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 smooth variations and local anisotropies via 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.

翻译：样条插值是一类广泛使用的插值方法，通过构造光滑插值函数来最小化涉及拉普拉斯算子的正则化能量泛函。尽管现有方法多聚焦于欧几里得域或球面，并依赖拉普拉斯算子的谱性质，本文通过将样条插值与克里金法等价转换，引入了一种适用于一般流形的方法。具体而言，该方法基于高斯马尔可夫随机场及三角化网格上拉普拉斯-贝尔特拉米算子的离散化，采用随机场的有限元逼近来定义流形上的随机场。该框架通过域形变实现对具有平滑变化和局部各向异性空间场的建模。该方法首先在球面上通过解析测试案例和污染相关研究进行验证，并与传统的球谐函数方法进行比较。在圆柱面上的额外实验进一步展示了该方法的通用性。