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Radial basis functions are popular for interpolating scattered data. In this context, the solution of the system of linear algebraic equations (SLAE) is the most time-consuming operation. Techniques fail with large point sets consisting of more than a few thousands points when direct methods and global support are used. In this paper we propose the use of wavelets to accelerate the solution of the SLAE that arise from the formulation of the problem of image interpolation from scattered data by means of compactly-supported radial basis functions. Examples demonstrate the superiority of the solution in the wavelet domain using preconditioned iterative methods.