A Convergent Fixed-Point Proximity Algorithm Accelerated by FISTA for the ℓ0 Sparse Recovery Problem

Zeng, Xueying; Shen, Lixin; Xu, Yuesheng

doi:10.1007/978-3-319-91274-5_2

Cited by 5 publications

(6 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We next consider the case when x / ∈ C I . By Item (ii) of Proposition 2.7, we conclude that there exists δ 2 > 0 such that The sufficient condition for a pair (x * , y * ) to be a local minimizer of the minimization problem (4.1) presented in Corollary 4.9 for a special example of convex function g was obtained in Proposition 2.3 of [32].…”

Section: This Together With the Fact Xmentioning

confidence: 78%

“…Form (4.2) relates to regularization by the envelope of the ℓ 0 norm and form (4.3) relates to regularization by the capped ℓ 1 norm [11]. For specific examples of f , see [31] for inverting incomplete Fourier transform, [32,33] for image/signal processing, [34] for medical image reconstruction and machine learning [16,17,21,29]. Employing the partition A j , j ∈ Z d+1 , of R d and the definition of • 0 , we have an alternative representation of function f :…”

Section: Sparse Regularization In the Spacial Domainmentioning

confidence: 99%

“…The aim of this work is to understand a global minimizer of regularization problems whose objective functions have the form of a fidelity term plus a regularization term involving the ℓ 0 norm. Regularization problems of this type appear frequently in recent studies of machine learning [16,17,21,29], computer graphics [8,27], signal processing [6,15,33], image processing [24,25,32], medical imaging [34] and statistics [9,35]. Many published results have demonstrated that the use of the ℓ 0 norm in regularization models promotes sparsity for the regularized solutions or the transformed regularized solutions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sparse Regularization with the $\ell_0$ Norm

Xu¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We consider a minimization problem whose objective function is the sum of a fidelity term, not necessarily convex, and a regularization term defined by a positive regularization parameter λ multiple of the ℓ 0 norm composed with a linear transform. This problem has wide applications in compressed sensing, sparse machine learning and image reconstruction. The goal of this paper is to understand what choices of the regularization parameter can dictate the level of sparsity under the transform for a global minimizer of the resulting regularized objective function. This is a critical issue but it has been left unaddressed. We address it from a geometric viewpoint with which the sparsity partition of the image space of the transform is introduced. Choices of the regularization parameter are specified to ensure that a global minimizer of the corresponding regularized objective function achieves a prescribed level of sparsity under the transform. Results are obtained for the spacial sparsity case in which the transform is the identity map, a case that covers several applications of practical importance, including machine learning, image/signal processing and medical image reconstruction.

show abstract

Section: This Together With the Fact Xmentioning

confidence: 78%

Section: Sparse Regularization In the Spacial Domainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Sparse Regularization with the $\ell_0$ Norm

Xu¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this subsection we propose subiteration-dependent preconditioners that originated from generalization of the momentum approach. The momentum is an acceleration technique widely used in optimization [25]- [27]. The Nesterov's momentum [25] has been combined with ordered-subsets (OS) by Kim et al [28] for CT image reconstruction.…”

Section: Bsrem With Subiteration-dependent Preconditionersmentioning

confidence: 99%

A Fast Convergent Ordered-Subsets Algorithm with Subiteration-Dependent Preconditioners for PET Image Reconstruction

Jianfeng¹,

Ross²,

Andrzej³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We investigated the imaging performance of a fast convergent ordered-subsets algorithm with subiterationdependent preconditioners (SDPs) for positron emission tomography (PET) image reconstruction. In particular, we considered the use of SDP with the block sequential regularized expectation maximization (BSREM) approach with the relative difference prior (RDP) regularizer due to its prior clinical adaptation by vendors. Because the RDP regularization promotes smoothness in the reconstructed image, the directions of the gradients in smooth areas more accurately point toward the objective function's minimizer than those in variable areas. Motivated by this observation, two SDPs have been designed to increase iteration stepsizes in the smooth areas and reduce iteration step-sizes in the variable areas relative to a conventional expectation maximization preconditioner. The momentum technique used for convergence acceleration can be viewed as a special case of SDP. We have proved the global convergence of SDP-BSREM algorithms by assuming certain characteristics of the preconditioner. By means of numerical experiments using both simulated and clinical PET data, we have shown that our SDP-BSREM substantially improved the convergence rate, as compared to conventional BSREM and a vendor's implementation as Q.Clear. Specifically, in numerical experiments with a synthetic brain phantom, both proposed algorithms outperformed the conventional BSREM by a factor of two, while in experiments with clinical whole-body J.

show abstract

“…The resulting model is always convex and can be solved efficiently with many available tools. However, compared with the convex 1 norm-based model, a nonconvex 0 norm-based model has certain advantages in the context of image processing [4,58]. Here, we propose a novel regularization model that combines the sparsity maximization and denoising task, and we introduce the Moreau envelope of the 0 norm to reformulate the problem.…”

Section: Introductionmentioning

confidence: 99%

Sparsity promoting regularization for effective noise suppression in SPECT image reconstruction

Zheng

Krol

et al. 2019

Inverse Problems

Self Cite

View full text Add to dashboard Cite

The purpose of this research is to develop an advanced reconstruction method for low-count, hence high-noise, single-photon emission computed tomography (SPECT) image reconstruction. It consists of a novel reconstruction model to suppress noise while conducting reconstruction and an efficient algorithm to solve the model. A novel regularizer is introduced as the nonconvex denoising term based on the approximate sparsity of the image under a geometric tight frame transform domain. The deblurring term is based on the negative log-likelihood of the SPECT data model. To solve the resulting nonconvex optimization problem a preconditioned fixed-point proximity algorithm (PFPA) is introduced. We prove that under appropriate assumptions, PFPA converges to a local solution of the optimization problem at a global O(1/k) convergence rate. Substantial numerical results for simulation data are presented to demonstrate the superiority of the proposed method in denoising, suppressing artifacts and reconstruction accuracy. We 7 Author to whom correspondence should be addressed.

show abstract

A Convergent Fixed-Point Proximity Algorithm Accelerated by FISTA for the ℓ0 Sparse Recovery Problem

Cited by 5 publications

References 24 publications

Sparse Regularization with the $\ell_0$ Norm

Sparse Regularization with the $\ell_0$ Norm

A Fast Convergent Ordered-Subsets Algorithm with Subiteration-Dependent Preconditioners for PET Image Reconstruction

Sparsity promoting regularization for effective noise suppression in SPECT image reconstruction

Contact Info

Product

Resources

About