Inverse Source Problem for a Host Medium Having Pointlike Inhomogeneities

The reconstruction of a source embedded within a multipath environment, which is created by inserting a grid of point scatterers in the scene, is addressed. In particular, the source Fourier spectrum is assumed known so that the focus here is on the reconstruction of the spatial support. As well documented, multipath can allow for resolution improvement. However, it also gives rise to artifacts when a backpropagation-like imaging is adopted. In this paper, we study in detail how resolution improvement and artifacts depend on the grid layout by employing a weighted backpropagation algorithm. More in detail, stationary phase arguments are used to predict the reconstruction leading order terms to which resolution improvement is linked. Moreover, it is shown that artifacts are mainly due to high-order terms and are dependent on the point scatterers' arrangement. The nature of such artifacts is studied, and a simple way to mitigate their role (without resolution loss) is introduced; it consists in a suitable nonuniform grid arrangement with a "hole in the center." Backpropagation is then compared with an inverse filtering imaging based on the truncated singular value decomposition (TSVD) of the radiation operator. It is shown that the TSVD is less prone to artifacts and can, in principle, allow for a higher resolution improvement. However, when model error (due to multiple scattering between the elements of the grid) and/or noise corrupt data, backpropagation performs definitely better. The theoretical findings are supported by an extensive numerical analysis. In particular, to keep the figures simple, we consider only 2-D cases.