Optimal Transport on the Lie Group of Roto-translations

The roto-translation group SE(2) has been of active interest in image analysis due to methods that lift the image data to multiorientation representations defined on this Lie group. This has led to impactful applications of crossing-preserving flows for image denoising, geodesic tracking, and roto-translation equivariant deep learning. In this paper, we develop a computational framework for optimal transportation over Lie groups, with a special focus on SE(2). We make several theoretical contributions (generalizable to matrix Lie groups) such as the nonoptimality of group actions as transport maps, invariance and equivariance of optimal transport, and the quality of the entropic-regularized optimal transport plan using geodesic distance approximations. We develop a Sinkhorn-like algorithm that can be efficiently implemented using fast and accurate distance approximations of the Lie group and GPU-friendly group convolutions. We report valuable advancements in the experiments on 1) image barycentric interpolation, 2) interpolation of planar orientation fields, and 3) Wasserstein gradient flows on SE(2). We observe that our framework of lifting images to SE(2) and optimal transport with left-invariant anisotropic metrics leads to equivariant transport along dominant contours and salient line structures in the image. This yields sharper and more meaningful interpolations compared to their counterparts on R^2.