Cosine transform preconditioners for high resolution image reconstruction

This paper studies the application of preconditioned conjugate gradient met hods in high resolution image reconstruction problelns. We consider reconst ructing high resolution images from multiple undersampled, shifted, degrade d frames with subpixel displacement errors, The resulting blurring matrices are spatially variant. The classical Tikhonov regularization and the Neuma nn boundary condition are used in the reconstruction process, The precondit ioners are derived by taking the cosine transform approximation of the blur ring matrices. We prove that when the L-2 or H-1 norm regularization functi onal is used, the spectra of the preconditioned normal systems are clustere d around 1 for sufficiently small subpixel displacement errors. Conjugate g radient methods will hence converge very quickly when applied to solving th ese preconditioned normal equations, Numerical examples are given to illust rate the fast convergence, (C) 2000 Elsevier Science Inc. All rights reserv ed.