Image reconstruction of compressed sensing MRI using graph-based redundant wavelet transform
Graphical abstract
Introduction
Magnetic resonance imaging (MRI) is widely used in medical diagnoses because of its high resolution and noninvasiveness. However, physical conditions, such as gradient amplitude and slew-rate, and physiological conditions such as nerve stimulation limit data acquisition speeds. Many researchers focus on accelerating MRI (Akcakaya et al., 2012, Chang and Ji, 2010, Chen et al., 2010, Ji and Liang, 2001, Singh et al., 2011, Ji et al., 2003). Compressed sensing MRI (CS-MRI) is promising for accelerating data acquisition by collecting fewer data than those required by the Nyquist sampling theorem (Jacob, 2009, Weller et al., 2013, Ying and Ji, 2011). The CS can be combined further with parallel imaging to achieve higher acceleration factors (Huang et al., 2012, Weller et al., 2011). In general, three requirements exist in a successful CS application: sparse representation (Wang et al., 2014, Zhang et al., 2012), incoherent undersampling artifacts (Greiser and von Kienlin, 2003, Tsai and Nishimura, 2000), and an effective nonlinear reconstruction algorithm (Aelterman et al., 2011, Lustig et al., 2008, Majumdar and Ward, 2011, Majumdar and Ward, 2012, Majumdar et al., 2013, Hu et al., 2012).
In CS-MRI, finding an optimally sparse representation for magnetic resonance (MR) images is important because the reconstruction error is usually lower if the image representation is sparser (Qu et al., 2012 , 2014 Qu et al., 2010, Ravishankar and Bresler, 2011). Conventional transforms, e.g., discrete cosine transform, total variation, and wavelet transform, have been used to sparsely represent MR images (Chaari et al., 2011, Khalidov et al., 2011). These methods represent an image sparsely using predefined bases, which exhibit no difference among images. The image reconstruction may become unsatisfactory when the data are highly undersampled because of the insufficiently sparse representations (Qu et al., 2012, Ravishankar and Bresler, 2011).
Recently, patch-based methods have attracted considerable interest in CS-MRI because adaptively sparse representations can be trained with easy manipulations on patches (Akcakaya et al., 2011, Akcakaya et al., 2014, Maggioni et al., 2013, Ning et al., 2013, Qu et al., 2012, Qu et al., 2014, Ravishankar and Bresler, 2011, Wang and Ying, 2014, Huang et al., 2014). For example, the geometric edge of a patch has been applied to train the adaptively sparse representations (Ning et al., 2013, Qu et al., 2012). Assuming that image patches are linear combinations of element patches, Aharon et al. used the dictionary learning algorithm K-SVD to train a patch-based dictionary (Aharon et al., 2006, Ravishankar and Bresler, 2011). Both methods significantly improve the image reconstruction over the predefined basis method (Ning et al., 2013, Qu et al., 2012, Ravishankar and Bresler, 2011). However, these methods sparsely represent each patch separately, and neither of them considers the relationship, e.g., the differences, among patches in the whole image. This relationship offers us an opportunity to take it into account and to further remove artifacts in CS-MRI.
A recent report demonstrated that a graph can be formed by viewing image patches as vertices and their differences as edges; the shortest path on the graph minimizes the total difference of all image patches (Ram et al., 2011, Ram et al., 2012). This graph enables us to produce a smoother signal if a proper shortest path can be estimated. In this article, we introduce this graph-based transform into CS-MRI to remove artifacts introduced by the k-space undersampling. Because smoother signals usually lead to sparser representation when wavelet coefficients are applied, the introduced transform is also called a graph-based redundant wavelet transform (GBRWT) because it combines graph and redundant wavelets.
In the present study, we assume that a proper reference image is available in CS-MRI. All patches are reordered according to a short path on the graph, and a sparse representation is achieved by applying redundant wavelets to the smooth signal organized by traveling along this short path. GBRWT is then incorporated into an l1-norm-based iterative MR image reconstruction.
The remainder of this article is organized as follows: Section 2 presents the implementation of GBRWT in MR images and the graph-based reconstruction model in CS-MRI. The main results on realistic MR imaging data will be given in Section 3. Discussions are presented in Section 4. Finally, Section 5 concludes this work and notes the future work.
Section snippets
Conventional CS-MRI
Undersampled k-space data from an MRI scanner can be denoted by where x is the discrete image to be reconstructed; is the acquired k-space data; denote noises; and is the undersampling and Fourier transform and directly relies on the undersampling scheme. An image is reconstructed by enforcing the sparsity of the image according to where ΨH transforms image x from spatial domain to sparse coefficients; ‖ · ‖1 stands for the l1 norm that
Results
To evaluate the performance of the presented method, we use three datasets in experiments. All k-space data were acquired on real MRI scanners. Dataset 1 (Fig. 5(a), (b), (e)–(h)) was acquired from a healthy volunteer in a 3T Siemens Trio Tim MRI scanner using the T2-weighted turo spin echo sequence (TR/TE=6100/99 ms, FOV = 220 × 220 mm, slice thickness = 3 mm). Dataset 2 (Fig. 5(c) and (d)) was acquired from another healthy volunteer in a 1.5T Philips MRI scanner with fast field echo sequence
Different undersampling rates
The GBRWT leads to lower reconstruction errors than PBDW and SIDWT, as shown in Fig. 8(a) and (b), when data are limited (i.e., the undersampling rate is lower than 35%). The improvement of reconstruction is more pronounced in Fig. 8(a) than in (b) for the proposed method. The reason for this difference may be because GBRWT makes use of smoothness more efficiently, because the corresponding source image in Fig. 5(a) is smoother than the latter in Fig. 5(f).
With added noise
To test the performance of the
Conclusions
In this article, a new image reconstruction method based on a graph-based redundant wavelet transform is proposed for CS-MRI. This method explores the graph structure to model images and images’ approximate coefficients in each wavelet decomposition level to minimize the total difference of all image patches. The input signal can then be smoothed by new orders estimated by solving the travelling salesman problem in the graph. Wavelet filtering of smoother signals leads to sparser
Acknowledgments
The authors would like thank Drs. Bingwen Zheng, Feng Huang and Xi Peng for providing data in Sections 3 and 4. The authors sincerely thank Drs. Michael Lustig, Junzhou Huang, Saiprasad Ravishankar and Yoram Bresler for sharing codes in the comparisons. The authors sincerely extend their appreciations to Professor Tim (Tien Mo) Shih for linguistic assistance. The authors are grateful to the reviewers for their thorough reviews. This work was supported by the NNSF of China (61201045, 11375147,
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