Weighted Nuclear Norm Minimization with Application to Image Denoising

@article{Gu2014WeightedNN,
  title={Weighted Nuclear Norm Minimization with Application to Image Denoising},
  author={Shuhang Gu and Lei Zhang and Wangmeng Zuo and Xiangchu Feng},
  journal={2014 IEEE Conference on Computer Vision and Pattern Recognition},
  year={2014},
  pages={2862-2869}
}
As a convex relaxation of the low rank matrix factorization problem, the nuclear norm minimization has been attracting significant research interest in recent years. The standard nuclear norm minimization regularizes each singular value equally to pursue the convexity of the objective function. However, this greatly restricts its capability and flexibility in dealing with many practical problems (e.g., denoising), where the singular values have clear physical meanings and should be treated… 

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References

SHOWING 1-10 OF 33 REFERENCES
Matrix completion by Truncated Nuclear Norm Regularization
TLDR
This paper proposes a novel matrix completion algorithm based on the Truncated Nuclear Norm Regularization (TNNR) by only minimizing the smallest N-r singular values, where N is the number of singular values and r is the rank of the matrix.
Accelerated low-rank visual recovery by random projection
TLDR
Theoretic analysis well justifies the proposed scheme, along with greatly reduced optimization complexity, and can serve as a general principal to accelerate many other nuclear norm oriented problems in numerous tasks.
Robust video denoising using low rank matrix completion
TLDR
The robustness and effectiveness of the proposed Denoising algorithm on removing mixed noise, e.g. heavy Gaussian noise mixed with impulsive noise, is validated in the experiments and the proposed approach compares favorably against some existing video denoising algorithms.
Nonlocal Image Restoration With Bilateral Variance Estimation: A Low-Rank Approach
TLDR
This paper takes a low-rank approach toward SSC and provides a conceptually simple interpretation from a bilateral variance estimation perspective, namely that singular-value decomposition of similar packed patches can be viewed as pooling both local and nonlocal information for estimating signal variances.
Practical low-rank matrix approximation under robust L1-norm
TLDR
This work proposes to add a convex trace-norm regularization term to improve convergence, without introducing too much heterogenous information, and customize a scalable first-order optimization algorithm to solve the regularized formulation on the basis of the augmented Lagrange multiplier (ALM) method.
Natural image denoising: Optimality and inherent bounds
TLDR
This paper takes a non parametric approach and represents the distribution of natural images using a huge set of 1010 patches and derives a simple statistical measure which provides a lower bound on the optimal Bayesian minimum mean square error (MMSE).
A Singular Value Thresholding Algorithm for Matrix Completion
TLDR
This paper develops a simple first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank, and develops a framework in which one can understand these algorithms in terms of well-known Lagrange multiplier algorithms.
Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries
TLDR
This work addresses the image denoising problem, where zero-mean white and homogeneous Gaussian additive noise is to be removed from a given image, and uses the K-SVD algorithm to obtain a dictionary that describes the image content effectively.
Patch Complexity, Finite Pixel Correlations and Optimal Denoising
TLDR
A law of diminishing return is presented, namely that with increasing patch size, rare patches not only require a much larger dataset, but also gain little from it, and this result suggests novel adaptive variable-sized patch schemes for denoising.
The phase transition of matrix recovery from Gaussian measurements matches the minimax MSE of matrix denoising
TLDR
Extensive experiments are reported showing that the phase transition in the first problem, matrix recovery from Gaussian measurements, coincides with the minimax risk curve in the second problem, Matrix denoising in Gaussian noise, for any rank fraction (at each common aspect ratio β).
...
...