Maximum principle-preserving, unconditionally energy-stable, and convergent method with second-order accuracy for the phase-field model of image inpainting

Image inpainting is a technique for reconstructing missing or damaged regions of an image. In this paper, we propose a novel linear numerical method with second-order accuracy in both space and time for solving the modified Allen–Cahn equation applied to image inpainting. The proposed method is conditionally maximum principle-preserving, second-order accurate, and unconditionally energy-stable. A leap-frog finite difference scheme is employed to discretize the modified Allen–Cahn equation. Additionally, we present a comprehensive stability analysis and provide an error estimate for the method. Numerical experiments validate the effectiveness of the proposed method, demonstrating its accuracy, stability, expandability, and efficiency.