  We propose the Learned Primal-Dual algorithm for tomographic reconstruction.
The algorithm accounts for a (possibly non-linear) forward operator in a deep
neural network by unrolling a proximal primal-dual optimization method, but
where the proximal operators have been replaced with convolutional neural
networks. The algorithm is trained end-to-end, working directly from raw
measured data and it does not depend on any initial reconstruction such as FBP.
  We compare performance of the proposed method on low dose CT reconstruction
against FBP, TV, and deep learning based post-processing of FBP. For the
Shepp-Logan phantom we obtain >6dB PSNR improvement against all compared
methods. For human phantoms the corresponding improvement is 6.6dB over TV and
2.2dB over learned post-processing along with a substantial improvement in the
SSIM. Finally, our algorithm involves only ten forward-back-projection
computations, making the method feasible for time critical clinical
applications.
