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

Abstract

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}.

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

“…2a. In previous works by various authors, it has been shown that PDE-based geometric tracking algorithms [4,9,10,12] perform well in M 2 . Here, one first calculates a distance map which is based on the image data.…”

“…Applying the TV-flow enhancement in M 2 leads to crossing-preserving contextual denoising that preserves crossings, and line structures, and removes noisy non-aligned structures from the optical enhancement. Altogether the scheme results in a vascular tracking algorithm that provides better results as wavefronts follow the complex branching vasculature better than the approach in [4], see Fig. 1.…”

“…2. The main contributions of this article are; 1) the development of a new asymmetric, data-driven left-invariant Finsler geometric model that includes contextual contrast enhancement via TV-flows on SE(2), 2) experiments that show that application of this new Finsler geometric model reduces many tracking errors compared to previous left-invariant models [2,9] and the recent datadriven model [4], 3) the new model performs very well on both realistic, unevenly illuminated retinal images and allows full vascular trees computations from a single distance map. The inclusion of the optical and TV-flow enhancements in the Finsler function no longer require a 2-step algorithm as in [4], but with the techniques in this work a single run of building the distance map suffices.…”

“…Then, the shortest path is obtained by a steepest descent algorithm applied on this distance map. In all experiments we use the Anisotropic Fast Marching [4,13] by J.-M. Mirebeau for distance map computations.…”

“…In the experimental section, we rely on the mixed metric tensor field, which is needed to avoid wrong exits at complex structures, see [4], and is given by:…”