A Fourier finite volume approach for the optical inverse problem of quantitative photoacoustic tomography

A new approach for solving the optical inverse problem of quantitative photoacoustic tomography is introduced, which interpolates between the well-known diffusion approximation and a radiative transfer equation based model. The proposed formulation combines a spatial finite volume scheme with a truncated Fourier expansion in the direction variable for the radiative transfer equation. The finite volume scheme provides a natural and simple approach for representing piecewise constant image data modelled using transport equations. The truncated Fourier expansion in the direction variable facilitates the interpolation between the diffusion approximation at low order, and the full radiative transfer model as the truncation limit $N\rightarrow\infty$. It is therefore possible to tune the precision of the model to the demands of the imaging application, taking $N=1$ for cases when the diffusion approximation would suffice and increasing the number of terms otherwise. We will then utilise the non-linear optimisation functionality of Matlab to address the corresponding large-scale nonlinear inverse problem using gradient based quasi-Newton minimisation via the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm. Numerical experiments for two test-cases of increasing complexity and resolution will be presented, and the effect of logarithmically rescaling the problem data on the accuracy of the reconstructed solutions will be investigated. We will focus on cases where the diffusion approximation is not sufficient to demonstrate that our approach can provide significant accuracy gains with only a modest increase in the number of Fourier terms included.