Manifold Curvature Descriptors from Hypersurface Integral Invariants

TL;DR

This paper introduces a method to estimate intrinsic and extrinsic curvature of hypersurfaces using integral invariants derived from PCA, applicable to high-dimensional manifolds and useful for geometry processing.

Contribution

It generalizes existing surface curvature estimation techniques to hypersurfaces in any dimension, providing a multi-scale approach based on integral invariants and PCA.

Findings

Eigenvalues and eigenvectors serve as multi-scale estimators of principal curvatures.

Asymptotic expansions relate integral invariants to curvature properties.

Method applicable to high-dimensional point cloud data for geometric analysis.

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 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 intersetions, 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.