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Image restoration plays an important role in video technology. In this paper, a robust image and video denoising method, based on random projection and partial sorted ℓ p -norm, is proposed. First, the input signal is decomposed into two components: a low rank component and a sparse component. The low rank component is approximated by random projection. Second, the sparse one is recovered by partial sorted ℓ p -norm. A generalized iterative thresholding shrinkage solver is developed for the resulting problem. Some theoretical results about sparse random projection are provided. Numerical experiments for mixed Gaussian and random value impulsive noise demonstrated that the proposed method outperforms some state-of-art restoration methods, both quantitatively and visually. [ABSTRACT FROM AUTHOR]