Fast algorithms enabling optimization and deep learning for photoacoustic tomography in a circular detection geometry

TL;DR

This paper introduces fast algorithms for computing forward and adjoint operators in photoacoustic tomography with circular detection, significantly accelerating iterative image reconstruction methods including deep learning approaches.

Contribution

The authors develop asymptotically fast algorithms with $ ext{O}(n^2 ext{log} n)$ complexity for the forward and adjoint operators in circular geometry, enhancing photoacoustic image reconstruction.

Findings

Algorithms achieve $ ext{O}(n^2 ext{log} n)$ complexity.

Numerical simulations demonstrate improved reconstruction performance.

Applicable to variational and deep learning-based methods.

Abstract

The inverse source problem arising in photoacoustic tomography and in several other coupled-physics modalities is frequently solved by iterative algorithms. Such algorithms are based on the minimization of a certain cost functional. In addition, novel deep learning techniques are currently being investigated to further improve such optimization approaches. All such methods require multiple applications of the operator defining the forward problem, and of its adjoint. In this paper, we present new asymptotically fast algorithms for numerical evaluation of the forward and adjoint operators, applicable in the circular acquisition geometry. For an $(n \times n)$ image, our algorithms compute these operators in $\mathcal{O}(n^2 \log n)$ floating point operations. We demonstrate the performance of our algorithms in numerical simulations, where they are used as an integral part of several iterative image reconstruction techniques: classic variational methods, such as non-negative least squares and total variation regularized least squares, as well as deep learning methods, such as learned primal dual. A Python implementation of our algorithms and computational examples is available to the general public.