Block Diagonal Preconditioners for an Image De-blurring Problem with Fractional Total Variation

Even though using the total fractional-order variation (TFOV) model reduces or eliminates the stair casing effect in image de-blurring problems, but in the same time it generates a dense and ill-conditioned linear system of equations. These properties lead to slow convergence of any iterative method such as Krylov subspace methods. One treatment of the slowness property is to apply the preconditioning technique. In this research work, we propose two block diagonal preconditioners to construct an image from a given blurred image using circulant matrices. These matrices allow us to use the fast Fourier transform (FFT) and the convolution theorem. Using FFT and the convolution theorem reduces the cost of the computations from O(n ² ) into O(n log n) operations in each iteration and also reduces the storages. Our proposed preconditioners are of Murphy, Golub, and Wathen type. Numerical examples are given to illustrate the efficiency of our preconditioners.