Efficient preconditioning for image reconstruction with radial basis functions

Authors:
Frédéric Magoulès;Luis A. Diago;Ichiro Hagiwara
Affiliations:
Applied Mathematics and Systems Laboratory, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay Malabry Cedex, France;Department of Mechanical Sciences and Engineering, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguroku, Tokyo 152-8852, Japan;Department of Mechanical Sciences and Engineering, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguroku, Tokyo 152-8852, Japan
Venue:
Advances in Engineering Software
Year:
2007

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Abstract

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.