The X-ray transform is one of the most fundamental integral operators in image processing and reconstruction. In this article, we revisit its mathematical formalism, and propose an innovative approach making use of Reproducing Kernel Hilbert Spaces (RKHS). Within this framework, the X-ray transform can be considered as a natural analogue of Euclidean projections. The RKHS framework considerably simplifies projection image interpolation, and leads to an analogue of the celebrated representer theorem for the problem of tomographic reconstruction. It leads to methodology that is dimension-free and stands apart from conventional filtered back-projection techniques, as it does not hinge on the Fourier transform. It also allows us to establish sharp stability results at a genuinely functional level, but in the realistic setting where the data are discrete and noisy. The RKHS framework is amenable to any reproducing kernel on a unit ball, affording a high level of generality. When the kernel is chosen to be rotation-invariant, one can obtain explicit spectral representations which elucidate the regularity structure of the associated Hilbert spaces, and one can also solve the reconstruction problem at the same computational cost as filtered back-projection.

翻译：X射线变换是图像处理与重建中最基本的积分算子之一。本文重新审视了其数学形式体系，并提出了一种利用再生核希尔伯特空间（RKHS）的创新方法。在该框架下，X射线变换可视为欧几里得投影的自然类比。RKHS框架显著简化了投影图像插值过程，并导出了断层重建问题中著名的表示定理的类似形式。该方法具有维度无关性，且不依赖傅里叶变换，因而与传统滤波反投影技术截然不同。它还能在数据离散且含噪声的现实场景中，在真正的函数层面上建立精确的稳定性结果。RKHS框架适用于单位球上的任意再生核，具有高度普适性。当选择旋转不变核时，可获取显式谱表示以阐明相关希尔伯特空间的正则结构，同时能以与滤波反投影相同的计算成本解决重建问题。