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Abstract:
Many 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
sought-after minimal surface 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 surface point coordinates as well as
surface orientation. We derive the Euler-Lagrange equation in arbitrary
dimensional space without the need for any surface parameterization,
generalizing existing proofs. Our work opens up the possibility of solving
problems involving minimal hypersurfaces in a dimension higher than three,
which were previously impossible to solve in practice. We also introduce two
applications of our new framework: We show how to reconstruct temporally
coherent geometry from multiple video streams, and we use the same framework
for the volumetric reconstruction of refractive and transparent natural
phenomena, here bodies of flowing water.
