Reconstructing vasculature in three dimensions is a challenging problem. Early approaches concentrated on coronary vasculature in X-ray images, recent work uses magnetic resonance imagery of cerebral vasculature. In both cases a priori information has been used, and often the way this is represented has proven limiting to the scope of applications supported. For example, a particular representation may be useful only for X-ray images. This paper addresses two issues: 1) representing a collection of vasculature and 2) the reconstruction of individual vasculature from images. Our representation learns the variations in branching structures and vessel shapes that occur between individuals. It supports a vascular catalogue containing three-dimensional (3-D) anatomical models. The representation is task independent; here we use it to reconstruct vasculature from images. Our algorithm has four features to which we draw attention: 1) it is not premised wholly upon X-ray images (though that is our focus here); 2) it produces several feasible solutions rather than one; 3) it can generalize from the catalogue to reconstruct instances not yet learned; 4) it exhibits polynomial time complexity, reasonable memory consumption, and is reliable. Both our representation and reconstruction algorithm are new and useful approaches. In support of these claims, we present results gathered from X-rays of both simulated and real vasculature.