scholarly journals Tube-Based Taut String Algorithms for Total Variation Regularization

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1141
Author(s):  
Artyom Makovetskii ◽  
Sergei Voronin ◽  
Vitaly Kober ◽  
Aleksei Voronin
Keyword(s):  
Exact Solutions ◽  
Total Variation ◽  
Total Variation Regularization ◽  
Geometric Description ◽  
String Method ◽  
String Algorithms ◽  
Practical Applications ◽  
Taut String ◽  
Tv Regularization ◽  
Regularization Problem

Removing noise from signals using total variation regularization is a challenging signal processing problem arising in many practical applications. The taut string method is one of the most efficient approaches for solving the 1D TV regularization problem. In this paper we propose a geometric description of the linearized taut string method. This geometric description leads to the notion of the “tube”. We propose three tube-based taut string algorithms for total variation regularization. Different weight functionals can be used in the 1D TV regularization that lead to different types of tubes. We consider uniform, vertically nonuniform, vertically and horizontally nonuniform tubes. The proposed geometric approach is used to speed-up TV regularization processing by dividing the tubes into subtubes and using parallel processing. We introduce the concept of a relatively convex tube and describe the relationship between the geometric characteristics of tubes and exact solutions to the TV regularization. The properties of exact solutions can also be used to design efficient algorithms for solving the TV regularization problem. The performance of the proposed algorithms is discussed and illustrated by computer simulation.

scholarly journals An Improved Particle Swarm Optimization Based on Total Variation Regularization and Projection Constraint with Applications in Ground-Penetrating Radar Inversion: A Model Simulation Study

2021 ◽  
Vol 13 (13) ◽  
pp. 2514
Author(s):  
Qianwei Dai ◽  
Hao Zhang ◽  
Bin Zhang
Keyword(s):  
Particle Swarm Optimization ◽  
Total Variation ◽  
Ground Penetrating Radar ◽  
Particle Swarm ◽  
Premature Convergence ◽  
Total Variation Regularization ◽  
Interface Model ◽  
Swarm Optimization ◽  
Tv Regularization ◽  
Ground Penetrating

The chaos oscillation particle swarm optimization (COPSO) algorithm is prone to binge trapped in the local optima when dealing with certain complex models in ground-penetrating radar (GPR) data inversion, because it inherently suffers from premature convergence, high computational costs, and extremely slow convergence times, especially in the middle and later periods of iterative inversion. Considering that the bilateral connections between different particle positions can improve both the algorithmic searching efficiency and the convergence performance, we first develop a fast single-trace-based approach to construct an initial model for 2-D PSO inversion and then propose a TV-regularization-based improved PSO (TVIPSO) algorithm that employs total variation (TV) regularization as a constraint technique to adaptively update the positions of particles. B by adding the new velocity variations and optimal step size matrices, the search range of the random particles in the solution space can be significantly reduced, meaning blindness in the search process can be avoided. By introducing constraint-oriented regularization to allow the optimization search to move out of the inaccurate region, the premature convergence and blurring problems can be mitigated to further guarantee the inversion accuracy and efficiency. We report on three inversion experiments involving multilayered, fluctuated terrain models and a typical complicated inner-interface model to demonstrate the performance of the proposed algorithm. The results of the fluctuated terrain model show that compared with the COPSO algorithm, the fitness error (MAE) of the TVIPSO algorithm is reduced from 2.3715 to 1.0921, while for the complicated inner-interface model the fitness error (MARE) of the TVIPSO algorithm is reduced from 1.9539 to 1.5674.

scholarly journals Image Compression Approach using Segmentation and Total Variation Regularization

2021 ◽  
Vol 15 ◽  
pp. 43-47
Author(s):  
Ahmad Shahin ◽  
Walid Moudani ◽  
Fadi Chakik
Keyword(s):  
Image Compression ◽  
Total Variation ◽  
Image Data ◽  
Smart Devices ◽  
Multimedia Content ◽  
Total Variation Regularization ◽  
Second Stage ◽  
Tv Regularization ◽  
Rof Model

