Vessels as 4D curves: Global minimal 4D paths to extract 3D tubular surfaces

In this paper, we propose an innovative approach to the segmentation of tubular or vessel-like structures which combines all the benefits of minimal path techniques (global minimizers, fast computation, powerful incorporation of user input) with some of the benefits of active surface techniques (representation of a full 3D tubular surface rather than a just curve). The key is to represent the trajectory of the vessel not as a 3D curve but to go up a dimension and represent the entire vessel as a 4D curve, where each 4D point represents a 3D sphere (three coordinates for the center point and one for the radius). The 3D vessel structure is then obtained as the envelope of the family of spheres traversed along this 4D curve. Because the 3D surface is simply a curve in 4D, we are able to fully exploit minimal path techniques to obtain global minimizing trajectories between two user supplied end-points in order to reconstruct vessels from noisy or low contrast 3D data without the sensitivity to local minima inherent in most active surface techniques. In contrast to standard purely spatial 3D minimal path techniques, however, we are able to represent the full vessel surface rather than just a curve which runs through its interior. Our representation also yields a natural notion of a vessel's "central curve", which is obtained by tracing the center points of the family of 3D spheres rather than its envelope. We demonstrate the utility of this approach on 2D images of roads as well as both 2D and 3D MR angiography and CT .