Ανάπτυξη υπολογιστικών μεθόδων επίλυσης του τρισδιάστατου προβλήματος αντίστροφης σκέδασης

Abstract

In the inverse scattering problem, the aim is to reconstruct the unknown spatial distribution of the indexes of wave propagation in an area of interest. The problem is of great practical interest in diverse scientific fields. The mathematical and computational complexity renders the development of efficient computational methods a demanding task. The problem is mathematically characterized by ill-posedness while the computational burden is heavy especially in the three dimensional case. In this work, the problem was addressed exclusively in its three dimensional formulation when the field can be described by the scalar wave equation. The aim was to develop efficient computational methods for solving the inverse scattering problem placing emphasis on the issue of the required data for an accurate reconstruction and on the computational requirements in order to ensure the applicability of the developed methods in practical problems. In the first part, the 3-D extensions of the Direct Fourier Interpolation (DFI) and Filtered Backpropagation (FBP) algorithms, are presented and the problem of the data needed for a full 3-D reconstruction is investigated. It is shown that the restriction of the plane wave illumination in a single plane, results in the omission of the contribution of certain spatial frequencies. The effect of this omission is studied by comparing the results of reconstruction with and without data obtained from other incident directions which fill the spatial frequency domain. It is concluded that the use of data obtained for incident direction only in a single plane is restrictive as regards the range of scatterers that can be imaged accurately. An improved version of the 3-D FBP algorithm is then presented which is able to incorporate data collected from more than one illumination planes and thus improving the quality of the reconstruction. In the second part, the inverse scattering problem in the time domain is examined. A novel algorithm based on a simplified version of the Rytov approximation is developed. The algorithm is based on a non-linear optimization iterative process where a discrepancy functional is minimized. In every iteration, a "slice" of the area of interest perpendicular to the propagation direction of the incident field is reconstructed. The algorithm has minimum data requirements regarding the number of the incident pulses and the points where the scattered field is measured. It is shown that it can reconstruct reasonably well the surface characteristics of strong scatterers while the accuracy of the results for deeper levels depends mainly on the contrast of the scattering objects.