scholarly journals Projected Wirtinger Gradient Descent for Digital Waves Reconstruction

2020 ◽  
Vol 37 (6) ◽  
pp. 919-927
Author(s):  
Suhui Liu ◽  
Lelai Shi ◽  
Weiyu Xu
Keyword(s):  
Gradient Descent ◽  
Hankel Matrix ◽  
Low Rank ◽  
Signal Recovery ◽  
Digital Signals ◽  
Target Signal ◽  
Structured Matrix ◽  
Iterative Shrinkage ◽  
Speed Up ◽  
Linear Property

This work attempts to recover digital signals from a few stochastic samples in time domain. The target signal is the linear combination of one-dimensional complex sine components with R different but continuous frequencies. These frequencies control the continuous values in the domain of normalized frequency [0, 1), contrary to the previous research into compressed sensing. To recover the target signal, the problem was transformed into the completion of a low-rank structured matrix, drawing on the linear property of the Hankel matrix. Based on the completion of the structured matrix, the authors put forward a feasible-point algorithm, analyzed its convergence, and speeded up the convergence with the fast iterative shrinkage-thresholding (FIST) algorithm. The initial algorithm and the speed up strategy were proved effective through repeated numerical simulations. The research results shed new lights on the signal recovery in various fields.

scholarly journals A sporadic decomposition of Hankel structured matrix in logarithmic and wavelet domain for impulse noise removal

2018 ◽  
Vol 7 (4) ◽  
pp. 2309
Author(s):  
Baby Victoria.L ◽  
Sathappan S
Keyword(s):  
Impulse Noise ◽  
Hankel Matrix ◽  
Noise Removal ◽  
Noise Model ◽  
Low Rank ◽  
Impulse Noise Removal ◽  
Structured Matrix ◽  
Filter Methods ◽  
Patch Image ◽  
Transform Domains

Noise removal from the color images is the most significant and challenging task in image processing. Among different conventional filter methods, a robust Annihilating filter-based Low-rank Hankel matrix (r-ALOHA) approach was proposed as an impulse noise removal algorithm that uses the sparse and low-rank decomposition of a Hankel structured matrix to decompose the sparse impulse noise components from an original image. However, in this algorithm, the patch image was considered as it was sparse in the Fourier domain only. It requires an analysis of noise removal performance by considering the other transform domains. Hence in this article, the r-ALOHA can be extended into other transform domains such as log and exponential. In the log and exponential domain, the logarithmic and exponential functions are used for modeling the multiplicative noise model. But, this model is used only for positive outcomes. Therefore, wavelet transform domain is applied to the noise model that localizes an image pixel in both frequency and time domain simultaneously. Moreover, it separates the most vital information in a given image. Thus, it is feasible for obtaining a better approximation of the considered function using few coefficients. Finally, the experimental results show the performance effectiveness of the proposed algorithm.  

scholarly journals Fast Algorithms for the Anisotropic LLT Model in Image Denoising

2011 ◽  
Vol 1 (3) ◽  
pp. 264-283 ◽  
Author(s):  
Zhi-Feng Pang ◽  
Li-Lian Wang ◽  
Yu-Fei Yang
Keyword(s):  
Convergence Rate ◽  
Image Denoising ◽  
Projection Method ◽  
Split Bregman Method ◽  
Split Bregman ◽  
Minimization Problems ◽  
Iterative Shrinkage ◽  
Speed Up ◽  
Thresholding Algorithm ◽  
Iterative Shrinkage Thresholding

AbstractIn this paper, we propose a new projection method for solving a general minimization problems with twoL1-regularization terms for image denoising. It is related to the split Bregman method, but it avoids solving PDEs in the iteration. We employ the fast iterative shrinkage-thresholding algorithm (FISTA) to speed up the proposed method to a convergence rateO(k−2). We also show the convergence of the algorithms. Finally, we apply the methods to the anisotropic Lysaker, Lundervold and Tai (LLT) model and demonstrate their efficiency.

scholarly journals Convergence of Gradient Descent for Low-Rank Matrix Approximation

