Tubular anisotropy for 2D vessel segmentation

In this paper, we present a new approach for segmentation of tubular structures in 2D images providing minimal interaction. The main objective is to extract centerlines and boundaries of the vessels at the same time. The first step is to represent the trajectory of the vessel not as a 2D curve but to go up a dimension and represent the entire vessel as a 3D curve, where each point represents a 2D disc (two coordinates for the center point and one for the radius). The 2D vessel structure is then obtained as the envelope of the family of discs traversed along this 3D curve. Since this 2D shape is defined simply from a 3D curve, we are able to fully exploit minimal path techniques to obtain globally minimizing trajectories between two or more user supplied points using front propagation. The main contribution of our approach consists on building a multi-resolution metric that guides the propagation in this 3D space. We have chosen to exploit the tubular structure of the vessels one wants to extract to built an anisotropic metric giving higher speed on the center of the vessels and also when the minimal path tangent is coherent with the vessel´s direction. This measure is required to be robust against the disturbance introduced by noise or adjacent structures with intensity similar to the target vessel. Indeed, if we examine the flux of the projected image gradient along a given direction on a circle of a given radius (or scale), one can prove that this flux is maximal at the center of the vessel, in its direction and with its exact radius. This approach is called optimally oriented flux. Combining anisotropic minimal paths techniques and optimally oriented flux we obtain promising results on noisy synthetic and real data.