A Gradient Descent Procedure for Variational Dynamic Surface Problems with Constraints

Many problems in image analysis and computer vision involving boundaries and regions can be cast in a variational formulation. This means that m-surfaces, e.g. curves and surfaces, are determined as minimizers of functionals using e.g. the variational level set method. In this paper we consider such variational problems with constraints given by functionals. We use the geometric interpretation of gradients for functionals to construct gradient descent evolutions for these constrained problems. The result is a generalization of the standard gradient projection method to an infinite-dimensional level set framework. The method is illustrated with examples and the results are valid for surfaces of any dimension.