Propagation of singularities for linearised hybrid data impedance tomography

Abstract

For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit propagating singularities under certain non-elliptic conditions, and the associated directions of propagation are precisely identified relative to the directions in which ellipticity is lost. The same result is found in the setting for the corresponding normal formulation of the scalar pseudo-differential equations. A numerical reconstruction procedure based of the least squares finite element method is derived, and a series of numerical experiments visualise exactly how the loss of ellipticity manifests itself as propagating singularities.

A fully non-linear optimization approach to acousto-electric tomography

“…With either of these choice of boundary conditions, u 1 and u 2 have no critical points and ∇u 1 , ∇u 2 are non-parallel in Ω [3]. This choice of boundary conditions is motivated by the linear reconstruction algorithms, where BC1 and BC2 lead to different qualitative behavior in the reconstructions [10]. Unless otherwise explicitly stated, the boundary condition in the numerical experiments is BC1; see (BC).…”

“…For the case s = 1, in the optimality system (10), the following reduced L 2 gradient components appear…”

“…Recall that at each iterative step, the update for σ is found by solving (10). In the case of H 1 regularization, since (10) involves an additional Laplacian term, the obtained update for σ is more regular compared to that with the L 2 regularization set up. Due to this, the artifacts with H 1 regularization are less pronounced leading to reconstructions with better contrast.…”

“…It can be shown that the linearized problem in R 2 is (microlocally) solvable in case of only two boundary conditions [8]. However, if the interior gradient fields ∇u 1 , ∇u 2 are somewhere orthogonal in the interior, then local instabilities occur and the inversion allows propagation of singularities [10]. Consequently, the particular choice of boundary conditions turns out to be crucial.…”

A New Optimization Approach to Sparse Reconstruction of Log-Conductivity in Acousto-Electric Tomography

“…However, the analysis and application of (3) relies on linearization of this model so that actually an approximation of the AET problem is considered. Nevertheless, based on the results in [4,7,31,32,9] one can conclude that the linearized inverse problem corresponding to (3) is elliptic in the sense of Douglis-Nirenberg, and thus solvable, if there are at least n sets of measurements H i (σ), i = 1, . .…”