Multi-step fixed-point proximity algorithms for solving a class of optimization problems arising from image processing

Qia, LI; Shen, Lixin; Xu, Yuesheng; Zhang, Na

doi:10.1007/s10444-014-9363-2

Cited by 39 publications

(100 citation statements)

References 40 publications

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“…So the additional cost is ignorable. In what follows, to simplify the presentation, we will not systematically compare the schemes developed here with those in [1,8,12,13] any more.…”

Section: C)mentioning

confidence: 99%

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A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization

Chen¹

2016

JCM

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We extend a primal-dual fixed point algorithm (PDFP) proposed in [5] to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrate how practical and fully decoupled schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also present. We demonstrate how different algorithms can be obtained by splitting the problems in different ways through the classic example of sparsity regularized least square model with constraint. In particular, for a class of linearly constrained problems, which are of great interest in the context of multi-block ADMM, can be solved by PDFP with a guarantee of convergence. Finally, some experiments are provided to illustrate the performance of several schemes derived by the PDFP algorithm.

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“…So the additional cost is ignorable. In what follows, to simplify the presentation, we will not systematically compare the schemes developed here with those in [1,8,12,13] any more.…”

Section: C)mentioning

confidence: 99%

“…Nevertheless, the computation cost increases with the addition of a symmetric step, which may double the work of each step. To avoid the disadvantage, we can also extend the scheme in [1,8,12,13] with the same treatment given above.…”

Section: Comparison To Admm-like Algorithmsmentioning

confidence: 99%

A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization

Chen¹

2016

JCM

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Section: Introductionmentioning

confidence: 99%

“…A number of algorithms have been proposed to address this, including those derived from the augmented Lagrangian framework, e.g., Alternating Direction Method of Multipliers (ADMM), [6][7][8] others derived from a primal-dual framework, e.g., Chambolle-Pock, [9][10][11] and those derived from a fixed-point proximity framework, 12,13 e.g., Preconditioned Alternating Projection Algorithm (PAPA). 5,14,15 PAPA, in particular, is suitable for optimization problems with three convex terms. † Furthermore, PAPA's explicit form is easily derived, unlike ADMM, where the minimizations of the alternating directions are unspecified and must be implemented by the user.…”

Section: Introductionmentioning

confidence: 99%

Relaxed ordered subset preconditioned alternating projection algorithm for PET reconstruction with automated penalty weight selection

Schmidtlein

Lin

et al. 2017

Medical Physics

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Acceleration of HOTV-PAPA was achieved using ROS. This was accompanied by an improved RMSE metric and perceptual image quality that were both superior to that obtained with either clinical or optimized OSEM. This may allow up to a four-fold reduction of the radiation dose to the patients in a PET study, as compared with current clinical practice. The proposed unsupervised parameter selection method provided useful estimates of the penalty weights for the selected phantoms' and patients' PET studies. In sum, the outcomes of this research indicate that ROS-HOTV-PAPA is an appropriate candidate for clinical applications and warrants further research.

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“…However, the numerical results show that the correction step is time consuming and the ADMM (LADMM) with Gaussian back substitution may require more iterations than the direct extension of ADMM (LADMM) to achieve the same objective function value. Therefore, in this paper, we dedicate to establishing convergent and efficient algorithms.As shown in [1,19,21,24], the notion of proximity operators provides a useful tool for the algorithmic development due to its firmly nonexpansive property. ADMM was shown in [19] a special case of the proximity algorithms.…”

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confidence: 99%

Two-Step Fixed-Point Proximity Algorithms for Multi-block Separable Convex Problems

Qia

Zhang

2016

J Sci Comput

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Multi-block separable convex problems recently received considerable attention. This class of optimization problems minimizes a separable convex objective function with linear constraints. The algorithmic challenges come from the fact that the classic alternating direction method of multipliers (ADMM) for the problem is not necessarily convergent. However, it is observed that ADMM outperforms numerically many of its variants with guaranteed theoretical convergence. The goal of this paper is to develop convergent and computationally efficient algorithms for solving multi-block separable convex problems. We first characterize the solutions of the optimization problems by proximity operators of the convex functions involved in their objective function. We then design a two-step fixed-point iterative scheme for solving these problems based on the characterization. We further prove convergence of the iterative scheme and show that it has O( 1 k ) convergence rate in the ergodic sense and the sense of the partial primal-dual gap, where k denotes the iteration number. Moreover, we derive specific two-step fixed-point proximity algorithms (2SFPPA) from the proposed iterative scheme and establish their global convergence. Numerical experiments for solving the sparse MRI problem demonstrate the numerical efficiency of the proposed 2SFPPA. 1The alternating direction method of multipliers (ADMM) [13] was originally proposed for solving problem (1.1) with s = 2, and was recently widely used in the area of image processing [4,14,30,35]. Since ADMM requires inner iterations to solve its subproblems of ADMM, its linearized version (LADMM) was proposed and was successfully used in applications [12]. As s ≥ 3, one can directly extend the original ADMM (LADMM) to problem (1.1). Without an additional assumption, however, it was recently shown in [8] that the direct extension of ADMM to multi-block convex problems is not necessarily convergent, although it may work well in practice. Very recently, there were some investigations [10,17,18,22] on convergence of the extension of ADMM under some additional assumptions. Some researchers dedicated to modify ADMM or LADMM to make it convergent. For instance, the Jacobian-type ADMM was proposed in [11] for parallel computing, the semi-proximal ADMM proposed in [18,32] is for convex quadratic programming and conic programming, the Gaussian back substitution technique was proposed in [15,16] to make ADMM and LADMM converge. It was shown in [15,16] the attractiveness of the Gaussian back substitution technique for theoretical analysis on convergence of ADMM-type algorithms. However, the numerical results show that the correction step is time consuming and the ADMM (LADMM) with Gaussian back substitution may require more iterations than the direct extension of ADMM (LADMM) to achieve the same objective function value. Therefore, in this paper, we dedicate to establishing convergent and efficient algorithms.As shown in [1,19,21,24], the notion of proximity operators provides a useful tool for the...

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Multi-step fixed-point proximity algorithms for solving a class of optimization problems arising from image processing

Cited by 39 publications

References 40 publications

A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization

A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization

Relaxed ordered subset preconditioned alternating projection algorithm for PET reconstruction with automated penalty weight selection

Two-Step Fixed-Point Proximity Algorithms for Multi-block Separable Convex Problems

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