Vessel enhancement and segmentation aim at (binary) per-pixel segmentation considering certain local features as probabilistic vessel indicators. We propose a new methodology to combine any local probability map with local directional vessel information. The resulting global vessel segmentation is represented as a set of discrete streamlines populating the vascular structures and providing additional connectivity and geometric shape information. The streamlines are computed by numerical integration of the directional vector field that is obtained from the eigenanalysis of the local Hessian indicating the local vessel direction. The streamline representation allows for sophisticated post-processing techniques using the additional information to refine the segmentation result with respect to the requirements of the particular application such as image registration. We propose different post-processing techniques for hierarchical segmentation, centerline extraction, and catheter removal to be used for X-ray angiograms. We further demonstrate how the global approach is able to significantly improve the segmentation compared to conventional local Hessian-based approaches.