scholarly journals Medical Image Denoising Via Matrix Norm Minimization Problems

2021 ◽  
Vol 24 (2) ◽  
pp. 72-77
Author(s):  
Zainab Abd-Alzahra ◽  
◽  
Basad Al-Sarray ◽  
Keyword(s):  
Minimization Problem ◽  
Matrix Completion ◽  
Nuclear Norm ◽  
Nuclear Norm Minimization ◽  
Minimization Problems ◽  
Completion Problem ◽  
Norm Minimization ◽  
Matrix Completion Problem ◽  
The Matrix ◽  
The Given

This paper presents the matrix completion problem for image denoising. Three problems based on matrix norm are performing: Spectral norm minimization problem (SNP), Nuclear norm minimization problem (NNP), and Weighted nuclear norm minimization problem (WNNP). In general, images representing by a matrix this matrix contains the information of the image, some information is irrelevant or unfavorable, so to overcome this unwanted information in the image matrix, information completion is used to comperes the matrix and remove this unwanted information. The unwanted information is handled by defining {0,1}-operator under some threshold. Applying this operator on a given matrix keeps the important information in the image and removing the unwanted information by solving the matrix completion problem that is defined by P. The quadratic programming use to solve the given three norm-based minimization problems. To improve the optimal solution a weighted exponential is used to compute the weighted vector of spectral that use to improve the threshold of optimal low rank that getting from solving the nuclear norm and spectral norm problems. The result of applying the proposed method on different types of images is given by adopting some metrics. The results showed the ability of the given methods.

Nuclear norm and indicator function model for matrix completion

2016 ◽  
Vol 24 (1) ◽  
Author(s):  
Juan Geng ◽  
Laisheng Wang ◽  
Xiuyu Wang
Keyword(s):  
Proximal Point Algorithm ◽  
Matrix Completion ◽  
Indicator Function ◽  
Nuclear Norm ◽  
Proximal Point ◽  
Completion Problem ◽  
Matrix Completion Problem ◽  
The Matrix ◽  
Completion Problems ◽  
Norm Model

AbstractIn the matrix completion problem, most methods to solve the nuclear norm model are relaxing it to the nuclear norm regularized least squares problem. In this paper, we propose a new unconstraint model for matrix completion problem based on nuclear norm and indicator function and design a proximal point algorithm (PPA-IF) to solve it. Then the convergence of our algorithm is established strictly. Finally, we report numerical results for solving noiseless and noisy matrix completion problems and image reconstruction.

scholarly journals Tensor theta norms and low rank recovery

2020 ◽  
Author(s):  
Holger Rauhut ◽  
Željka Stojanac
Keyword(s):  
Convex Relaxation ◽  
Nuclear Norm ◽  
Low Rank ◽  
Nuclear Norm Minimization ◽  
Computational Algebraic Geometry ◽  
Matrix Recovery ◽  
Norm Minimization ◽  
The Matrix ◽  
Higher Order Tensors ◽  
Tensor Nuclear Norm

AbstractWe study extensions of compressive sensing and low rank matrix recovery to the recovery of tensors of low rank from incomplete linear information. While the reconstruction of low rank matrices via nuclear norm minimization is rather well-understand by now, almost no theory is available so far for the extension to higher order tensors due to various theoretical and computational difficulties arising for tensor decompositions. In fact, nuclear norm minimization for matrix recovery is a tractable convex relaxation approach, but the extension of the nuclear norm to tensors is in general NP-hard to compute. In this article, we introduce convex relaxations of the tensor nuclear norm which are computable in polynomial time via semidefinite programming. Our approach is based on theta bodies, a concept from real computational algebraic geometry which is similar to the one of the better known Lasserre relaxations. We introduce polynomial ideals which are generated by the second-order minors corresponding to different matricizations of the tensor (where the tensor entries are treated as variables) such that the nuclear norm ball is the convex hull of the algebraic variety of the ideal. The theta body of order k for such an ideal generates a new norm which we call the θk-norm. We show that in the matrix case, these norms reduce to the standard nuclear norm. For tensors of order three or higher however, we indeed obtain new norms. The sequence of the corresponding unit-θk-norm balls converges asymptotically to the unit tensor nuclear norm ball. By providing the Gröbner basis for the ideals, we explicitly give semidefinite programs for the computation of the θk-norm and for the minimization of the θk-norm under an affine constraint. Finally, numerical experiments for order-three tensor recovery via θ1-norm minimization suggest that our approach successfully reconstructs tensors of low rank from incomplete linear (random) measurements.

