Geometric Tracking and Grouping of Complex Vasculature in Medical Images

The retina contains rich information about an individual’s health, enabling non-invasive diagnostics through retinal imaging. This thesis focuses on improving automated methods for analyzing these images, specifically in tracking blood vessels, identifying connected components, and classifying vessels as arteries or veins.

Tracking blood vessels
We present three advances in vessel tracking:
- Data-driven Cartan connections: We introduce a new geodesic tracking model on the space of positions and orientations M_2:= R^2 x S^1, which allows globally optimal tracking of complex vasculature. The model automatically adapts to curvature and misalignment and is computed via a modified anisotropic fast-marching method. We formulate a theorem that describes all shortest and straight curves with respect to the new data-driven connection. The experiments confirm that the geodesic tracking method uses a single, flexible, transparent, data-driven geodesic model, providing globally optimal curves that correctly follow highly complex vascular structures in retinal images.
- Optical and TV-flow enhancements: We provide a single, new, more effective Finsler function on M_2 that combines PDE-based tracking with PDE-based enhancements and optical corrections (e.g., dehazing and illumination), improving performance on challenging clinical images.
- Crossing-preserving tracking on spherical images: We recognize that the retina is not flat, but spherical. We propose a method to compute cusp-free, crossing-preserving geodesics in the space of spherical positions and orientations W_2 on wide-field images. The results show clear advantages compared to the non-crossing-preserving tracking in W_2 and are comparable to the tracking results in M_2.

Identifying connected components
We develop a new algorithm to extract connected components from a compact set I in Lie groups with a left-invariant Riemannian distance. For a given delta>0, the algorithm finds the largest cover of I such that all sets in the cover are separated by at least distance delta. Using morphological dilations, the presented method groups well-aligned structures and distinguishes at crossings. We prove convergence in a finite number of steps. Furthermore, we propose to use specific affinity matrices, post-processing the delta-connected components by grouping them based on their local proximity and alignment. Applying the delta-connected component method to retinal images via the orientation scores on SE(2) outperforms standard R^2 approaches in identifying vascular branches.

Classifying vessels as arteries and veins
We enhance artery-vein classification by combining convolutional neural networks with our delta-connected component framework. This ensures consistent labeling along entire vascular trees, improving reliability in clinical applications.