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_2: Matrix completion that estimates missing values in visual data is an important topic in computer vision. Most of the recent studies focused on the low rank matrix approximation via the nuclear norm. However, the visual data, such as images, is rich in texture which may not be well approximated by low rank constraint. In this paper, we propose a novel matrix completion method, which combines the nuclear norm with the local geometric regularizer to solve the problem of matrix completion for redundant texture images. And in this paper we mainly consider one of the most commonly graph regularized parameters: the total variation norm which is a widely used measure for enforcing intensity continuity and recovering a piecewise smooth image. The experimental results show that the encouraging results can be obtained by the proposed method on real texture images compared to the state-of-the-art methods.
summary: Additionally, in @cite_2 , a modified linear total variation was defined as: which leads to a smooth low-rank matrix completion problem: