Metric learning in diffeomorphic registration and a new fidelity measure based on unbalanced optimal transport.

Diffeomorphic registration is an ill-posed inverse problem which is
often solved via the minimization of an energy functional which
involves a control on the deformation and a fidelity measure.
Motivated by the design of these two terms which is crucial for
practical applications, my talk will be divided into two parts: the
first one will focus on the design of the regularization of the
deformation in the LDDMM (large deformation by diffeomorphisms)
setting. Starting from the simple sum of kernels, we show how to
extend it to spatially varying metrics using left-invariant
metrics. We then present a method to estimate this metric in a
template/population context. The second part of the talk will
introduce the use of optimal transport as a fidelity measure. After
recalling standard optimal transport, we present an extension of
optimal transport to the case of unbalanced measures. Building on
recent numerical advances, we discuss the use of fast scaling
algorithms to approximate the problem via entropic
regularization. Preliminary results on synthetic data are shown.