Efficient Nonparametric Bayesian Inference For X-Ray Transforms
Abstract
We consider the statistical inverse problem of recovering a function f: M 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.