  In deformable registration, the geometric framework - large deformation
diffeomorphic metric mapping or LDDMM, in short - has inspired numerous
techniques for comparing, deforming, averaging and analyzing shapes or images.
Grounded in flows, which are akin to the equations of motion used in fluid
dynamics, LDDMM algorithms solve the flow equation in the space of plausible
deformations, i.e. diffeomorphisms. In this work, we make use of deep residual
neural networks to solve the non-stationary ODE (flow equation) based on a
Euler's discretization scheme. The central idea is to represent time-dependent
velocity fields as fully connected ReLU neural networks (building blocks) and
derive optimal weights by minimizing a regularized loss function. Computing
minimizing paths between deformations, thus between shapes, turns to find
optimal network parameters by back-propagating over the intermediate building
blocks. Geometrically, at each time step, ResNet-LDDMM searches for an optimal
partition of the space into multiple polytopes, and then computes optimal
velocity vectors as affine transformations on each of these polytopes. As a
result, different parts of the shape, even if they are close (such as two
fingers of a hand), can be made to belong to different polytopes, and therefore
be moved in different directions without costing too much energy. Importantly,
we show how diffeomorphic transformations, or more precisely bilipshitz
transformations, are predicted by our algorithm. We illustrate these ideas on
diverse registration problems of 3D shapes under complex topology-preserving
transformations. We thus provide essential foundations for more advanced shape
variability analysis under a novel joint geometric-neural networks
Riemannian-like framework, i.e. ResNet-LDDMM.
