Raymond M. Fitzgerald

Image restoration and resolution enhancement

The ill-posed problem of restoring object information from finitely many measurements of its spectrum can be solved by using the best approximation in Hilbert spaces appropriately designed to include a priori information about object extent and shape and noise statistics. The procedures that are derived are noniterative, the linear ones extending the minimum-energy band-limited extrapolation methods (and thus related to Gerchberg–Papoulis iteration) and the nonlinear ones generalizing Burg’s maximum-entropy reconstruction of nonnegative objects.

Spectral Estimators that Extend the Maximum Entropy and Maximum Likelihood Methods

The theory of best linear approximation in weighted

RECONSTRUCTION FROM PARTIAL INFORMATION WITH APPLICATIONS TO TOMOGRAPHY

L^2

An approximation theoretic approach to maximum entropy spectral analysis

spaces is used to obtain a general procedure, the PDFT, for linearly reconstructing the Fourier transform from sampled data. The PDFT can be used either directly to reduce sidelobe structure and to extrapolate the data or indirectly to obtain high resolution spectral estimators. The direct and indirect PDFT include as special cases many of the commonly used spectral techniques, including Burg’s maximum entropy method, Capon’s maximum likelihood method, the spectral estimators based on bandlimited extrapolation, the eigenvalue/eigenvector methods for detecting sinusoids in noise (Pisarenko method, Schmidt’s MUSIC, eigenvector power beamforming), and the best linear unbiased estimator (BLUE) for regression coefficients. By exploiting their relationship to the linear PDFT, these nonlinear techniques can be analyzed in terms of linear approximation theory. In addition to providing a unifying formulation for many different spectral estimators, the PDFT...

Spectral distortion in sampling rate conversion by zero-order polynomial interpolation

The problem of reconstruction from projections in Hilbert space is treated. An axiomatic basis is considered which leads to techniques which provide improved reconstructions by incorporating prior knowledge to tailor the Hilbert space to the problem at hand. When applied to reconstruction of the Fourier transform of a function sampled at finitely many discrete points, the procedures lead to previously derived optimal estimation techniques. When applied to x-ray tomography, the procedures lead to new reconstruction techniques which are shown to include as a special case the minimum energy reconstruction of Logan and Shepp [Duke Math. J., 42 (1975), pp. 645–659].

Linear and nonlinear estimators for one- and two-dimensional Fourier transforms

It is shown that the theory of best approximation in weighted mean square can be used to provide a mathematical foundation for the analysis and extension of the maximum entropy method of spectral analysis.

Time-limited sampling theorems for band-limited signals (Corresp.)

For such applications as digital beamforming in sonar and sampling rate conversion in digital audio systems, there is interest in fast, approximate interpolation techniques. Computationally, the simplest of these is to approximate the desired time series sample by the available sample which is nearest in time. This procedure is variously known as sample-and-hold interpolation, zero-order polynomial interpolation, or selective subsampling. The effect of this approximation on signal-to-noise ratio has already been addressed in the literature. This correspondence extends those previous analyses by deriving the spectral characteristics of the noise generated. It is shown that, because the timing errors involved in zero-order polynomial interpolation are not random, the effect on the input spectrum is not simply to broaden it by the transfer of power from narrowband signal components to broadband noise, from narrowband signals to narrowband noise or distortion components. >

High‐resolution beamforming with oversampled arrays

Previously presented estimators for one-dimensional Fourier transforms are modified to provide two-dimensional analogs which are computationally feasible for large systems of equations. Near-optimal, two-dimensional estimators are obtained by an iterative approximation method which involves the inversion of block lower triangular matrices.

Image Restoration And Resolution Enhancement

A procedure is given for the recovery of a band-limited signal from any set of samples by which it is uniquely determined. Containing, as a special case, Shannons sampling theorem, it applies even to situations, such as time-limited sampling, for which no sampling basis exists, and extends to those eases the essential features of the Shannon representation needed for approximation from finite data. A second reconstruction, based on sequences of time-limited equispaced sample values, is also presented, and the extension to stochastic processes is discussed.

An approximation theoretic approach to synthetic aperture processing

Interpreting plane‐wave beamforming as wave‐vector spectrum estimation, we show that the maximum entropy spectrum estimation technique (MEM) indicates the presence of false targets when used to beamform a spatially oversampled array. The source of this instability lies in the presence in the MEM technique of the prior estimate that the wave‐vector spectrum is white over the Nyquist band. By the use of a new, adaptive, high‐resolution spectral estimator, which includes MEM as a special case, the instability is removed. The results are of particular significance for short arrays operated at low frequencies.