New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)

Portegies, Jorg M.; Duits, Remco

doi:10.48550/arxiv.1604.03843

Cited by 1 publication

(3 citation statements)

References 27 publications

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“…In both quantitative and visual comparison it was found that the approximations accurately follow the true sub-Riemannian distances, a conclusion which was further supported by the equal performance in quantitative perceptual grouping experiments. We also numerically showed that the weighted logarithmic norms used in this paper provide a more accurate approach for approximating the heat kernel and fundamental solution of the sub-Laplacian on SE(n), compared to previous approaches [22,47,12].…”

Section: Discussionmentioning

confidence: 84%

“…We reason that the Folland-Kaplan-Korányi provides an accurate approximation to the fundamental solution on SE(n) as well, as it provides the exact fundamental solution on the Heisenberg type approximation (SE(n)) 0 . As such, we provide an approach to approximating the heat kernel and fundamental solution of the sub-Laplacian on SE(n), as an alternative to the works [22,47,12].…”

Section: Nilpotent Approximationmentioning

confidence: 99%

“…Also here, the Folland-Kaplan-Korányi-type norm can be used to approximate the fundamental solutions of the sub-Laplacian on SE(3). The norm ||c|| ξ,ζ with ζ = 1 was for example used in [23] approximations of the heat kernel and the fundamental solution on SE(3), of which only recently exact solutions were found in [47]. In the context of this paper, we can approximate the exact solutions of the sub-Laplacian on SE(3) by c 2−Q 1,16 , with homogeneous dimensions Q = 9, as the exact solution of the sub-Laplacian on the approximation group (SE(3)) 0 .…”

Section: Sub-riemannian Geometry In Se(3)mentioning

confidence: 99%

See 2 more Smart Citations

Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

2018

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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 R n and SE(n).

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Section: Discussionmentioning

confidence: 84%

Section: Nilpotent Approximationmentioning

confidence: 99%