Stabilizing inverse problems by internal data. II. Non-local internal data and generic linearized uniqueness

TL;DR

This paper extends a stabilization technique for inverse problems using internal data to non-local data scenarios like acousto-optical tomography, demonstrating generic linearized uniqueness and stability improvements in hybrid imaging modalities.

Contribution

It adapts the stabilization method to non-local internal data involving Green's functions and establishes generic linearized uniqueness in various hybrid imaging contexts.

Findings

Technique successfully applied to acousto-optical tomography.

Achieved generic linearized uniqueness results.

Enhanced stability analysis for hybrid imaging modalities.

Abstract

In the previous paper "Stabilizing Inverse Problems by Internal Data", the authors introduced a simple procedure that allows one to detect whether and explain why internal information arising in several novel coupled physics (hybrid) imaging modalities could turn extremely unstable techniques, such as optical tomography or electrical impedance tomography, into stable, good-resolution procedures. It was shown that in all cases of interest, the Frechet derivative of the forward mapping is a pseudo-differential operator with an explicitly computable principal symbol. If one can set up the imaging procedure in such a way that the symbol is elliptic, this would indicate that the problem was stabilized. In the cases when the symbol is not elliptic, the technique suggests how to change the procedure (e.g., by adding extra measurements) to achieve ellipticity. In this article, we consider the situation arising in acousto-optical tomography (also called ultrasound modulated optical tomography), where the internal data available involves the Green's function, and thus depends globally on the unknown parameter(s) of the equation and its solution. It is shown that the technique of "Stabilizing Inverse Problems by Internal Data" can be successfully adopted to this situation as well. We also obtain results on generic uniqueness for the linearized problem in a variety of situations, including those arising in acousto-electric and quantitative photoacoustic tomography.