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Hyperspectral image (HSI) denoising based on nonlocal subspace representation has attracted a lot of attention recently. However, most of the existing works mainly focus on refining the representation coefficient images (RCIs) using certain nonlocal denoiser but ignore the understanding why these pseudoimages have a similar spatial structure as the original HSI. In this work, we revisit such vein from the respective of principal component analysis (PCA). Inspired by an alternative sparse PCA, we propose a spectral sparse subspace representation strategy to simultaneously learn low-dimensional spectral subspace and novel RCIs with sparse loadings. It turns out that the resulting RCIs possess a more significant spatial structure due to the adaptive sparse combination of spectral bands. A simple nonlocal low-rank approximation is then employed to further remove the residual noise of the RCIs. Finally, the entire denoised HSI is obtained by inverse spectral sparse PCA. Extensive experiments on the simulated and real HSI datasets show that the proposed nonlocal spectral sparse subspace representation method, dubbed as NS3R, has excellent performance both in denoising effect and running time compared with many other state-of-the-art methods.

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A plethora of previous studies indicates that making full use of multifarious intrinsic properties of primordial data is a valid pathway to recover original images from their degraded observations. Typically, both low-rankness and local-smoothness broadly exist in real-world tensor data such as hyperspectral images and videos. Modeling based on both properties has received a great deal of attention, whereas most studies concentrate on experimental performance, and theoretical investigations are still lacking. In this paper, we study the tensor compressive sensing problem based on the tensor correlated total variation, which is a new regularizer used to simultaneously capture both properties existing in the same dataset. The new regularizer has the outstanding advantage of not using a trade-off parameter to balance the two properties. The obtained theories provide a robust recovery guarantee, where the error bound shows that our model certainly benefits from both properties in ground-truth data adaptively. Moreover, based on the ADMM update procedure, we design an algorithm with a global convergence guarantee to solve this model. At last, we carry out experiments to apply our model to hyperspectral image and video restoration problems. The experimental results show that our method is prominently better than many other competing ones. Our code and Supplementary Material are available at https://github.com/fsliuxl/cs-tctv.