scholarly journals Weighted Schatten p-Norm Low Rank Error Constraint for Image Denoising

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 158
Author(s):  
Jiucheng Xu ◽  
Yihao Cheng ◽  
Yuanyuan Ma
Keyword(s):  
Image Denoising ◽  
Block Matching ◽  
Low Rank ◽  
Self Similarity ◽  
Data Set ◽  
Noisy Images ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Non Local ◽  
Low Rank Matrix

Traditional image denoising algorithms obtain prior information from noisy images that are directly based on low rank matrix restoration, which pays little attention to the nonlocal self-similarity errors between clear images and noisy images. This paper proposes a new image denoising algorithm based on low rank matrix restoration in order to solve this problem. The proposed algorithm introduces the non-local self-similarity error between the clear image and noisy image into the weighted Schatten p-norm minimization model using the non-local self-similarity of the image. In addition, the low rank error is constrained by using Schatten p-norm to obtain a better low rank matrix in order to improve the performance of the image denoising algorithm. The results demonstrate that, on the classic data set, when comparing with block matching 3D filtering (BM3D), weighted nuclear norm minimization (WNNM), weighted Schatten p-norm minimization (WSNM), and FFDNet, the proposed algorithm achieves a higher peak signal-to-noise ratio, better denoising effect, and visual effects with improved robustness and generalization.

scholarly journals Multichannel Color Image Denoising via Weighted Schatten p-norm Minimization

2020 ◽  
Author(s):  
Xinjian Huang ◽  
Bo Du ◽  
Weiwei Liu
Keyword(s):  
Image Denoising ◽  
Color Image ◽  
Low Rank ◽  
Norm Minimization ◽  
Alternating Direction ◽  
Rank Matrix ◽  
Color Image Denoising ◽  
Non Local ◽  
Comparison Algorithms ◽  
Low Rank Matrix

The R, G and B channels of a color image generally have different noise statistical properties or noise strengths. It is thus problematic to apply grayscale image denoising algorithms to color image denoising. In this paper, based on the non-local self-similarity of an image and the different noise strength across each channel, we propose a MultiChannel Weighted Schatten p-Norm Minimization (MCWSNM) model for RGB color image denoising. More specifically, considering a small local RGB patch in a noisy image, we first find its nonlocal similar cubic patches in a search window with an appropriate size. These similar cubic patches are then vectorized and grouped to construct a noisy low-rank matrix, which can be recovered using the Schatten p-norm minimization framework. Moreover, a weight matrix is introduced to balance each channel’s contribution to the final denoising results. The proposed MCWSNM can be solved via the alternating direction method of multipliers. Convergence property of the proposed method are also theoretically analyzed . Experiments conducted on both synthetic and real noisy color image datasets demonstrate highly competitive denoising performance, outperforming comparison algorithms, including several methods based on neural networks.

scholarly journals Clipped Matrix Completion: A Remedy for Ceiling Effects

2019 ◽  
Vol 33 ◽  
pp. 5151-5158
Author(s):  
Takeshi Teshima ◽  
Miao Xu ◽  
Issei Sato ◽  
Masashi Sugiyama
Keyword(s):  
Matrix Completion ◽  
Low Rank ◽  
Trace Norm ◽  
Minimization Algorithm ◽  
Benchmark Data ◽  
Hinge Loss ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Recovery Error ◽  
Low Rank Matrix

We consider the problem of recovering a low-rank matrix from its clipped observations. Clipping is conceivable in many scientific areas that obstructs statistical analyses. On the other hand, matrix completion (MC) methods can recover a low-rank matrix from various information deficits by using the principle of low-rank completion. However, the current theoretical guarantees for low-rank MC do not apply to clipped matrices, as the deficit depends on the underlying values. Therefore, the feasibility of clipped matrix completion (CMC) is not trivial. In this paper, we first provide a theoretical guarantee for the exact recovery of CMC by using a trace-norm minimization algorithm. Furthermore, we propose practical CMC algorithms by extending ordinary MC methods. Our extension is to use the squared hinge loss in place of the squared loss for reducing the penalty of overestimation on clipped entries. We also propose a novel regularization term tailored for CMC. It is a combination of two trace-norm terms, and we theoretically bound the recovery error under the regularization. We demonstrate the effectiveness of the proposed methods through experiments using both synthetic and benchmark data for recommendation systems.

