Thai D H, Gottschlich C
Department of Statistical Science, Duke University, Box 90251, Durham, NC 27708-0251, USA.
Institute for Mathematical Stochastics, University of Goettingen, Goldschmidtstrasse 7, 37077 Göttingen, Germany.
R Soc Open Sci. 2018 Jul 25;5(7):171176. doi: 10.1098/rsos.171176. eCollection 2018 Jul.
We consider the very challenging task of restoring images (i) that have a large number of missing pixels, (ii) whose existing pixels are corrupted by noise, and (iii) that ideally contain both cartoon and texture elements. The combination of these three properties makes this inverse problem a very difficult one. The solution proposed in this manuscript is based on directional global three-part decomposition (DG3PD) (Thai, Gottschlich. 2016 , 1-20 (doi:10.1186/s13640-015-0097-y)) with a directional total variation norm, directional G-norm and ℓ-norm in the curvelet domain as key ingredients of the model. Image decomposition by DG3PD enables a decoupled inpainting and denoising of the cartoon and texture components. A comparison with existing approaches for inpainting and denoising shows the advantages of the proposed method. Moreover, we regard the image restoration problem from the viewpoint of a Bayesian framework and we discuss the connections between the proposed solution by function space and related image representation by harmonic analysis and pyramid decomposition.
我们考虑恢复图像这一极具挑战性的任务，这些图像具有以下特点：(i) 存在大量缺失像素；(ii) 现有像素被噪声破坏；(iii) 理想情况下同时包含卡通和纹理元素。这三个特性的组合使得这个逆问题非常困难。本手稿中提出的解决方案基于方向全局三部分分解（DG3PD）（泰国人，戈特施利希。2016 年，1 - 20（doi:10.1186/s13640-015-0097-y）），在曲波域中使用方向全变差范数、方向 G 范数和 ℓ 范数作为模型的关键要素。通过 DG3PD 进行图像分解能够对卡通和纹理分量进行解耦的图像修复和去噪。与现有的图像修复和去噪方法进行比较，显示了所提出方法的优势。此外，我们从贝叶斯框架的角度看待图像恢复问题，并讨论了通过函数空间提出的解决方案与通过调和分析和金字塔分解的相关图像表示之间的联系。
