Fast image inpainting strategy based on the space-fractional modified Cahn-Hilliard equations

Abstract The solution strategy of the space-fractional modified Cahn-Hilliard equation as a tool for the gray value image inpainting model is studied. The existing strategies solve the convexity splitting scheme of the vector-valued Cahn-Hilliard model by Fourier spectral method. In this paper, we constructed a fast solver for the discretized linear systems possessing the saddle-point structure within block-Toeplitz-Toeplitz-block (BTTB) structure arising from the 2D space-fractional modified Cahn-Hilliard equation. The new solver enjoys computational advantage since circulant approximation and fast Fourier transforms (FFTs) can be used for solving the involved linear subsystems. Theoretical analysis shows the spectrum of the preconditioned matrix clusters around 1, which implies the fast convergence rate of the proposed preconditioner. Numerical examples are given to confirm the effectiveness of our method.