  The goal of quantitative photoacoustic tomography is to determine optical and
acoustical material properties from initial pressure maps as obtained, for
instance, from photoacoustic imaging. The most relevant parameters are
absorption, diffusion and Grueneisen coefficients, all of which can be
heterogeneous. Recent work by Bal and Ren shows that in general, unique
reconstruction of all three parameters is impossible, even if multiple
measurements of the initial pressure (corresponding to different laser
excitation directions at a single wavelength) are available.
  Here, we propose a restriction to piecewise constant material parameters. We
show that in the diffusion approximation of light transfer, piecewise constant
absorption, diffusion and Gr\"uneisen coefficients can be recovered uniquely
from photoacoustic measurements at a single wavelength. In addition, we
implemented our ideas numerically and tested them on simulated
three-dimensional data.
