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Mammone Rutgers University 25 Signal Recovery from Partial Information Christine Podilchuk Introduction •Formulation of the Signal Recovery Problem•Least Squares Solutions•Signal Recover

Inverse Problems and

Signal Reconstruction

Richard J Mammone

Rutgers University

25 Signal Recovery from Partial Information Christine Podilchuk

Introduction •Formulation of the Signal Recovery Problem•Least Squares Solutions•Signal Recovery using Projection onto Convex Sets (POCS) •Row-Based Methods•Block-Based Methods

•Image Restoration Using POCS

Introduction •The Reconstruction Problem•Transform Methods•Filtered Backprojection (FBP)

•The Linogram Method•Series Expansion Methods•Algebraic Reconstruction Techniques (ART)

•Expectation Maximization (EM)•Comparison of the Performance of Algorithms

Zhang

Introduction •Speech Production and Spectrum-Related Parameterization • Template-Based Speech Processing •Robust Speech Processing•Affine Transform•Transformation of Predic-tor Coefficients •Affine Transform of Cepstral Coefficients•Parameters of Affine Transform• Correspondence of Cepstral Vectors

28 Inverse Problems, Statistical Mechanics and Simulated Annealing K Venkatesh

Prasad

Background •Inverse Problems in DSP•Analogies with Statistical Mechanics•The Simulated Annealing Procedure

Introduction •The EM Algorithm•Some Fundamental Problems•Applications•Experimental Results •Summary and Conclusion

30 Inverse Problems in Array Processing Kevin R Farrell

Introduction •Background Theory•Narrowband Arrays•Broadband Arrays•Inverse Formula-tions for Array Processing •Simulation Results•Summary

31 Channel Equalization as a Regularized Inverse Problem John F Doherty

Introduction •Discrete-Time Intersymbol Interference Channel Model•Channel Equalization Filtering •Regularization•Discrete-Time Adaptive Filtering•Numerical Results•Conclusion

Introduction: Dereverberation Using Microphone Arrays •Simple Delay-and-Sum Beamformers

•Matched Filtering•Diophantine Inverse Filtering Using the Multiple Input-Output (MINT) Model •Results•Summary

33 Synthetic Aperture Radar Algorithms Clay Stewart and Vic Larson

Introduction •Image Formation•SAR Image Enhancement•Automatic Object Detection and Classification in SAR Imagery

34 Iterative Image Restoration Algorithms Aggelos K Katsaggelos

Introduction •Iterative Recovery Algorithms•Spatially Invariant Degradation•Matrix-Vector Formulation •Matrix-Vector and Discrete Frequency Representations•Convergence•Use of Constraints •Class of Higher Order Iterative Algorithms•Other Forms of8(x)•Discussion

THERE ARE MANY SITUATIONS where a desired signal cannot be measured directly The

measurement might be degraded by physical limitations of the signal source and/or by the measurement device itself The acquired signal is thus a transformation of the desired signal The inversion of such transformations is the subject of the present chapter In the following sections

we will review several inverse problems and various methods of implementation of the inversion or recovery process The methods differ in the ability to deal with the specific limitations present in each

application For example, the a priori constraint of non-negativity is important for image recovery,

but not so for adaptive array processing The goal of the following sections is to present the basic approaches of inversion and signal recovery Each section focuses on a particular application area and describes the appropriate methods for that area

The first chapter, 25, is entitled “Signal Recovery from Partial Information” by Christine Podilchuk This section reviews the basic problem of signal recovery The idea of projection onto convex sets (POCs) is introduced as an elegant solution to the signal recovery problem The inclusion of linear and non-linear constraints are addressed The POCs method is shown to be a subset of the set theoretic approach to signal estimation The application of image of restoration is described in detail

Chapter 26 is entitled “Algorithms for Computed Tomography” by Gabor T Herman This section presents methods to reconstruct the interiors of objects from data collected based on transmitted or emitted radiation The problem occurs in a wide range of application areas The computer algorithms used for achieving the reconstructions are discussed The basic techniques of image reconstruction from projections are classified into “Transform Methods” (including Filtered Backprojection and the Linogram Methods) and “Series Expansion Methods” (including, in particular, the Algebraic Recon-struction Techniques and the method of Expectation Maximization) In addition, a performance comparison of the various algorithms for computed tomography is given

