Multiscale Hierarchical Image Decomposition and Refinements: Qualitative and Quantitative Results
Wen Li, Elena Resmerita, Luminita A. Vese · SIAM Journal on Imaging Sciences · 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.