Geometric Kernel Interpolation and Regression

TL;DR

This paper introduces a kernel-based framework connecting interpolation and regression for geometric data, enabling numerical computation of geometric quantities and operators on manifolds, with applications to point clouds and hypersurfaces.

Contribution

It presents a novel variational interpretation linking kernel interpolation and regression, facilitating geometric analysis and operator approximation on manifolds from point cloud data.

Findings

Kernel regression provides regularization for ill-posed interpolation.

Geometric quantities like tangent planes and curvatures can be numerically estimated.

The method effectively approximates geometric operators such as the Laplace-Beltrami.

Abstract

Exploiting the variational interpretation of kernel interpolation we exhibit a direct connection between interpolation and regression, where interpolation appears as a limiting case of regression. By applying this framework to point clouds or samples of smooth manifolds (hypersurfaces, in particular), we show how fundamental geometric quantities such as tangent plane and principal curvatures can be computed numerically using a kernel based (approximate) level set function (often a defining function) for smooth hypersurfaces. In the case of point clouds, the approach generates an interpolated hypersurface, which is an approximation of the underlying manifold when the cloud is a sample of it. It is shown how the geometric quantities obtained can be used in the numerical approximation/computation of geometric operators like the surface gradient or the Laplace-Beltrami operator in the spirit of kernel based meshfree methods. Kernel based interpolation can be extremely ill-posed, especially when using smooth kernels, and the regression approximation offers a natural regularization that proves also quite useful when dealing with geometric or functional data that are affected by errors or noise.