Recovering images corrupted by multiplicative noise is a well known challenging task. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, we adapt a variety of both classical and new multiplicative noise removing models to the MHDM form. On the basis of previous work, we further present a tight and a refined version of the corresponding multiplicative MHDM. We discuss existence and uniqueness of solutions for the proposed models, and additionally, provide convergence properties. Moreover, we present a discrepancy principle stopping criterion which prevents recovering excess noise in the multiscale reconstruction. Through comprehensive numerical experiments and comparisons, we qualitatively and quantitatively evaluate the validity of all proposed models for denoising and deblurring images degraded by multiplicative noise. By construction, these multiplicative multiscale hierarchical decomposition methods have the added benefit of recovering many scales of an image, which can provide features of interest beyond image denoising.

翻译：恢复受乘性噪声干扰的图像是一项众所周知的挑战性任务。受多尺度层次分解方法（MHDM）在图像处理中取得成功的启发，我们将多种经典和新型乘性噪声去除模型适配至MHDM框架。基于前期工作，我们进一步提出了对应乘性MHDM的紧致版本与精炼版本。讨论了所提出模型解的存在性与唯一性，并给出了收敛性质。此外，我们提出了一个差异原则停止准则，用于防止多尺度重建中噪声的过度恢复。通过全面的数值实验与对比，我们从定性与定量角度评估了所有模型在去除乘性噪声导致的图像降质与模糊方面的有效性。从构造上看，这些乘性多尺度层次分解方法具有恢复图像多尺度信息的附加优势，能够提供除图像去噪之外感兴趣的特征。