Three-Dimensional Microwave Imaging in $L^{p}$ Banach Spaces: Numerical and Experimental Results

Microwave Imaging by Means of Lebesgue-Space Inversion: An Overview

“…Compressive sensing strategies, which allow to retrieve sparse solutions, have been proposed recently to mitigate this drawback. More recently, regularization methods developed in the framework of the regularization theory in the more general Banach spaces also have been introduced in the mathematical literature [27][28][29] and investigated for microwave imaging applications [30][31][32][33].…”

“…imaging applications [30][31][32][33]. Due to the geometrical properties of specific Banach spaces, these regularization methods allow one to obtain solutions endowed with lower over-smoothness and ringing effects, which results in a better localization of targets and restoration of the discontinuities between different media.…”

Microwave Imaging of 3D Dielectric Structures by Means of a Newton-CG Method in lp Spaces

“…Such inversion procedure has been firstly proposed for microwave imaging of 2D structures in [42], and it is extended for the first time to 3D settings in this paper. The main advantage of the regularization in the more general Banach spaces is that they allow, for 1 < p < 2, to reduce the oversmoothing and ringing effects that are usually associated to the conventional reconstructions in l 2 Hilbert spaces [40,44]. The key point for extending the standard CG inversion scheme to Banach spaces is represented by the duality maps J C , J E , and J C * that are used in the update formulas in Figure 2 [51].…”

“…It is important to remark that the main difference with respect to the approach presented in [44] is related to the adoption of the CG inner solver in each Newton linearization step instead of the basic Landweber algorithm. Such a modification allows to exploit the faster convergence and good regularization properties of the CG method, which still hold in l p spaces [43].…”

“…Despite the increased complexity of the l p space approaches, the obtained results in both 2D [41,42] and 3D configurations [44] are promising. So far, only the Landweber solver has been extensively validated for the use inside l p space inexact Newton techniques, but its convergence speed sometimes appears very low, leading to a considerable number of required iterations of the method.…”