Computerized Tomography and Reproducing Kernels

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.