The roto-translation group SE2 has been of active interest in image analysis
due to methods that lift the image data to multi-orientation representations
defined on this Lie group. This has led to impactful applications of
crossing-preserving flows for image de-noising, 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 SE2. We make several theoretical contributions (generalizable
to matrix Lie groups) such as the non-optimality 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 barycenters, 2) interpolation of planar orientation
fields, and 3) Wasserstein gradient flows on SE2. We observe that our framework
of lifting images to SE2 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 $\mathbb{R}^2$

在这篇论文中，我们针对 SE2 提出了一种计算框架，用于李群中的最优输运，并报告了在图像重心、平面方向场插值和 SE2 上的 Wasserstein 梯度流方面的宝贵进展。