  In this paper, we derive a neural network architecture based on an analytical
formulation of the parallel-to-fan beam conversion problem following the
concept of precision learning. The network allows to learn the unknown
operators in this conversion in a data-driven manner avoiding interpolation and
potential loss of resolution. Integration of known operators results in a small
number of trainable parameters that can be estimated from synthetic data only.
The concept is evaluated in the context of Hybrid MRI/X-ray imaging where
transformation of the parallel-beam MRI projections to fan-beam X-ray
projections is required. The proposed method is compared to a traditional
rebinning method. The results demonstrate that the proposed method is superior
to ray-by-ray interpolation and is able to deliver sharper images using the
same amount of parallel-beam input projections which is crucial for
interventional applications. We believe that this approach forms a basis for
further work uniting deep learning, signal processing, physics, and traditional
pattern recognition.
