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

Authors:
Jan Erik Solem;Niels Chr. Overgaard
Affiliations:
Applied Mathematics Group, School of Technology and Society, Malmö University, Sweden;Applied Mathematics Group, School of Technology and Society, Malmö University, Sweden
Venue:
Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
Year:
2005

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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.