scholarly journals On the nuclear norm heuristic for a Hankel matrix completion problem

Automatica ◽  
2015 ◽  
Vol 51 ◽  
pp. 268-272 ◽  
Author(s):  
Liang Dai ◽  
Kristiaan Pelckmans
Keyword(s):  
Matrix Completion ◽  
Hankel Matrix ◽  
Nuclear Norm ◽  
Completion Problem ◽  
Matrix Completion Problem

scholarly journals Speech Recognition: B

2021 ◽  
pp. 217-242
Author(s):  
Jean Walrand
Keyword(s):  
Matrix Completion ◽  
Gradient Projection ◽  
Stochastic Gradient ◽  
Projection Algorithm ◽  
General Technique ◽  
Gradient Projection Algorithm ◽  
Completion Problem ◽  
Matrix Completion Problem ◽  
Online Learning Algorithms ◽  
The Matrix

AbstractOnline learning algorithms update their estimates as additional observations are made. Section 12.1 explains a simple example: online linear regression. The stochastic gradient projection algorithm is a general technique to update estimates based on additional observations; it is widely used in machine learning. Section 12.2 presents the theory behind that algorithm. When analyzing large amounts of data, one faces the problems of identifying the most relevant data and of how to use efficiently the available data. Section 12.3 explains three examples of how these questions are addressed: the LASSO algorithm, compressed sensing, and the matrix completion problem. Section 12.4 discusses deep neural networks for which the stochastic gradient projection algorithm is easy to implement.

Line survey joint denoising via low-rank minimization

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. V21-V32 ◽  
Author(s):  
Zhao Liu ◽  
Jianwei Ma ◽  
Xueshan Yong
Keyword(s):  
Minimization Problem ◽  
Nuclear Norm ◽  
Low Rank ◽  
Single Shot ◽  
Rank Minimization ◽  
Nuclear Norm Minimization ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Line Survey ◽  
Low Rank Matrix

Prestack seismic data denoising is an important step in seismic processing due to the development of prestack time migration. Reduced-rank filtering is a state-of-the-art method for prestack seismic denoising that uses predictability between neighbor traces for each single frequency. Different from the original way of embedding low-rank matrix based on the Hankel or Toeplitz transform, we have developed a new multishot gathers joint denoising method in a line survey, which used a new way of rearranging data to a matrix with low rank. Inspired by video denoising, each single-shot record in the line survey can be viewed as a frame in the video sequence. Due to high redundancy and similar event structure among the shot gathers, similar patches can be selected from different shot gathers in the line survey to rearrange a low-rank matrix. Then, seismic denoising is formulated into a low-rank minimization problem that can be further relaxed into a nuclear-norm minimization problem. A fast algorithm, called the orthogonal rank-one matrix pursuit, is used to solve the nuclear-norm minimization. Using this method avoids the computation of a full singular value decomposition. Our method is validated using synthetic and field data, in comparison with [Formula: see text] deconvolution and singular spectrum analysis methods.

scholarly journals Enhancing Matrix Completion Using a Modified Second-Order Total Variation

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Wendong Wang ◽  
Jianjun Wang
Keyword(s):  
Total Variation ◽  
Optimization Problem ◽  
State Of The Art ◽  
Matrix Completion ◽  
Second Order ◽  
Low Rank ◽  
Superior Performance ◽  
Completion Problem ◽  
Matrix Completion Problem ◽  
The Matrix

In this paper, we propose a new method to deal with the matrix completion problem. Different from most existing matrix completion methods that only pursue the low rank of underlying matrices, the proposed method simultaneously optimizes their low rank and smoothness such that they mutually help each other and hence yield a better performance. In particular, the proposed method becomes very competitive with the introduction of a modified second-order total variation, even when it is compared with some recently emerged matrix completion methods that also combine the low rank and smoothness priors of matrices together. An efficient algorithm is developed to solve the induced optimization problem. The extensive experiments further confirm the superior performance of the proposed method over many state-of-the-art methods.

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