Local approximation of scalar functions on 3D shapes and volumetric data

In this paper, we tackle the problem of computing a map that locally interpolates or approximates the values of a scalar function, which have been sampled on a surface or a volumetric domain. We propose a local approximation with radial basis functions, which conjugates different features such as locality, independence of any tessellation of the sample points, and approximation accuracy. The proposed approach handles maps defined on both 3D shapes and volumetric data and has extrapolation capabilities higher than linear precision methods and moving least-squares techniques with polynomial functions. It is also robust with respect to data discretization and computationally efficient through the solution of a small and well-conditioned linear system. With respect to previous work, it allows an easy control on the preservation of local details and smoothness through both interpolating and least-squares constraints. The main application we consider is the approximation of maps defined on grids, 3D shapes, and volumetric data.