Hyperspectral image compressive sensing reconstruction (HSI-CSR) is an important issue in remote sensing, and has recently been investigated increasingly by the sparsity prior based approaches. However, most of the available HSI-CSR methods consider the sparsity prior in spatial and spectral vector domains via vectorizing hyperspectral cubes along a certain dimension. Besides, in most previous works, little attention has been paid to exploiting the underlying nonlocal structure in spatial domain of the HSI. In this paper, we propose a nonlocal tensor sparse and low-rank regularization (NTSRLR) approach, which can encode essential structured sparsity of an HSI and explore its advantages for HSI-CSR task. Specifically, we study how to utilize reasonably the l1 -based sparsity of core tensor and tensor nuclear norm function as tensor sparse and low-rank regularization, respectively, to describe the nonlocal spatial-spectral correlation hidden in an HSI. To study the minimization problem of the proposed algorithm, we design a fast implementation strategy based on the alternative direction multiplier method (ADMM) technique. Experimental results on various HSI datasets verify that the proposed HSI-CSR algorithm can significantly outperform existing state-of-the-art CSR techniques for HSI recovery
