Regularization by nonlinear low-pass filtering

The solution of an inverse problem means describing algorithmic steps for recovering the unknown object approximately from given noisy data. Furthermore, the recovery process has to be stabilized against measurement noise and modelling errors. Such stabilization is called regularization. The picture below shows the conceptual situation for the case of EIT.

Here are the official requirements that a regularization scheme must satisfy for being properly stabilized. In short, the worst-case reconstruction error should become small in the asymptotic limit of small noise.

Regularization of EIT is only partially understood currently. There are three main methodologies, namely Bayesian inversion, variational regularization, and problem-specific regularization, of which two latter ones are discussed here. Variational regularization struggles with convergence rates (in theory) and local minima (in practice). Problem-specific regularization (with explicit convergence speed in a Banach space norm) has been proved for twice differentiable conductivities, but the goal would be discontinuous conductivities.

Knudsen K, Lassas M, Mueller J L and Siltanen S 2009,
Regularized D-bar method for the inverse conductivity problem.
Inverse Problems and Imaging 3(4), pp. 599-624. PDF (408 KB)

This and the following slides explain problem-specific regularization of EIT based on low-pass filtering a nonlinear Fourier transform.

Adrian Nachman showed in his legendary 1996 Annals of Mathematics article that one can use the infinite-precision EIT data, namely the Dirichlet-to-Neumann map, to compute the nonlinear Fourier transform of a conductivity. The process involves solving a boundary integral equation (BIE).

Jennifer Mueller, David Isaacson and I implemented Nachman's proof as a computer algorithm and observed that the evaluation of the nonlinear Fourier transform breaks down for high frequencies because of measurement noise. This leads to a bad reconstruction. Therefore, a low-pass filtering is needed, as shown in the next slide.

Siltanen S, Mueller J L and Isaacson D 2000,
An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem,
Inverse Problems 16, pp. 681-699. PDF 845 KB

Erratum: An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem,
Inverse problems 17, pp. 1561-1563. PDF (54 KB)

Knudsen K, Lassas M, Mueller J L and Siltanen S 2009,
Regularized D-bar method for the inverse conductivity problem.
Inverse Problems and Imaging 3(4), pp. 599-624. PDF (408 KB)