  In computed tomography, data consist of measurements of the attenuation of
X-rays passing through an object. The goal is to reconstruct the linear
attenuation coefficient of the object's interior. For each position of the
X-ray source, characterized by its angle with respect to a fixed coordinate
system, one measures a set of data referred to as a view. A common assumption
is that these view angles are known, but in some applications they are known
with imprecision. We propose a framework to solve a Bayesian inverse problem
that jointly estimates the view angles and an image of the object's attenuation
coefficient. We also include a few hyperparameters that characterize the
likelihood and the priors. Our approach is based on a Gibbs sampler where the
associated conditional densities are simulated using different sampling schemes
- hence the term hybrid. In particular, the conditional distribution associated
with the reconstruction is nonlinear in the image pixels, non-Gaussian and
high-dimensional. We approach this distribution by constructing a Laplace
approximation that represents the target conditional locally at each Gibbs
iteration. This enables sampling of the attenuation coefficients in an
efficient manner using iterative reconstruction algorithms. The numerical
results show that our algorithm is able to jointly identify the image and the
view angles, while also providing uncertainty estimates of both. We demonstrate
our method with 2D X-ray computed tomography problems using fan beam
configurations.

Computation