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）最初被引入作为全变差正则化的迭代方法，旨在从受加性或乘性噪声污染的图像中恢复不同尺度的细节。鉴于其在图像复原之外的广泛应用，我们将MHDM迭代进行扩展，以解决巴拿赫空间中更广泛类型的线性不适定问题。为此，我们针对更一般的凸罚函数甚至非凸罚函数定义了MHDM，并给出了数据保真项的收敛性结果。我们还提出了一种灵活的方法版本，使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——例如在一维全变差去噪中即为此情形。最后，我们研究了几种特例，并通过数值实验指出MHDM的鲁棒行为。