A Banach Space Regularization Approach for Multifrequency Microwave Imaging

Abstract

A method for microwave imaging of dielectric targets is proposed. It is based on a tomographic approach in which the field scattered by an unknown target (and collected in a proper observation domain) is inverted by using an inexact-Newton method developed inLpBanach spaces. In particular, the extension of the approach to multifrequency data processing is reported. The mathematical formulation of the new method is described and the results of numerical simulations are reported and discussed, analyzing the behavior of the multifrequency processing technique combined with the Banach spaces reconstruction method.

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

“…, F. Subsequently, a subscript ω is added to the frequency-dependent functions and operators to specify at which frequency they refer. However, since the contrast function depends upon the frequency, it is necessary to modify the problem formulation [31,41]. To explain, let us assume that the dielectric permittivity and the electric conductivity do not depend on the frequency (i.e., dispersion is neglected).…”

“…Taking the point of view of applications, a sort of compromise between these trends has to be found. Some indications have been derived from numerical and experimental analyses, in which this parameter has been studied with regard to different target typologies, their dielectric properties, size, and the amount of data signal-to-noise ratio [30,41]. Despite this, with the tools described so far, only an a-posteriori selection of the optimal p has been done.…”

The Cylindrical Wave Approach for the Electromagnetic Scattering by Targets behind a Wall

“…The results in Figure 5b show the reduction in amplitude for lateral positions of the source, for a wall with L = 20 and εr1 = 4. This extension of the method to a line source excitation allows modeling of the multistatic acquisition employed in multifrequency approaches [24], as explored in recent algorithms for the imaging of targets in a through-wall environment [25].…”

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

“…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.…”

“…Another key issue is the accuracy of the provided solution, frequently affected by ringing phenomena and oversmoothing effects when the reconstruction procedure is formulated in the standard mathematical framework, that is, in the framework of Hilbert spaces. In order to overcome the latter problem, compressive sensing and sparsitypromoting techniques [36][37][38][39], as well as Banach space formulations, have been proposed, the last with both Landweber- [40,41] and conjugate gradient-based inner solvers [42]. As regards Banach space methods and in particular l p space approaches, the lack of a dot product inducing a complete space for p ≠ 2 does not allow to use spectral tools such as the singular value decomposition, and therefore, much more involving convex analysis tools need to be used to characterize the inversion procedures.…”