An Iterative Support Shrinking Algorithm for Non-Lipschitz Optimization in Image Restoration

Zeng, Chao; Jia, Rui; Wu, Chunlin

doi:10.1007/s10851-018-0830-0

Cited by 21 publications

(57 citation statements)

References 46 publications

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“…Also, the global convergence of the iterative sequence is provided, and the limit is a critical point of the original objective functional. These results, which can be regarded in some sense as extensions of that in [60], improve those in [55,57] for a special case of single channel image restoration; see Remark 5.1.…”

Section: Introductionmentioning

confidence: 57%

“…(i) We list some potential functions satisfying Assumption 2.1: φ 1 (t) = t p , 0 < p < 1 [14,30,55,57]; φ 2 (t) = ln(at p +1), a > 0, 0 < p < 1 [55,57].…”

Section: Remark 22mentioning

confidence: 99%

“…Although there are several recent good advances [31,37,51], the gradient operator ∇ in 2D image restoration problems does not fit their assumptions on the linear operator. Very recently, an iterative support shrinking strategy has been derived and used with the iteratively reweighted ℓ 1 [10,24] or least squares [18,34] algorithmic structure, for various signal and image restoration problems [23,38,49,53,55,57]. In particular, the parallel [55,57] considered the gradient compounded non-Lipschitz variational model (1.3) in the discrete setting with q = 1,2 for the single channel case, i.e., M = 1, and constructed the iterative support shrinking algorithm with proximal linearization (ISS-APL) with convergence results established by Kurdyka-Łojasiewicz (KL) property [1][2][3].…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, an iterative support shrinking strategy has been derived and used with the iteratively reweighted ℓ 1 [10,24] or least squares [18,34] algorithmic structure, for various signal and image restoration problems [23,38,49,53,55,57]. In particular, the parallel [55,57] considered the gradient compounded non-Lipschitz variational model (1.3) in the discrete setting with q = 1,2 for the single channel case, i.e., M = 1, and constructed the iterative support shrinking algorithm with proximal linearization (ISS-APL) with convergence results established by Kurdyka-Łojasiewicz (KL) property [1][2][3]. It was therein proved that, under an exact inner loop, the sequence generated by ISSAPL converges to a critical point of some median auxiliary problem, instead of the original one.…”

Section: Introductionmentioning

confidence: 99%

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A General Non-Lipschitz Joint Regularized Model for Multi-Channel/Modality Image Reconstruction

ChunlinWu¹

2021

CSIAM-AM

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Multi-channel/modality image joint reconstruction has gained much research interest in recent years. In this paper, we propose to use a nonconvex and non-Lipschitz joint regularizer in a general variational model for joint reconstruction under additive measurement noise. This framework has good ability in edge-preserving by sharing common edge features of individual images. We study the lower bound theory for the non-Lipschitz joint reconstruction model in two important cases with Gaussian and impulsive measurement noise, respectively. In addition, we extend previous works to propose an inexact iterative support shrinking algorithm with proximal linearization for multi-channel image reconstruction (InISSAPL-MC) and prove that the iterative sequence converges globally to a critical point of the original objective function. In a special case of single channel image restoration, the convergence result improves those in the literature. For numerical implementation, we adopt primal dual method to the inner subproblem. Numerical experiments in color image restoration and two-modality undersampled magnetic resonance imaging (MRI) reconstruction show that the proposed non-Lipschitz joint reconstruction method achieves considerable improvements in terms of edge preservation for piecewise constant images compared to existing methods.

