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 barycentric interpolation, 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 R^2

翻译：旋转平移群SE2在图像分析领域一直备受关注，这主要归因于将图像数据提升至该李群上定义的多方向表示的方法。这已促成保持交叉特性的流在图像去噪、测地线追踪及旋转平移等变深度学习中的广泛应用。本文针对李群（特别聚焦于SE2）上的最优传输问题，构建了一个计算框架。我们在理论层面做出了若干贡献（可推广至矩阵李群），包括：群作用作为传输映射的非最优性、最优传输的不变性与等变性，以及利用测地线距离近似进行熵正则化最优传输方案的质量评估。我们开发了一种类Sinkhorn算法，该算法可通过李群的快速精确距离近似与GPU友好的群卷积实现高效计算。实验部分取得了重要进展，主要体现在：1）图像重心插值，2）平面方向场插值，3）SE2上的Wasserstein梯度流。我们观察到，将图像提升至SE2并结合左不变各向异性度量的最优传输框架，能够沿图像中主导轮廓与显著线结构实现等变传输。与R^2上的对应方法相比，这产生了更清晰且更具语义意义的插值结果。