A hybrid finite element/finite difference method for reconstruction of dielectric properties of conductive objects

TL;DR

This work addresses the coefficient inverse problem of reconstructing spatially distributed dielectric properties in conductive media from time-dependent boundary measurements by solving Maxwell's equations. It develops a hybrid FE/FD domain-decomposition framework combined with a Tikhonov-regularized, Lagrangian optimization and an adaptive conjugate gradient algorithm to efficiently recover the permittivity $\varepsilon(x)$ and, with future work, the conductivity $\sigma(x)$. Key contributions include a stable forward-model formulation with FE/FD coupling, a variational approach to obtain gradients via adjoint equations, and an adaptive mesh refinement strategy (ACGA) that focuses computational effort where $|h\varepsilon|+|h\sigma|$ is large. Demonstrations on anatomically realistic Wisconsin breast phantoms show qualitative and quantitative recovery of relative permittivity under known conductivity, highlighting potential for microwave breast imaging and tumor detection.

Abstract

The aim of this article is to present a hybrid finite element/finite difference method which is used for reconstructions of electromagnetic properties within a realistic breast phantom. This is done by studying the mentioned properties' (electric permittivity and conductivity in this case) representing coefficients in a constellation of Maxwell's equations. This information is valuable since these coefficient can reveal types of tissues within the breast, and in applications could be used to detect shapes and locations of tumours. Because of the ill-posed nature of this coefficient inverse problem, we approach it as an optimization problem by introducing the corresponding Tikhonov functional and in turn Lagrangian. These are then minimized by utilizing an interplay between finite element and finite difference methods for solutions of direct and adjoint problems, and thereafter by applying a conjugate gradient method to an adaptively refined mesh.

Figures (4)

• Figure 1: Picture of domain decomposition $\Omega := {\Omega_{\mathrm{FDM}}} \cup {\Omega_{\mathrm{FEM}}}$.
• Figure 2: 2D presentation of media numbers for breast phantom with $\rm ID\_012204$ of database wisconsin. Left figure presents media numbers shown on the original mesh. Right figure shows media numbers on the sampled mesh where the mesh size was $8h$ , $h$ is the mesh size of the original mesh.
• Figure 3: Dielectric properties at 6GHz of breast phantom with $\rm ID\_012204$ of wisconsin taken in our computations. Left images: $\varepsilon_r$, right images: $\sigma$. Top row presents original images on the mesh with number of nodes $34 036 992$. Bottom row presents sampled values of $\varepsilon_r$ and $\sigma$ on the mesh with number of nodes $63492$, where the mesh size was $8h$ , $h$ is the mesh size of the original mesh.
• Figure 4: Left figure: exact isosurface of $\varepsilon_r$ and right figure: finite element reconstruction $\varepsilon_{r_h}$ (outlined in yellow color) corresponding to tissue types fibroconnective/glandular 2,3 obtained on two times refined finite element mesh ${K_{h_2}}$. The noise level in the data is $\delta= 10\%$.