Brain Connectivity Measures via Direct Sub-Finslerian Front Propagation on the 5D Sphere Bundle of Positions and Directions

Portegies, Jorg M.; Meesters, Stephan; Ossenblok, Pauly; Fuster, Andrea; Florack, Luc; Duits, Remco

doi:10.1007/978-3-030-05831-9_24

Cited by 5 publications

(3 citation statements)

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“…where Γ 𝑇 was deőned in (9). Moreover, the őeld of control sets given by p ↦ → B F 𝑈 (p) recall (10) remains continuous when using F 𝑈 or G 𝑈 (instead of F or G), which directly follows by [29,Lemma 6] in conjunction with the important coercivity property (21). The nice thing in practice is that in-place rotations are automatically placed at optimal locations by the eikonal PDE system (solved by the anisotropic fast-marching algorithm that we discuss in the next section).…”

Section: Asymmetric Finsler Extension To Automatically Deal With Bifu...mentioning

confidence: 81%

“…1. they coincide with the usual left-invariant frame if the data is locally constant, 2. they are more stable as they are constructed by coercive metric tensor őelds, recall (21).…”

Section: Discussionmentioning

confidence: 99%

“…This has motivated methods that aim to lift the image function to higher dimensional spaces. For example, the space of positions and orientations [20,21] or radius-lifted spaces [22] where the lifting yields the beneőt of disentangling seemingly complex crossing structures in the retinal images. In this article, we focus on the methods [13, 22ś24] that perform the geodesic tracking in the 3D-space of positions and orientations M 2 .…”

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: Asymmetric Finsler Extension To Automatically Deal With Bifu...mentioning

confidence: 81%

“…1. they coincide with the usual left-invariant frame if the data is locally constant, 2. they are more stable as they are constructed by coercive metric tensor őelds, recall (21).…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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