High contrast microwave tomography using topology optimization techniques
Journal of Computational and Applied Mathematics
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Microwave tomography is a technique in which microwaves illuminate a specimen, and measurements of the scattered electrical field are used to determine and depict the specimen's dielectric and conductive properties. This article presents a new method to perform the reconstruction. The reconstruction method is illustrated by assuming time harmonic scattering in two space dimensions in a setup tailored for medical applications. We prove that the resulting constrained nonlinear least-squares problem admits a solution. The governing Helmholtz equation is discretized by using the finite-element method, and the dielectric properties are allowed to attain different values at each element within a given region. The reconstruction algorithm uses methodologies borrowed from topology optimization of linearly elastic structures. Numerical examples illustrate the reconstruction method in a parameter range typical for human tissue and for the challenging case where the size of the object is in the same order as the wavelength. A reasonable estimate of the dielectric properties is obtained by using one observation per 20 unknowns when the permittivity is allowed to vary continuously within a given interval. Using a priori information that the permittivity attains only certain values results in a good estimate and a sharp image. As opposed to topology optimization for structures, there is no indication of mesh dependency and checkerboarding when forcing the permittivity to attain discrete values.