Inverse Diffusion Theory of Photoacoustics

TL;DR

This paper develops a mathematical framework for uniquely and stably reconstructing diffusion and absorption parameters in photoacoustic imaging using internal data and complex geometrical optics solutions.

Contribution

It introduces a novel method for parameter reconstruction in elliptic equations using internal data, with stability results under geometric conditions and explicit boundary condition characterization.

Findings

Unique determination of parameters from two internal data sets.

Stability guaranteed under strict convexity or with 2n internal data.

Characterization of boundary conditions via complex geometrical optics solutions.

Abstract

This paper analyzes the reconstruction of diffusion and absorption parameters in an elliptic equation from knowledge of internal data. In the application of photo-acoustics, the internal data are the amount of thermal energy deposited by high frequency radiation propagating inside a domain of interest. These data are obtained by solving an inverse wave equation, which is well-studied in the literature. We show that knowledge of two internal data based on well-chosen boundary conditions uniquely determines two constitutive parameters in diffusion and Schroedinger equations. Stability of the reconstruction is guaranteed under additional geometric constraints of strict convexity. No geometric constraints are necessary when $2n$ internal data for well-chosen boundary conditions are available, where $n$ is spatial dimension. The set of well-chosen boundary conditions is characterized in terms of appropriate complex geometrical optics (CGO) solutions.