Quantitative thermo-acoustic imaging: An exact reconstruction formula

TL;DR

This paper develops an exact analytical formula for reconstructing the absorption coefficient in quantitative thermo-acoustic imaging from electromagnetic data, addressing instability issues with noise regularization and optimization techniques.

Contribution

It introduces the first analytical reconstruction formula involving derivatives of data and improves accuracy through an optimal control-based correction.

Findings

Derived an explicit formula for absorption coefficient reconstruction

Addressed instability by incorporating noise regularization

Enhanced resolution via an optimal control correction step

Abstract

This paper aims to mathematically advance the field of quantitative thermo-acoustic imaging. Given several electromagnetic data sets, we establish for the first time an analytical formula for reconstructing the absorption coefficient from thermal energy measurements. Since the formula involves derivatives of the given data up to the third order, it is unstable in the sense that small measurement noises may cause large errors. However, in the presence of measurement noise, the obtained formula, together with a noise regularization technique, provides a good initial guess for the true absorption coefficient. We finally correct the errors by deriving a reconstruction formula based on the least square solution of an optimal control problem and prove that this optimization step reduces the errors occurring and enhances the resolution.