Related papers: Thinplate Splines on the Sphere

To predict smooth physical phenomena from observations, spline interpolation provides an interpretable framework by minimizing an energy functional associated with the Laplacian operator. This work proposes a methodology to construct a…

We provide a new approach for computing integrals over hypersurfaces in the level set framework. The method is based on the discretization (via simple Riemann sums) of the classical formulation used in the level set framework, with the…

Computing the envelope of deforming planar domains is a significant and challenging problem with a wide range of potential applications. We approximate the envelope using circular arc splines, curves that balance geometric flexibility and…

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation…

This note concerns the interpolation problem with two parametrized families of splines related to polynomial spline interpolation. We address the questions of uniqueness and establish basic convergence rates for splines of the form $…

A thin plate spline surface for interpolation of smooth transfinite data prescribed along concentric circles was recently proposed by Bejancu, using Kounchev's polyspline method. The construction of the new `Beppo Levi polyspline' surface…

This article gives a ``fundamental solution'' based energy-norm harmonic interpolation approach for two half-space settings of interest: the upper-half $\mathbb{R}^n$ plane, where fundamental solutions satisfy Laplace's equation, and the…

We present Neural Splines, a technique for 3D surface reconstruction that is based on random feature kernels arising from infinitely-wide shallow ReLU networks. Our method achieves state-of-the-art results, outperforming recent neural…

This letter aims at resolving the issues raised in the recent short communication [1] and answered by [2] by proposing a systematic approximation scheme based on non-mapped shape functions, which both allows to fully exploit the unique…

We address the problem of approximating an unknown function from its discrete samples given at arbitrarily scattered sites. This problem is essential in numerical sciences, where modern applications also highlight the need for a solution to…

We consider piecewise linear interpolation from the perspective of kernel interpolation and quadrature. If the Sobolev space $W_2^1(0, 1)$ is equipped with a suitable inner product, its reproducing kernel is piecewise linear and gives rise…

Representing complex shapes with simple primitives in high accuracy is important for a variety of applications in computer graphics and geometry processing. Existing solutions may produce suboptimal samples or are complex to implement. We…

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way,…

This survey gives an overview of three central algebraic themes related to the study of splines: duality, group actions, and homology. Splines are piecewise polynomial functions of a prescribed order of smoothness on some subdivided domain…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda + \sum_{k = 1}^d [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are linear forms in…

The thin plate spline is a popular tool for the interpolation and smoothing of scattered data. In this paper we propose a novel stabilized mixed finite element method for the discretization of thin plate splines. The mixed formulation is…

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…

In this note we prove almost sure unisolvence of RBF interpolation on randomly distributed sequences by a wide class of polyharmonic splines (including Thin-Plate Splines), without polynomial addition.

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda_0 + \sum_{k = 1}^d \lambda_k [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are…

We extend the classical theory of variational interpolating splines to the case of compact Riemannian manifolds. Our consideration includes in particular such problems as interpolation of a function by its values on a discrete set of points…