  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 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 $I_a$. 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 $I_a^*I_a$ 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.