2015 ◽  
Vol 61 (8) ◽  
pp. 4451-4457 ◽  
Author(s):  
Renaud-Alexandre Pitaval ◽  
Wei Dai ◽  
Olav Tirkkonen
Keyword(s):  
Gradient Descent ◽  
Low Rank ◽  
Matrix Approximation ◽  
Rank Matrix ◽  
Low Rank Matrix ◽  
Low Rank Matrix Approximation

Enhanced low-rank matrix estimation for simultaneous denoising and reconstruction of five-dimensional seismic data

Geophysics ◽  
2021 ◽  
pp. 1-96
Author(s):  
Yapo Abolé Serge Innocent Oboué ◽  
Yangkang Chen
Keyword(s):  
Seismic Data ◽  
Hankel Matrix ◽  
Low Rank ◽  
The Novel ◽  
Proximity Function ◽  
Rank Reduction ◽  
Matrix Estimation ◽  
Useful Signal ◽  
Truncated Svd ◽  
Noisy Observation

Noise and missing traces usually influence the quality of multidimensional seismic data. It is, therefore, necessary to e stimate the useful signal from its noisy observation. The damped rank-reduction (DRR) method has emerged as an effective method to reconstruct the useful signal matrix from its noisy observation. However, the higher the noise level and the ratio of missing traces, the weaker the DRR operator becomes. Consequently, the estimated low-rank signal matrix includes a unignorable amount of residual noise that influences the next processing steps. This paper focuses on the problem of estimating a low-rank signal matrix from its noisy observation. To elaborate on the novel algorithm, we formulate an improved proximity function by mixing the moving-average filter and the arctangent penalty function. We first apply the proximity function to the level-4 block Hankel matrix before the singular value decomposition (SVD), and then, to singular values, during the damped truncated SVD process. The relationship between the novel proximity function and the DRR framework leads to an optimization problem, which results in better recovery performance. The proposed algorithm aims at producing an enhanced rank-reduction operator to estimate the useful signal matrix with a higher quality. Experiments are conducted on synthetic and real 5-D seismic data to compare the effectiveness of our approach to the DRR approach. The proposed approach is shown to obtain better performance since the estimated low-rank signal matrix is cleaner and contains less amount of artifacts compared to the DRR algorithm.

scholarly journals Performance Analysis of Structured Matrix Decomposition with Contour Based Spatial Prior for Extracting Salient Object from Complex Scene

2019 ◽  
Vol 15 (2) ◽  
pp. 133-140
Author(s):  
Ramesh Bhandari ◽  
Sharad Kumar Ghimire
Keyword(s):  
Computer Vision ◽  
Research Work ◽  
Similarity Index ◽  
Matrix Decomposition ◽  
Low Rank ◽  
Salient Object ◽  
Complex Scene ◽  
Decomposition Model ◽  
Structured Matrix ◽  
Segmentation Image

 Automatically extracting most conspicuous object from an image is useful and important for many computer vision related tasks. Performance of several applications such as object segmentation, image classification based on salient object and content based image editing in computer vision can be improved using this technique. In this research work, performance of structured matrix decomposition with contour based spatial prior is analyzed for extracting salient object from the complex scene. To separate background and salient object, structured matrix decomposition model based on low rank matrix recovery theory is used along with two structural regularizations. Tree structured sparsity inducing regularization is used to capture image structure and to enforce the same object to assign similar saliency values. And, Laplacian regularization is used to enlarge the gap between background part and salient object part. In addition to structured matrix decomposition model, general high level priors along with biologically inspired contour based spatial prior is integrated to improve the performance of saliency related tasks. The performance of the proposed method is evaluated on two demanding datasets, namely, ICOSEG and PASCAL-S for complex scene images. For PASCAL-S dataset precision recall curve of proposed method starts from 0.81 and follows top and right-hand border more than structured matrix decomposition which starts from 0.79. Similarly, structural similarity index score, which is 0.596654 and 0.394864 without using contour based spatial prior and 0.720875 and 0.568001 using contour based spatial prior for ICOSEG and PASCAL-S datasets shows improved result.

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