A simple method using Morozov's discrepancy principle for solving inverse scattering problems

Abstract

This paper is a continuation of earlier research in which a simple inversion scheme was given for inverse scattering problems in the resonance region which is easy to implement and is relatively independent of the geometry and physical properties of the scatterer. The purpose of the paper is to give new and improved theorems establishing the mathematical basis of this method and to show how noisy data can be treated using Morozov's discrepancy principle where the regularization parameter is a function of an auxiliary parameter appearing in the inversion scheme.

An Approximation Property of Importance in Inverse Scattering Theory

“…A recently developed method for solving this inverse scattering problem is the linear sampling method [1,[3][4][5]. In this method, a parameter y ∈ R 2 is introduced and, for each fixed y, a solution g = g(·, y) ∈ L 2 (Ω) is sought to the far-field equation …”

Shape Reconstruction via Equivalence Principles,constrained Inverse Source Problems and Sparsity Promotion

“…Notably, there is no need of processing the data separately for each frequency and putting together the different indicators, as usually done e.g., in LSM or FM. With respect to the latter, which just entail the computation of the singular value decomposition of the data matrix [10][11][12], the proposed method is more onerous from a computational point of view. In fact, it involves the solution of a constrained optimization problem which relies on an iterative procedure.…”

“…In fact, sampling methods, such as LSM and FM [11][12][13][14], require a single frequency multiview (multistatic) configuration, while the Physical Optics based approach in [19,42] is based on single-view multifrequency experiments.…”

“…To test the performances of the method, we compare the obtained results with the standard LSM indicator, widely adopted in microwave imaging [11,12].…”

“…Among them, the most popular qualitative approaches are the multiple signal classification (MUSIC) for point-like targets [5][6][7], decomposition of time reversal operator (DORT) and time reversal (TR) based methods for small size objects [8,9], linear sampling method (LSM) [10][11][12], factorization method (FM) [13,14] and point source method [15] for generic scatterers. These methods rely on an indicator function computed on the (properly sampled) imaging domain through an auxiliary linear problem, whose values determine whether the tested point lies inside or outside the scatter.…”