On the Connection between Dynamical Optimal Transport and Functional Lifting

Functional lifting methods provide a tool for approximating solutions of difficult non-convex problems by embedding them into a larger space. In this work, we investigate a mathematically rigorous formulation based on embedding into the space of pointwise probability measures over a fixed range $\Gamma$. Interestingly, this approach can be derived as a generalization of the theory of dynamical optimal transport. Imposing the established continuity equation as a constraint corresponds to variational models with first-order regularization. By modifying the continuity equation, the approach can also be extended to models with higher-order regularization.

Deformable Groupwise Image Registration using Low-Rank and Sparse Decomposition

Low-rank and sparse decompositions and robust PCA (RPCA) are highly successful techniques in image processing and have recently found use in groupwise image registration. In this paper, we investigate the drawbacks of the most common RPCA-dissimi\-larity metric in image registration and derive an improved version. In particular, this new metric models low-rank requirements through explicit constraints instead of penalties and thus avoids the pitfalls of the established metric. Equipped with total variation regularization, we present a theoretically justified multilevel scheme based on first-order primal-dual optimization to solve the resulting non-parametric registration problem. As confirmed by numerical experiments, our metric especially lends itself to data involving recurring changes in object appearance and potential sparse perturbations. We numerically compare its peformance to a number of related approaches.