A Preconditioning Technique for First‐Order Primal‐Dual Splitting Method in Convex Optimization

Wen, Meng; Peng, Jigen; Tang, Yuchao; Zhu, Chuanxi; Yue, Shigang

doi:10.1155/2017/3694525

Cited by 14 publications

(5 citation statements)

References 18 publications

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“…Finally, we wish to note that the constrained TV model (4) can also be derived using other iterative algorithms, such as the primal-dual Chambolle-Pock algorithm [15], the alternating direction method of multipliers [29,38,39], and the preconditioned primal-dual algorithm [40,41]. We have not presented the corresponding numerical results here.…”

Section: Discussionmentioning

confidence: 99%

Iterative Methods for Computing the Resolvent of the Sum of a Maximal Monotone Operator and Composite Operator with Applications

Bao

Tang

2019

Mathematical Problems in Engineering

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Total variation image denoising models have received considerable attention in the last two decades. To solve constrained total variation image denoising problems, we utilize the computation of a resolvent operator, which consists of a maximal monotone operator and a composite operator. More precisely, the composite operator consists of a maximal monotone operator and a bounded linear operator. Based on recent work, in this paper we propose a fixed-point approach for computing this resolvent operator. Under mild conditions on the iterative parameters, we prove strong convergence of the iterative sequence, which is based on the classical Krasnoselskii–Mann algorithm in general Hilbert spaces. As a direct application, we obtain an effective iterative algorithm for solving the proximity operator of the sum of two convex functions, one of which is the composition of a convex function with a linear transformation. Numerical experiments on image denoising are presented to illustrate the efficiency and effectiveness of the proposed iterative algorithm. In particular, we report the numerical results for the proposed algorithm with different step sizes and relaxation parameters.

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Section: Discussionmentioning

confidence: 99%

Iterative Methods for Computing the Resolvent of the Sum of a Maximal Monotone Operator and Composite Operator with Applications

Bao

Tang

2019

Mathematical Problems in Engineering

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“…The basic principle of primal-dual splitting method was introduced by Condat [42], Vu [43] simultaneously, and subsequently by Combettes and Pesquet [44,45], and preconditioning technique is also used for primal-dual splitting method [46]. Denote H be a real Hilbert space with its inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖ 1/2 .…”

Section: First-order Primal-dual Splitting Methods For Hotvsmentioning

confidence: 99%

Simultaneously spatial and temporal higher-order total variations for noise suppression and motion reduction in DCE and IVIM

Ding

Mohamed

et al. 2020

Medical Imaging 2020: Image Processing

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In many applications based on kinetic evaluation analysis and model fitting, quantitative mapping retrieved on data series from modalites such as MRI is completed on a voxel-by-voxel basis, where motion and low signal to noise ratio (SNR) would considerably degenerate the reliability of estimations. The coherence of image series in space and time can be used as prior knowledge to mitigate this occurrence. In this study, spatial and temporal higher-order total variations (HOTVs) are applied on a data series of MRI signal (e.g. dynamic contrast-enhanced (DCE) MRI and intravoxel incoherent motion (IVIM) MRI) to exploit the coherence of signal in space and time to minimize the variabilities caused by motion as well as improving quality of images with low SNR while retaining the physical details of original data properly. Simultaneously applying spatial and temporal HOTVs on images is non-trivial in implementation since it is a non-smooth optimization problem with multiple regularizers. Therefore, we use the proximal gradient method as well as a primal-dual split proximal mechanism to address the problem properly. In addition to increase the reliability of quantitative parametric map estimations, this preprocessing procedure can be included into many existing map estimation algorithms and pipelines effortlessly. We demonstrate our method on the parametric maps estimation for DCE MRI and IVIM MRI.

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“…In fact, Vu's algorithm [28] is equivalent to the Condat's algorithm [1]. Some generalizations of the Condat-Vu algorithm [1,28] can be found in [29,30]. Combettes et al [31] further pointed out that the primal-dual version of the Condat-Vu algorithm [1,28] can be derived from the variable metric forward-backward splitting algorithm [32] framework.…”

Section: Introductionmentioning

confidence: 99%

A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem

Tang¹

2019

JCM

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The forward-backward operator splitting algorithm is one of the most important methods for solving the optimization problem of the sum of two convex functions, where one is differentiable with a Lipschitz continuous gradient and the other is possibly nonsmooth but proximable. It is convenient to solve some optimization problems in the form of dual or primal-dual problems. Both methods are mature in theory. In this paper, we construct several efficient first-order splitting algorithms for solving a multi-block composite convex optimization problem. The objective function includes a smooth function with a Lipschitz continuous gradient, a proximable convex function that may be nonsmooth, and a finite sum of a composition of a proximable function and a bounded linear operator. To solve such an optimization problem, we transform it into the sum of three convex functions by defining an appropriate inner product space. On the basis of the dual forward-backward splitting algorithm and the primal-dual forward-backward splitting algorithm, we develop several iterative algorithms that involve only computing the gradient of the differentiable function and proximity operators of related convex functions. These iterative algorithms are matrix-inversion-free and completely splitting algorithms. Finally, we employ the proposed iterative algorithms to solve a regularized general prior image constrained compressed sensing (PICCS) model that is derived from computed tomography (CT) image reconstruction under sparse sampling of projection measurements. Numerical results show that the proposed iterative algorithms outperform other algorithms.

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A Preconditioning Technique for First‐Order Primal‐Dual Splitting Method in Convex Optimization

Cited by 14 publications

References 18 publications

Iterative Methods for Computing the Resolvent of the Sum of a Maximal Monotone Operator and Composite Operator with Applications

Iterative Methods for Computing the Resolvent of the Sum of a Maximal Monotone Operator and Composite Operator with Applications

Simultaneously spatial and temporal higher-order total variations for noise suppression and motion reduction in DCE and IVIM

A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem

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