We propose a novel hybrid approach for automatic 3D segmentation and quantification of high-resolution 7 Tesla magnetic resonance angiography (MRA) images of the human cerebral vasculature. Our approach consists of two main steps. First, a 3D model-based approach is used to segment and quantify thick vessels and most parts of thin vessels. Second, remaining vessel gaps of the first step in low-contrast and noisy regions are completed using a 3D minimal path approach, which exploits directional information. We present two novel minimal path approaches. The first is an explicit approach based on energy minimization using probabilistic sampling, and the second is an implicit approach based on fast marching with anisotropic directional prior. We conducted an extensive evaluation with over 2300 3D synthetic images and 40 real 3D 7 Tesla MRA images. Quantitative and qualitative evaluation shows that our approach achieves superior results compared with a previous minimal path approach. Furthermore, our approach was successfully used in two clinical studies on stroke and vascular dementia.