Abstract:

We consider the statistical inverse problem of recovering a function f:M??, where M is a smooth compact Riemannian manifold with boundary, from measurements of general Xray transforms I(f) of f, corrupted by additive Gaussian noise. For M equal to the unit disk with `flat' geometry 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. We prove Bernsteinvon Mises theorems that entail that posteriorbased inferences such as credible sets are valid and optimal from a frequentist point of view for a large family of semiparametric aspects of f. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which is shown to attain the semiparametric Cram\'erRao information bound.
