Minimum-norm interpolation for unknown surface reconstruction

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 the 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.