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Abstract

Many interesting problems in computer vision can be formulated as a minimization problem for an {\em 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 {\em 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 practical applications, the surface evolution which converges to a solution of the Euler-Lagrange equation can be implemented using level set techniques. The well-known transition to a level set evolution equation, which we briefly review in this paper, works in the general case as well. That way, problems involving minimal hypersurfaces in dimensions higher than three, which were previously impossible to solve in practice, can now be introduced and handled by generalized versions of existing algorithms. As one example, we sketch a novel idea how to reconstruct temporally coherent geometry from multiple video streams.