In this paper we present a hybrid model for image compression based on segmentation and total variation regularization. The main motivation behind our approach is to offer decode image with immediate access to objects/features of interest. We are targeting high quality decoded image in order to be useful on smart devices, for analysis purpose, as well as for multimedia content-based description standards. The image is approximated as a set of uniform regions: The technique will assign well-defined members to homogenous regions in order to achieve image segmentation. The Adaptive fuzzy c-means (AFcM) is a guide to cluster image data. A second stage coding is applied using entropy coding to remove the whole image entropy redundancy. In the decompression phase, the reverse process is applied in which the decoded image suffers from missing details due to the coarse segmentation. For this reason, we suggest the application of total variation (TV) regularization, such as the Rudin-Osher-Fatemi (ROF) model, to enhance the quality of the coded image. Our experimental results had shown that ROF may increase the PSNR and hence offer better quality for a set of benchmark grayscale images.

scholarly journals TV2++: A Novel Spatial-Temporal Total Variation for Super Resolution with Exponential-type Norm

2020 ◽  
Author(s):  
Lizhen Deng ◽  
Zhetao Zhou ◽  
Guoxia Xu ◽  
Hu Zhu ◽  
Bing-Kun Bao
Keyword(s):  
Total Variation ◽  
Exponential Type ◽  
Optimization Problems ◽  
Super Resolution ◽  
Low Rank ◽  
Good Effect ◽  
Total Variation Regularization ◽  
Time Dimension ◽  
Tv Regularization ◽  
Image Super Resolution

Abstract Recently, many super-resolution algorithms have been proposed to recover high resolution images to improve visualization and help better analyze images. Among them, total variation regularization (TV) methods have been proven to have a good effect in retaining image edge information. However, these TV methods do not consider the temporal correlation between images. Our algorithm designs a new TV regularization (TV2++) to take advantage of the time dimension information of the images, further improving the utilization of useful information in the images. In addition, the union of global low rank regularization and TV regularization further enhances the image super resolution recovery. And we extend the exponential-type penalty (ETP) function on singular values of a matrix to enhance low-rank matrix recovery. A novel image super-resolution algorithm based on the ETP norm and TV2++ regularization is proposed. And the alternating direction method of multipliers (ADMM) is applied to solve the optimization problems effectively. Numerous experimental results prove that the proposed algorithm is superior to other algorithms.

scholarly journals TV2++: A Novel Spatial-Temporal Total Variation for Super Resolution with Exponential-type Norm

2020 ◽  
Author(s):  
Lizhen Deng ◽  
Zhetao Zhou ◽  
Guoxia Xu ◽  
Hu Zhu ◽  
Bing-Kun Bao
Keyword(s):  
Total Variation ◽  
Exponential Type ◽  
Optimization Problems ◽  
Super Resolution ◽  
Low Rank ◽  
Good Effect ◽  
Total Variation Regularization ◽  
Time Dimension ◽  
Tv Regularization ◽  
Image Super Resolution

Abstract Recently, many super-resolution algorithms have been proposed to recover high resolution images to improve visualization and help better analyze images. Among them, total variation regularization (TV) methods have been proven to have a good effect in retaining image edge information. However, these TV methods do not consider the temporal correlation between images. Our algorithm designs a new TV regularization (TV2++) to take advantage of the time dimension information of the images, further improving the utilization of useful information in the images. In addition, the union of global low rank regularization and TV regularization further enhances the image super resolution recovery. And we extend the exponential-type penalty (ETP) function on singular values of a matrix to enhance low-rank matrix recovery. A novel image super-resolution algorithm based on the ETP norm and TV2++ regularization is proposed. And the alternating direction method of multipliers (ADMM) is applied to solve the optimization problems effectively. Numerous experimental results prove that the proposed algorithm is superior to other algorithms.

