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因其将图像数据提升至该李群上多方向表示的方法，在图像分析领域持续引发研究兴趣。这催生了交叉保持流在图像去噪、测地线追踪以及旋转平移等变深度学习中的重要应用。本文构建了李群上最优输运的计算框架，重点关注SE2群。我们提出了多项理论贡献（可推广至矩阵李群），包括群作用作为输运映射的非最优性、最优输运的等变与不变性，以及基于测地距离近似的熵正则化最优输运方案质量分析。我们开发了类似Sinkhorn的算法，该算法可利用李群快速精确的距离近似和GPU友好型群卷积高效实现。在以下实验中报告了有价值的进展：1）图像重心插值，2）平面方向场插值，3）SE2群上的Wasserstein梯度流。我们观察到，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，能够实现沿图像主导轮廓与显著线条结构的等变输运。与R²空间上的对应方法相比，该方法可产生更清晰且更具语义性的插值结果。