Cosine transform preconditioners for high resolution image reconstruction

This paper studies the application of preconditioned conjugate gradient methods in high resolution image reconstruction problems. We consider reconstructing high resolution images from multiple undersampled, shifted, degraded frames with subpixel displacement errors. The resulting blurring matrices are spatially variant. The classical Tikhonov regularization and the Neumann boundary condition are used in the reconstruction process. The preconditioners are derived by taking the cosine transform approximation of the blurring matrices. We prove that when the L₂ or H₁ norm regularization functional is used, the spectra of the preconditioned normal systems are clustered around 1 for sufficiently small subpixel displacement errors. Conjugate gradient methods will hence converge very quickly when applied to solving these preconditioned normal equations. Numerical examples are given to illustrate the fast convergence.