 Research article
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Joint sparse reconstruction of multicontrast MRI images with graph based redundant wavelet transform
BMC Medical Imagingvolume 18, Article number: 7 (2018)
Abstract
Background
Multicontrast images in magnetic resonance imaging (MRI) provide abundant contrast information reflecting the characteristics of the internal tissues of human bodies, and thus have been widely utilized in clinical diagnosis. However, long acquisition time limits the application of multicontrast MRI. One efficient way to accelerate data acquisition is to undersample the kspace data and then reconstruct images with sparsity constraint. However, images are compromised at high acceleration factor if images are reconstructed individually. We aim to improve the images with a jointly sparse reconstruction and Graphbased redundant wavelet transform (GBRWT).
Methods
First, a sparsifying transform, GBRWT, is trained to reflect the similarity of tissue structures in multicontrast images. Second, joint multicontrast image reconstruction is formulated as a ℓ_{2, 1} norm optimization problem under GBRWT representations. Third, the optimization problem is numerically solved using a derived alternating direction method.
Results
Experimental results in synthetic and in vivo MRI data demonstrate that the proposed joint reconstruction method can achieve lower reconstruction errors and better preserve image structures than the compared joint reconstruction methods. Besides, the proposed method outperforms single image reconstruction with joint sparsity constraint of multicontrast images.
Conclusions
The proposed method explores the joint sparsity of multicontrast MRI images under graphbased redundant wavelet transform and realizes joint sparse reconstruction of multicontrast images. Experiment demonstrate that the proposed method outperforms the compared joint reconstruction methods as well as individual reconstructions. With this high quality image reconstruction method, it is possible to achieve the high acceleration factors by exploring the complementary information provided by multicontrast MRI.
Background
Multicontrast images in magnetic resonance imaging (MRI) provide abundant contrast information reflecting the characteristics of the internal tissues of human bodies, and thus have been utilized in clinical diagnosis. However, long acquisition time limits the application of multicontrast MR imaging.
Undersampling the kspace data and reconstructing images with sparsity constraint is one efficient way to accelerate MRI sampling [1,2,3,4,5]. However, the data acquisition factor is limited since images are compromised when images are reconstructed individually. The previous work [6] suggested to use another fullysampled contrast image to train an adaptive sparse representation with Graphbased redundant wavelet transform (GBRWT) and then greatly improve the reconstructed images [7]. This approach, however, cannot reduce the overall acceleration factor in data acquisition because of the full sampling in another contrast images [6]. Thus, to further accelerate multicontrast MRI, undersampling all multicontrast images, e.g.T1 weighted (T1W), T2 weighted (T2W) and proton density weighted (PDW) MRI images, and maintain high quality image reconstruction are expected.
The MRI image structures under different contrast settings are the same due to the multiple acquisitions of the same anatomical cross section [6, 8,9,10,11,12]. Thus, nonzero coefficients may occur at the same spatial locations in the sparsifying transform domains, e.g. finite difference, wavelet transform [2] and patchbased sparse transformations [13,14,15,16]. Therefore, it is possible to improve the image reconstruction if this extra information is incorporated into sparse image reconstruction [17].
Sparse representation capability plays a key role in sparse MRI reconstruction. The GBRWT [6, 7] transform was verified to have good sparsification capability for MRI images. The main step of GBRWT transform is to construct a graph to find new permutations adaptive to target image structures, and then to obtain the sparser transformation with wavelet filters acting on the permutated smooth signals. However, if high acceleration factor is set, very limited information will be provided for single image thus the reconstruction will be compromised. Thus, the combining merits of joint reconstruction and GBRWT are expected.
In this study, we propose to reconstruct the multicontrast MRI with adaptive GBRWT sparse representations and joint sparsity among multicontrast images. An alternating direction method with continuity (ADMC) [18] algorithm is introduced to solve the joint ℓ_{2, 1}norm minimization problem. The proposed approach will be compared with the joint sparse reconstruction method based on shift invariant discrete wavelet transform (SIDWT) [17] and Bayesian compressed sensing (BCS) [19].
Methods
The undersampled kspace data of multicontrast MRI images are expressed as
