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

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.