Weighted Minimal Hypersurface Reconstruction

@article{citeulike:1562618, 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.}, author = {Goldlucke, Bastian and Ihrke, Ivo and Linz, Christian and Magnor, Marcus }, booktitle = {Pattern Analysis and Machine Intelligence, IEEE Transactions on}, citeulike-article-id = {1562618}, journal = {Pattern Analysis and Machine Intelligence, IEEE Transactions on}, keywords = {levelset, variational}, number = {7}, pages = {1194--1208}, posted-at = {2007-08-15 15:08:46}, priority = {2}, title = {Weighted Minimal Hypersurface Reconstruction}, url = {http://ieeexplore.ieee.org/xpls/abs\_all.jsp?arnumber=4204162}, volume = {29}, year = {2007} }

TY - JOUR ID - citeulike:1562618 L3 - citeulike-article-id:1562618 TI - Weighted Minimal Hypersurface Reconstruction T2 - Pattern Analysis and Machine Intelligence, IEEE Transactions on JF - Pattern Analysis and Machine Intelligence, IEEE Transactions on VL - 29 IS - 7 SP - 1194 EP - 1208 N2 - 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. KW - levelset KW - variational AU - Goldlucke, Bastian AU - Ihrke, Ivo AU - Linz, Christian AU - Magnor, Marcus PY - 2007/// UR - http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4204162 ER -