Weighted Minimal Hypersurfaces and Their Applications in Computer Vision

Abstract. Many interesting problems in computer vision can be formulated as a minimization problem for an energy functional. If this functional is given as an integral of a scalar-valued weight function over an unknown hypersurface, then the minimal surface we are looking for can be determined as a solution of the functional's Euler-Lagrange equation. This paper deals with a general class of weight functions that may depend on the surface point and normal. By making use of a mathematical tool called the method of the moving frame, we are able to derive the EulerLagrange equation in arbitrary-dimensional space and without the need for any surface parameterization. Our work generalizes existing proofs, and we demonstrate that it yields the correct evolution equations for a variety of previous computer vision techniques which can be expressed in terms of our theoretical framework. In addition, problems involving minimal hypersurfaces in dimensions higher than three, which were previo...

• Type: Conference
• Year: 2004
• Where: ECCV
• Authors: Bastian Goldlücke, Marcus A. Magnor
• Keywords: Computer Vision, Correct Evolution Equations, ECCV 2004, Many Interesting Problems, Minimal Surface, Scalar-valued Weight Function, Surface Parameterization