A multi-scale approach to 3D scattered data interpolation with compactly supported basis functions

We propose a hierarchical approach to 3D scattered data interpolation with compactly supported basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarse-to-fine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point cloud distributed along a surface, we first use spatial down sampling to construct a coarse-to-fine hierarchy of point sets. Then we interpolate the sets starting from the coarsest level. We interpolate a point set of the hierarchy, as an offsetting of the interpolating function computed at the previous level. An original point set and its coarse-to-fine hierarchy of interpolated sets is presented. According to our numerical experiments, the method is essentially faster than the state-of-the-art scattered data approximation with globally supported RBFs (Carr et al., 2001) and much simpler to implement.