Analysis of Reconstructions Obtained Solving $l^p$-Penalized Minimization Problems

Most of the inverse problems arising in applied electromagnetics come from an underdetermined direct problem, this is the case, for instance, of spatial resolution enhancement. This implies that no unique inverse operator exists; therefore, additional constraints must be imposed on the sought solution. When dealing with microwave remote sensing, among the possible choices, the minimum p-norm constraint, with 1 <; p ≤ 2, allows obtaining reconstructions in Hilbert (p = 2) and Banach (1 <; p <; 2) subspaces. Recently, it has been experimentally proven that reconstructions in Banach subspaces mitigate the oversmoothing and the Gibbs oscillations that typically characterize reconstructions in Hilbert subspaces. However, no fair intercomparison among the different reconstructions has been done. In this paper, a mathematical framework to analyze reconstructions in Hilbert and Banach subspaces is provided. The reconstruction problem is formulated as the solution of a p-norm constrained minimization problem. Two signals are considered that model abrupt and spot-like discontinuities. The study, undertaken in both the noise-free and the noisy cases, demonstrates that l
^p
reconstructions for 1 <; p <; 2 significantly outperform the l
²
ones when spot-like discontinuities are considered; when dealing with abrupt discontinuities, l
²
and lp reconstructions are characterized by similar performance; however, l
^p
reconstructions exhibit oscillations when the background is not properly accounted for.