Chapter 27 is entitled “Robust Speech Processing as an Inverse Problem” by Richard J Mammone and Xiaoyu Zhang The performance of speech and speaker recognition systems is significantly affected by the acoustic environment The background noise level, the filtering effects introduced by the microphone and the communication channel dramatically affect the performance of recognition systems It is therefore critical that these speech recognition systems be capable of detecting the ambient acoustic environment continue and inverse their effects from the speech signal This is the inverse problem in robust speech processing that will be addressed in this section A general approach to solving this inverse problem is presented based on an affine transform model in the cepstrum domain

Chapter 28 is entitled “Inverse Problems, Statistical Mechanics and Simulated Annealing” by K Venkatesh Prasad In this section, a computational approach to 3-D coordinate restoration is pre-sented The problem is to obtain high-resolution coordinates of 3-D volume-elements (voxels) from observations of their corresponding 2-D picture-elements (pixels) The problem is posed as a com-binatorial optimization problem and borrowing from our understanding of statistical mechanics,

we show how to adapt the tool of simulated annealing to solve this problem This method is highly amenable to parallel and distributed processing

Chapter 29 is entitled “Image Recovery Using the EM Algorithm” by Jun Zhang and Aggelos K

Katsaggelos In this section, the image recovery/reconstruction problem is formulated as a maximum-likelihood (ML) problem in which the image is recovered by maximizing an appropriately defined likelihood function These likelihood functions are often highly non-linear and when some of the variables involved are not directly observable, they can only be specified in integral form (i.e., aver-aging over the “hidden variables”) The EM (expectation-maximization) algorithm is revised and applied to some typical image recovery problems Examples include image restoration using the Markov random field model and single and multiple channel image restoration with blur identifica-tion

Chapter 30 is entitled “Inverse Problems In Array Processing” by Kevin R Farrell Array processing uses multiple sensors to improve signal reception by reducing the effects of interfering signals that originate from different spatial locations Array processing algorithms are generally implemented via narrowband and broadband arrays, both of which are discussed in this chapter Two classical approaches, namely sidelobe canceler and Frost beam formers, are reviewed These algorithms are formulated as an inverse problem and an iterative approach for solving the resulting inverse problem

is provided

Chapter 31 is entitled “Channel Equalization as a Regularized Inverse Problem” by John F Doherty

In this section, the relationship between communication channel equalization and the inversion of

a linear system of equations is examined A regularized method of inversion is an inversion process

in which the noise dominated modes of the restored signal are attenuated Channel equalization is the process that reduces the effects of a band-limited channel at the receiver of a communication system A regularized method of channel equalization is presented in this section Although there are many ways to accomplish this, the method presented uses linear and adaptive filters, which makes the transition to matrix inversion possible

Chapter 32 is entitled “Inverse Problems in Microphone Arrays” by A.C Surendran The response

of an acoustic enclosure is, in general, a non-minimum phase function and hence not invertible In this section, we discuss techniques using microphone arrays that attempt to recover speech signals degraded by the filtering effect of acoustic enclosures by either approximately or exactly “inverting” the room response The aim of such systems is to force the impulse response of the overall system, after de-reverberation, to be an impulse function Beamforming and matched-filtering techniques (that approximate this ideal case) and the Diophantine inverse filtering method (a technique that provides an exact inverse) are discussed in detail

Chapter 33 is entitled “Synthetic Aperture Radar Algorithms” by Clay Stewart and Vic Larson

A synthetic aperture radar (SAR) is a radar sensor that provides azimuth resolution superior to that achievable with its real beam by synthesizing a long aperture by platform motion This section presents an overview of the basics of SAR phenomenology and the associated algorithms that are used

to form the radar image and to enhance it The section begins with an overview of SAR applications, historical development, fundamental phenomenology, and a survey of modern SAR systems It also presents examples of SAR imagery This is followed by a discussion of the basic principles of SAR image formation that begins with side looking radar, progresses to unfocused SAR, and finishes with focused SAR A discussion of SAR image enhancement techniques, such as the polarimetric whitening filters, follows Finally, a brief discussion of automatic target detection and classification techniques

is offered

Chapter 34 is entitled “Iterative Image Restoration Algorithms” by Aggelos K Katsaggelos In this section, a class of iterative restoration algorithms is presented Such algorithms provide solutions

to the problem of recovering an original signal or image from a noisy and blurred observation of it This situation is encountered in a number of important applications, ranging from the restoration

of images obtained by the Hubble space telescope, to the restoration of compressed images The successive approximation methods form the basis of the material presented in this section

The sample of applications and methods described in this chapter are meant to be representative

of the large volume of work performed in this field There is no claim of completeness, any omissions

of significant contributors or other errors are solely the responsibility of the section editor, and all praiseworthy contributions are due solely to the chapter authors