Vessel Tracking via Sub-Riemannian Geodesics on the Projective Line Bundle

Bekkers, EJ Erik; Duits, Remco; Mashtakov, Alexey; Sachkov, Yu. L.

doi:10.1007/978-3-319-68445-1_89

Cited by 11 publications

(11 citation statements)

References 24 publications

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“…In a flat geometry on R 2 × S 1 the distance between the red and green arrow and the source would be equal, and the geodesics straight lines (see dashed lines). In sub-Riemannian geometry on SE (2) the green arrow has a shorter distance to the source. The left image shows 2D projections of the sub-Riemannian geodesics in solid black, the right image shows their paths in SE (2).…”

Section: Nilpotent Approximationmentioning

confidence: 99%

“…In sub-Riemannian geometry on SE (2) the green arrow has a shorter distance to the source. The left image shows 2D projections of the sub-Riemannian geodesics in solid black, the right image shows their paths in SE (2).…”

Section: Nilpotent Approximationmentioning

confidence: 99%

“…The distance associated with the Folland-Kaplan-Korányi type norm on (SE(n)) 0 is locally equivalent to the sub-Riemannian distance on SE (2), as was formally proven in full generality in the seminal work by Nagel, Stein and Wainger [43]. In this paper, we show by qualitative and quantitative comparison that the norm on (SE(n)) 0 indeed provides a sharp local approximation of the sub-Riemannian distances on SE(n).…”

Section: Nilpotent Approximationmentioning

confidence: 99%

“…with group elements g, g ∈ SE (2). The group acts on the (coupled) space of positions and orientations R 2…”

Section: Se(2)mentioning

confidence: 99%

“…First, key points are generated (from a single source point) using minimal path tracking with key points [5]. Second, the automatically generated key points, with estimated orientations, are grouped based on an adaption of the perceptual grouping algorithm [13] with the use of sub-Riemannian distances on SE (2). The result on the right is obtained with the nilpotent approximations of the sub-Riemannian distances in SE (2).…”

Section: The Lie Algebra Exponential Map and Commutatorsmentioning

confidence: 99%

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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: Nilpotent Approximationmentioning

confidence: 99%

Section: Nilpotent Approximationmentioning

confidence: 99%

Section: Nilpotent Approximationmentioning

confidence: 99%

“…with group elements g, g ∈ SE (2). The group acts on the (coupled) space of positions and orientations R 2…”

Section: Se(2)mentioning

confidence: 99%

Section: The Lie Algebra Exponential Map and Commutatorsmentioning

confidence: 99%

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

2018

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Recent Geometric Flows in Multi-orientation Image Processing via a Cartan Connection

Duits¹,

Smets²,

Wemmenhove³

et al. 2021

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

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Optimal Paths for Variants of the 2D and 3D Reeds–Shepp Car with Applications in Image Analysis

et al. 2018

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We present a PDE-based approach for finding optimal paths for the Reeds–Shepp car. In our model we minimize a (data-driven) functional involving both curvature and length penalization, with several generalizations. Our approach encompasses the two- and three-dimensional variants of this model, state-dependent costs, and moreover, the possibility of removing the reverse gear of the vehicle. We prove both global and local controllability results of the models. Via eikonal equations on the manifold we compute distance maps w.r.t. highly anisotropic Finsler metrics, which approximate the singular (quasi)-distances underlying the model. This is achieved using a fast-marching (FM) method, building on Mirebeau (Numer Math 126(3):515–557, 2013 ; SIAM J Numer Anal 52(4):1573–1599, 2014 ). The FM method is based on specific discretization stencils which are adapted to the preferred directions of the Finsler metric and obey a generalized acuteness property. The shortest paths can be found with a gradient descent method on the distance map, which we formalize in a theorem. We justify the use of our approximating metrics by proving convergence results. Our curve optimization model in with data-driven cost allows to extract complex tubular structures from medical images, e.g., crossings, and incomplete data due to occlusions or low contrast. Our work extends the results of Sanguinetti et al. (Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications LNCS 9423, 2015 ) on numerical sub-Riemannian eikonal equations and the Reeds–Shepp car to 3D, with comparisons to exact solutions by Duits et al. (J Dyn Control Syst 22(4):771–805, 2016 ). Numerical experiments show the high potential of our method in two applications: vessel tracking in retinal images for the case and brain connectivity measures from diffusion-weighted MRI data for the case , extending the work of Bekkers et al. (SIAM J Imaging Sci 8(4):2740–2770, 2015 ). We demonstrate how the new model without reverse gear better handles bifurcations.

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Vessel Tracking via Sub-Riemannian Geodesics on the Projective Line Bundle

Cited by 11 publications

References 24 publications

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

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

Recent Geometric Flows in Multi-orientation Image Processing via a Cartan Connection

Optimal Paths for Variants of the 2D and 3D Reeds–Shepp Car with Applications in Image Analysis

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