Gradient Vector Flow Models for Boundary Extraction in 2D Images
  The Gradient Vector Flow (GVF) is a vector diffusion approach based on Partial Differential Equations (PDEs). This method has been applied together with snake models for boundary extraction medical images segmentation. The key idea is to use a diffusion-reaction PDE to generate a new external force field that makes snake models less sensitivity to initialization as well as improves the snake's ability to move into boundary concavities. In this paper, we firstly review basic results about convergence and numerical analysis of usual GVF schemes. We point out that GVF presents numerical problems due to discontinuities image intensity. This point is considered from a practical viewpoint from which the GVF parameters must follow a relationship in order to improve numerical convergence. Besides, we present an analytical analysis of the GVF dependency from the parameters values. Also, we observe that the method can be used for multiply connected domains by just imposing the suitable boundary condition. In the experimental results we verify these theoretical points and demonstrate the utility of GVF on a segmentation approach that we have developed based on snakes. 
Gilson A. Giraldi, Leandro S. Marturelli, Paulo S. Rodrigues