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Spline Interpolation on Compact Riemannian Manifolds

Charlie Sire, Mike Pereira, Thomas Romary

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 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.

Spline Interpolation on Compact Riemannian Manifolds

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 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.

Paper Structure

This paper contains 50 sections, 7 theorems, 100 equations, 12 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Theory of splines on manifolds
  3. RKHS on a compact manifold
  4. Spline interpolation on a compact manifold
  5. Smoothing splines on a compact manifold
  6. Finite element approximation of spline predictors
  7. Theory of finite element splines on manifolds
  8. Discretization of Z
  9. Interpolation formulas
  10. Intrinsic GMRF
  11. First scenario: computation of $\mathbf{u}_{0, \alpha}(\mathbf{x})$ for $\mathbf{x} \in \mathbb{R}^n$
  12. Second scenario : computation of $\mathbf{u}_{\tau, \alpha}(\mathbf{x})$ for $\mathbf{x} \in \mathbb{R}^n$
  13. Discussion on $\alpha$
  14. Anisotropic splines
  15. Local anisotropy through space deformation
  16. ...and 35 more sections

Key Result

Theorem 1

If the set of points $\mathcal{S}$ is such that the matrix $\mathbf{K}_1 = \left[K_1(\mathbf{s}_i,\mathbf{s}_j)\right]_{1 \leq i,j \leq n}$ is positive definite, then the solution $u^{\star}$ to the minimization problem is given, for all $\mathbf{s} \in \mathcal{M}$, by where $a_0 \in \mathbb{R}$, $\mathbf{b} = (b_i)_{i=1}^n \in \mathbb{R}^n$, and $\mathbf{k}_1(\mathbf{s}) = \left(K_1(\mathbf{s}_

Figures (12)

  • Figure 1: Results for the analytical function on the sphere in the first scenario. $n=10$ observation points are shown as black dots. Top left: true function. Top right: prediction with anisotropies induced in the local charts defined by spherical coordinates $(\theta, \phi)$. Bottom left: isotropic splines (no space deformation). Bottom right: splines using a kernel based on spherical harmonics.
  • Figure 2: Boxplots of absolute prediction errors at the triangulation nodes for the analytical function on the sphere. Each boxplot corresponds to a different prediction method: isotropic splines, anisotropic splines, and spherical harmonics.
  • Figure 3: Computation time of predictions at the triangulation points as the number of observation points increases from $10$ to $2000$. Red: our finite-element-based method. Blue: classical kriging with spherical harmonics.
  • Figure 4: Results for the normalized concentration of carbon dioxide on the sphere in the first scenario. $n = 50$ observation points are shown as black dots. Top left: true function. Top right: prediction with anisotropies induced in the local charts defined by spherical coordinates $(\theta, \phi)$. Bottom left: isotropic splines (no space deformation). Bottom right: splines using a kernel based on spherical harmonics.
  • Figure 5: Boxplots of absolute prediction errors at the triangulation nodes for the real-world data on the sphere. Each boxplot corresponds to a different prediction method: isotropic splines, anisotropic splines, and spherical harmonics.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Theorem 1: Interpolating splines wahbabook
  • Theorem 2: Smoothing splines wahbabook
  • Theorem 3: Finite element approximation of spline predictors.
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition I.1: Error Bound on the Eigenvalue
  • proof