There's an neat technique for fitting RBF representations to dense point clouds. Instead of the impractical matrix inversion, the trick is to solve the linear system with a fast multipole method, similar to the Barnes-Hut algorithm for the n-body problem.

Carr, J.C. et al, "Reconstruction and Representation of 3D Objects with Radial Basis Functions", Siggraph 2001

http://dl.acm.org/citation.cfm?id=383266