Michael A. Fiddy

The Phase Problem

The paper discusses the use of the theory of entire functions for solving the phase problem. In all practical cases only three forms of logarithmic Hilbert transform could possibly be required. The paper defines them and analyses their applicability. A generating form is also put forward for cases of possible theoretical interest. The uniqueness of the phase obtained from a logarithmic Hilbert transform is investigated and the difficulties due to the presence of zeros in the complex plane are discussed. Methods are put forward for both the removal of the zeros and, when this is not possible, for locating them in order to include their effect. The paper analyses known experimental methods for phase determination from the point of view of the theory presented and highlights their unique character.

Regularized image reconstruction using SVD and a neural network method for matrix inversion

Two methods of matrix inversion are compared for use in an image reconstruction algorithm. The first is based on energy minimization using a Hopfield neural network. This is compared with the inverse obtained using singular value decomposition (SVD). It is shown for a practical example that the neural network provides a more useful and robust matrix inverse. >

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.

Detector for an optical-fiber acoustic sensor using dynamic holographic interferometry.

It is shown that the acoustically induced phase modulation of coherent light propagating within a multimode optical fiber may be linearly detected by an interferometer employing an unmodulated replica of the fiber beam as the reference. It is proposed that a dynamic hologram provide such a reference. This technique allows a performance comparable with that of a monomode interferometric sensor to be obtained, but with considerably relaxed alignment tolerances.

Waveguide hologram fingerprint entry device

The recording and entry of fingerprints into the local and na- tional databases is increasingly relying on optics to simplify and speed up this process. A prototype device for the entry of fingerprints that uses a waveguide hologram as part of the scheme for illuminating the finger is described. The use of the waveguide hologram enables reduction in the size, weight, and energy consumption of a fingerprint entry device (or live scan device). The entry device then becomes highly portable and thus useful in many office and field applications. Coupled with electronic or optical processing and storage along with telephone or radio trans- mission of the captured fingerprints, rapid identification of individuals be- comes a realizable goal. The components required for the prototype ho- lographic fingerprint entry device (HoloFED) including the light source, the illumination waveguide hologram, the imaging system and the stor- age, and processing system are discussed. Examples of fingerprints captured are shown. The trade-offs necessary for the implementation of the prototype are object-to-image distance, optical efficiency, weight, and cost.

Reconstruction of the Refractive Index from Experimental Backscattering Data Using a Globally Convergent Inverse Method

The problem to be studied in this work is within the context of coefficient identification problems for the wave equation. More precisely, we consider the problem of reconstruction of the refractive index (or equivalently, the dielectric constant) of an inhomogeneous medium using one backscattering boundary measurement. The goal of this paper is to analyze the performance of the globally convergent algorithm of Beilina and Klibanov on experimental data collected using a microwave scattering facility at the University of North Carolina at Charlotte. The main challenge in working with experimental data is the huge misfit between these data and computationally simulated data. We present data preprocessing steps to make the former somehow look similar to the latter. Results of both nonblind and blind targets are shown that indicate good reconstructions even for high contrasts between the targets and the background medium.

Denoising during optical coherence tomography of the prostate nerves via wavelet shrinkage using dual-tree complex wavelet transform

The dual-tree complex wavelet transform (CDWT) is a relatively recent enhancement to the discrete wavelet transform (DWT), with important additional properties. It is nearly shift-invariant and directionally selective in two and higher dimensions. In this letter, a locally adaptive denoising algorithm is applied to reduce speckle noise in time-domain optical coherence tomography (OCT) images of the prostate. The algorithm is illustrated using DWT and CDWT. Applying the CDWT provides improved results for speckle noise reduction in OCT images. The cavernous nerve and prostate gland can be separated from discontinuities due to noise, and image quality metrics improvements with a signal-to-noise ratio increase of 14 dB are attained.

Compressive Imaging Sensors

This paper describes a compressive sensing strategy developed under the Compressive Optical MONTAGE Photography Initiative. Multiplex and multi-channel measurements are generally necessary for compressive sensing. In a compressive imaging system described here, static focal plane coding is used with multiple image apertures for non-degenerate multiplexing and multiple channel sampling. According to classical analysis, one might expect the number of pixels in a reconstructed image to equal the total number of pixels across the sampling channels, but we demonstrate that the system can achieve up to 50% compression with conventional benchmarking images. In general, the compression rate depends on the compression potential of an image with respect to the coding and decoding schemes employed in the system.

Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm

An approximately globally convergent numerical method for a 1D coefficient inverse problem for a hyperbolic PDE is applied to image dielectric constants of targets from blind experimental data. The data were collected in the field by the Forward Looking Radar of the US Army Research Laboratory. A posteriori analysis has revealed that computed and tabulated values of dielectric constants are in good agreement. Convergence analysis is presented.

Image estimation from scattered field data

The purpose of this paper is to reappraise the linearizing methods frequently used to solve inverse scattering problems. We describe inversion algorithms based on the Born and the Rytov approximations and the nature of the distortions obtained in the reconstructions when using them. We present extensions of these methods, namely, the distorted‐wave Born and the distorted‐wave Rytov approximations, which incorporate prior knowledge about part of the scattering structure. A method for inverting scattered field data using these distorted‐wave approximations is described, which retains the computational simplicity of the Born and the Rytov techniques. Some examples of their use with simulated and real data are given. A further extension of our distorted‐wave formalism, which leads to improvements of the reconstructed image, is suggested. This entails a spectral estimation procedure also based on the incorporation of prior knowledge about the scatterer. This spectral estimation procedure can be useful for interpolation of scattered field data as well as resolution enhancement.