Total variation regularization for depth-to-basement estimate: Part 2 — Physicogeologic meaning and comparisons with previous inversion methods

Geophysics ◽  
2011 ◽  
Vol 76 (1) ◽  
pp. I13-I20 ◽  
Author(s):  
Williams A. Lima ◽  
Cristiano M. Martins ◽  
João B. Silva ◽  
Valeria C. Barbosa
Keyword(s):  
Total Variation ◽  
Regularization Parameter ◽  
Synthetic Data ◽  
Gravity Inversion ◽  
Total Variation Regularization ◽  
Discontinuous Solutions ◽  
Mathematical Basis ◽  
Inversion Methods ◽  
Tv Regularization ◽  
Entropic Regularization

We applied the mathematical basis of the total variation (TV) regularization to analyze the physicogeologic meaning of the TV method and compared it with previous gravity inversion methods (weighted smoothness and entropic Regularization) to estimate discontinuous basements. In the second part, we analyze the physicogeologic meaning of the TV method and compare it with previous gravity inversion methods (weighted smoothness and entropic regularization) to estimate discontinuous basements. Presenting a mathematical review of these methods, we show that minimizing the TV stabilizing function favors discontinuous solutions because a smooth solution, to honor the data, must oscillate, and the presence of these oscillations increases the value of the TV stabilizing function. These three methods are applied to synthetic data produced by a simulated 2D graben bordered by step faults. TV regularization and weighted smoothness are also applied to the real anomaly of Steptoe Valley, Nevada, U.S.A. In all applications, the three methods perform similarly. TV regularization, however, has the advantage, compared with weighted smoothness, of requiring no a priori information about the maximum depth of the basin. As compared with entropic regularization, TV regularization is much simpler to use because it requires, in general, the tuning of just one regularization parameter.

scholarly journals TV2++: a novel spatial-temporal total variation for super resolution with exponential-type norm

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Lizhen Deng ◽  
Zhetao Zhou ◽  
Guoxia Xu ◽  
Hu Zhu ◽  
Bing-Kun Bao
Keyword(s):  
Total Variation ◽  
Exponential Type ◽  
Optimization Problems ◽  
Super Resolution ◽  
Low Rank ◽  
Good Effect ◽  
Total Variation Regularization ◽  
Time Dimension ◽  
Tv Regularization ◽  
Image Super Resolution

Abstract Recently, many super-resolution algorithms have been proposed to recover high-resolution images to improve visualization and help better analyze images. Among them, total variation regularization (TV) methods have been proven to have a good effect in retaining image edge information. However, these TV methods do not consider the temporal correlation between images. Our algorithm designs a new TV regularization (TV2++) to take advantage of the time dimension information of the images, further improving the utilization of useful information in the images. In addition, the union of global low rank regularization and TV regularization further enhances the image super-resolution recovery. And we extend the exponential-type penalty (ETP) function on singular values of a matrix to enhance low-rank matrix recovery. A novel image super-resolution algorithm based on the ETP norm and TV2++ regularization is proposed. And the alternating direction method of multipliers (ADMM) is applied to solve the optimization problems effectively. Numerous experimental results prove that the proposed algorithm is superior to other algorithms.

3D Dix inversion using bound-constrained total variation regularization

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. R311-R320 ◽  
Author(s):  
Ali Gholami ◽  
Ehsan Zabihi Naeini
Keyword(s):  
Total Variation ◽  
Real Data ◽  
Iterative Refinement ◽  
Total Variation Regularization ◽  
Interval Velocity ◽  
Tv Regularization ◽  
Regularization Techniques ◽  
Refinement Strategy ◽  
Memory Efficient

Given the ill-conditioned nature of Dix inversion, the resultant Dix interval-velocity field is often unrealistic, noisy, and highly dependent on the quality of the provided root-mean-square velocities. While the classic least-squares regularization techniques, e.g., various forms of Tikhonov regularization, lead to somewhat suboptimal stability, we formulated the Dix inversion as a new constrained optimization problem. This enables one to incorporate prior knowledge as soft and/or hard bounds for the optimization, effectively treating it as a denoising problem. The solution to the problem is achieved by a bound-constrained total variation (TV) regularization. TV regularization has the advantage of being able to recover the discontinuities in the model, but it often comes with a large memory and compute requirements. Therefore, we have developed a simple and memory-efficient algorithm using iterative refinement strategy. The quality of the new algorithm is also cross-examined against different strategies, which are currently used in practice. Overall, we observe that the proposed method outperforms classic Dix inversion methods on the synthetic and real data examples.

