NON-ITERATIVE EIGENFUNCTION-BASED INVERSION (NIEI) ALGORITHM FOR 2D HELMHOLTZ EQUATION

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 whole-domain 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).