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

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.

Automated summary

This study investigated the solution strategy for the space-fractional modified Cahn-Hilliard equation as a tool for gray value image inpainting model. A fast solver was developed for solving the linear systems arising from the 2D space-fractional modified Cahn-Hilliard equation. The theoretical analysis showed a fast convergence rate of the proposed preconditioner, which was confirmed in numerical examples.