  Deep learning (DL) has shown unprecedented performance for many image
analysis and image enhancement tasks. Yet, solving large-scale inverse problems
like tomographic reconstruction remains challenging for DL. These problems
involve non-local and space-variant integral transforms between the input and
output domains, for which no efficient neural network models have been found. A
prior attempt to solve such problems with supervised learning relied on a
brute-force fully connected network and applied it to reconstruction for a
$128^4$ system matrix size. This cannot practically scale to realistic data
sizes such as $512^4$ and $512^6$ for three-dimensional data sets. Here we
present a novel framework to solve such problems with deep learning by casting
the original problem as a continuum of intermediate representations between the
input and output data. The original problem is broken down into a sequence of
simpler transformations that can be well mapped onto an efficient hierarchical
network architecture, with exponentially fewer parameters than a generic
network would need. We applied the approach to computed tomography (CT) image
reconstruction for a $512^4$ system matrix size. To our knowledge, this enabled
the first data-driven DL solver for full-size CT reconstruction without relying
on the structure of direct (analytical) or iterative (numerical) inversion
techniques. The proposed approach is applicable to other imaging problems such
as emission and magnetic resonance reconstruction. More broadly, hierarchical
DL opens the door to a new class of solvers for general inverse problems, which
could potentially lead to improved signal-to-noise ratio, spatial resolution
and computational efficiency in various areas.
