Stabilizing Inverse Problems by Internal Data

TL;DR

This paper presents a general technique to stabilize inverse problems in hybrid imaging by analyzing when interior data makes the linearized problem elliptic, thereby ensuring stability and Fredholm properties.

Contribution

It introduces a unified approach to determine which interior data stabilizes inverse problems, applicable across various hybrid imaging methods.

Findings

Interior data can stabilize inverse problems by making the linearized operator elliptic.

The technique applies to multiple hybrid imaging modalities.

Stability is achieved through Fredholm properties, not necessarily local uniqueness.

Abstract

Several newly developing hybrid imaging methods (e.g., those combining electrical impedance or optical imaging with acoustics) enable one to obtain some auxiliary interior information (usually some combination of the electrical conductivity and the current) about the parameters of the tissues. This information, in turn, happens to stabilize the exponentially unstable and thus low resolution optical and electrical impedance tomography. Various known instances of this effect have been studied individually. We show that there is a simple general technique (covering all known cases) that shows what kind of interior data stabilizes the reconstruction, and why. Namely, we show when the linearized problem becomes elliptic pseudo-differential one, and thus stable. Stability here is meant as the problem being Fredholm, so the local uniqueness is not shown and probably does not hold in such generality.