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

Duits, Remco; Smets, Bart M. N.; Wemmenhove, A. J.; Portegies, Jim; Bekkers, Erik J.

doi:10.1007/978-3-030-03009-4_101-1

Cited by 3 publications

(6 citation statements)

References 80 publications

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“…( 20) is used to deőne the dual of the data-driven left-invariant frame via diagonalisation. This deviates from LADs in previous work [38,44]. We now have the advantage of coercivity…”

Section: Remark 9 (Advantages Of Our Data-driven Metric and Frame)contrasting

confidence: 78%

“…It has led to exact formulas [27,64] for speciőc settings (uniform cost case in the left-invariant model G given in (14), 𝐶 = 1 which was introduced on M 2 ≡ 𝑆𝐸 (2) by Citti and Sarti [65] and deeply analyzed by Sachkov [66]). Remarkably, the Cartan plus connection ∇ [+] has torsion resulting in different 'straight curves' (parallel velocity) and 'shortest curves' (parallel momentum), and it underlies many multi-orientation image analysis applications [44].…”

Section: Main Theoremsmentioning

confidence: 99%

“…First, we introduce the general deőnition of Cartan connections, after which we also introduce the Cartan plus connection [44]. In this article, we will introduce a datadriven version of the Cartan plus connection, leading to a generalization of the existing theory on shortest and straight curves in M 2 .…”

Section: Cartan Connectionsmentioning

confidence: 99%

“…We discretize the eikonal PDE (43), which has an asymmetric quadratic structure (44), using adaptive őnite differences on the Cartesian grid Ω ℎ := Ω ∩ ℎZ 3 of the domain Ω, where ℎ > 0 denotes the grid scale. Note that 2𝜋/ℎ must be an integer, for consistency with the periodic boundary conditions in the angular coordinate.…”

Section: Discretization and Consistencymentioning

confidence: 99%

“…Proposition 2 suggests the following discretization of F * (d𝑊 (p)), as in (44), for arbitrary 𝐷 ∈ 𝑆 ++ 3 , 𝜂 ∈ R 3 , 𝜀 > 0:…”

Section: Discretization and Consistencymentioning

confidence: 99%

See 4 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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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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“…( 20) is used to deőne the dual of the data-driven left-invariant frame via diagonalisation. This deviates from LADs in previous work [38,44]. We now have the advantage of coercivity…”

Section: Remark 9 (Advantages Of Our Data-driven Metric and Frame)contrasting

confidence: 78%

Section: Main Theoremsmentioning

confidence: 99%

Section: Cartan Connectionsmentioning

confidence: 99%

Section: Discretization and Consistencymentioning

confidence: 99%

“…Proposition 2 suggests the following discretization of F * (d𝑊 (p)), as in (44), for arbitrary 𝐷 ∈ 𝑆 ++ 3 , 𝜂 ∈ R 3 , 𝜀 > 0:…”

Section: Discretization and Consistencymentioning

confidence: 99%

See 3 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 of Retinal Vascular Trees with Optical and TV-Flow Enhancement in SE(2)

Berg

Zhang

Smets

et al. 2023

Lecture Notes in Computer Science

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

Cited by 3 publications

References 80 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 of Retinal Vascular Trees with Optical and TV-Flow Enhancement in SE(2)

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

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