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This article presents a novel global gradient sparse and nonlocal low-rank tensor decomposition model with a hyper-Laplacian prior for hyperspectral image (HSI) superresolution to produce a high-resolution HSI (HR-HSI) by fusing a low-resolution HSI (LR-HSI) with an HR multispectral image (HR-MSI). Inspired by the investigated hyper-Laplacian distribution of the gradients of the difference images between the upsampled LR-HSI and latent HR-HSI, we formulate the relationship between these two datasets as a $\ell _{p}$ $(0 < p < 1)$-norm term to enforce spectral preservation. Then, the relationship between the HR-MSI and latent HR-HSI is built using a tensor-based fidelity term to recover the spatial details. To effectively capture the high spatio-spectral-nonlocal similarities of the latent HR-HSI, we design a novel nonlocal low-rank Tucker decomposition to model the 3-D regular tensors constructed from the grouped nonlocal similar HR-HSI cubes. The global spatial-spectral total variation regularization is then adopted to ensure the global spatial piecewise smoothness and spectral consistency of the reconstructed HR-HSI from nonlocal low-rank cubes. Finally, an alternating direction method of multipliers-based algorithm is designed to efficiently solve the optimization problem. Experiments on both the synthetic and real datasets collected by different sensors show the effectiveness of the proposed method, from visual and quantitative assessments.

To reconstruct natural images from compressed sensing (CS) measurements accurately and effectively, a CS image reconstruction algorithm based on hybrid-weighted total variation (HWTV) and nonlocal low-rank (NLR) is proposed. It considers the local smoothness and nonlocal self-similarity (NSS) in image, improves traditional hybrid total variation (TV) model, and constructs a new edge detection operator with mean curvature to adaptively select the TV. The HWTV combines the advantages of first-order TV and second-order TV to preserve the edges of the image and avoid the staircase effect in the smooth areas. And NLR can effectively reduce the redundant information and retain the structural information of the image. In addition, the proposed algorithm constructs prior regularization terms with improved HWTV model and NLR model, and utilizes soft threshold function and smooth but non-convex function to solve the TV and low-rank optimization problems, respectively. Finally, the alternative direction multiplier method (ADMM) iterative strategy is used to separate the target model into several sub-problems, and the most efficient methods are adopted to solve each sub-problem. Experimental results show that, compared with the state-of-the-art CS reconstruction algorithms, the proposed algorithm can achieve higher reconstruction quality, especially in the case of low sampling rates.