# Computerized Tomography and Reproducing Kernels

arXiv: https://arxiv.org/abs/2311.07465 · HTML: https://mlsanity.com/paper/2311.07465/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 50 references that are in the map.
- First posted on arXiv 13 Nov 2023.

## Builds on (most cited first, 6 of 6)

- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [A unifying representer theorem for inverse problems and machine learning](https://mlsanity.com/paper/1903.00687/) (2019)
- [Fast Steerable Principal Component Analysis](https://mlsanity.com/paper/1412.0781/) (2014)
- [Efficient Nonparametric Bayesian Inference For X-Ray Transforms](https://mlsanity.com/paper/1708.06332/) (2017)
- [Limited View Tomographic Reconstruction Using a Deep Recurrent Framework with Residual Dense Spatial-Channel Attention Network and Sinogram Consistency](https://mlsanity.com/paper/2009.01782/) (2020)

## Abstract (arXiv)

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.