Global Optimality of Gradient Vector Flow

Abstract

In [1, 2], Xu and Prince introduced gradient vector flow (GVF), a class of vector fields derived from images, that can be used as external forces for deformable models [3]. Figure 1 illustrates the use of GVF in a deformable model to extract a two-dimensional U-shape object. GVF can be defined through either a variational formulation or a partial differential equation. In this paper, we are concerned with the variational formulation introduced in [2]. The solution to this variational formulation was obtained in [2] by first deriving the necessary condition, the Euler-Lagrange Equation (ELE), and then solving the ELE numerically. Here, we prove the convexity of the GVF variational formulation using the convexity analysis described in [4] and point out that the corresponding ELE is in fact a sufficient condition for globally minimizing the variational energy formulation.