Abstract
The inpainting of damaged images has a wide range of applications, and many different mathematical methods have been proposed to solve this problem. Inpainting with the help of Cahn--Hilliard models has been particularly successful, and it turns out that Cahn--Hilliard inpainting with the double obstacle potential can lead to better results compared to inpainting with a smooth double well potential. However, a mathematical analysis of this approach is missing so far. In this paper we give first analytical results for a Cahn--Hilliard double obstacle inpainting model regarding existence of global solutions to the time-dependent problem and stationary solutions to the time-independent problem without constraints on the parameters involved. With the help of numerical results we show the effectiveness of the approach for binary and grayscale images.
Abstract (translated)
受损图像的修复具有广泛的应用，并且已经提出了许多不同的数学方法来解决该问题。在Cahn-Hilliard模型的帮助下修复已经特别成功，并且事实证明，Cahn-Hilliard修复双重障碍潜力可以带来更好的结果，而不是修复平滑的双井潜力。然而，迄今为止缺少对这种方法的数学分析。在本文中，我们给出了Cahn - Hilliard双障碍修复模型的第一个分析结果，该模型关于时间依赖性问题的全局解的存在性和时间无关问题的平稳解，而不受参数的约束。在数值结果的帮助下，我们展示了该方法对二值和灰度图像的有效性。
URL
https://arxiv.org/abs/1801.05527
