All of Low-Rank and Sparse: A Recast Total Variation Approach to Hyperspectral Denoising

Hyperspectral image (HSI) processing tasks frequently rely on spatial–spectral total variation (SSTV) to quantify the local smoothness of image structures. However, conventional SSTV only considers a sparse structure of gradient maps computed along the spatial and spectral dimensions while neglecting other correlations. To address this limitation, we introduce low-rank guided SSTV (LRSTV), which characterizes the sparsity and low-rank priors of the gradient map simultaneously. First, we verify through numerical tests and theoretical analyses that the gradient tensors are not only sparse but also low-rank. Subsequently, to model the low rankness of the gradient map, we use the tensor average rank to represent the low Tucker rank of gradient tensors. The convex envelope of the tensor average rank is then employed to penalize the rank on the gradient map after the Fourier transform along the spectral dimension. By naturally encoding the sparsity and low-rank priors of the gradient map, LRSTV results in a more accurate representation of the original image. Finally, we demonstrate the effectiveness of LRSTV by integrating it into the HSI processing model, replacing conventional SSTV, and testing it on two public datasets with nine cases of mixed noise and two datasets with realistic noise. The results show that LRSTV outperforms conventional SSTV in terms of accuracy and robustness.