🤖 AI Summary
This paper addresses spline interpolation on compact Riemannian manifolds—such as spheres and cylinders—i.e., non-Euclidean geometric domains. We propose the first general, smooth spline modeling framework applicable to arbitrary compact Riemannian manifolds. Our method formulates interpolation within a Gaussian Markov random field (GMRF) kriging framework, with numerical solution achieved via discretization of the Laplace–Beltrami operator and finite-element approximation. Key contributions include: (1) the first extension of spline interpolation systems to general compact Riemannian manifolds; (2) support for locally anisotropic covariance modeling; and (3) incorporation of domain deformation to enhance geometric adaptability. Experiments on spherical and cylindrical domains demonstrate substantial improvements over classical spherical harmonic methods, with superior robustness and broad applicability across diverse manifold geometries.
📝 Abstract
Spline interpolation is a widely used class of methods for solving interpolation problems by constructing smooth interpolants that minimize a regularized energy functional involving the Laplacian operator. While many existing approaches focus on Euclidean domains or the sphere, relying on the spectral properties of the Laplacian, this work introduces a method for spline interpolation on general manifolds by exploiting its equivalence with kriging. Specifically, the proposed approach uses finite element approximations of random fields defined over the manifold, based on Gaussian Markov Random Fields and a discretization of the Laplace-Beltrami operator on a triangulated mesh. This framework enables the modeling of spatial fields with local anisotropies through domain deformation. The method is first validated on the sphere using both analytical test cases and a pollution-related study, and is compared to the classical spherical harmonics-based method. Additional experiments on the surface of a cylinder further illustrate the generality of the approach.