Structural total variation regularization: theory and applications in signal and image recovery

Abstract Real-world data often have multiple sparsity structures that are essential for robust recovery of signals and images from compressed measurements. In this paper, we first propose a novel structural total variation (TV) regularization named block TV (BTV) to simultaneously regularize traditional sparsity, gradient sparsity and block sparsity structures. Then, we investigate mathematical theories of the BTV method under certain block restricted isometry property (RIP) condition. Specifically, within a unified theoretical framework, we prove that the BTV method outperforms the traditional $\ell_1$, TV, and $\ell_{2,1}$ methods under mild conditions. Lastly, we design a numerical algorithm for the regularized BTV method within the framework of alternating direction method of multipliers. Experiments on the recovery of synthetic signals, biomedical ECG and EEG signals, gray images, and MRI images verified our theories, showing that the BTV method can always be superior to many competing ones.