Medical & Biological Engineering & Computing

, Volume 56, Issue 7, pp 1211–1225 | Cite as

A singular K-space model for fast reconstruction of magnetic resonance images from undersampled data

  • Jianhua Luo
  • Zhiying Mou
  • Binjie Qin
  • Wanqing Li
  • Philip Ogunbona
  • Marc C. Robini
  • Yuemin Zhu
Original Article


Reconstructing magnetic resonance images from undersampled k-space data is a challenging problem. This paper introduces a novel method of image reconstruction from undersampled k-space data based on the concept of singularizing operators and a novel singular k-space model. Exploring the sparsity of an image in the k-space, the singular k-space model (SKM) is proposed in terms of the k-space functions of a singularizing operator. The singularizing operator is constructed by combining basic difference operators. An algorithm is developed to reliably estimate the model parameters from undersampled k-space data. The estimated parameters are then used to recover the missing k-space data through the model, subsequently achieving high-quality reconstruction of the image using inverse Fourier transform. Experiments on physical phantom and real brain MR images have shown that the proposed SKM method constantly outperforms the popular total variation (TV) and the classical zero-filling (ZF) methods regardless of the undersampling rates, the noise levels, and the image structures. For the same objective quality of the reconstructed images, the proposed method requires much less k-space data than the TV method. The SKM method is an effective method for fast MRI reconstruction from the undersampled k-space data.

Graphical abstract

Two Real Images and their sparsified images by singularizing operator


Total variation (TV) Magnetic resonance imaging (MRI) Undersampled k-space data Singular k-space model Image reconstruction 


Funding information

This work was supported in part by China Aviation Industry under the project (No. cxy204SHJD22, 2015), the National Natural Science Foundation of China (61271320 and 60872102), Medical Engineering Cross Fund of Shanghai Jiao Tong University (YG2014MS29), the Region Auvergne-Rhône-Alpes of France under the project CMIRA COOPERA/EXPLORA PRO 2016 and the 2010 UIC International Linkage Grant of University of Wollongong, Australia.


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Copyright information

© International Federation for Medical and Biological Engineering 2017

Authors and Affiliations

  1. 1.School of Aeronautics and AstronauticsShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.China National Aeronautical Radio Electronics Research InstituteShanghaiPeople’s Republic of China
  3. 3.School of Biomedical EngineeringShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  4. 4.School of Computing and Information TechnologyUniversity of WollongongWollongongAustralia
  5. 5.INSA Lyon, CNRS, Inserm, CREATIS UMR 5220, U1206, F-69621Université LyonLyonFrance

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