Image restoration by cosine transform-based iterative regularization

We consider an ill-posed deconvolution problem with a noise-contaminated observation, and a known convolution kernel. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting blurring matrices are block Toeplitz-plus-Hankel matrices with Toeplitz-plus-Hankel blocks. We study the application of the preconditioned iterative regularization scheme for solving these linear systems, where the blurring matrices are approximated by cosine transform preconditioners. We give a simple approach for finding these preconditioners and show how iterations can be effectively and efficiently regularized for solving ill-posed problems by using the spectral decomposition of the preconditioner.