where x = [x_{1}; ⋯; x_{ T }] denotes the column stacked multicontrast images, T the number of contrasts. y = [y_{ 1 }; ⋯; y_{ T }] the column stacked undersampled kspace data, ε noises in the sampled data. The UF is the undersampling operator in the Fourier space, which can be expressed as
Each U_{ i }F_{ i }, (i = 1, ⋯, T) acts on one of the multicontrast images. We adopt different sampling patterns, i.e. U_{1} ≠ ⋯ ≠ U_{ i } ≠ ⋯ ≠ U_{ T }, and the same Fourier transform bases, i.e. F_{1} = ⋯ = F_{ i } = ⋯ = F_{ T } for each image of individual contrast.
The flowchart of the proposed joint sparse reconstruction is shown in Fig. 1. Reconstructed image based on SIDWT [17] is adopted as reference image to train the GBRWT from the undersampled kspace data, because SIDWT can mitigate the blocky artifacts introduced by orthogonal wavelet transform and better preserve the structures in the target images [15, 16]. With GBRWT as the sparse representation, multicontrast images can be simultaneously reconstructed by implementing joint sparsity constraints on these transformation coefficients.
Graphbased redundant wavelet transform
Given a reference image, the GBRWT is achieved by carrying out redundant wavelet transform on permuted signals of new orders [7]. The new orders are found in weighted graph constructed from the reference image, in which image patches collected by a sliding window serve as the vertex and the patch similarities computed using w_{m, n} = w(b_{ m }, b_{ n }) = ‖b_{ m } − b_{ n }‖_{2} (where b_{ m } and b_{ n } denote the m^{th} and the n^{th} patches) serve as the weight. The new orders are obtained by finding the shortest possible path on the patchbased graph [7, 20]. Then, redundant wavelet transform is performed on permuted pixels to achieve sparse representation.
The process of permutation and wavelet filtering in GBRWT is shown in Fig. 2. In the l^{th} level decomposition, the input signal a_{ l } will be first reordered by permutation matrix P_{ l }, whose inverse process is \( {\mathbf{P}}_l^{\mathrm{H}} \) and satisfied \( {\mathbf{P}}_l^{\mathrm{H}}{\mathbf{P}}_l=\mathbf{I} \). Then, nondecimated wavelet transformation Φ_{ l }, whose inverse process is \( {\boldsymbol{\Phi}}_l^{\mathrm{H}} \) and satisfied \( {\boldsymbol{\Phi}}_l^{\mathrm{H}}{\boldsymbol{\Phi}}_l=\mathbf{I} \), are performed on the reordered pixels. The output a_{l + 1} and d_{l + 1} of l^{th} level nondecimated decomposition will be of the same size with the input signal \( {\tilde{a}}_l \). Let Φ_{ l }P_{ l } be the l^{th} level decomposition of GBRWT, and \( {\mathbf{P}}_l^{\mathrm{H}}{\boldsymbol{\Phi}}_l^{\mathrm{H}} \) be corresponding composition process, and then Φ_{ l }P_{ l } satisfies the following property
where c denotes the redundancy of GBRWT transform. It has been verified that GBRWT provides sparser representations than traditional wavelet transform, thus can improve the MRI image reconstruction [7].
Joint sparsity of multicontrast image coefficients
Multicontrast MRI images are obtained by different parameter settings, but share the same anatomical cross section [6,7,8]. The image structures corresponding to tissue locations remain unchanged with contrast varied, which leads to spatial positionrelated coefficients. Joint sparsity means that, under appropriate sparsifying transforms, the positions of nonzero coefficients correspond directly to same spatial locations in multiple images. Figure 3 shows that the nonzero transform coefficients of two contrast images occur at the same positions in the Haar wavelets transform and GBRWT domains. Thus, the joint sparsity of GBRWT provides extra information on images and may further improve the reconstruction of multicontrast images.
Problem formulation
The joint sparsity promoting problem in multicontrast MRI image reconstruction with GBRWT is solved using the mixed ℓ_{2, 1} norm minimization [9, 21, 22]:
where, \( \boldsymbol{\Psi} =\left[\begin{array}{ccc}{\boldsymbol{\Psi}}_g& 0& 0\\ {}0& \ddots & 0\\ {}0& 0& {\boldsymbol{\Psi}}_g\end{array}\right] \) and Ψ_{g} denotes the GBRWT representation, in which l^{th} level decomposition be expressed as Φ_{ l }P_{ l }. Let α = Ψx be the corresponding coefficients, then for an image set which includes T kinds of contrasts, the column stacked coefficients can be expressed as: α = [α_{1}; ⋯; α_{ T }]. The role of grouping operator G is to reshape the column stacked coefficients of multicontrast MRI images into a matrix as shown in Fig. 4. Then, one column of Gα stands for coefficients of one image, and one row forms a group.
The ℓ_{2, 1} norm is defined as
where, G is the group operator, N is the number of transform coefficient and T the number of contrast.
Numerical algorithm
The alternating direction method with continuation [18] is incorporated in the ℓ_{2, 1} norm optimization. Let α = Ψx, the objective in Eq. (4) can be rewritten as
Furthermore, the objective function in (6) can be overrelaxed to be unconstraint as
The λ is a parameter to balance the sparsity and data fidelity. The β is fixed in the inner loops and changes continuously to achieve optimal reconstruction in the outer loops. When β → ∞, the solution of Eq. (7) approaches to that of Eq. (6). When β is fixed, x and α will be computed alternatively by the following two steps:

