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

Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer review

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Uittreksel

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).

TaalEngels
Pagina's882-899
Aantal pagina's18
TijdschriftJournal of Mathematical Imaging and Vision
Volume60
Nummer van het tijdschrift6
DOI's
StatusGepubliceerd - jan 2018

Vingerafdruk

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

Bibliografische nota

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

Trefwoorden

    Citeer dit

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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, Nr. 6, 01.2018, blz. 882-899.

    Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer review

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