Efficient nonparametric Bayesian inference for $X$-ray transforms

Abstract

We consider the statistical inverse problem of recovering a function $f:M\to \mathbb{R}$, where $M$ is a smooth compact Riemannian manifold with boundary, from measurements of general $X$-ray transforms $I_{a}(f)$ 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 study a nonparametric Bayesian inference method 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 $I_{a}$. We prove Bernstein–von Mises theorems for a large family of one-dimensional linear functionals of $f$, and they entail that posterior-based inferences such as credible sets are valid and optimal from a frequentist point of view. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which attains the semiparametric information lower bound. The proofs rely on an invertibility result for the “Fisher information” operator $I_{a}^{*}I_{a}$ 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.