Nilpotent approximations of sub-Riemannian distances for fast perceptual grouping of blood vessels in 2D and 3D

Erik J. Bekkers, Da Chen, Jorg M. Portegies

Research output: Contribution to journalArticleAcademicpeer-review

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Abstract

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 (Formula presented.) and SE(n).

Original languageEnglish
Pages (from-to)882-899
Number of pages18
JournalJournal of Mathematical Imaging and Vision
Volume60
Issue number6
DOIs
Publication statusPublished - Jan 2018

Fingerprint

Perceptual Grouping
Blood Vessels
blood vessels
Blood vessels
Geometry
Approximation
approximation
norms
Grouping
Experiments
Sub-Riemannian Geometry
Sub-Laplacian
Norm
Nilpotent Group
Fundamental Solution
Vessel
vessels
Logarithmic
Curve
curves

Bibliographical note

18 pages, 9 figures, 3 tables, in review at JMIV

Keywords

  • Geodesic vessel tracking
  • Nilpotent approximation
  • Perceptual grouping
  • Roto-translation group
  • SE(2)
  • SE(3)
  • Sub-Riemannian geometry

Cite this

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title = "Nilpotent approximations of sub-Riemannian distances for fast perceptual grouping of blood vessels in 2D and 3D",
abstract = "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 (Formula presented.) and SE(n).",
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Nilpotent approximations of sub-Riemannian distances for fast perceptual grouping of blood vessels in 2D and 3D. / Bekkers, Erik J.; Chen, Da; Portegies, Jorg M.

In: Journal of Mathematical Imaging and Vision, Vol. 60, No. 6, 01.2018, p. 882-899.

Research output: Contribution to journalArticleAcademicpeer-review

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