text: Here we study the extreme visual recovery problem, in which over 90 of pixel values in a given image are missing. Existing low rank-based algorithms are only effective for recovering data with at most 90 missing values. Thus, we exploit visual data's smoothness property to help solve this challenging extreme visual recovery problem. Based on the Discrete Cosine Transformation (DCT), we propose a novel DCT norm that involves all pixels and produces smooth estimations in any view. Our theoretical analysis shows that the total variation (TV) norm, which only achieves local smoothness, is a special case of the proposed DCT norm. We also develop a new visual recovery algorithm by minimizing the DCT and nuclear norms to achieve a more visually pleasing estimation. Experimental results on a benchmark image dataset demonstrate that the proposed approach is superior to state-of-the-art methods in terms of peak signal-to-noise ratio and structural similarity.
  @cite_9: The CT limited-angle problem has been discussed for many years because of the strong demand in medical imaging applications. However as a severely ill-posed problem, it is hard to achieve accurate reconstruction if no additional prior information is given. Recently, an ART+TV reconstruction method was proposed. With the prior knowledge of image sparsity, the ART+TV gives high quality reconstructions far beyond other methods. In our work, we further explored the ART+TV method on limited-angle problems and delivered a new ART+ATV method. In our new method, the actual range of view angles is considered as another prior information and according to it an anisotropic total variation (ATV) norm is designed to perform L1-minimization. Meanwhile inherited from the ART+TV method, the prior knowledge of image sparsity is considered as well. With the combination of the two priors, the ART+ATV method shows a high-quality and fast-convergence reconstruction on limited-angle problems.
summary: Since the @math is isotropic and not differentiable, an anisotropic version is proposed in @cite_9 that is easier to minimize: