A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

TL;DR

This paper introduces a variational approach to solve the inverse problem in quantitative photoacoustic tomography, achieving unique solutions for piecewise constant coefficients using a Mumford-Shah-like functional approximation.

Contribution

It presents a novel variational method with Ambrosio-Tortorelli approximation for reconstructing piecewise constant optical coefficients in photoacoustic tomography.

Findings

Method successfully reconstructs coefficients from simulated data.

Unique solutions are obtained for piecewise constant coefficients.

Numerical implementation demonstrates effectiveness on 2D data.

Abstract

In this article, we consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients from a single measurement of the absorbed energy (in the steady-state diffusion approximation of light transfer). This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. We show that when the coefficients are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of the coefficients, we suggest a variational method based based on an Ambrosio-Tortorelli-approximation of a Mumford-Shah-like functional, which we implemented numerically and tested on simulated two-dimensional data.