A polynomial fit preconditioner for band Toeplitz matrices in image reconstruction

The Preconditioned Conjugate Gradient is often applied in image reconstruction as a regularizing method. When the blurring matrix has Toeplitz structure, the modified circulant preconditioner and the inverse Toeplitz preconditioner have been shown to be effective. We introduce here a preconditioner for symmetric positive definite Toeplitz matrices based on a trigonometric polynomial fit which has the same effectiveness of the previous ones but has a lower cost when applied to band matrices. The case of band block Toeplitz matrices with band Toeplitz blocks (BTTB) corresponding to separable point spread functions is also considered.