A family of modified regularizing circulant preconditioners for image reconstruction problems.

Customary circulant preconditioners of Chan, Strang, and inverse-Toeplitz type aremodified for the regularized solution of ill-conditioned Toeplitz systems. Given a symmetric two-level Toeplitz matrix A, and two parameters ? and ?, consider the customary preconditioner C and modify it by replacing its eigenvalues with modulus less than ? with the value ?. The circulant matrix obtained in this way is used as a preconditioner. We analyze the distribution of the eigenvalues of the preconditioned matrix and the regularization properties. Numerical experiments, shown for the image restoration problem, validate the theoretical results.