Magnetic resonance image reconstruction from undersampled measurements using a patch-based nonlocal operator

Xiaobo Qu^{a,*}, Yingkun Hou^{b}, Fan Lam^{c}, Di Guo^{d}, Jianhui Zhong^{e}, Zhong Chen^{a,*}

a Department of Electronic Science, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, State Key Laboratory of Physical Chemistry of Solid Surfaces, Xiamen University, Xiamen 361005, China

b School of Information Science and Technology, Taishan University, Taian 271021, China

c Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

d School of Computer and Information Engineering, Xiamen University of Technology, Xiamen 361024, China

e Department of Imaging Sciences, University of Rochester, Box 648, Elmwood Avenue, Rochester, NY 14642-8648, USA

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

Abstract:

Compressed sensing MRI (CS-MRI) has shown great potential in reducing data acquisition time in MRI. Sparsity or compressibility plays an important role to reduce the image reconstruction error. Conventional CS-MRI typically uses a pre-defined sparsifying transform such as wavelet or finite difference, which sometimes does not lead to a sufficient sparse representation for the image to be reconstructed. In this paper, we design a patch-based nonlocal operator (PANO) to sparsify magnetic resonance images by making use of the similarity of image patches. The definition of PANO results in sparse representation for similar patches and allows us to establish a general formulation to trade the sparsity of these patches with the data consistency. It also provides feasibility to incorporate prior information learnt from undersampled data or another contrast image, which leads to optimized sparse representation of images to be reconstructed. Simulation results on in vivo data demonstrate that the proposed method achieves lower reconstruction error and higher visual quality than conventional CS-MRI methods.

The proposed method assumes that a guide image is available to give a good estimate of nonlocal similarity for image patches. This nonlocal similarity information is integrated as prior information into the proposed patch-based nonlocal operator (PANO), which is utilized to establish a general reconstruction formulation. The essential component of the proposed method is the design of the PANO.

Fig. 1. Group image patches. (a) An image with 6 6 pixels; (b) four groups of patches; (c) the patch and group dimension.

Fig. 2. Illustration of the similar patches found via block matching and the sparsity results in. (a) A search region X with and the reference patch T with ; (b) Q = 16 similar patches found by the norm distance measure with patch size L = 8; (c) 3D array stacked from the similar patches, and (d) curves for decay of pixel values, 2D and 3D wavelet coefficients.

Fig. 3. Comparison of different guide images. (a) variable density Cartesian sampling pattern with sampling rate 0.40; (b) locations of low-frequency k-space data; (c) the fully sampled MR image; (d) reconstructed image from only the low-frequency k-space data; (e) reconstructed image from zero-filled k-space data; (f) reconstructed image from SIDWT.

Fig. 4. Flowchart of the proposed PANO-based MRI reconstruction from undersampled data.

Fig. 5. Images reconstructed using proposed method with the different guide images (a) fully sampled image in Fig. 3(c), (b) low-resolution image in Fig. 3(d), (c) zero-filling image in Fig. 3(e), and (d) conventional CS-MRI reconstruction in Fig. 3(f). The RLNEs of (a)–(d) are 0.077, 0.087, 0.083, 0.081.

Conclusions:

A new MR image reconstruction method is presented. The proposed method exploits nonlocal similarity of image patches by establishing a patch-based nonlocal operator, PANO, which effectively produces sparse vectors by operating on grouped similar patches of the image. A reconstruction formulation is proposed to incorporate a sparsity constraint on PANO-produced coefficients, which can be considered as a generalization of previously proposed patch-based reconstruction methods. Simulation results based on fully sampled experimental data demonstrated consistent improvement in reconstruction accuracy of the proposed method over conventional CS-MRI reconstructions and several alternative CS-based reconstructions. We have also shown that the similarity information required by PANO can be iteratively learnt from a guide image reconstructed from undersampled k-space data and the proposed method is not sensitive to the initial guide image. In general, only learning the similarity twice is sufficiently enough to maximize the performance of the proposed method for the tested images. When the data are highly undesampled, learning the similarity from another contrast image with fully sampled data greatly improve the reconstruction.

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