<p>Numerous photodetectors in hyperspectral imagers with different spectral reflectance responses to pixels result in the obtained hyperspectral image (HSI) being prone to producing stripes distortions along-track direction. Many variation-based methods show sufficient destriping performance. However, these destriping algorithms have limitations in removing multidirectional stripes and lack robustness. Therefore, the HSI destriping algorithm with spectral tensor sparse approximation is proposed in this article, which integrates comprehensively prior information from HSI and directional structured stripes. Specifically, for image component, we apply the unidirectional spectral total variation regularization to enhance the spectral continuity of image. Moreover, for stripes component, we introduce group sparsity regularization to maintain the special direction and spatial linear characteristics of vertical and horizontal stripes. Subsequently, the split Bregman is designed to solve the proposed optimization model. Further, the restored image is obtained by the inverse matrix of the diagonal matrix after singular value decomposition on the block circulant matrix. Finally, the experimental results in simulated and real datasets validate the superiority and robustness of the proposed method, which is superior to other state-of-the-art algorithms in both visual and quantitative indicators. Especially, the spectral curves of the recovery HSI by the proposed algorithm are much smoother as the stripes in the scenes become gradually denser.</p>