A kernel based method is proposed for the construction of signature (defining) functions of subsets of $\mathbb{R}^d$. The subsets can range from full dimensional manifolds (open subsets) to point clouds (a finite number of points) and include bounded (closed) smooth manifolds of any codimension. The interpolation and analysis of point clouds are the main application. Two extreme cases in terms of regularity are considered, where the data set is interpolated by an analytic surface, at the one extreme, and by a Hölder continuous surface, at the other. The signature function can be computed as a combination of translated kernels, the coefficients of which are the solution of a Fredholm integral equation (matrix equation in the finite dimensional case). Once it is obtained, it can be used to estimate the dimension as well as the normal and the curvatures of the interpolated manifold. The method is global and does not require the data set to be organized or structured in any particular way. It admits a variational formulation with a natural regularized counterpart, that proves useful in dealing with data sets corrupted by numerical error or noise. The underlying analytical structure of the approach is presented in general before it is applied to the case of point clouds.

翻译：本文提出了一种基于核的方法，用于构造 $\mathbb{R}^d$ 子集的签名（定义）函数。这些子集的范围可以从全维流形（开子集）到点云（有限个点），并包括任何余维的有界（闭）光滑流形。点云的插值与分析是其主要应用。我们考虑了两种正则性方面的极端情况：在一种极端下，数据集由解析曲面插值；在另一种极端下，则由 Hölder 连续曲面插值。签名函数可以计算为平移核的组合，其系数是 Fredholm 积分方程（在有限维情况下为矩阵方程）的解。一旦获得该函数，即可用于估计插值流形的维数、法向量及曲率。该方法是全局性的，不要求数据集以任何特定方式组织或结构化。它允许具有自然正则化对应项的变分公式，这在处理受数值误差或噪声污染的数据集时被证明是有用的。在将其应用于点云情况之前，我们首先总体介绍了该方法的底层分析结构。