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 $\mathbb{R}^2$

翻译：旋转平移群SE2在图像分析领域引起广泛关注，因该方法将图像数据提升至定义在该李群上的多方向表示。这催生了跨保存流在图像去噪、测地线追踪和旋转平移等变深度学习中的突破性应用。本文构建了李群最优输运的计算框架，特别聚焦于SE2群。我们提出了若干理论贡献（可推广至矩阵李群），包括群作用作为输运映射的非最优性、最优输运的不变性与等变性，以及基于测地距离近似的熵正则化最优输运计划的质量。发展了可高效实现的Sinkhorn类算法，该算法利用李群快速精确的距离近似与GPU友好的群卷积。在三个实验领域报告了重要进展：1）图像重心；2）平面方向场插值；3）SE2上的Wasserstein梯度流。观察到将图像提升至SE2并采用左不变各向异性度量的最优输运框架，可沿图像主导轮廓与显著线条结构实现等变输运，相较于$\mathbb{R}^2$上的对应方法产生更清晰且更有意义的插值结果。