A shape optimization approach for electrical impedance tomography with point measurements

TL;DR

This paper develops a shape optimization method for electrical impedance tomography using point measurements, extending the adjoint method to Banach spaces and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a novel shape derivative computation involving point evaluations in EIT, enabling improved reconstruction algorithms.

Findings

The method effectively reconstructs conductivities from point measurements.

Numerical results show the influence of measurement patterns on reconstruction quality.

The approach extends shape optimization techniques to handle Dirac measures in EIT.

Abstract

Working within the class of piecewise constant conductivities, the inverse problem of electrical impedance tomography can be recast as a shape optimization problem where the discontinuity interface is the unknown. Using Gr\"oger's $W^{1}_p$-estimates for mixed boundary value problems, the averaged adjoint method is extended to the case of Banach spaces, which allows to compute the derivative of shape functionals involving point evaluations. We compute the corresponding distributed expression of the shape derivative and show that it may contain Dirac measures in addition to the usual domain integrals. We use this distributed shape derivative to devise a numerical algorithm, show various numerical results supporting the method, and based on these results we discuss the influence of the point measurements patterns on the quality of the reconstructions.