1)
With x fixed, α will be computed by solve the objective:
Algorithm 1: Joint multicontrast MRI reconstruction based on GBRWT  
Parameters: λ Input:  
kspace data y = [y_{1}; ⋯; y_{ T }]; g levels of permutation orders P_{ j }, {j = 1, ⋯, g}; regularization parameter λ; tolerance of inner loop η = 10^{−4}.  
Initialization: x = (UF)^{H}y, x_{ previous } = x, β = 2^{6}.  
Main:  
While β ≤ 2^{12}  
(1) Given x, solving α by computing Eq. (10) for each group of coefficients α^{i}, {i = 1, ⋯, N};  
(2) Applying α into Eq. (12) to obtain the solution x; (3) If ‖Δx‖ = ‖x_{ previous } − x‖ > η, then x_{ previous } = x, go to step (1); Otherwise: go to step (4); (4) \( \widehat{\mathbf{x}}=\mathbf{x} \), β = 2β, go to step (1);  
End while  
Output \( \widehat{\mathbf{x}} \) 
To find the extreme of objective function in Eq. (8), firstly the equivalent transformation \( {\left\Vert \boldsymbol{\upalpha} \boldsymbol{\Psi} \mathbf{x}\right\Vert}_2^2={\left\Vert \mathbf{G}\boldsymbol{\upalpha } \mathbf{G}\left(\boldsymbol{\Psi} \mathbf{x}\right)\right\Vert}_F^2 \) is taken; then, the coefficients in rows of Gα (each group) are computed separately by solving least square method. Let α^{i} = (Gα)_{i, :}, (Ψx)^{i} = (Ψx)_{i, :} denote the ith group of Gα and Ψx respectively, we find solution by
Then, α can be obtained via each group computing:

