  This paper proposes a new two-step procedure for sparse-view tomographic
image reconstruction. It is called RISING, since it combines an early-stopped
Rapid Iterative Solver with a subsequent Iteration Network-based Gaining step.
So far, regularized iterative methods have widely been used for X-ray computed
tomography image reconstruction from low-sampled data, since they converge to a
sparse solution in a suitable domain, as upheld by the Compressed Sensing
theory. Unfortunately, their use is practically limited by their high
computational cost which imposes to perform only a few iterations in the
available time for clinical exams. Data-driven methods, using neural networks
to post-process a coarse and noisy image obtained from geometrical algorithms,
have been recently studied and appreciated for both their computational speed
and accurate reconstructions. However, there is no evidence, neither
theoretically nor numerically, that neural networks based algorithms solve the
mathematical inverse problem modeling the tomographic reconstruction process.
In our two-step approach, the first phase executes very few iterations of a
regularized model-based algorithm whereas the second step completes the missing
iterations by means of a neural network. The resulting hybrid deep-variational
framework preserves the convergence properties of the iterative method and, at
the same time, it exploits the computational speed and flexibility of a
data-driven approach. Experiments performed on a simulated and a real data set
confirm the numerical and visual accuracy of the reconstructed RISING images in
short computational times.
