English

Weak and approximate curvatures of a measure: a varifold perspective

Classical Analysis and ODEs 2020-01-29 v2 Numerical Analysis Functional Analysis Numerical Analysis

Abstract

By revisiting the notion of generalized second fundamental form originally introduced by Hutchinson for a special class of integral varifolds, we define a weak curvature tensor that is particularly well-suited for being extended to general varifolds of any dimension and codimension through regularization. The resulting approximate second fundamental forms are defined not only for piecewise-smooth surfaces, but also for datasets of very general type (like, e.g., point clouds). We obtain explicitly computable formulas for both weak and approximate curvature tensors, we exhibit structural properties and prove convergence results, and lastly we provide some numerical tests on point clouds that confirm the generality and effectiveness of our approach.

Keywords

Cite

@article{arxiv.1904.05930,
  title  = {Weak and approximate curvatures of a measure: a varifold perspective},
  author = {Blanche Buet and Gian Paolo Leonardi and Simon Masnou},
  journal= {arXiv preprint arXiv:1904.05930},
  year   = {2020}
}

Comments

35 pages, 5 figures