Implementable multi-dimensional inverse scattering theory

Abstract

Mathematical and engineering aspects of direct and inverse scattering and diffraction problems posed in more than one dimension are considered. A unified approach is in­troduced. Descriptions of scalar linear wave motion that are commonly invoked when treating inverse problems are summarised and extended. Several original contributions (developed in association with others) to branches of inverse theory are presented. Two aspects of computed tomography (CT) are treated : first, previously neglected con­ sequences of the inevitable sampling of data are examined; second, account is taken of finite (and variable) detector resolution as well as attenuation (the coefficient being assumed constant, as is common) of the radiation on its passage through the body to the detectors, for single photon emission CT. A generalised volume source formulation of scattering/diffraction is developed; it is shown to lead to a sequence of useful approx­imate formulations which are potentially suitable bases for inversion algorithms. Two particular implementations of the Newton-Kantorovich approach to inverse scattering (in which the form of the scattering is iteratively refined, with the direct problem being solved at each iteration) are developed: first; the null-field method (or extended bound­ary condition) is treated; second, a numerical algorithm is devised and implemented for a global inverse theory (previously formulated only analytically). Four aspects of Fourier phase retrieval are investigated : first, the effect of choice of image support for complex images is examined; second, it is shown in an inverse scattering con­text, that extra information, likely to be available in practice, permits complex-valued image-forms to be readily recovered; third, an extension of the electron-microscopal Gerchberg-Saxton algorithm, adapted for the radio engineering problem of deducing an antenna's aperture distribution from only the magnitude of its far-field radiation pattern, is presented; fourth, a reconstruction is reported of a structure consistent with one of the widely-heralded recently-recorded (by Shechtman) electron diffraction patterns suggestive of quasi-crystalline specimens not exhibiting crystalline symmetry.