Fast reconstruction approaches for photoacoustic tomography with smoothing Sobolev/Matérn priors

In photoacoustic tomography (PAT), the computation of the initial pressure distribution within an object from its time-dependent boundary measurements over time is considered. This problem can be approached from two well-established points of view: deterministically using regularization methods, or stochastically using the Bayesian framework. Both approaches frequently require the solution of a variational problem. In this paper, we elaborate the connection between these approaches by establishing the equivalence between a smoothing Matérn class of covariance operators and Sobolev embedding operator $ E_s: H^s \hookrightarrow L^2 $. We further discuss the use of a Wavelet-based implementation of the adjoint operator $ E_s^* $, which also allows for efficient evaluations for certain Matérn covariance operators, leading to efficient implementations both in terms of computational effort as well as memory requirements. The proposed methods are validated with reconstructions for the photoacoustic problem.