models for many image formation processes including: quantum limited CCD arrays; gamma ray scintillation detectors; electron microscopy; stereology of porous materials; and bio-assay instrumentation. Each of these image formation processes leads to a measurement composed of a superposition of random shapes at random spatial locations, a signal called a filtered Poisson process (FPP), and additive measurement noise, reasonably assumed to be Gaussian in many cases. As the statistical distribution of the resulting images is not closed form an optimal estimator of signal parameters cannot be directly implemented. In this paper we present MSE bounds and iterative algorithms for estimating signal parameters in the Gaussian noise contaminated filtered Poisson image model. Two models of interest will be considered: (1) the case of a spatial shot noise process where shape superposition is non-occluding (linear); and (2) the case of a spatial coverage process where the superposition is occluding (non-linear). The key to the development of the bound and the iterative estimator is a decomposition of the measurement model into a cascade of a discrete event Poisson process channel (the complete data channel) and a continuous Gaussian waveform channel (the incomplete data channel). The decomposition allows us to classify estimator performance into two regimes: the Poisson limited regime and the Gaussian limited regime. We will provide illustrations for non-occluding and occluding FPP models in two respective applications: source position estimation using a photo-sensitive CCD array (photo-positioning) and estimation of parameters of pore shape

distribution in a random porous medium (granulometry).

Joseph A. O'Sullivan, Ph.D.

Washington University

There are two key properties of relative entropy that are analogous to the orthogonal projection property for squared error. The first property corresponds to minimizing relative entropy over the first variable when that variable is restricted to a linear family. The second property corresponds to minimizing relative entropy over the second variable when that variable is restricted to an exponential family. Maximum likelihood estimation for exponential families of densities is characterized as a double minimization of relative entropy where the first entry is minimized over a linear family and the second over an exponential family. This double minimization yields the expectation- maximization algorithm. General properties of such alternating minimization algorithms are discussed including convergence issues. Special cases of this general algorithm include expectation-maximization algorithms derived and analyzed by Don Snyder and colleagues.

Stefano Soatto

Washington University ESSRL

We formulate the problem of reconstructing the three-dimensional shape and radiance of a scene as the minimization of the Information-divergence between blurred images, and propose an algorithm that is provably convergent and guarantees that the solution is admissible, in the sense of corresponding to a positive radiance and imaging kernel. The motivation for the use of Information-divergence comes from the work of Csizar, while the fundamental elements of the proof of convergence come from work by Snyder et al.

Joint work with Paolo Favaro.

Arthur B. Baggeroer

Departments of Ocean Engineering and
Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Cambridge, MA 02139

Arrays are used extensively in virtually all sonar systems. The frequency bands range from below 50 Hz for long range passive sonar and acoustic tomography to above 30 kHz for acoustic imaging and telemetry. The ocean environment dominates all aspects of the design of a sonar array system including the array itself, the signal models used, the ambient noise and the processing algorithms. There is a wide gap between the theories now advanced and the state of the art now practiced in operational sonar array processing. Many algorithms now developed are appropriate for high signal to noise ratios, plane wave signal models, ideally calibrated arrays and white noise environments. Typically, passive sonars operate against low signal to noise sources, propagating through a complex medium where non-stationary clutter from shipping dominates. Active sonars operate in highly reverberant and bandlimited environments. While typically operating at high frequencies, imaging and telemetry systems still confront complex propagation which introduce spreading in travel time and frequency as well as spatial spreading and diversity. All of these issues are now being magnified because we are fielding large arrays with hundred plus sensors. In this presentation we give an overview of several array processing problems now confronting the design of operational sonars and attempt to couple them to some of the relevant theoretical issues.

Richard E. Blahut

Coordinated Science Laboratory
Department of Electrical Engineering
University of Illinois

Huygens' Principle, which underlies much of optics and antenna theory, was introduced qualitatively in 1678, and given a mathematical formulation by Fresnel in 1818 and by Kirchhoff in 1882. Poincare (1892) and Sommerfeld (1894) found inconsistencies in the underlying assumptions and provided a new derivation of a slightly different statement. This history suggests that the accepted formulation may not be as accessible as it purports to be.

The principle is usually taught as a part of optics or electromagnetic theory, although it is more general and applies as well to acoustics and ultrasonics. It should be thought of as a central principle of signal theory. However, there is no satisfactory derivation of Huygens' principle in the signal processing literature. The standard approach uses Green's functions and often intermingles with the development a discussion of physical realizability of consistent boundary conditions.

This talk presents an alternative derivation of Huygens' principle as an exercise in two-dimensional signal processing. First, a nontrivial but rather appealing two-dimensional Fourier transform pair is derived. Then the relationship between the values taken by a monochromatic monodirectional plane wave in two successive planes it passes through is written down. Finally, because a signal in any plane can be decomposed into spatial frequencies, the relationship between the values taken by an arbitrary wave as it passes through two successive planes can be written down in the two-dimensional spatial Fourier transform domain. The convolution theorem together with the two-dimensional Fourier transform pair that began the talk then provide a statement of the Huygens-Fresnel principle and the form of the Huygens-Fresnel point-spread function of free space. The entire derivation uses the methods of signal processing. Happy birthday, Don.

John C. Schotland, M.D., Ph.D.

Washington University

There is considerable interest in the development of optical methods for medical imaging. It is anticipated that such methods would provide fundamentally new diagnostic capabilities while complementing existing imaging modalities. This talk will review recent work on initial clinical applications to mammography and peripheral vascular disease.

|Imaging in Radio Astronomy via an Expectation-Maximization Algorithm for Structured Covariance Estimation

Aaron D. Lanterman, Ph.D.

University of Illinois at Urbana-Champaign

Image restoration in radio astronomy is often posed as a problem of reconstructing a nonnegative function from sparse samples of its Fourier transform. We explore an alternative approach which reformulates the problem in terms of estimating the entries of a diagonal covariance matrix from Gaussian data. Maximum-likelihood estimates of the covariance cannot be readily computed analytically; hence we explore an iterative expectation-maximization algorithm originally proposed by Snyder, O'Sullivan, and Miller in the context of radar imaging. One of the advantages of maximum-entropy techniques in traditional radio astronomy formulations is that the entropy functional ensures nonnegative estimates; in the maximum-likelihood approach, nonnegativity is automatically guaranteed by the form of the EM iterations.

The resulting maximum-likelihood estimates tend to be unacceptably rough due to the ill-posed nature of maximum-likelihood estimation of functions from limited data, so some kind of regularization is needed. We will explore penalized likelihoods based on entropy functionals, a roughness penalty proposed by Silverman, and an information-theoretic formulation of Good's roughness penalty crafted by O'Sullivan.