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Hyperspectral image (HSI) has some advantages over natural image for various applications due to the extra spectral information. During the acquisition, it is often contaminated by severe noises including Gaussian noise, impulse noise, deadlines, and stripes. The image quality degeneration would badly effect some applications. In this paper, we present a HSI restoration method named smooth and robust low rank tensor recovery. Specifically, we propose a structural tensor decomposition in accordance with the linear spectral mixture model of HSI. It decomposes a tensor into sums of outer matrix vector products, where the vectors are orthogonal due to the independence of endmember spectrums. Based on it, the global low rank tensor structure can be well exposited for HSI denoising. In addition, the 3D anisotropic total variation is used for spatial spectral piecewise smoothness of HSI. Meanwhile, the sparse noise including impulse noise, deadlines and stripes, is detected by the l1 norm regularization. The Frobenius norm is used for the heavy Gaussian noise in some real world scenarios. The alternating direction method of multipliers is adopted to solve the proposed optimization model, which simultaneously exploits the global low rank property and the spatial spectral smoothness of the HSI. Numerical experiments on both simulated and real data illustrate the superiority of the proposed method in comparison with the existing ones.