A Well-Conditioned Non-Iterative Approach to Solution of the Inverse Problem

Abstract

A novel methodology leading to well-conditioned formulation of the inverse scattering problem is proposed. The solution of the inverse scattering problem is made possible through elimination of the inherent ill-posedness of the inverse source problem. The latter is achieved by staging of the imaging experiment in a medium with the Green's function exhibiting focusing properties. The method is shown to cast the inverse source problem into a well-conditioned matrix equation without addition of the non-physical regularization term to the pertinent integral equation. This is contrary to the conventional iterative optimization-like methods applied to regularized integral equation with non-directional Green's function. The method is shown to be applicable in both traditional diffraction limited imaging where the evanescent waves do not participate in the information transfer from the object to the observation region as well as in the metamaterial media where focusing goes beyond the diffraction limit with the aid of the evanescent wave propagation. The diffraction limited imaging prototype is implemented using a parabolic reflector providing desired focusing properties of the Green's function. The Raleigh criteria for diffraction limited resolution is revisited to demonstrate that upon availability of the medium providing sufficient focusing properties the resolution in the conventional diffraction limited microwave tomography can go up to half-wavelength. The implementation of algorithm in medium supporting evanescent wave propagation is demonstrated using the focusing Green's function of the Veselago lens.

Superlens Enhanced 2-D Microwave Tomography With Contrast Source Inversion Method

“…The full-rank matrix F S indicates well-posedness of the underlying inverse scattering problem formulation. In [20], it was shown that such well-posed formulation of the inverse problem allows for its direct solution in analogous manner to the solution of the forward scattering problem. For comparison, in Fig.…”

“…the boundedness of the solution in Tikhonov regularization [19]). In [20], [21], it was shown, however, that the rank-deficiency of the discretized inverse scattering operator and thus the ill-posedness of the inverse scattering problem can be eliminated in rigorous compliance with Maxwell's Equations. For that purpose, the kernel of the forward scattering operator must be the Green's function of a focusing media and the sensor positions must be properly situated with respect to the locations of the sources in the imaging domain.…”

Analysis of the Super‐Resolution Effect on Microwave Tomography

“…Okhmatovski et al () used an original method to noniteratively solve a nonlinear inverse scattered problem. They examined the advantage of using a Veselago lens in the imaging experiment to observe the super‐resolution using far‐field measured data only.…”

“…They used the following explanation for this effect: "even though … only scattered waves corresponding to propagating waves can be measured, the scattered waves contain high resolution information about the scatterer because of the evanescent-propagating waves conversion in a multiply scattered field." Okhmatovski et al (2012) used an original method to noniteratively solve a nonlinear inverse scattered problem. They examined the advantage of using a Veselago lens in the imaging experiment to observe the super-resolution using far-field measured data only.…”

The Effect of Antenna Incident Field Distribution on Microwave Tomography Reconstruction

“…Within the framework of these two approaches, some of the utilized techniques are (i ) increasing the number of antennas (or, probes) [1], (ii ) using multiple-frequency data sets [2], (iii ) using different boundary conditions [3], (iv ) using an appropriate Green's function [4], (v ) simultaneous use of transverse magnetic and electric data sets [5], (vi ) using appropriate data calibration techniques [6], (vii ) using more effective inversion algorithms and regularization techniques [7,8], (viii ) accurate MWT system modeling [9], (ix ) using a priori information [10], etc.…”