Accurate analysis of the hepatic vasculature is of great importance for many medical applications, such as liver surgical planning and diagnosis of tumors and/or vascular diseases. Vessel segmentation is a pivotal step for the morphological and topological analysis of the vascular systems. Physical imaging limitations together with the inherent geometrical complexity of the vessels make the problem challenging. In this paper, we propose a series of methods and techniques that separate and segment the portal vein and the hepatic vein from CT images, and extract the centerlines of both vessel trees. We compare the results obtained with our iterative segmentation-and-reconnection approach with those obtained with a traditional region growing method, and we show that our results are substantially better.