2)
Fix α, x can be computed by solving
The minimization with respect to x can be solved by finding the extreme of least square problem in Eq. (11). Finally, we get
where, c is the redundancy caused by GBRWT transform. The numerical algorithm pseudocode is listed in Algorithm 1.
Results
The image reconstruction was performed on a server with E52637 v3 (3.5G Hz) *2 CPU, 8 GB memory. The proposed method, Joint sparse reconstruction based on GBRWT (JGBRWT), is compared with the Joint sparse reconstruction method based on SIDWT (JSIDWT) [15,16,17], that replacing Ψ_{g} with SIDWT in Eq. (4) and Joint reconstruction with Bayesian Compressed Sensing (JBCS) [19], which is a stateoftheart joint multicontrast image reconstruction that jointly explores the gradient coefficients of multiple images. The comparison with GBRWTbased single image reconstruction [7] is also included to demonstrate the advantage of the joint reconstruction. The parameter values for JBCS are taken as the same in the cods shared by the authors (http://martinos.org/~berkin/software.html). For the proposed method and JSIDWT, λ is set as 10^{4}.
The relative ℓ_{2} norm error (RLNE) defined as \( e\left(\overset{\frown }{\mathbf{x}}\right)={\left\Vert \overset{\frown }{\mathbf{x}}\tilde{\mathbf{x}}\right\Vert}_{\mathbf{2}}/{\left\Vert \tilde{\mathbf{x}}\right\Vert}_{\mathbf{2}} \) (in which \( \tilde{\mathbf{x}} \) is ground truth and \( \overset{\frown }{\mathbf{x}} \) is the reconstructed image) and mean structure similarity index measure (MSSIM) [23] served as the criteria for assessing the quality of reconstructed image quality. Smaller RLNE means lower reconstructed error and higher MSSIM indicates better structure preservation capability.
The Brainweb images (http://brainweb.bic.mni.mcgill.ca/) [24, 25] (Fig. 5) as well as the in vivo multicontrast images were used to validate the efficiency of the proposed method. The multicontrast knee images (Fig. 6) were acquired from GE 3 Tesla scanner (Discovery MR750W, USA) with parameters (T1W: FSE, TR/TE = 499 ms/9.63 ms; T2W: TR/TE = 2435 ms/49.98 ms, Proton density weighted image: TR/TE = 2253 ms/31.81 ms; FOV = 180 × 180 mm^{2}, slice thickness = 4 mm). The multicontrast brain images (Fig. 7) were acquired from SIMENS 3 Tesla scanner (MAGNETOM Trio Tim, Germany) with parameters (T2W: TSE, TR/TE = 3000 ms/66 ms,; T1W, FLAIR: TR/TE = 3900 ms/9.3 ms,; FOV = 200 × 200 mm^{2}, slice thickness = 5 mm) for Fig. 7(A) and TSE: TR = 4000 ms, FOV = 192 × 192 mm^{2}, slice thickness = 3 mm, ∆TE =8 ms for the multiecho data in Fig. 7(B).
Fully sampled multicontrast MRI images shown in Figs 5, 6 and 7 are used for the experiment of undersampling and joint sparse reconstruction. Reconstruction errors shown in Figs. 8, 9 and 10 reveal that the proposed method outperforms the JSIDWT and JBCS. The lower error of the proposed method indicates better fidelity and edgepreserving capabilities compared with JSIDWT and JBCS. Besides, the reconstruction errors were reduced when comparing the proposed method with single image reconstruction based on the same GBRWT transform, implying that the improvement obtained by joint reconstruction over single image reconstruction.
Tables 1, 2 and 3 show RLNEs and MSSIMs of reconstructed images. These criteria indicate that the proposed method gained the highest MSSIM and the lowest RLNE, and thus recovered the images most faithfully.
One typical brain image reconstruction with 27% sampled data are shown in Fig. 10. In the zoomedin area of the 2nd row, the sulcus of the T2W image appears in the middle of the fully sampled and JGBRWT reconstructed images, but nearly disappears for JBCS reconstruction. In the marked region of the 3rd, the proposed method leads to more consistent reconstruction with the fully sampled image than other methods. These improvements are also confirmed by the error images in the last two rows.
2D undersampling
The 2Dundersampling patterns (Fig. 11) was explored to demonstrate the potential applications of the proposed method in 3D imaging, in which 2D phase encoding plane can be undersampled.
Brainweb reconstructed errors shown in Fig. 12 demonstrate that on the simulated database, the lowest reconstruction errors were obtained with the proposed method. The corresponding RLNE/MSSIM are shown in Table 4. Figure 13 implies that the proposed method led to the lowest brightness in the error images and thus maintained fidelity best. The criteria listed in Tables 4 and 5 indicate that the proposed method achieved the highest MSSIM and the lowest RLNE on the tested dataset.
Different sampling rates
The curves in Fig. 14 show that the RLNEs decreased with sampling rate increased. The RLNE line of the proposed JGBRWT method (dark green line) is lower than that of GBRWT (or contrastbycontrast reconstruction, black line) with the same GBRWT representation, indicating benefits are achieved by utilizing joint sparsity among multicontrast images. The JGBRWT also outperforms other joint reconstruction method, including JSIDWT (red line) and JBCS (blue line), in terms of lower RLNEs at all sampling rates.
The same sampling patterns
The proposed method is compatible to same or different sampling patterns. Reconstruction criteria in Table 6 show that the proposed method outperforms the compared ones under the same sampling patterns. Besides, at the same sampling rate, using different sampling patterns lead to better evaluation criteria than using same sampling patterns (Table 6 vs. Table 1).
Discussions
Limitations on choosing image to train the graph
Choose an arbitrary prereconstructed image as reference will lead to reconstruction errors (RLNEs) slightly change as shown in Fig. 15. But the RLNEs are still much lower than single image reconstruction. A possible way in the future work is to train a GBRWT jointly from all the undersampled multicontrast images to make full use of the common/complementary information of multicontrast images.
Limitations on unregistered images
Unregistered multicontrast images will go against the joint sparsity assumption, and thus affect joint reconstruction performance. Reconstructed images of aligned and misaligned multicontrast images (we simulate misalignment by rotating Fig. 16(a) with 10 degrees) shown in Fig. 16 demonstrate that misalignment will make the detail reconstruction deteriorated. RLNE obviously increased in sparse reconstruction of misaligned multicontrast images. Improved image reconstruction is expected by incorporating the registration into image reconstruction process as it was done in [6], which would be interesting as a future work.
Computation complexity
The main step of numerical algorithm to solve the proposed joint reconstruction problem include a soft thresholding to solve α and a onestep computation to solve x, which is with the same computation complexity as single contrast image reconstruction, but with more data to compute, and thus no obvious additional computational burden.
Program at our platform (E5–2637 v3 (3.5G Hz) *2 CPU, 8 GB memory) shows that, the SIDWTbased single image reconstruction need 20 s, and SIDWTbased joint reconstruction need 100 with 4 different contrast images at low sampling rate. The GBRWTbased single image reconstruction need 200 s and GBRWTbased joint reconstruction need 103 s with 4 different contrast images at low sampling rate.
Experiment with noise
Multicontrast images in Fig. 7(A) in the manuscript are used in noise experiment. Noisy data are simulated by adding Gaussian white noise with variance σ^{2} = 0.02 on real and imaginary part of kspace data. Figure 17 demonstrate that the proposed method outperforms the compared ones in preserving image structures as well as removing noise. According to Table 7 the proposed method achieves lowest RLNEs, highest MSSIMs and highest SNRs. The signal to noise rate (SNR) is defined as SNR = 10log_{10}(μ/σ), where u is the mean of image density and δ is the standard deviation of the noise extracted from the image background.
Parameters
Two noise level are considered (Gaussian white noise with variance σ^{2} = 0.02 and σ^{2} = 0.03) in testing λ. The optimal λ for σ^{2} = 0.02 and σ^{2} = 0.03 are 600 and 400 respectively on the tested data according the curve shown in Fig. 18.
The parameters of GBRWT include patchsize and decomposition levels, which have been discussed in [7]. The suggested patchsize in GBRWT are from 4 × 4 to 7 × 7, and suggested decomposition level is 3–5 level. We use the patchsize 7 × 7 and do 5 level decomposition in this experiment.
Conclusions
A new approach is proposed to simultaneously explore the adaptive sparse image representation under graphbased redundant wavelet transform and the joint sparse reconstruction of multicontrast MRI images. Experimental results in synthetic and in vivo MRI data demonstrate that the proposed method can achieve lower reconstruction errors than the compared methods. With this high quality image reconstruction method, it is possible to achieve the high acceleration factors by exploring the complementary information provided by multicontrast MRI.
Abbreviations
 ADMC:

