Using the shape gradient for active contour segmentation:: from the continuous to the discrete formulation

A variational approach to image or video segmentation consists in defining an energy depending on local or global image characteristics, the minimum of which being reached for objects of interest. This study focuses on energies written as an integral on a domain of a function which can depend on this domain. The derivative of the energy with respect to the domain, the so-called shape derivative, is a function of a velocity field applied to the domain boundary. For a given, non-optimal domain, the velocity should be chosen such that the shape derivative is negative, thus indicating a way to deform the domain in order to decrease its energy. Minimizing the energy through an iterative deformation process is known as the active contour method. In the continuous framework, setting the velocity to the opposite of the gradient associated with the L-2 inner product is a common practice. In this paper, it is noted that the negativity of the shape derivative is not preserved, in general, by the discretization of this velocity required by implementation. In order to guarantee that the negativity condition holds in the discrete framework, it is proposed to choose the velocity as a linear combination of pre-defined velocities. This approach also gives more flexibility to the active contour process by allowing to introduce some a priori knowledge about the optimal domain. Some experimental results illustrate the differences between the classical and the proposed approach.