A sparsity-based nonlinear reconstruction method for two-photon photoacoustic tomography

TL;DR

This paper introduces a nonlinear optimization method for sparse reconstruction of absorption coefficients in photoacoustic tomography, utilizing $L^1$ regularization and a proximal algorithm to improve image reconstruction accuracy.

Contribution

The paper develops a novel nonlinear optimization framework with $L^1$ regularization for sparse reconstruction in PAT, including a new proof of PDE solution uniqueness.

Findings

Effective reconstruction of absorption coefficients demonstrated

Proximal method with PDE solver shows promising results

Framework applicable to other inverse problems in imaging

Abstract

We present a new nonlinear optimization approach for the sparse reconstruction of single-photon absorption and two-photon absorption coefficients in photoacoustic tomography (PAT). This framework comprises of minimizing an objective functional involving a least squares fit of the interior pressure field data corresponding to two boundary source functions, where the absorption coefficients and the photon density are related through a semi-linear elliptic partial differential equation (PDE) arising in PAT. Further, the objective functional consists of an $L^1$ regularization term that promotes sparsity patterns in absorption coefficients. The motivation for this framework primarily comes from some recent works related to solving inverse problems in acousto-electric tomography and current density impedance tomography. We provide a new proof of existence and uniqueness of a solution to the semi-linear PDE. Further, a proximal method, involving a Picard solver for the semi-linear PDE and its adjoint, is used to solve the optimization problem. Several numerical experiments are presented to demonstrate the effectiveness of the proposed framework.