We present a deep learning-based computational algorithm for inversion of
circular Radon transforms in the partial radial setup, arising in photoacoustic
tomography. We first demonstrate that the truncated singular value
decomposition-based method, which is the only traditional algorithm available
to solve this problem, leads to severe artifacts which renders the
reconstructed field as unusable. With the objective of overcoming this
computational bottleneck, we train a ResBlock based U-Net to recover the
inferred field that directly operates on the measured data. Numerical results
with augmented Shepp-Logan phantoms, in the presence of noisy full and limited
view data, demonstrate the superiority of the proposed algorithm.

我们提出了一种基于深度学习的计算算法来破解局部径向设定下的圆形 Radon 变换，该变换在光声断层扫描中出现。我们首先证明了截断奇异值分解方法，这是唯一可用于解决此问题的传统算法，会导致严重的伪影从而使重建场景没有用武之地。为了克服这个计算瓶颈，我们训练了一个基于 ResBlock 的 U-Net，以直接通过测量数据恢复推断场景。在噪声全视图和有限视图数据存在的情况下，使用增强 Shepp-Logan 幻影的数值结果，证明了所提出算法的优越性。

学习偏径向设置下圆形 Radon 变换的端到端反演

Learning end-to-end inversion of circular Radon transforms in the  partial radial setup

Let (M,g) be an analytic, compact, Riemannian manifold with boundary, of
dimension n >= 2. We study a class of generalized Radon transforms, integrating
over a family of hypersurfaces embedded in M, satisfying the Bolker condition
[23]. Using analytic microlocal analysis, we prove a microlocal regularity
theorem for generalized Radon transforms on analytic manifolds defined on an
analytic family of hypersurfaces. We then show injectivity and stability for an
open, dense subset of smooth generalized Radon transforms satisfying the Bolker
condition, including the analytic ones.

本文研究了在满足 Bolker 条件的嵌入于解析紧黎曼流形的解析超曲面族上广义 Radon 变换的微局部正则性，进而证明了在该条件下开放稠密子集上广义 Radon 变换的单性和稳定性。