Multi-scale and Adaptive CS-RBFs for Shape Reconstruction from Clouds of Points

We describe a multi-scale approach for interpolation and approximation of a point set surface by compactly supported radial basis functions. Given a set of points scattered over a surface, we first use down-sampling to construct a point set hierarchy. Then starting from the coarsest level, for each level of the hierarchy, we use compactly supported RBFs to approximate the set of points at the level as an offset of the RBF approximation computed at the previous level. A simple RBF centre reduction scheme combined with the multi-scale approach accelerates the latter and allows us to achieve high quality approximations using relatively small number of RBF centres.

We also develop an adaptive RBF fitting prsocedure for which the RBF centres are randomly chosen from the set of points of the level. The randomness is controlled by the density of points and geometric characteristic of the set. The support size of the RBF we use to approximate the point set at a vicinity of a point depends on the local density of the set at that point. Thus parts with complex geometry are approximated by dense RBFs with small supports.

Numerical experiments demonstrate high speed and good performance of the proposed methods in processing irregularly sampled and/or incomplete data.