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.

中文翻译：

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无条件稳定算法，具有独特的可解性，用于使用惩罚 Allen-Cahn 方程进行图像修复


图像修复是一种利用来自周围区域的信息来恢复损坏或缺失部分的技术。要使用数学工具和数值技术实现二进制图像修复，有效的数学模型和高效、稳定的数值求解器是必不可少的。这项工作旨在提出一种实用且无条件稳定的图像修复数值算法。惩罚 Allen-Cahn 方程是使用变分方法从自由能推导出来的。所提出的数学模型通过在周围图像值的约束下消除受损区域来实现修复。算子拆分策略用于将原始模型拆分为两个子问题。第一个是经典的 Allen-Cahn 方程，第二个是惩罚方程。对于 Allen-Cahn 方程，采用线性和强守稳性因式分解方案来计算中间解。然后，从简单的校正步骤显式更新最终解决方案。控制方程在空间中使用有限差分法离散化。我们分析证明，所提出的算法是无条件稳定的，并且是唯一可求解的。在数值模拟中，我们首先通过几个简单的基准来验证效率和稳定性。通过比较现在和以前的结果来验证二进制图像修复的能力。通过稍微调整控制方程，所提方法可以很好地实现各种灰度值图像的图像修复。最后，将所提方法推广到三维空间中，以展示其在修复受损三维物体方面的有效性。 主要的科学贡献是：（i） 开发了一种高效实用的图像修复数值方法;（ii） 已对无条件稳定性和独特可解性进行了分析估计;（iii） 实施了大量的数值实验，验证了所提方法的稳定性和能力;（iv） 目前的方法可以直接扩展以实现 3D 修复。为了方便感兴趣的读者进行相关研究，我们在附录 A-D 中提供了基本的计算代码。