Zongying Lai^{1}, Xinlin Zhang^{1}, Di Guo^{2}, Xiaofeng Du^{2}, Zhong Chen^{1}, and Xiaobo Qu^{1}

Multi-contrast images in magnetic resonance imaging (MRI) are widely applied in clinical applications, since an abundant contrast information reflects the characteristics of the internal tissue of human body, providing an effective reference for clinical diagnosis. However, long acquisition time limits the application of magnetic resonance multi contrast imaging. Under-sampling the k-space data and reconstructing images with sparsity constraint is one efficient way to accelerate magnetic resonance imaging sampling. In this work, multi-contrast undersampled MRI images are jointly reconstructed under the sparse representation using graph wavelets. Experiment results demonstrate that the proposed method outperforms the compared state-of-the-art methods.

**Purpose**

**Methods**

The under-sampling of k-space data in MRI imaging can be expressed as: $$$\mathbf{y}={{\mathbf{F}}_{\mathbf{U}}}\mathbf{x}+\boldsymbol{\varepsilon }$$$, where $$$\mathbf{\varepsilon }$$$ denotes the noise contained in k-space data; $$$\mathbf{x}={{\left[ {{\mathbf{x}}_{1}},\cdots {{\mathbf{x}}_{C}} \right]}^{T}}$$$ the column vector formed by multi-contrast images; $$$\mathbf{y}={{\left[ {{\mathbf{y}}_{1}},\cdots ,{{\mathbf{y}}_{C}} \right]}^{T}}$$$ the under-sampled k-space data formed in same rule; $$${{\mathbf{F}}_{\mathbf{U}}}=diag\left( {{\mathbf{U}}_{1}}\mathbf{F},\cdots ,{{\mathbf{U}}_{C}}\mathbf{F} \right)$$$ the operator that performs Fourier transform and then under sampling on each image.

By constraining the joint sparsity of multi-contrast images with $$${{l}_{2,1}}$$$-norm, the multi-contrast images joint reconstruction optimization function can be formulated as: $$\underset{\mathbf{x}}{\mathop{\min }}\,{{\left\| \mathbf{G\Psi x} \right\|}_{2,1}}\ \ \ s.t.\ \ \ \left\| {{\mathbf{F}}_{\mathbf{U}}}\mathbf{x}-\mathbf{y} \right\|_{2}^{2}\le {{\sigma }^{2}}$$, where $$$\mathbf{\Psi }=diag\left( {{\mathbf{\Psi }}_{g}},\cdots ,{{\mathbf{\Psi }}_{g}} \right)$$$, and $$${{\mathbf{\Psi }}_{g}}$$$ denotes graph-based wavelets transform [4] performed on each image, accordingly, $$$\boldsymbol{\alpha}={{\mathbf{\Psi }}_{g}}\mathbf{x}$$$ denotes the coefficients; $$$\mathbf{G}$$$ denotes the grouping operator satisfying $$$\mathbf{G}\boldsymbol {\alpha }=\left[ \begin{matrix} {{\alpha }_{11}} & \cdots & {{\alpha }_{1C}} \\ \vdots & \ddots & \vdots \\ {{\alpha }_{N1}} & \cdots & {{\alpha }_{NC}} \\ \end{matrix} \right]$$$, where the column of $$$\mathbf{G\alpha }$$$ denotes transformation coefficients of one contrast image. Therefore, the $$${{l}_{2,1}}$$$-norm of $$$\mathbf{G}\boldsymbol {\alpha }$$$ can be defined as $$${{\left\|\mathbf{G}\boldsymbol{\alpha} \right\|}_{2,1}}=\sum\limits_{i=1}^{N}{{{\left( \sum\limits_{j=1}^{T}{{{\left| {{\boldsymbol{\alpha}}_{ij}} \right|}^{2}}} \right)}^{{1}/{2}\;}}}$$$. We name the proposed method as joint GBRWT (JGBRWT) reconstruction.

**Results**

**Conclusion**

1. M. Lustig, D. Donoho, and J. M. Pauly, "Sparse MRI: The application of compressed
sensing for rapid MR imaging," *Magnetic
Resonance in Medicine*, vol. 58, pp. 1182-1195, 2007.

2. X. Qu, D. Guo, B. Ning, Y. Hou, Y. Lin, S. Cai, and Z. Chen, "Undersampled MRI reconstruction with patch-based
directional wavelets," *Magnetic
Resonance Imaging*, vol. 30, pp. 964-977, 2012.

3. X.
Qu, Y. Hou, F. Lam, D. Guo, J. Zhong, and Z. Chen, "Magnetic resonance
image reconstruction from undersampled measurements using a patch-based
nonlocal operator,"* Medical Image
Analysis*, vol. 18, pp. 843-856, 2014.

4. Z.
Lai, X. Qu, Y. Liu, D. Guo, J. Ye, Z. Zhan, and Z. Chen, "Image reconstruction of compressed sensing MRI using
graph-based redundant wavelet transform," *Medical Image Analysis*, vol. 27, pp. 93-104, 2016.

5. C.
A. Cocosco, V. Kollokian, Kwan, and A. C. Evans, "BrainWeb: Online interface
to a 3D MRI simulated brain database," *Neuroimage*,
vol. 5, pp. 425, 1997.

6. B.
Bilgic, V. K. Goyal, and E. Adalsteinsson, "Multi-contrast reconstruction
with Bayesian compressed sensing," *Magnetic
Resonance in Medicine*, vol. 66, pp. 1601-1615, 2011.