NDF and PSF Analysis in Inverse Source and Scattering Problems for Circumference Geometries

Abstract

This paper aims at discussing the resolution achievable in the reconstruction of both circumference sources from their radiated far-field and circumference scatterers from their scattered far-field observed for the 2D scalar case. The investigation is based on an inverse problem approach, requiring the analysis of the spectral decomposition of the pertinent linear operator by the Singular Value Decomposition (SVD). The attention is focused upon the evaluation of the Number of Degrees of Freedom (NDF), connected to singular values behavior, and of the Point Spread Function (PSF), which accounts for the reconstruction of a point-like unknown and depends on both the NDF and on the singular functions. A closed-form evaluation of the PSF relevant to the inverse source problem is first provided. In addition, an approximated closed-form evaluation is introduced and compared with the exact one. This is important for the subsequent evaluation of the PSF relevant to the inverse scattering problem, which is based on a similar approximation. In this case, the approximation accuracy of the PSF is verified at least in its main lobe region by numerical simulation since it is the most critical one as far as the resolution discussion is concerned. The main result of the analysis is the space invariance of the PSF when the observation is the full angle in the far-zone region, showing that resolution remains unchanged over the entire source/investigation domain in the considered geometries. The paper also poses the problem of identifying the minimum number and the optimal directions of the impinging plane waves in the inverse scattering problem to achieve the full NDF; some numerical results about it are presented. Finally, a numerical application of the PSF concept is performed in inverse scattering, and its relevance in the presence of noisy data is outlined.