  In this paper we introduce CorticalFlow, a new geometric deep-learning model
that, given a 3-dimensional image, learns to deform a reference template
towards a targeted object. To conserve the template mesh's topological
properties, we train our model over a set of diffeomorphic transformations.
This new implementation of a flow Ordinary Differential Equation (ODE)
framework benefits from a small GPU memory footprint, allowing the generation
of surfaces with several hundred thousand vertices. To reduce topological
errors introduced by its discrete resolution, we derive numeric conditions
which improve the manifoldness of the predicted triangle mesh. To exhibit the
utility of CorticalFlow, we demonstrate its performance for the challenging
task of brain cortical surface reconstruction. In contrast to current
state-of-the-art, CorticalFlow produces superior surfaces while reducing the
computation time from nine and a half minutes to one second. More
significantly, CorticalFlow enforces the generation of anatomically plausible
surfaces; the absence of which has been a major impediment restricting the
clinical relevance of such surface reconstruction methods.
