Exploiting sparsity and low-rank structure for the recovery of multi-slice breast MRIs with reduced sampling error
It has been shown that, magnetic resonance images (MRIs) with sparsity representation in a transformed domain, e.g. spatial finite-differences (FD), or discrete cosine transform (DCT), can be restored from undersampled k-space via applying current compressive sampling theory. The paper presents a model-based method for the restoration of MRIs. The reduced-order model, in which a full-system-response is projected onto a subspace of lower dimensionality, has been used to accelerate image reconstruction by reducing the size of the involved linear system. In this paper, the singular value threshold (SVT) technique is applied as a denoising scheme to reduce and select the model order of the inverse Fourier transform image, and to restore multi-slice breast MRIs that have been compressively sampled in k-space. The restored MRIs with SVT for denoising show reduced sampling errors compared to the direct MRI restoration methods via spatial FD, or DCT. Compressive sampling is a technique for finding sparse solutions to underdetermined linear systems. The sparsity that is implicit in MRIs is to explore the solution to MRI reconstruction after transformation from significantly undersampled k-space. The challenge, however, is that, since some incoherent artifacts result from the random undersampling, noise-like interference is added to the image with sparse representation. These recovery algorithms in the literature are not capable of fully removing the artifacts. It is necessary to introduce a denoising procedure to improve the quality of image recovery. This paper applies a singular value threshold algorithm to reduce the model order of image basis functions, which allows further improvement of the quality of image reconstruction with removal of noise artifacts. The principle of the denoising scheme is to reconstruct the sparse MRI matrices optimally with a lower rank via selecting smaller number of dominant singular values. The singular value threshold algorithm is performed by minimizing the nuclear norm of difference between the sampled image and the recovered image. It has been illustrated that this algorithm improves the ability of previous image reconstruction algorithms to remove noise artifacts while significantly improving the quality of MRI recovery.
KeywordsSingular value threshold Model order reduction Sparse sensing MRI k-space Low rank matrix Finite differences Discrete cosine transform
The MRI datasets were afforded by Q. Yang, Apollo Medical Imaging Technology Pty. Ltd. North Melbourne, VIC 3051, Australia, and A. Pitman, Department of Anatomy and Cell Biology, The University of Melbourne and Sydney School of Medicine, The University of Notre Dame, Australia. This work was supported in part by the Australian Research Council (ARC) Discovery Project funding scheme—Project No. DP0988064. This project was also carried out under an Australian Research Council (ARC) Linkage Grant—LP0775463.
- 2.Ljunggren S (1983) A simple graphical representation of fourier-based imaging methods. J Magn Reson 54(2):338–343Google Scholar
- 7.Wiener N (1949) Extrapolation, interpolation, and smoothing of stationary time series. Wiley, New YorkGoogle Scholar
- 11.Osher S, Mao Y, Dong B, Yin W (2010) Fast linearized Bregman iteration for compressive sensing and sparse denoising. Commun Math Sci 8(1):93–111Google Scholar
- 13.Golub GH, Van Loan CF (1996) Matrix computations. The John Hopkins University Press, Baltimore, MDGoogle Scholar