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：多尺度层次分解方法（MHDM）最初被提出作为全变分正则化的迭代方法，旨在从受加性噪声或乘性噪声污染的图像中恢复不同尺度上的细节。鉴于其在图像恢复之外的广泛应用，我们将MHDM迭代扩展以解决Banach空间中更大类别的线性不適定问题。因此，我们针对更一般的凸甚至非凸惩罚项定义了MHDM，并为数据保真项提供了收敛性结果。我们还提出了一种灵活版本的方法，采用自适应凸泛函进行正则化，并展示了数据的有趣多尺度分解。这一分解结果在可表示为自适应MHDM的Bregman迭代方法中尤为突出。此外，我们给出了MHDM迭代与变分Tikhonov正则化一致的必要和充分条件，例如在一维全变分去噪中便是如此。最后，我们研究了几个具体实例，并进行了数值实验，展示了MHDM的稳健性能。