2017/07/10 by Erik J. Bekkers, Da Chen, Bekkers, Erik J. +3
Computer Science · Medicine · #Advanced Image and Video Retrieval Techniques #FOS: Mathematics #Group Theory (math.GR) #Medical Image Segmentation Techniques #Retinal Imaging and Analysis
paper · pdf · doi:10.48550/arxiv.1707.02811
openalex publication_date 2017/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose an efficient approach for the grouping of local orientations\n(points on vessels) via nilpotent approximations of sub-Riemannian distances in\nthe 2D and 3D roto-translation groups SE(2) and SE(3). In our distance\napproximations we consider homogeneous norms on nilpotent groups that locally\napproximate SE(n), and which are obtained via the exponential and logarithmic\nmap on SE(n). In a qualitative validation we show that the norms provide\naccurate approximations of the true sub-Riemannian distances, and we discuss\ntheir relations to the fundamental solution of the sub-Laplacian on SE(n).\nThe quantitative experiments further confirm the accuracy of the\napproximations. Quantitative results are obtained by evaluating perceptual\ngrouping performance of retinal blood vessels in 2D images and curves in\nchallenging 3D synthetic volumes. The results show that 1) sub-Riemannian\ngeometry is essential in achieving top performance and 2) that grouping via the\nfast analytic approximations performs almost equally, or better, than\ndata-adaptive fast marching approaches on \ℝn and SE(n).\n