Virtual experiments and compressive sensing for subsurface microwave tomography

This chapter describes how to take advantage together from the compressive sensing (CS) theory and the virtual experiments (VE)-based linear inversion approach. The overall imaging strategy is made up of two steps. The first one is concerned with the solution of a "design equation" which recast the original scattering experiments into a set of "virtual" equivalent ones. The second one tackles the solution of a linearized inverse scattering problem by means of a CS-inspired approach. The chapter provides the mathematical formulation of the inverse scattering problem underlying the microwave imaging for the two dimensional scalar problem. It addresses the VE framework with emphasis on the adopted internal field approximation and on the possibility to counteract difficulties arising from aspect limitation of the measurement setup. The chapter presents basics of CS for microwave imaging formulation and outlines reconstruction results toward simulated data in two measurements setups, the surface ground-penetrating radar and the cross-borehole configurations, corresponding to reflected and transmitted field measurements.