  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.
