Efficient preconditioning for image reconstruction with radial basis functions

Radial basis functions are a popular basis for interpolating scattered data during the image reconstruction process in graphic analysis. In this context, the solution of a linear system of equations represents the most time-consuming operation. In this paper an efficient preconditioning technique is proposed to solve these linear systems of equations. This algorithm consists of an iterative method which enforces at each iteration a projection of the residual onto a suitable subspace called coarse space. This constraint is ensured by solving an auxiliary problem at each iteration without regularisation. As increasing the number of the coarse space basis functions increases the computational cost of the algorithm, correct selection of coarse space basis is addressed in the paper. Numerical results illustrate the convergence properties of the proposed method with wavelet-like basis functions and regular distributed radial basis functions for image reconstruction.