Fast reconstruction approaches for photoacoustic tomography with smoothing Sobolev/Mat\'ern priors

TL;DR

This paper introduces fast, efficient reconstruction methods for photoacoustic tomography by linking Bayesian smoothing priors with deterministic regularization, utilizing wavelet-based implementations for improved computational performance.

Contribution

It establishes the equivalence between Matérn covariance operators and Sobolev embeddings, enabling efficient Bayesian and deterministic reconstruction methods in PAT.

Findings

Efficient wavelet-based implementation of the adjoint operator.

Validation of the proposed methods with photoacoustic reconstructions.

Reduction in computational effort and memory usage.

Abstract

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 regularisation methods, or stochastically using the Bayesian framework. Both approaches frequently require the solution of a variational problem. In the paper we elaborate the connection between these approaches by establishing the equivalence between a smoothing Mat{\'e}rn 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 Mat{\'e}rn 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.