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 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 H\"older continuous surface, at the other. The signature function can be computed as a linear combination of translated kernels, the coefficients of which are the solution of a finite dimensional linear problem. Once it is obtained, it can be used to estimate the dimension as well as the normal and the curvatures of the interpolated surface. The method is global and does not require explicit knowledge of local neighborhoods or any other structure present in the data set. It admits a variational formulation with a natural ``regularized'' counterpart, that proves to be 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：提出一种基于核的方法，用于构造 $\mathbb{R}^d$ 子集的签名（定义）函数。这些子集的范围涵盖全维流形（开子集）到点云（有限个点），并包括任意余维的有界光滑流形。点云的插值与分析是其主要应用方向。本研究考虑了正则性方面的两个极端情形：数据集分别由解析曲面和赫尔德连续曲面进行插值。签名函数可表示为平移核的线性组合，其系数通过求解有限维线性问题获得。该函数一旦求得，即可用于估计插值曲面的维数、法向量及曲率。该方法具有全局性，无需显式了解数据集中的局部邻域或其他结构。方法具有变分形式，其自然的“正则化”版本在处理受数值误差或噪声污染的数据集时尤为有效。在将该方法应用于点云案例之前，先对其底层分析框架进行一般性阐述。