Undersampled MRI reconstruction with patch-based directional wavelets

Xiaobo Qu

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

^{2}Department of Communication Engineering, Xiamen University, Xiamen, 361005, China

^{3}School of Information Science and Technology, Taishan University, Taian 271021, China

Citations: **Xiaobo Qu**, Di Guo, Bende Ning, Yingkun Hou, Yulan Lin, Shuhui Cai, Zhong Chen. Undersampled MRI reconstruction with patch-based directional wavelets, *Magnetic Resonance Imaging*, in press, 2012.

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Download the code: Toolbox_sparseMRI_PBDW

**Abstract**:

Compressed sensing has shown great potential in reducing data acquisition time in magnetic resonance imaging (MRI). In traditional compressed sensing MRI methods, an image is reconstructed by enforcing its sparse representation with respect to a preconstructed basis or dictionary. In this paper, patch-based directional wavelets are proposed to reconstruct images from undersampled k-space data. A parameter of patch-based directional wavelets, indicating the geometric direction of each patch, is trained from the reconstructed image using conventional compressed sensing MRI methods and incorporated into the sparsifying transform to provide the sparse representation for the image to be reconstructed. A reconstruction formulation is proposed and solved via an efficient alternating direction algorithm. Simulation results on phantom and in vivo data indicate that the proposed method outperforms conventional compressed sensing MRI methods in preserving the edges and suppressing the noise. Besides, the proposed method is not sensitive to the initial image when training directions.

Keywords: MRI; Fast imaging; Compressed sensing; Wavelet; Directional transform

In PBDW, the image is divided into patches, and the geometric direction is selected out among the candidate directions to provide the sparsest representation for each patch. Then, pixels in each patch are arranged parallel to the geometric direction and formed into a column vector. Finally, wavelet transform is performed on the column vector of each patch. The flowchart of PBDW is illustrated in Fig.1, and an example shown in Fig. 2 further illustrates the process.

Fig. 1. Flowchart of the patch-based directional wavelets. The process of geometric direction estimation in a patch is shown in the dashed rectangle.

Fig. 2. Enhanced sparsity by rearranging the pixels parallel to optimal geometric direction. (A) A patch with all candidate directions; (B) projecting a pixel to the axis which is orthogonal to a given direction; (C) the indexes of the arranged pixels; (D) a patch marked with two candidate directions; (E) rearranged pixels (1D vectors) parallel to the two directions, and the horizontal axis is the index of the pixel in the new vector; (F) sorted magnitude of 1D Haar wavelet coefficients and (G) approximation error by preserving S largest coefficients.

Fig. 5. Flowchart of the proposed method.

Main results:

1. Reconstruction with phantom

A simulated phantom with nonpiecewise constant image features is adopted in simulation [26]. The sampling pattern with sampling rate 0.30 is shown in Fig. 6A.

Fig. 6. Reconstructed phantom images. (A) The Cartesian sampling pattern with a sampling rate 0.30; (B) fully sampled image; (C–H) reconstructed images using ODWT+TV, TV, PBW, SIDWT, PBDW with directions estimated from the SIDWT-based reconstruction and PBDW with directions estimated from the fully sampled image.

2. Reconstruction with in vivo data

Fig. 7. Reconstructed lemon images. (A) The Cartesian sampling pattern with a sampling rate 0.40; (B) fully sampled image; (C–H) reconstructed images using ODWT+TV, TV, PBW, SIDWT, PBDW with directions estimated from the SIDWT-based reconstruction and PBDW with directions estimated from the fully sampled image.

Fig. 8. Reconstructed brain images. (A) The Cartesian sampling pattern with a sampling rate 0.45; (B) fully sampled image; (C)-(H) reconstructed images using ODWT+TV, TV, PBW, SIDWT, PBDW with directions estimated from the SIDWT-based reconstruction, PBDW with directions estimated from the fully sampled image.

Fig. 10. Reconstructed lemon images with added noise. (A) The Cartesian sampling pattern with a sampling rate of 0.50; (B) fully sampled image; (C–H) reconstructed images using ODWT+TV, TV, PBW, SIDWT, PBDW with directions estimated from the SIDWT-based reconstruction and PBDW with directions estimated from the fully sampled image.

Fig. 11. Reconstructed brain images with added noise. (A) The Cartesian sampling pattern with a sampling rate of 0.60; (B) fully sampled image; (C–H) reconstructed images using ODWT+TV, TV, PBW, PBDW with directions estimated from the SIDWT-based reconstruction and PBDW with directions estimated from the fully sampled image.

Fig. 12. Effect of the initial reconstruction in PBDW for brain image. (A) RLNE versus times of estimation directions at a sampling rate of 0.45, (B) RLNE versus the sampling rates when the times of estimation directions are two. Note: The “0” in the horizontal axis in panel (A) corresponds to the first image used to train the directions. This image is obtained using conventional CS-MRI methods.

Conclusion:

In this paper, PBDW-based undersampled MRI reconstruction is proposed. Geometric directions in PBDW are trained from the reconstructed image using conventional CSMRI methods and integrated into the final undersampled MRI reconstruction. Simulation results on phantom and in vivo data indicate that the proposed method can better preserve the edges and suppress the noise than traditional CS-MRI methods do. The PBDW is suggested to follow conventional CS-MRI reconstructions to improve the reconstruction if computational burden is acceptable for the applications. The limitation of PBDW is that edges are blurred at low sampling rates. Further improvement on the reconstruction is possible if better direction estimation is obtained. Finding a reasonable initial image at low sampling rate is still open. In order to avoid introducing an explicit initial image to train the directions, one may incorporate the direction training in the reconstruction formulation so that geometric directions totally rely on the undersampled data, rather than the image reconstructed using conventional CS-MRI methods. Extension of this work with the weighted ℓ 1 norm optimization [28] and nonconvex optimization [29] is being considered to further improve the reconstruction.

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