In the past several years researchers have become increasingly more interested in effective numerical methods for image reconstruction which would deal with highly scattered radiation, as opposed to the conventional techniques of computed tomography. A major practical attraction of such methods lies in a number of important applications which range from the early female breast cancer diagnosis and underwater mine search ( i.e. diffuse imaging) to the ``classical'' inverse problems in geophysics (i.e. acoustical imaging).
These methods are expected to work with n-dimensional (n=2,3), rather than with 1-dimensional Inverse Scattering Problems (ISP). Obviously, the multi-dimensionality is a major obstacle in the development of such algorithms.
In this talk, we will present a novel approach to this challenging problem derived recently by ourselves [1,2]. We call this procedure ``Carleman's Weight Method'' (CWM). CWM works with ISPs for both hyperbolic and parabolic equations which model perfectly acoustical and diffuse imaging respectively. One of the attractive features of CWM is that rigorous global convergence results are proven.
Also, we will present our numerical resutls for a locally convergent version of CWM (i.e. Newton's Method) which deals with diffuse image reconstruction. It should be pointed out that these results were obtained for two ranges of parameters which perfectly model the situation of both an early breast cancer diagnosis and underwater mine search using ultrafast laser pulse propagation. Our computational experience shows a good potential of CWM. In particular, its computational complexity is of order several magnitudes less than that of many competing algorithms.
This is joint work with Thomas R. Lucas and Robert M. Frank.