We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups SE(2) and SE(3). In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate SE(n), and which are obtained via the exponential and logarithmic map on SE(n). In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on SE(n). The quantitative experiments further confirm the accuracy of the approximations. Quantitative results are obtained by evaluating perceptual grouping performance of retinal blood vessels in 2D images and curves in challenging 3D synthetic volumes. The results show that (1) sub-Riemannian geometry is essential in achieving top performance and (2) grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on \(\mathbb {R}^n\) and SE(n).

中文翻译：

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用于 2D 和 3D 血管快速感知分组的亚黎曼距离的幂零近似。

我们提出了一种通过 2D 和 3D 旋转平移组 SE(2) 和 SE(3) 中亚黎曼距离的幂等近似来对局部方向（血管上的点）进行分组的有效方法。在我们的距离逼近中，我们考虑了局部逼近 SE( n ) 的幂零群上的齐次范数，这些范数是通过 SE( n ) 上的指数和对数映射获得的。在定性验证中，我们表明范数提供了真实亚黎曼距离的准确近似，我们讨论了它们与 SE( n ) 上亚拉普拉斯算子基本解的关系）。定量实验进一步证实了近似的准确性。通过评估 2D 图像中视网膜血管的感知分组性能和具有挑战性的 3D 合成体积中的曲线来获得定量结果。结果表明，(1) 亚黎曼几何对于实现最佳性能至关重要，(2) 通过快速解析近似进行分组的性能​​几乎与\(\mathbb {R}^上的数据自适应快速行进方法相同或更好。 n\)和 SE( n )。