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Section: Introductionmentioning

confidence: 57%

“…(i) We list some potential functions satisfying Assumption 2.1: φ 1 (t) = t p , 0 < p < 1 [14,30,55,57]; φ 2 (t) = ln(at p +1), a > 0, 0 < p < 1 [55,57].…”

Section: Remark 22mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A General Non-Lipschitz Joint Regularized Model for Multi-Channel/Modality Image Reconstruction

ChunlinWu¹

2021

CSIAM-AM

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“…Actually reweighted methods reformulate the original non-Lipschitz q -2 to Lipschitz ones by a de-singularizing parameter. Very recently, [25,39] developed methods by successively shrinking the support of the variables to overcome non-Lipschitz property, in which [25] considered the non-group case with r = ∞ and [39] focused on the image restoration with r = 2. To the best of our knowledge, we note that most of the references considered only r = 2 in these methods.For the group sparse optimization problem (1.1), most algorithms were proposed only in the case of r = 2 as well.…”

mentioning

confidence: 99%

An efficient and globally convergent algorithm for $\ell_{p,q} - \ell_r$ model in group sparse optimization

Xue

Feng

2020

Communications in Mathematical Sciences

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Group sparsity combines the underlying sparsity and group structure of the data in problems. We develop a proximally linearized algorithm InISSAPL for the non-Lipschitz group sparse p,qr optimization problem. The algorithm gives a unified framework for all the parameters p ≥ 1, 0 < q < 1, 1 ≤ r ≤ ∞, which is applicable to different kinds of measurement noise. In particular, it includes the addition of the non-smooth 1,q regularization term and the non-smooth 1 / ∞ fidelity term as special cases. It allows an inexact inner loop accessible to the implementation of scaled ADMM, and still has global convergence. The algorithm is efficient and fast with computation only on the shrinking group support set. Many numerical experiments are presented for the algorithm with diversity of parameters p, q, r. The comparisons show that our algorithm is superior to others in the existing works.Laplace noise or heavy-tailed noise such as impulsive noise, the 1 fidelity term (r = 1) is a good choice. For the noise by uniform distribution or quantization error, the ∞ fidelity term (r = ∞) suits.There are many references to study the sparse optimization problem without group structure in it, i.e. the non-group model in which the number of groups g equals N . Then the p,q term in (1.1) is degenerated to q (0 < q < 1) regularization one. One class of methods is smoothing approximate methods [5,[12][13][14]24]. By a smoothing function ϕ(x, θ), the non-Lipschitz property of the objective function can be removed. The second class of methods is general iterative shrinkage-thresholding algorithms (GISA) for q -2 problem [9,34,40]. GISA was inspired by the great success of soft thresholding and iterative shrinkage-thresholding algorithms (ISTA) [3,16] for convex 1 -2 problem. The third class of methods is the iterative reweighted minimization methods for q -2 minimization problem; see, e.g. [10,15,23,27]. Actually reweighted methods reformulate the original non-Lipschitz q -2 to Lipschitz ones by a de-singularizing parameter. Very recently, [25,39] developed methods by successively shrinking the support of the variables to overcome non-Lipschitz property, in which [25] considered the non-group case with r = ∞ and [39] focused on the image restoration with r = 2. To the best of our knowledge, we note that most of the references considered only r = 2 in these methods.For the group sparse optimization problem (1.1), most algorithms were proposed only in the case of r = 2 as well. Hu et al. [21] investigated this problem via p,q regularization, others developed algorithms for 2,1 regularized least squares, e.g. group Lasso [8,17,38]. As noted before, it is important and necessary to develop algorithm for general 1 ≤ r ≤ ∞. This will bring the difficulty to universally handle the noise parameter r with regularized parameters p, q in the group structure. In addition, the regularization term p,q with parameters p ≥ 1, 0 < q < 1 in the objective function E in (1.1) leads to a non-convex, non-Lipschitz optimization problem. This non-smoothness becom...

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Effective Two-Stage Image Segmentation: A New Non-Lipschitz Decomposition Approach with Convergent Algorithm

Guo

Xue

2021

J Math Imaging Vis

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An Iterative Support Shrinking Algorithm for Non-Lipschitz Optimization in Image Restoration

Cited by 21 publications

References 46 publications

A General Non-Lipschitz Joint Regularized Model for Multi-Channel/Modality Image Reconstruction

A General Non-Lipschitz Joint Regularized Model for Multi-Channel/Modality Image Reconstruction

An efficient and globally convergent algorithm for $\ell_{p,q} - \ell_r$ model in group sparse optimization

Effective Two-Stage Image Segmentation: A New Non-Lipschitz Decomposition Approach with Convergent Algorithm

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