Data-Driven Sub-Riemannian Geodesics in SE(2)

Bekkers, EJ Erik; Duits, Remco; Mashtakov, Alexey; Sanguinetti, GR Gonzalo

doi:10.1007/978-3-319-18461-6_49

Cited by 9 publications

(15 citation statements)

References 19 publications

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“…Here we describe the domains of parameter ξ > 0, which correspond to different dynamics of the Hamiltonian system (39). It is well known (see, e.g., [22]) that this system has two integrals of motion that depend only on momentum components h i (see illustration in Fig.…”

Section: Classification Of Different Types Of Solutionsmentioning

confidence: 99%

“…Thus we obtained the vertical part of (49). Doing the same for the horizontal part in (39), we obtain the full system (49), where the Hamiltonian H(s) = ξ −2 h 2 1 (s)+h 2 2 (s) 2 = 1 2 together with the cuspless constraint h 1 (s) > 0 allows us to write h 1 (s) = ξ 1 − h 2 2 (s). Finally, the boundary conditions (50) follow from (40), since t = 0 ⇔ s = 0.…”

Section: Sub-riemannian Geodesics In So(3) With Cuspless Spherical Prmentioning

confidence: 99%

“…In this comparison, we use the SR arclength parameterization t, where the geodesic γ SO(3) (·) is given by exponential map Exp(h 0 , ·), recall (47), i.e., by the projection in SO(3) of the solution of the Hamiltonian system (39), with initial condition h(0) = h 0 , γ SO(3) (0) = e = (0, 0, 0). To establish the comparison of wavefronts we switch to polar coordinates (β, c), recall (43), where the Hamiltonian system in SO(3) reads as:…”

Section: Comparison Of Sr Wavefronts In So(3) and Se(2) For C =mentioning

confidence: 99%

“…To integrate the Hamiltonian system (39) we follow the idea of V. Jurdjevic [24], where one can employ left invariance and introduce initial rotation D 0 of momentum space, such that the initial momentum transforms to h(0) = (0, 0, M ), solve the problem in this simple case (i.e., to find the trajectorỹ R(t) that corresponds to h(0) = (0, 0, M )), and obtain general solution R(t) by a backward transformation R(t) = D −1 0R (t). Proof of Theorem 4.1: Substitution of (36) to (31) giveṡ…”

Section: B Integration Of the Hamiltonian System P 1 Mecmentioning

confidence: 99%

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Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in $${\text {SO(3)}}$$ SO(3)

et al. 2017

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In order to detect salient lines in spherical images, we consider the problem of minimizing the functional l 0 C(γ (s)) ξ 2 + k 2 g (s) ds for a curve γ on a sphere with fixed boundary points and directions. The total length l is free, s denotes the spherical arclength, and k g denotes the geodesic curvature of γ . Here the smooth external cost C ≥ δ > 0 is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group SO(3) and show that the spherical projection of certain SR geodesics provides a solution to our curve optimization problem. In fact, this holds only for the geodesics whose spherical projection does not exhibit a cusp. The problem is a spherical extension of a well-known contour perception model, where we extend the model by Boscain and Rossi to the general case ξ > 0, C = 1. For C = 1, we derive SR geodesics and evaluate the first cusp time. We show that these curves have a simpler expression when they are parameterized by spherical arclength rather than by sub-Riemannian arclength. For case C = 1 (data-driven SR geodesics), we solve via a SR Fast Marching method. Finally, we show an experiment of vessel tracking in a spherical image of the retina and study the effect of including the spherical geometry in analysis of vessels curvature.

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Section: Classification Of Different Types Of Solutionsmentioning

confidence: 99%

Section: Sub-riemannian Geodesics In So(3) With Cuspless Spherical Prmentioning

confidence: 99%

Section: Comparison Of Sr Wavefronts In So(3) and Se(2) For C =mentioning

confidence: 99%

Section: B Integration Of the Hamiltonian System P 1 Mecmentioning

confidence: 99%

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Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in $${\text {SO(3)}}$$ SO(3)

et al. 2017

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“…This paper is devoted to usage of sub-Riemannian (SR) geometry in image processing and modeling of the human visual system. It summarizes the results of joint works [1][2][3][4][5]. In recent research in psychology of vision [6,7] it was shown that SR geodesics appear as natural curves that model a mechanism of the primary visual cortex V1 of a human brain for completion of contours that are partially corrupted or hidden from observation, see Fig.…”

Section: Introductionmentioning

confidence: 92%

Sub-Riemannian Geometry in Image Processing and Modeling of the Human Visual System

Mashtakov¹

2019

Nelin. Dinam.

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This paper summarizes results of a sequence of works related to usage of sub-Riemannian (SR) geometry in image processing and modeling of the human visual system. In recent research in psychology of vision (J. Petitot, G. Citti, A. Sarti) it was shown that SR geodesics appear as natural curves that model a mechanism of the primary visual cortex V1 of a human brain for completion of contours that are partially corrupted or hidden from observation. We extend the model to include data adaptivity via a suitable external cost in the SR metric. We show that data adaptive SR geodesics are useful in real image analysis applications and provide a refined model of V1 that takes into account the presence of a visual stimulus.

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