High-order Variation of Active Contours and Its Application

Two central questions in medical image segmentation raised by C. A. Davatyikos and J. L. Prince are: will an active contour algorithm converges to a unique solution, and if so, will this solution be near the truth? This paper had showed a way to calculate the second order variation of the energy functional of an active contour, which was used to determine the convexity of the energy functional and partially answered the above questions, because a sufficient condition for ensuring a critical path to be a true minimizer of the energy functional is such that ensures the energy functional to be convex near the path. This paper had also pointed out that a solution to the Euler-Lagrange equation of an energy functional may not be a minimizer of the functional, but it was taken for granted as the minimizer in many papers about medical image segmentations. In addition, this paper had introduced a general formula for the calculation of the Hesse of an energy functional and showed by example how to use it to induce other conditions for the case of normal geodesics.