Alternating direction method with continuity
 GBRWT:

Graphbased redundant wavelet transform
 JBCS:

Joint reconstruction based on Bayesian compressed sensing
 JGBRWT:

Joint sparse reconstruction based on GBRWT
 JSIDWT:

Joint sparse reconstruction based on SIDWT
 MRI:

Magnetic resonance imaging
 MSSIM:

Mean structure similarity index measure
 PDW:

Proton density weighted MRI image
 RLNE:

Relative ℓ_{2} norm error
 SIDWT:

Shift invariant discrete wavelet transform
 T1W:

T1 weighted MRI image
 T2W:

T2 weighted MRI image
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Acknowledgments
The authors would like to thank Dr. Xi Peng for providing multicontrast brain MRI data in Fig. 7 and Dr. Ying Chen for language editing in this work.
Funding
This work was partially supported by National Key R&D Program of China (2017YFC0108703), National Natural Science Foundation of China (61571380, 6171101498, U1632274, 61672335, 61601276 and 61302174), Natural Science Foundation of Fujian Province of China (2018J06018, 2016J05205 and 2016J01327), Fundamental Research Funds for the Central Universities (20720180056), Foundation of Fujian Educational Committee (JAT160358) and Important Joint Research Project on Major Diseases of Xiamen City (3502Z20149032). These funding bodies do not play any role in the design of the study and in collection, analysis, and interpretation of data and in writing the manuscript.
Availability of data and materials
The datasets used in the study will be publicly available at the authors’ website (http://csrc.xmu.edu.cn/project/GraphWavelet_JointSparseMRI/Toolbox_GraphWavelet_JointSparseMRI.zip).
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Contributions
XQ designed the joint multicontrast MRI reconstruction method and ZL implemented this method. Algorithm development and data analysis were carried out by ZL, XZ, DG, XD, YY, GG, ZC and XQ. All authors have been involved in drafting and revising the manuscript and approved the final version to be published. All authors read and approved the final manuscript.
Corresponding author
Correspondence to Xiaobo Qu.
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Ethics approval and consent to participate
This study (joint reconstruction of knee images) was approved by Institutional Review Board of No.2 Hospital Xiamen (ethical approval number 2014001). This retrospective study (joint reconstruction of brain MRI images) was approved by Institute Review Board of Shenzhen Institutes of Advanced Technology, Chinese Academy of Science (ethical approval number SIATIRB130315H0024). Participant for all images have informed consent that he knew the risks and agreed to participate in the research.
Competing interests
One author, Xiaobo Qu, works as a Section Editor for the BMC Medical Imaging. The other authors declare that they have no competing interests.
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Keywords
 Magnetic resonance imaging
 Fast imaging
 Image reconstruction
 Sparsity