Qualitative Methods in Inverse Scattering Theory

Abstract

A major problem in the use of ultrasound or microwaves for purposes of nondestructive testing or medical imaging is the computational complexity of solving the inverse scattering problem that arises in such applications. This is due to the fact that in order to achieve satisfactory resolution and sufficient penetration of the wave into the material it is often necessary to use frequencies in the resonance region. In this case the inverse scattering problem is not only improperly posed but also nonlinear and even in the case of two dimensions the time needed to solve such problems can be prohibitive. To date the time consuming nature of the problem has mainly been dealt with by the introduction of various innovative schemes that avoid the use of volume integral equations and instead rely on finite difference or finite element methods (cf. [5], [8]). However, for large scale problems (for example those involving imaging of the human body) the problem of computational complexity remains a serious problem for any practitioner. In this paper we would like to propose a different approach to this problem that, although still in its infancy, has the promise of providing rapid solutions to a number of inverse scattering problems of practical significance.

This research was supported in part by a grant from the Air Force Office of Scientific Research.