  Quantitative photo-acoustic tomography (QPAT) seeks to reconstruct a
distribution of optical attenuation coefficients inside a sample from a set of
time series of pressure data that is measured outside the sample. The
associated inverse problems involve two steps, namely acoustic and optical,
which can be solved separately or as a direct composite problem. We adopt the
latter approach for realistic acoustic media that possess heterogeneous and
often not accurately known distributions for sound speed and ambient density,
as well as an attenuation following a frequency power law that is evident in
tissue media. We use a Diffusion Approximation (DA) model for the optical
portion of the problem. We solve the corresponding composite inverse problem
using three total variation (TV) regularised optimisation approaches.
Accordingly, we develop two Krylov-subspace inexact-Newton algorithms that
utilise the Jacobian matrix in a matrix-free manner in order to handle the
computational cost. Additionally, we use a gradient-based algorithm that
computes a search direction using the L-BFGS method, and applies a TV
regularisation based on the Alternating Direction Method of Multipliers (ADMM)
as a benchmark, because this method is popular for QPAT and direct QPAT. The
results indicate the superiority of the developed inexact Newton algorithms
over gradient-based Quasi-Newton approaches for a comparable computational
complexity.
