Compressed Sensing with Frames and Sparsity in Levels Class

Recently, lots of studies demonstrated that the signals are not only sparse in some system (e.g. shearlets) but also reveal a certain structure such as sparsity in levels. Therefore, sampling strategy is designed as a variable subsampling strategy in order to use this extra structure, for example magnetic resonance imaging (MRI) and etc. In this paper, we investigate the uniform recovery guarantees on the signals which possess sparsity in levels with respect to a general dual frame. First, we prove that the stable and robust recovery is possible when the weighted l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l<^>{2} $\end{document}-robust null space property in levels is satisfied. Second, we establish sufficient conditions under which subsampled isometry satisfies the weighted l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l<^>{2} $\end{document}-robust null space property in levels.