The Multiscale Hierarchical Decomposition Method (MHDM) was introduced as an iterative method for total variation regularization, with the aim of recovering details at various scales from images corrupted by additive or multiplicative noise. Given its success beyond image restoration, we extend the MHDM iterates in order to solve larger classes of linear ill-posed problems in Banach spaces. Thus, we define the MHDM for more general convex or even non-convex penalties, and provide convergence results for the data fidelity term. We also propose a flexible version of the method using adaptive convex functionals for regularization, and show an interesting multiscale decomposition of the data. This decomposition result is highlighted for the Bregman iteration method that can be expressed as an adaptive MHDM. Furthermore, we state necessary and sufficient conditions when the MHDM iteration agrees with the variational Tikhonov regularization, which is the case, for instance, for one-dimensional total variation denoising. Finally, we investigate several particular instances and perform numerical experiments that point out the robust behavior of the MHDM.
翻译:----
多尺度分层分解方法(MHDM)最初作为迭代方法用于总变差正则化,旨在从受到加性或乘性噪声污染的图像中恢复各种尺度的细节。考虑到该方法不仅限于图像修复领域,在巴拿赫空间中解决更大的线性病态问题,我们扩展了MHDM迭代次数。因此,我们定义了更一般的凸或甚至非凸惩罚,为正则化提供了灵活的方法,并为数据保真度提供了收敛结果。同时,我们提出了一种使用自适应凸函数对方法的柔性版本,以及表明数据的有趣多尺度分解的方法。此分解结果针对可表示为自适应MHDM的Bregman迭代方法进行了突出。此外,我们阐述了当MHDM迭代与变分Tikhonov正则化相同时的必要和充分条件,例如,对于一维总变差去噪。最后,我们调查了几个特定的实例,并进行了数值实验,表明MHDM的鲁棒行为。