scholarly journals A unified approach for a 1D generalized total variation problem

2021 ◽  
Author(s):  
Cheng Lu ◽  
Dorit S. Hochbaum
Keyword(s):  
Total Variation ◽  
Total Variation Regularization ◽  
Signal Denoising ◽  
Unified Approach ◽  
Worst Case ◽  
Worst Case Complexity ◽  
Regularization Problem ◽  
General Convex ◽  
Convex Regularization ◽  
Optimization Solvers

AbstractWe study a 1-dimensional discrete signal denoising problem that consists of minimizing a sum of separable convex fidelity terms and convex regularization terms, the latter penalize the differences of adjacent signal values. This problem generalizes the total variation regularization problem. We provide here a unified approach to solve the problem for general convex fidelity and regularization functions that is based on the Karush–Kuhn–Tucker optimality conditions. This approach is shown here to lead to a fast algorithm for the problem with general convex fidelity and regularization functions, and a faster algorithm if, in addition, the fidelity functions are differentiable and the regularization functions are strictly convex. Both algorithms achieve the best theoretical worst case complexity over existing algorithms for the classes of objective functions studied here. Also in practice, our C++ implementation of the method is considerably faster than popular C++ nonlinear optimization solvers for the problem.

scholarly journals Total Variation Regularization Algorithms for Images Corrupted with Different Noise Models: A Review

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Paul Rodríguez
Keyword(s):  
Total Variation ◽  
Blind Deconvolution ◽  
Numerical Algorithms ◽  
Total Variation Regularization ◽  
General Technique ◽  
Denoising Method ◽  
Tv Regularization ◽  
Mixed Noise ◽  
Map Estimator ◽  
Noise Models

Total Variation (TV) regularization has evolved from an image denoising method for images corrupted with Gaussian noise into a more general technique for inverse problems such as deblurring, blind deconvolution, and inpainting, which also encompasses the Impulse, Poisson, Speckle, and mixed noise models. This paper focuses on giving a summary of the most relevant TV numerical algorithms for solving the restoration problem for grayscale/color images corrupted with several noise models, that is, Gaussian, Salt & Pepper, Poisson, and Speckle (Gamma) noise models as well as for the mixed noise scenarios, such the mixed Gaussian and impulse model. We also include the description of the maximum a posteriori (MAP) estimator for each model as well as a summary of general optimization procedures that are typically used to solve the TV problem.

scholarly journals An Efficient Image Reconstruction Framework Using Total Variation Regularization with Lp-Quasinorm and Group Gradient Sparsity

Information ◽  
2019 ◽  
Vol 10 (3) ◽  
pp. 115 ◽  
Author(s):  
Fan Lin ◽  
Yingpin Chen ◽  
Lingzhi Wang ◽  
Yuqun Chen ◽  
Wei Zhu ◽  
...  
Keyword(s):  
Image Reconstruction ◽  
Total Variation ◽  
Random Noise ◽  
Matrix Multiplication ◽  
Structural Similarity ◽  
Total Variation Regularization ◽  
Reconstruction Method ◽  
Image Gradient ◽  
Large Matrix ◽  
Tv Regularization

The total variation (TV) regularization-based methods are proven to be effective in removing random noise. However, these solutions usually have staircase effects. This paper proposes a new image reconstruction method based on TV regularization with Lp-quasinorm and group gradient sparsity. In this method, the regularization term of the group gradient sparsity can retrieve the neighborhood information of an image gradient, and the Lp-quasinorm constraint can characterize the sparsity of the image gradient. The method can effectively deblur images and remove impulse noise to well preserve image edge information and reduce the staircase effect. To improve the image recovery efficiency, a Fast Fourier Transform (FFT) is introduced to effectively avoid large matrix multiplication operations. Moreover, by introducing accelerated alternating direction method of multipliers (ADMM) in the method to allow for a fast restart of the optimization process, this method can run faster. In numerical experiments on standard test images sourced form Emory University and CVG-UGR (Computer Vision Group, University of Granada) image database, the advantage of the new method is verified by comparing it with existing advanced TV-based methods in terms of peak signal-to-noise ratio (PSNR), structural similarity (SSIM), and operational time.

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