Weighted Minimal Hypersurface Reconstruction

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 to solve problems involving minimal hypersurfaces in 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.