# Efficient Nonparametric Bayesian Inference For X-Ray Transforms

arXiv: https://arxiv.org/abs/1708.06332 · HTML: https://mlsanity.com/paper/1708.06332/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Aug 2017.

## Built on by (most cited first, 4 of 4)

- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023)
- [On free energy barriers in Gaussian priors and failure of cold start MCMC for high-dimensional unimodal distributions](https://mlsanity.com/paper/2209.02001/) (2022)
- [Computerized Tomography and Reproducing Kernels](https://mlsanity.com/paper/2311.07465/) (2023)

## Abstract (arXiv)

We consider the statistical inverse problem of recovering a function f: M → mathbb R, where M is a smooth compact Riemannian manifold with boundary, from measurements of general X-ray transforms Ia(f) of f, corrupted by additive Gaussian noise. For M equal to the unit disk with `flat' geometry and a=0 this reduces to the standard Radon transform, but our general setting allows for anisotropic media M and can further model local `attenuation' effects -- both highly relevant in practical imaging problems such as SPECT tomography. We propose a nonparametric Bayesian inference approach based on standard Gaussian process priors for f. The posterior reconstruction of f corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty that does not require the calculation of the singular value decomposition of the forward operator Ia. We prove Bernstein-von Mises theorems that entail that posterior-based inferences such as credible sets are valid and optimal from a frequentist point of view for a large family of semi-parametric aspects of f. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which is shown to attain the semi-parametric Cram'er-Rao information bound. The proofs rely on an invertibility result for the `Fisher information' operator Ia^*Ia between suitable function spaces, a result of independent interest that relies on techniques from microlocal analysis. We illustrate the performance of the proposed method via simulations in various settings.