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Many interesting problems in computer vision can be formu- lated as a minimization problem for an energy functional. If this func- tional 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 Euler- Lagrange 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 pre- viously impossible to solve in practice, can now be introduced and han- dled by generalized versions of existing algorithms. As one example, we sketch a novel idea how to reconstruct temporally coherent geometry from multiple video streams.

Weighted Minimal Hypersurfaces and Their Applications in Computer Vision