Extrinsic Curvature Estimation by Closest Point Projection of Rays

Abstract

We propose a parametrization-free framework for estimating the second fundamental form and the principal curvatures of a submanifold of $\mathbb{R}^d$. The method uses the closest point projections of a few short random rays and recovers the curvature from the curves that these projections induce on the submanifold, without coordinate charts or local fitting. We analyze the geometry of the induced curves and characterize when they are geodesics. For hypersurfaces, these results give two algorithms with error bounds: a least squares reconstruction of the second fundamental form from the curvatures of the induced curves, and a direct estimate of all principal curvatures from the eigenvalues of a $d\times d$ matrix built from $d$ rays, which needs no normal information and costs $\mathcal{O}(d^3)$ besides the projections. In higher codimension, we can also recover the second fundamental form one normal direction at a time. On point clouds, where the projection becomes a nearest neighbor query, we bound the additional error. Experiments confirm the predicted rates for hypersurfaces in up to $128$ dimensions, for surfaces of codimension up to $98$, and on point clouds, demonstrating the effectiveness of the proposed framework in practice.