Non-Iterative Eigenfunction-Based Inversion (Niei) Algorithm for 2d Helmholtz Equation

Abstract

A non-iterative inverse-source solver is introduced for the 2D Helmholtz boundary value problem (BVP). Microwave imaging within a chamber having electrically conducting walls is formulated as a time-harmonic 2D electromagnetic field problem that can be modelled by such a BVP. The novel inverse-source solver, which solves for contrast sources, is the first step in a two-stage process that recovers the complex permittivity of an object of interest in the second step. The unknown contrast sources, as well as the (permittivity) contrast, are represented using the eigenfunction basis associated with the chamber's shape; canonical shapes allowing for analytically defined eigenfunctions. This wholedomain eigenfunction basis allows the imposition of constraints on the contrast-source expansion at virtual spatial points or contours outside the imaging domain. These constraints effectively regularize the inverse-source problem and the result is a well-conditioned matrix equation for the contrast-source coefficients that is solved in a least-squares sense. The contrast-source coefficients corresponding to different illuminating fields are then utilized to recover the contrast expansion coefficients using one more well-conditioned matrix inversion. The performance of this algorithm is studied using a series of synthetic test problems. The results of this study are promising as they compare very well with, and at times out-perform, state-of-the-art inversion algorithms (both in terms of reconstruction quality and computation time).

Demonstration of Quantitative Microwave Imaging Using an Ideal Veselago Lens

“…Most ISTP research has focused on introducing some form of regularization in the mathematical formulation or algorithmic solution of the problem [1], [2], [20]- [24]. The non-linearity of the ISTP is mainly handled by utilizing iterative algorithms, although some non-iterative methods can be considered as typically low-resolution techniques, [25]. The ISTP iterative algorithms attempt to minimize some cost-functional that incorporates the measured data with a model that predicts contrast and/or contrast sources.…”

MiPhDUO: microwave imaging via physics-informed deep unrolled optimization

“…The non-linearity of ISPs is instead related to mutual and self-interactions between the unknown scatterers. To face these difficulties, different methods have been proposed in the literature [9][10][11][12][13][14][15][16][17][18][19][20][21]. By leaving aside the so-called qualitative methods [22], classical quantitative reconstruction methods can be roughly classified into approximated methods (usually based on linearizations) and 'full-wave' methods, which mean to deal with the problem in its full non-linearity.…”

Analysis of a Nonlinear Technique for Microwave Imaging of Targets Inside Conducting Cylinders

“…As to the solution of microwave inverse scattering problems, innovative techniques are continuously proposed by the research community [16][17][18][19][20][21][22]. It is also worth noting that some specific approaches have been investigated for the imaging inside metallic enclosures, such as eigenfunction-based inversion algorithms [23,24]. Among the various possible inversion schemes, Newton-based methods look promising in many contexts, for their effectiveness in dealing with the intrinsic nonlinearity of the problem at hand [25][26][27].…”