The Variational Origin of Motion by Gaussian Curvature

A variational formulation of an image analysis problem has the nice feature that it is often easier to predict the e ect of minimizing a certain energy functional than to interpret the corresponding EulerLagrange equations. For example, the equations of motion for an active contour usually contains a mean curvature term, which we know will regularizes the contour because mean curvature is the rst variation of curve length, and shorter curves are typically smoother than longer ones. In some applications it may be worth considering Gaussian curvature as a regularizing term instead of mean curvature. The present paper provides a variational principle for this: We show that Gaussian curvature of a regular surface in three-dimensional Euclidean space is the rst variation of an energy functional de ned on the surface. Some properties of the corresponding motion by Gaussian curvature are pointed out, and a simple example is given, where minimization of this functional yields a nontrivial solution.