@article{10902/28915,
year = {2020},
month = {10},
url = {https://hdl.handle.net/10902/28915},
abstract = {Integral invariants obtained from Principal Component Analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms. We generalize to hypersurfaces in any dimension major results known for surfaces in space, which in turn yield a method to estimate the extrinsic and intrinsic curvature tensor of an embedded Riemannian submanifold of general codimension. In particular, integral invariants are defined by the volume, barycenter, and the EVD of the covariance matrix of the domain. We obtain the asymptotic expansion of such invariants for a spherical volume component delimited by a hypersurface and for the hypersurface patch created by ball intersections, showing that the eigenvalues and eigenvectors can be used as multi-scale estimators of the principal curvatures and principal directions. This approach may be interpreted as performing statistical analysis on the underlying point-set of a submanifold in order to obtain geometric descriptors at scale with potential applications to Manifold Learning and Geometry Processing of point clouds.},
organization = {We would like to thank Louis Scharf for very helpful discussions during the writing of this paper. This paper is based on research partially supported by the National Science Foundation under Grants No. DMS-1513633, and DMS-1322508.},
publisher = {Elsevier Inc.},
publisher = {Linear Algebra and Its Applications, 2020, 602, 179-205},
title = {Manifold curvature learning from hypersurface integral invariants},
author = {Álvarez Vizoso, Javier and Kirby Michael and Peterson, Chris},
}