Extracting Manifold Information from Point Clouds

TL;DR

This work tackles the problem of extracting intrinsic geometry (dimension, normals, curvatures) of a manifold from a point cloud using a global, kernel-based framework. It builds a defining function $u_M$ through two extremal regularity routes: a minimal-regularity variational formulation and a high-regularity heat-equation approach, with discrete counterparts yielding a computable signature $u_X$. Normal and curvature information are obtained from ambient derivatives of the interpolant, while symmetry, interpolation on hypersurfaces, and error estimates are developed to connect samples to geometry. Numerically, the method demonstrates robust normal and curvature estimation on circles, spheres, and higher-codimension objects, and its Gaussian-process interpretation provides a principled means to set regularization in noisy data scenarios.

Abstract

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 Hölder 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.

Figures (16)

• Figure 1: The top row depicts level lines of the signature function of the 30 white points and the implied normals obtained using the Gaussian kernel. The second row depicts the same for the Laplace kernel. The experiments in the first column correspond to no regularization ($\alpha=0$), where $\alpha=10^{-10}$ for the ones in the right column.
• Figure 2: The top row depicts level lines of the signature function of the 48 white points and the implied normals obtained using the Gaussian kernel. The second row depicts the same for the Laplace kernel. The experiments in the first column correspond to no regularization ($\alpha=0$), whereas $\alpha=10^{-10}$ for the ones in the right column.
• Figure 3: The circle example revisited by randomly displacing the original exact sample by 5% noise. Depicted are the level lines and implied normals obtained by means of the Gauss kernel based signature function compute with different regularization levels $\alpha=0,10^{-10},.01,.05,.1,.25$ in increasing order from top left to bottom right.
• Figure 4: The original set of data points is shown in all images as red diamonds. The blue dots show the noise-perturbed data points at the different noise levels: 5%, 10%, and 50% from left to right.
• Figure 5: The circle example revisited by randomly displacing the original exact sample by 10% noise. Depicted are the level lines and implied normals obtained by means of the Gauss kernel based signature function compute with different regularization levels $\alpha=0,10^{-10},.01,.05,.1,.25$ in increasing order from top left to bottom right.

Theorems & Definitions (30)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Remark 2.5
• Remark 2.6