Extremal Controls for the Duits Car

Mashtakov, Alexey

doi:10.1007/978-3-030-80209-7_9

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…This explains the deőnition of 𝑣(𝑊) below (36). A more explicit integration formula for (36) can be obtained in a similar way as in [27,64] (where exact solutions are derived for 𝐶 = 𝑈 = 1) via momentum preservation laws.…”

Section: Remark 14 (Assumptions On Point G In Backtracking (36))mentioning

confidence: 89%

“…The shortest curve 𝛾 : [0, 1] → 𝐺 with 𝛾(0) = g and 𝛾(1) = g 0 may be computed by steepest descent backtracking on the distance map 𝑊 (g) = 𝑑 G 𝑈 (g, g 0 ) 𝛾(𝑡) := 𝛾 𝑈 g,g 0 (𝑡) = Exp g (𝑣(𝑊)) (36) where Exp integrates the following vector field on 𝐺:…”

Section: Theorem 1 (Straight and Shortest Curves: Parallel Velocity A...mentioning

confidence: 99%

“…Eq. (36) follows by (34). Here 𝜆 = d𝑊 implies we must indeed set the following momentum components: λ𝑘 = A 𝑈 𝑘 𝑊 (𝛾(•)) , for all 𝑘 = 1, .…”

Section: Remark 24mentioning

confidence: 99%

“…Similarly to the plus-control variant of previous geometric curve optimization models [29] we replace symmetric, anisotropic, (sub-)Riemannian geodesic tracking by asymmetric, anisotropic Finslerian models that avoid cusps and allow for automatic placement of in-place rotations (recall Fig. 18 and see [29,36]). As a result, our model automatically accounts for bifurcations, thereby reducing the number of anchor points.…”

Section: Introductionmentioning

confidence: 99%

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Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

Berg

Smets

Pai

et al. 2022

Preprint

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We introduce a data-driven version of the plus Cartan connection on the homogeneous space $\mathbb{M}_2$ of 2D positions and orientations. We formulate a theorem that describes all shortest and straight curves (parallel velocity and parallel momentum, respectively) with respect to this new data-driven connection and corresponding Riemannian manifold. Then we use these shortest curves for geodesic tracking of complex vasculature in multi-orientation image representations defined on $\mathbb{M}_{2}$. The data-driven Cartan connection characterizes the Hamiltonian flow of all geodesics. It also allows for improved adaptation to curvature and misalignment of the (lifted) vessel structure that we track via globally optimal geodesics. We compute these geodesics numerically via steepest descent on distance maps on $\mathbb{M}_2$ that we compute by a new modified anisotropic fast-marching method. Our experiments range from tracking single blood vessels with fixed endpoints to tracking complete vascular trees in retinal images. Single vessel tracking is performed in a single run in the multi-orientation image representation, where we project the resulting geodesics back onto the underlying image. The complete vascular tree tracking requires only two runs and avoids prior segmentation, placement of extra anchor points, and dynamic switching between geodesic models. Altogether we provide a geodesic tracking method using a single, flexible, transparent, data-driven geodesic model providing globally optimal curves which correctly follow highly complex vascular structures in retinal images. All experiments in this article can be reproduced via documented \emph{Mathematica} notebooks available at \cite{githubNicky}.

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Section: Remark 14 (Assumptions On Point G In Backtracking (36))mentioning

confidence: 89%

Section: Theorem 1 (Straight and Shortest Curves: Parallel Velocity A...mentioning

confidence: 99%

“…Eq. (36) follows by (34). Here 𝜆 = d𝑊 implies we must indeed set the following momentum components: λ𝑘 = A 𝑈 𝑘 𝑊 (𝛾(•)) , for all 𝑘 = 1, .…”

Section: Remark 24mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

Berg

Smets

Pai

et al. 2022

Preprint

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Geodesic Tracking via New Data-Driven Connections of Cartan Type for Vascular Tree Tracking

van den Berg,

Smets,

Pai

et al. 2024

J Math Imaging Vis

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We introduce a data-driven version of the plus Cartan connection on the homogeneous space $${\mathbb {M}}_2$$ M 2 of 2D positions and orientations. We formulate a theorem that describes all shortest and straight curves (parallel velocity and parallel momentum, respectively) with respect to this new data-driven connection and corresponding Riemannian manifold. Then we use these shortest curves for geodesic tracking of complex vasculature in multi-orientation image representations defined on $${\mathbb {M}}_{2}$$ M 2 . The data-driven Cartan connection characterizes the Hamiltonian flow of all geodesics. It also allows for improved adaptation to curvature and misalignment of the (lifted) vessel structure that we track via globally optimal geodesics. We compute these geodesics numerically via steepest descent on distance maps on $${\mathbb {M}}_2$$ M 2 that we compute by a new modified anisotropic fast-marching method.Our experiments range from tracking single blood vessels with fixed endpoints to tracking complete vascular trees in retinal images. Single vessel tracking is performed in a single run in the multi-orientation image representation, where we project the resulting geodesics back onto the underlying image. The complete vascular tree tracking requires only two runs and avoids prior segmentation, placement of extra anchor points, and dynamic switching between geodesic models. Altogether we provide a geodesic tracking method using a single, flexible, transparent, data-driven geodesic model providing globally optimal curves which correctly follow highly complex vascular structures in retinal images. All experiments in this article can be reproduced via documented Mathematica notebooks available at van den Berg (Data-driven left-invariant tracking in Mathematica, 2022).

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Extremal Trajectories in a Time-Optimal Problem on the Group of Motions of a Plane with Admissible Control in a Circular Sector

Mashtakov,

Sachkov

2023

Proc. Steklov Inst. Math.

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Extremal Controls for the Duits Car

Cited by 4 publications

References 14 publications

Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

Geodesic Tracking via New Data-Driven Connections of Cartan Type for Vascular Tree Tracking

Extremal Trajectories in a Time-Optimal Problem on the Group of Motions of a Plane with Admissible Control in a Circular Sector

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