scholarly journals Recovery analysis of damped spectrally sparse signals and its relation to MUSIC

2020 ◽  
Author(s):  
Shuang Li ◽  
Hassan Mansour ◽  
Michael B Wakin
Keyword(s):  
Classical Music ◽  
Frequency Estimation ◽  
Low Rank ◽  
Sparse Signals ◽  
Music Algorithm ◽  
Matrix Recovery ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

Abstract One of the classical approaches for estimating the frequencies and damping factors in a spectrally sparse signal is the MUltiple SIgnal Classification (MUSIC) algorithm, which exploits the low-rank structure of an autocorrelation matrix. Low-rank matrices have also received considerable attention recently in the context of optimization algorithms with partial observations, and nuclear norm minimization (NNM) has been widely used as a popular heuristic of rank minimization for low-rank matrix recovery problems. On the other hand, it has been shown that NNM can be viewed as a special case of atomic norm minimization (ANM), which has achieved great success in solving line spectrum estimation problems. However, as far as we know, the general ANM (not NNM) considered in many existing works can only handle frequency estimation in undamped sinusoids. In this work, we aim to fill this gap and deal with damped spectrally sparse signal recovery problems. In particular, inspired by the dual analysis used in ANM, we offer a novel optimization-based perspective on the classical MUSIC algorithm and propose an algorithm for spectral estimation that involves searching for the peaks of the dual polynomial corresponding to a certain NNM problem, and we show that this algorithm is in fact equivalent to MUSIC itself. Building on this connection, we also extend the classical MUSIC algorithm to the missing data case. We provide exact recovery guarantees for our proposed algorithms and quantify how the sample complexity depends on the true spectral parameters. In particular, we provide a parameter-specific recovery bound for low-rank matrix recovery of jointly sparse signals rather than use certain incoherence properties as in existing literature. Simulation results also indicate that the proposed algorithms significantly outperform some relevant existing methods (e.g., ANM) in frequency estimation of damped exponentials.

Non-Local Image Inpainting Using Low-Rank Matrix Completion

2014 ◽  
Vol 34 (6) ◽  
pp. 111-122 ◽  
Author(s):  
Wei Li ◽  
Lei Zhao ◽  
Zhijie Lin ◽  
Duanqing Xu ◽  
Dongming Lu
Keyword(s):  
Matrix Completion ◽  
Image Inpainting ◽  
Low Rank ◽  
Rank Matrix ◽  
Non Local ◽  
Low Rank Matrix Completion ◽  
Low Rank Matrix ◽  
Local Image

Recovery of Low Rank Symmetric Matrices via Schatten p Norm Minimization

2016 ◽  
Vol 33 (01) ◽  
pp. 1650003
Author(s):  
Li Cui ◽  
Lu Liu ◽  
Di-Rong Chen ◽  
Jian-Feng Xie
Keyword(s):  
Symmetric Matrix ◽  
Linear Measurement ◽  
Low Rank ◽  
Symmetric Matrices ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Matrix Formula ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix ◽  
Recovery Problem

In this paper, we give an application of the perturbation inequality to the low rank matrix recovery problem and provide a condition on the linear map of underdetermined linear system that every minimal rank symmetric matrix [Formula: see text] can be exactly recovered from the linear measurement [Formula: see text] via some Schatten [Formula: see text] norm minimization. Moreover it is shown that the explicit bound on exponent [Formula: see text] in the Schatten [Formula: see text] norm minimization can be exactly extracted.

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