  The direct problem of optoacoustic signal generation in biological media
consists of solving the inhomogeneous optoacoustic wave equation for an initial
acoustic stress profile. In contrast, the mathematically challenging inverse
problem requires the reconstruction of the initial stress profile from a proper
set of observed signals. In this article, we consider the particular case of a
Gaussian transverse irradiation source profile in the paraxial approximation of
the wave equation, for which the direct problem along the beam axis can be cast
into a linear Volterra integral equation of the second kind. This integral
equation can be used in two ways: as a forward solver to predict optoacoustic
signals in terms of the direct problem, and as an inverse solver for which we
here devise highly efficient numerical schemes used for the reconstruction of
initial pressure profiles from observed signals, constituting a methodical
progress of computational aspects of optoacoustics. In this regard, we explore
the validity as well as the limits of the inversion scheme via numerical
experiments, with parameters geared towards actual optoacoustic problem
instances. The considered inversion input consists of synthetic data, obtained
by means of forward solvers based on the Volterra integral, and, more
generally, the optoacoustic Poisson integral. Regarding the latter, we
numerically invert signals that correspond to different detector-to-sample
distances and assess the convergence to the true initial stress profiles upon
approaching the far-field. Finally, we also address the effect of noise on the
quality of the reconstructed pressure profiles.
