Image Reconstruction of Compressed Sensing MRI Using Graph-based Redundant Wavelet Transform (中文，English)

Zongying Lai^{1,2}, Xiaobo Qu^{2,*}, Yunsong Liu^{2}, Di Guo^{3}, Jing Ye^{2}, Zhifang Zhan^{2}, Zhong Chen^{2,*}

^{1} Department of Communication Engineering and Electronic Science, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Xiamen University, Xiamen 361005, China.

^{2} Department of Electronic Science, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Xiamen University, Xiamen 361005, China.

^{3} School of Computer and Information Engineering, Xiamen University of Technology, Xiamen 361024, China.

Xiaobo Qu's Email: quxiaobo <at> xmu.edu.cn or quxiaobo2009 <at> gmail.com.

**Citations**:

Zongying Lai#, Xiaobo Qu#, Yunsong Liu, Di Guo, Jing Ye, Zhifang Zhan, Zhong Chen. Image reconstruction of compressed sensing MRI using graph-based redundant wavelet transform, Medical Image Analysis, 27: 93-104, 2016. (# denotes co-first authorship)

A free access to full text ( http://authors.elsevier.com/a/1S1Ie4rfPllVGc ) is provided by the publisher that valid for 50 days.

The code is shared at (http://www.quxiaobo.org/project/CS_MRI_GraphWavelet/Toolbox_GBRWT_MRI.zip)

**Abstract**:

Compressed sensing magnetic resonance imaging(CS-MRI) has shown great capacity for accelerating magnetic resonance imaging if an image can be sparsely represented. How the image is sparsified seriously affects its reconstruction quality. In the present study, a graph-based redundant wavelet transform (GBRWT) is introduced to sparsely represent magnetic resonance images in iterative image reconstructions. Using the l1 norm regularized formulation of the problem solved by an alternating-direction minimization with continuation algorithm, the experimental results demonstrate that the proposed method outperforms several state-of-the-art reconstruction methods in removing artifacts and achieves fewer reconstruction errors on the tested datasets.

**Keywords**: Compressed sensing, graph, wavelet, MRI, image reconstruction

**Methods**:

In GBRWT, image is divided to be patches with overlaps. We construct a weighted graph by viewing image patches as vertices and their differences as edges, and the shortest path on the graph minimizes the total difference of all image patches, e.g. the shortest-path-visit makes pixels ranges smoother. Redundant wavelet transform is carried out on the smooth signal (1D) to obtain the sparse representation. An example of patch-graph construction is shown in Fig. 1. The flowchart of GBRWT is illustrated in Fig.2 to illustrate the process of GBRWT-based CS-MRI.

Fig. 1 Patch-based graph

Fig. 2 Flow chart of GBRWT-based MRI reconstruction. The top right block diagram indicates the flow chart for finding the shortest path in patch-graphs to find new orders.

**Main result:**

1. Reconstruction with phantom

Fig. 3 Reconstructed images and errors in a phantom experiment using Cartesian sampling with 27% data.(a) The fully sampled image;

(b-e) reconstructed images based on WaTMRI, DLMRI, PBDW and GBRWT, respectively; (f)undersampling pattern;(g-j) reconstruction errors (scaled 5x).

2. Reconstruction in vivo data

Fig. 4 Reconstructed images and errors with 20% data were used in 2D random sampling.(a) The fully sampled image;

(b-e) reconstructed images based on WaTMRI, DLMRI, PBDW and GBRWT, respectively; (f) undersamplingpattern; (g-j) reconstruction errors (scaled 5x).

Fig. 5 Reconstructed images and errors using Cartesian sampling with 31% data.(a) The fully sampled image;

(b-e) reconstructed images based on WaTMRI, DLMRI, PBDW and GBRWT, respectively; (f) undersampling pattern; (g-j) reconstruction errors (scaled 5x).

Fig. 6 RLNEs change with the undersampling rate. The reference image of PBDW and GBRWT is the same SIDWT-based reconstructed image.

Conclusion:

In this paper, 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 representations of MR images, thus improving the reconstruction. When compared with conventional shift-invariant wavelets and several state-of-the-art CS-MRI image reconstruction methods, including PBDW, DLMRI and WaTMRI, reconstructed images using the proposed method are more consistent with fully sampled images in terms of image intensity and detail. Further improvements can be obtained by optimizing permutation orders trained from the patch-based graph and by further smoothing signals. Finally, parallel processes are expected to reduce the computation time caused by redundancy in future work.

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