A geometric formulation of gradient descent for variational problems with moving surfaces

Abstract

Developments within the computer vision community have led to the formulation of many interesting problems in a variational setting. This paper introduces the manifold of admissible surfaces and a scalar product on its tangent spaces. This makes it possible to properly define gradients and gradient descent procedures for variational problems involving m-surfaces. These concepts lead to a geometric understanding of current state of the art evolution methods and steepest descent evolution equations. By geometric reasoning, common procedures within the variational level set framework are explained and justified. Concrete computations for a general class of functionals are presented and applied to common variational problems for curves and surfaces. © Springer-Verlag Berlin Heidelberg 2005.