Direct regularized reconstruction for the three-dimensional Calder\'on problem

TL;DR

This paper introduces a regularized, stable, and direct reconstruction method for the 3D Calderón problem in Electrical Impedance Tomography, effectively handling noisy data and demonstrating promising numerical results.

Contribution

It develops a practical regularization strategy for the 3D Calderón problem using a truncated scattering transform, advancing towards real-world applications.

Findings

Method is robust to small data perturbations

Numerical tests show improved performance over theoretical predictions

Reconstruction remains stable with noisy data

Abstract

Electrical Impedance Tomography gives rise to the severely ill-posed Calder\'on problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The uniqueness and stability questions for the three-dimensional problem were largely answered in the affirmative in the 1980's using complex geometrical optics solutions, and this led further to a direct reconstruction method relying on a non-physical scattering transform. In this paper, the reconstruction problem is taken one step further towards practical applications by considering data contaminated by noise. Indeed, a regularization strategy for the three-dimensional Calder\'on problem is presented based on a suitable and explicit truncation of the scattering transform. This gives a certified, stable and direct reconstruction method that is robust to small perturbations of the data. Numerical tests on simulated noisy data illustrate the feasibility and regularizing effect of the method, and suggest that the numerical implementation performs better than predicted by theory.