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.

翻译：样条插值是一类广泛使用的插值方法，通过构造光滑的插值函数来最小化涉及拉普拉斯算子的正则化能量泛函。尽管现有方法大多关注欧几里得域或球面，并依赖于拉普拉斯算子的谱特性，但本文通过利用其与克里金法的等价性，提出了一种在一般流形上进行样条插值的方法。具体而言，所提出的方法基于高斯马尔可夫随机场以及在三角网格上对拉普拉斯-贝尔特拉米算子的离散化，使用定义在流形上的随机场的有限元近似。该框架通过区域变形实现了具有局部各向异性的空间场建模。该方法首先在球面上通过解析测试案例和一项污染相关研究进行验证，并与经典的基于球谐函数的方法进行了比较。在圆柱表面上的进一步实验进一步说明了该方法的普适性。