| A mature and difficult problem, still preoccupying many research communities in different application areas, is the recovery of a quantitative image of some unknown penetrable strongly scattering object. In most fields, such as ground penetrating radar, seismic and medical applications, the problem is compounded by the availability of only limited angle and noisy data. One of the more common approximate solution methods is based on diffraction tomography that relies on the first Born approximation method, which limits applications to weakly scattering situations. More sophisticated methods are typically iterative in nature, computationally intense and may not converge. We have studied an alternative nonlinear filtering approach and developed a new way to implement it, as well as evaluating different filter functions to find an optimal form. We have applied this approach to a number of classes of objects and developed a user-friendly scattered field simulator as a resource for this and related inverse scattering problems. We also re-investigated the widely accepted limitations of the first Born approximation and found that when close to a scattering resonance, the first Born approximation can yield a good estimate of the object's scattering cross section. Tied to all of these imaging applications is the issue of limited data: how many sources and how many receivers are required for a given quality and reliability of the resulting image. We took a fundamental look at this issue in terms of the number of degrees of freedom of the entire source-measurement domain and deduced clear guidelines on the minimum data sets necessary that should be measured, in order to expect a reasonable image. |
