On the analysis of image reconstruction based on direct and inverse scattering methods

This thesis is focused on Fourier based direct and inverse scattering methods. Specifically we have considered: (1) Studying further more traditional approximate methods for direct and inverse scattering namely Born, Rytov and some of their consequences such as: (a) Wait and Born approximation for the scattered field from a low contrast cylinder of infinite length at normal incidence, (b) application of the optical theorem to the Born series in the direct scattering problem to arrive at a derived sequence such that scattering cross section vanishes for N terms of the sequence, even though the total scattering cross section is non-vanishing. (2) Introduction of a new procedure for inverse scattering which we call "Arc shifting" giving data in an annulus in the spatial frequency domain and some of its consequences; such as the number and region of distribution of data points, and band-pass filtering. An example of how to optimally locate the annulus to recover the index of refraction of a homogeneous scatterer surrounded by a homogeneous background illuminated by a plane wave is illustrated. (3) Providing an overview and analysis of the exact inverse scattering method proposed by Ramm for the three dimensional reconstruction of compactly supported scatterer and some of its consequences, such as the domain of applicability of his method, as a means of classifying scattering potentials.; Also it is intended to point out some of the unresolved issues and open problems, associated with the above topics, which require further research and investigation. The emphasis of (1) is more on the direct problem and its implications for the inverse problem. The emphasis of (2) & (3) are on the inverse problem.