Computerized Tomography and Reproducing Kernels

Abstract

The X-ray transform is one of the most fundamental integral operators in image processing and reconstruction. In this article, we revisit the formalism of the X-ray transform by considering it as an operator between Reproducing Kernel Hilbert Spaces (RKHS). Within this framework, the X-ray transform can be viewed as a natural analogue of Euclidean projection. 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 (i.e. without recourse to discretization), but in the realistic setting where the data are discrete and noisy. The RKHS framework is versatile, accommodating any reproducing kernel on a unit ball, affording a high level of generality. When the kernel is chosen to be rotation-invariant, explicit spectral representations can be obtained, elucidating the regularity structure of the associated Hilbert spaces. Moreover, the reconstruction problem can be solved at the same computational cost as filtered back-projection.