Minimum-norm interpolation for unknown surface reconstruction

Abstract

We study algorithms to estimate geometric properties of raw point cloud data through implicit surface representations. Given that any level-set function with a constant level set corresponding to the surface can be used for such estimations, numerical methods need not specify a unique target function for these domain-type interpolation problems. In this paper, we focus on kernel-based interpolation by radial basis functions (RBF) and reformulate the uniquely solvable interpolation problem into a constrained optimization model. This model minimizes some user-defined norm while enforcing all interpolation conditions. To enable nontrivial feasible solutions, we propose to enhance the trial space with 1D kernel basis functions inspired by Kolmogorov-Arnold Networks (KANs). Numerical experiments demonstrate that our proposed mixed-dimensional trial space significantly improves surface reconstruction from raw point clouds. This is particularly evident in the precise estimation of surface normals, outperforming traditional RBF trial spaces including the one for Hermite interpolation. This framework not only enhances processing of raw point cloud data but also shows potential for further contributions to computational geometry. We demonstrate this with a point cloud processing example.

Figures (11)

• Figure 1: Local Stencils and Ghost Points in Point Cloud Data. (a) Illustrates the point cloud data $P$ shown in blue, with one local stencil highlighted in red. (b) Displays the same local stencil in red, augmented with off-surface (ghost) points $P_\pm$, depicted in green, which are generated using the offset parameter ${\delta}$ and estimated normals ${\hat{\mathbf{n}}}$.
• Figure 2: Schematic representation of the KAN architecture for the anisotropic RBF Neural Network, featuring the newly inspired KAN-RBF basis functions.
• Figure 3: Decomposition of a 3D data set into 1D projections: (a) shows the original data in the xyz-axis. (b) through (d) illustrate projections onto the xz, yz, and zx planes, respectively, with further projections onto the x, y, and z axes.
• Figure 4: Point clouds of $N=1000$ points for an ellipsoid, a torus, and a sphube. These point clouds are generated using parametric equations with points allocated through a Halton distribution to ensure low discrepancy.
• Figure 5: Example 1: Trial Center Distributions in KAN's First Layer: Config 1 uses the original data points, $\Xi$, as trial centers. Config 2 applies a uniform regrid, transforming $\Xi$ to $\mathcal{R}\Xi$. Config 3 stretches $\Xi$ to $\mathcal{S}\Xi$, adjusting the interval around the data center. Config 4 combines both stretching and uniform regrid techniques.

Theorems & Definitions (4)

• Remark 2.1
• Definition 4.1: KAN-inspired trial space
• Definition 4.2: HRBF trial space
• Remark 5.1