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 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.

A new nonlinear reconstruction method based on total variation regularization of neutron penumbral imaging

2011 ◽  
Vol 82 (9) ◽  
pp. 093504 ◽  
Author(s):  
Weixin Qian ◽  
Shuangxi Qi ◽  
Wanli Wang ◽  
Jinming Cheng ◽  
Dongbing Liu
Keyword(s):  
Total Variation ◽  
Total Variation Regularization ◽  
Reconstruction Method ◽  
Penumbral Imaging

Magnetic inversion to recover the subsurface block structures based on L1 norm and total variation regularization

2021 ◽  
Author(s):  
Mitsuru Utsugi
Keyword(s):  
Total Variation ◽  
Magnetic Data ◽  
Preconditioned Conjugate Gradient ◽  
Inversion Method ◽  
Total Variation Regularization ◽  
Subsurface Structure ◽  
L1 Norm ◽  
Magnetic Inversion ◽  
Original Definition ◽  
Primal Dual

Summary This paper presents a new sparse inversion method based on L1 norm regularization for 3D magnetic data. In isolation, L1 norm regularization yields model elements which are unconstrained by the input data to be exactly zero, leading to a sparse model with compact and focused structure. Here, we complement the L1 norm with a penalty minimizing total variation, the L1 norm of the model gradients; it is expected that the sharp boundaries of the subsurface structure are not compromised by incorporating this penalty. Although this penalty is widely used in the geophysical inversion studies, it is often replaced by an alternative quadratic penalty to ease solution of the penalized inversion problem; in this study, the original definition of the total variation, i.e., form of the L1 norm of the model gradients, is used. To solve the problem with this combined penalty of L1 norm and total variation, this study introduces alternative direction method of multipliers (ADMM), which is a primal-dual optimization algorithm that solves convex penalized problems based on the optimization of an augmented Lagrange function. To improve the computational efficiency of the algorithm to make this method applicable to large-scale magnetic inverse problems, this study applies matrix compression using the wavelet transform and the preconditioned conjugate gradient method. The inversion method is applied to both synthetic tests and real data, the synthetic tests demonstrate that, when subsurface structure is blocky, it can be reproduced almost perfectly.

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