Inversion formulas for the linearized impedance tomography problem

TL;DR

This paper derives explicit inversion formulas for the linearized electrical impedance tomography problem in 2D, enabling reconstruction of conductivity from boundary measurements, even with partial data, using a trigonometric and polynomial basis approach.

Contribution

It introduces a novel explicit solution formula for the linearized inverse impedance tomography problem, including cases with partial boundary data, by expanding coefficients into trigonometric and Legendre-M"untz polynomials.

Findings

Explicit inversion formulas for the linearized problem.

Unique determination of conductivity from partial boundary data.

Lower-triangular representation of the parameter-to-data mapping.

Abstract

We consider the linearized electrical impedance tomography problem in two dimensions on the unit disk. By a linearization around constant coefficients and using a trigonometric basis, we calculate the linearized Dirichlet-to-Neumann operator in terms of moments of the conduction coefficient of the problem. By expanding this coefficient into angular trigonometric functions and Legendre-M\"untz polynomials in radial coordinates, we can find a lower-triangular representation of the parameter to data mapping. As a consequence, we find an explicit solution formula for the corresponding inverse problem. Furthermore, we also consider the problem with boundary data given only on parts of the boundary while setting homogeneous Dirichlet values on the rest. We show that the conduction coefficient is uniquely determined from incomplete data of the linearized Dirichlet-to-Neumann operator with an explicit solution formula provided.