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.

Graphical abstractLevel-sets of a Laplacian eigenfunction f:P→Rf:P→R on a surface mesh and iso-surfaces of the corresponding local approximations F:R3→RF:R3→R with interpolating constraints. Figure optionsDownload full-size imageDownload high-quality image (506 K)Download as PowerPoint slideHighlights► We tackle the problem of computing the map underlying a discrete map defined on a discrete surface or a volumetric domain. ► The underlying map locally interpolates/approximates the values of the input scalar function, through meshless techniques. ► The proposed approach requires neither a parameterization domain nor a volumetric tessellation. ► The method provides a good approximation and extrapolation of the function values, without over-fitting the input data. ► The computation is stable with respect to data sampling, connectivity, noise, and local perturbations.