Unconditionally stable algorithm with unique solvability for image inpainting using a penalized Allen–Cahn equation

Abstract

Image inpainting is a technique that utilizes information from surrounding areas to restore damaged or missing parts. To achieve binary image inpainting with mathematical tools and numerical techniques, an effective mathematical model and an efficient, stable numerical solver are essential. This work aims to propose a practical and unconditionally stable numerical algorithm for image inpainting. A penalized Allen–Cahn equation is derived from a free energy using a variational approach. The proposed mathematical model achieves inpainting by eliminating the damaged region with the constraint of surrounding image values. The operator splitting strategy is used to split the original model into two subproblems. The first one is the classical Allen–Cahn equation, and the second one is a penalization equation. For the Allen–Cahn equation, a linear and strong stability-preserving factorization scheme is adopted to calculate the intermediate solution. Then, the final solution is explicitly updated from a simple correction step. The governing equation is discretized in space using the finite difference method. We analytically prove that the proposed algorithm is unconditionally stable and uniquely solvable. In the numerical simulations, we first verify the efficiency and stability via several simple benchmarks. The capability of binary image inpainting is validated by comparing the present and previous results. By slightly adjusting the governing equation, the proposed method can work well in achieving image inpainting of various gray-valued images. Finally, the proposed method is extended into three-dimensional space to show its effectiveness in restoring damaged 3D objects. The main scientific contributions are: (i) an efficient and practical numerical method is developed for image inpainting; (ii) the unconditional stability and unique solvability have been analytically estimated; (iii) extensive numerical experiments are implemented to validate the stability and capability of the proposed method; (iv) the present method can be straightforwardly extended to achieve 3D restoration. To facilitate interested readers in developing related research, we provide the basic computational codes in Appendices A-D.