SIAM Journal on Imaging Sciences, Volume 14, Issue 2, Page 844-877, January 2021.
The multiscale hierarchical decomposition method (MHDM) proposed in [E. Tadmor, S. Nezzar, and L. Vese, Multiscale Model. Simul., 2 (2004), pp. 554--579] has been proven very appropriate for denoising images with features at different scales and for scale separation. Extensions of it to image deblurring or to time-dependent settings [E. Tadmor and P. Athavale, Inverse Probl. Imaging, 35 (2009), pp. 693--710] have also been considered, showing convergence properties and more applications. The recent paper [K. Modin, A. Nachman, and L. Rondi, Adv. Math., 346 (2019), pp. 1009--1066] fills in further qualitative results even for nonlinear problems and introduces a tighter version of MHDM with better convergence properties. The contribution of the present work is as follows. First, we derive novel error estimates for MHDM and its tighter version. Second, we provide rules for early stopping of the algorithms in the case of perturbed data, while still ensuring stable approximations of the true image. Last but not least, we propose a refined version of the tighter MHDM, which allows recovering structured images and promotes different features of the components, as compared to the entire image. The theoretical results are validated by numerous numerical experiments for image denoising and deblurring, which also assess the analyzed methods in terms of rate of convergence, stopping rule, and quality of restoration.

中文翻译：

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多尺度分层图像分解和细化：定性和定量结果

SIAM 成像科学杂志，第 14 卷，第 2 期，第 844-877 页，2021 年 1 月。
[E. 中提出的多尺度层次分解方法（MHDM）。Tadmor、S. Nezzar 和 L. Vese，多尺度模型。Simul., 2 (2004), pp. 554--579] 已被证明非常适合具有不同尺度特征的图像去噪和尺度分离。将其扩展到图像去模糊或时间相关设置 [E. Tadmor 和 P. Athavale，逆问题。Imaging, 35 (2009), pp. 693--710] 也被考虑在内，显示了收敛特性和更多应用。最近的论文 [K. Modin, A. Nachman 和 L. Rondi, Adv. Math., 346 (2019), pp. 1009--1066] 甚至为非线性问题填充了进一步的定性结果，并引入了具有更好收敛特性的更严格的 MHDM 版本。目前工作的贡献如下。首先，我们为 MHDM 及其更严格的版本推导出新的误差估计。其次，我们提供了在数据受到扰动的情况下提前停止算法的规则，同时仍然确保真实图像的稳定近似。最后但并非最不重要的是，我们提出了更严格的 MHDM 的改进版本，与整个图像相比，它允许恢复结构化图像并提升组件的不同特征。理论结果通过图像去噪和去模糊的大量数值实验得到验证，这些实验还评估了收敛速度、停止规则和恢复质量方面的分析方法。与整个图像相比，它允许恢复结构化图像并提升组件的不同特征。理论结果通过图像去噪和去模糊的大量数值实验得到验证，这些实验还评估了收敛速度、停止规则和恢复质量方面的分析方法。与整个图像相比，它允许恢复结构化图像并提升组件的不同特征。理论结果通过图像去噪和去模糊的大量数值实验得到验证，这些实验还评估了收敛速度、停止规则和恢复质量方面的分析方法。