Recovery of small perturbations of an interface for an elliptic inverse problem via linearization

Abstract

Electrical impedance tomography (EIT) is used to find the conductivity distribution inside a region using electrostatic measurements collected on the boundary of the region. We study a simple version of the general EIT problem, in which the medium has constant conductivity but might contain a buried object of unknown shape and different, but also constant, conductivity. Our linearization about an approximate shape of the buried object follows Kaup and Santosa and has the advantage that it is valid for large contrast in the conductivity. We present a procedure to reconstruct the object boundary in the case where we know the conductivities and the centre and radius of a good circular approximation of the object boundary, using analytic solutions to the forward problem for circular objects with known conductivity. Assuming that the unknown object boundary is star-shaped with respect to the centre of the circle and a small perturbation of the approximating circle, we develop a linearized relation between the output voltage data that result from fixed input currents, entering as parameters, and the interface, entering as variables. This relation is used to find the Fourier coefficients of the perturbation of the interface. At least two measurements are needed to determine all coefficients, and more can be used for a least-squares fit. The quality of the recovered perturbation depends on the input frequency and the frequencies of the perturbation. Low frequencies work best as input, and are most easily recovered, in that case even where the amplitude of the perturbation is not very small. Several dipoles, corresponding to very close pairs of electrodes to induce the current, can be used successfully as input as well.

The Explicit-Jump Immersed Interface Method: Finite Difference Methods for PDEs with Piecewise Smooth Solutions

“…the one-sided interpolations on the correct side of the interface. Let satisfies ∆u = 0 with [u] = 0 and u + n = ρu − n at the interface r = s. This is a special case of a general analytic solution for circular interfaces found in [14]. Figure 7 shows the exact solutions on the square [0, 1] × [0, 1] for ρ = 5000 and ρ = 1/5000, with s = 0.25 in both cases.…”

Reconstructing small perturbations of scatterers from electric or acoustic far‐field measurements

“…We consider only the two-dimensional case, the extension to three dimensions being obvious. In connection with our work, we should also mention the paper by Kaup and Santosa [6] on detecting corrosion from steady-state voltage boundary perturbations and the work by Tolmasky and Wiegmann [10] on the reconstruction of small perturbations of an interface for the inverse conductivity problem.…”

Conductivity interface problems. Part I: Small perturbations of an interface

“…Since the expansion carries precise information on the shape of the inclusion, we will show how it can be efficiently exploited for designing significantly better algorithms. In connection with the reconstruction of interfaces, we refer readers to [6,12,17]. This paper is organized as follows: In section 2 we derive higher-order terms in the asymptotic expansion of a certain boundary integral operator which appears in the representation of the steady-state voltage potential.…”