Multimedia Tools and Applications

, Volume 78, Issue 10, pp 12749–12782 | Cite as

Undersampled CS image reconstruction using nonconvex nonsmooth mixed constraints

  • Ryan Wen LiuEmail author
  • Wei Yin
  • Lin Shi
  • Jinming Duan
  • Simon Chun Ho Yu
  • Defeng Wang


Compressed sensing magnetic resonance imaging (CS-MRI) has attracted considerable attention due to its great potential in reducing scanning time and guaranteeing high-quality reconstruction. In conventional CS-MRI framework, the total variation (TV) penalty and L1-norm constraint on wavelet coefficients are commonly combined to reduce the reconstruction error. However, TV sometimes tends to cause staircase-like artifacts due to its nature in favoring piecewise constant solution. To overcome the model-dependent deficiency, a hybrid TV (TV1,2) regularizer is introduced in this paper by combining TV with its second-order version (TV2). It is well known that the wavelet coefficients of MR images are not only approximately sparse, but also have the property of tree-structured hierarchical sparsity. Therefore, a L0-regularized tree-structured sparsity constraint is proposed to better represent the measure of sparseness in wavelet domain. In what follows, we present our new CS-MRI framework by combining the TV1,2 regularizer and L0-regularized tree-structured sparsity constraint. However, the combination makes CS-MRI problem difficult to handle due to the nonconvex and nonsmooth natures of mixed constraints. To achieve solution stability, the resulting composite minimization problem is decomposed into several simpler subproblems. Each of these subproblems has a closed-form solution or could be efficiently solved using existing numerical method. The results from simulation and in vivo experiments have demonstrated the good performance of our proposed method compared with several conventional MRI reconstruction methods.


Compressed sensing Magnetic resonance imaging Total variation Tree sparsity Fast composite splitting algorithm 

List of Abbreviations


Compressed sensing


Magnetic resonance imaging


Total variation


Multichannel total variation


Total generalized variation


Nonlocal total variation


Structure-guided total variation


Higher-degree total variation


Conjugate gradient


Fast iterative shrinkage/threshold algorithm


Mean doubly augmented Lagrangian


Fast composite splitting algorithm


Composite splitting denoising


Peak signal-to-noise ratio


Mean structural similarity


Relative L2 norm error



This work was partially supported by the National Natural Science Foundation of China (No.: 51609195), the Open Project Program of Key Laboratory of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education (No.: JYB201704), and the Wuhan University of Technology Excellent Dissertation Cultivation Fund (2017-YS-071). The first author would like to thank Mr. Quandang Ma for his helpful suggestions on manuscript revision.


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Authors and Affiliations

  1. 1.School of NavigationWuhan University of TechnologyWuhanChina
  2. 2.Key Laboratory of Intelligent Perception and Systems for High-Dimensional Information of Ministry of EducationNanjing University of Science and TechnologyNanjingChina
  3. 3.Department of Imaging and Interventional RadiologyThe Chinese University of Hong KongShatinChina
  4. 4.Biomedical Image Analysis Group & MRC London Institute of Medical SciencesImperial College LondonLondonUK

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