Directional analysis of 3D tubular structures via isotropic well-localized atoms

Accurate segmentation of 3D vessel-like structures is a major challenge in medical imaging. In this paper, we introduce a novel approach for the detection of 3D tubular structures that is particularly suited to capture the geometry of vessel-like networks, such as dendritic trees and vascular systems. Even though our approach relies on a system of isotropic multiscale analyzing atoms, we prove that their interaction via convolution with a tubular structure is equivalent to a set of directional atoms at various scales, automatically oriented along any possible direction and with cylindrical symmetry. This result sets the theoretical groundwork for the design of efficient discrete algorithms aiming at extracting the geometry of vessel-like structures in 3D medical images.