  The aim of this paper is to tackle the nonlinear optical reconstruction
problem. Given a set of acousto-optic measurements, we develop a mathematical
framework for the reconstruction problem in the case where the optical
absorption distribution is supposed to be a perturbation of a piecewise
constant function. Analyzing the acousto-optic measurements, we establish a new
equation in the sense of distributions for the optical absorption coefficient.
For doing so, we introduce a weak Helmholtz decomposition and interpret in a
weak sense the cross-correlation measurements using the spherical Radon
transform. We next show how to find an initial guess for the unknown
coefficient and finally construct the true coefficient by providing a Landweber
type iteration and proving that the resulting sequence converges to the
solution of the system constituted by the optical diffusion equation and the
new equation mentioned above. Our results in this paper generalize the
acousto-optic process proposed in [Ammari, Garnier, Nguyen and Seppecher, 2012]
for piecewise smooth optical absorption distributions.
