Vessels as 4-D Curves: Global Minimal 4-D Paths to Extract 3-D Tubular Surfaces and Centerlines

Abstract

In this paper, we propose an innovative approach to the segmentation of tubular structures. This approach combines all of the benefits of minimal path techniques such as global minimizers, fast computation, and powerful incorporation of user input, while also having the capability to represent and detect vessel surfaces directly which so far has been a feature restricted to active contour and surface techniques. The key is to represent the trajectory of a tubular structure not as a 3-D curve but to go up a dimension and represent the entire structure as a 4-D curve. Then we are able to fully exploit minimal path techniques to obtain global minimizing trajectories between two user supplied endpoints in order to reconstruct tubular structures from noisy or low contrast 3-D data without the sensitivity to local minima inherent in most active surface techniques. In contrast to standard purely spatial 3-D minimal path techniques, however, we are able to represent a full tubular surface rather than just a curve which runs through its interior. Our representation also yields a natural notion of a tube's "central curve." We demonstrate and validate the utility of this approach on magnetic resonance (MR) angiography and computed tomography (CT) images of coronary arteries.

Geodesic voting for the automatic extraction of tree structures. Methods and applications

“…The authors of [37] introduced a method to segment a tubular shape by finding a geodesic in a space augmented with the radius dimension in order to find both a centerline and the local radius of the tubular shape. As shown in Figure 6-(a), a 2D centerline curve and its local radius can be represented …”

“…The potential function proposed by [37] is adapted to the task of finding centerline and boundary as a minimal path. This potential is defined bỹ…”

Minimum Average-Cost Path for Real Time 3D Coronary Artery Segmentation of CT Images

“…Similar to [5,7,11], the cross sections of coronary arteries are assumed to be circular as an approximation which is reasonable for thin vessels such as coronaries. The 4D graph node X t = (p t , r t ) (Fig.…”

“…As described in [4], a variety of methods have been proposed for vessel segmentation with the aim of extracting centerlines in 3D space by using classical minimal path algorithms. Some exceptions can be found in [5,7,11] and they share the same idea of incorporating the cross sectional radius as an additional dimension together with the coordinates of the voxels. Thus when the tracking procedure ends, the radii of the vessel is also obtained.…”

“…Similar to [5,7,11], our approach represents the coronary artery centerline with its radii as a 4D tree branch with 3D coordinates of the centerline point and one addition dimension for associated radii. Given the start point, the end point and a point inside the distal part of an artery, our method can trace the centerline with associated radii efficiently without taking the risk of tracing a shortcut path, which is a classical issue of minimal path techniques.…”