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.

翻译：光声层析成像及多种其他耦合物理成像模态中的逆源问题常通过迭代算法求解。此类算法基于特定代价函数的最小化。此外，当前正在研究新型深度学习技术以进一步改进此类优化方法。所有这类方法均需多次应用定义正问题的算子及其伴随算子。本文针对圆形采集几何，提出了用于正算子和伴随算子数值计算的新型渐近快速算法。对于$(n \\times n)$图像，我们的算法以$\\mathcal{O}(n^2 \\log n)$次浮点运算完成算子计算。我们通过数值模拟展示了算法在多种迭代图像重建技术中的性能：经典变分方法（如非负最小二乘法和全变分正则化最小二乘法）以及深度学习方法（如学习型原始对偶算法）。算法的Python实现及计算示例已向公众开放。