Phase-Constrained Reconstruction of High-Resolution Multi-shot Diffusion Weighted Image ( [Chinese] )

Yiman Huang1, Xinlin Zhang1, Hua Guo2, Huijun Chen2, Di Guo3, Feng Huang4, Qin Xu4, Xiaobo Qu1*


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

2Center for Biomedical Imaging Research, Department of Biomedical Engineering, Tsinghua University, Beijing, 100084, China.

3School of Computer and Information Engineering, Fujian Provincial University Key Laboratory of Internet of Things Application Technology, Xiamen University of Technology, Xiamen 361024, China.

4Neusoft Medical System, Shanghai 200241, China.

Contact: quxiaobo<|at|>


Citations:  Yiman Huang, Xinlin Zhang, Hua Guo, Huijun Chen, Di Guo, Feng Huang, Qin Xu, Xiaobo Qu*, Phase-constrained reconstruction of high-resolution multi-shot diffusion weighted magnetic resonance image, Journal of Magnetic Resonance, DOI: 10.1016/j.jmr.2020.106690, 2020.

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Diffusion weighted imaging (DWI) is a unique examining method in tumor diagnosis, acute stroke evaluation. Single-shot echo planar imaging is currently conventional method for DWI. However, single-shot DWI suffers from image distortion, blurring and low spatial resolution. Although multi-shot DWI improves image resolution, it brings phase variations among different shots at the same time. We introduce a smooth phase constraint of each shot image into multi-shot navigator-free DWI reconstruction by imposing the low-rankness of Hankel matrix constructed from the k-space data. Furthermore, we exploit the partial sum minimization of singular values to constrain the low-rankness of Hankel matrix. Results on brain imaging data show that the proposed method outperforms the state-of-the-art methods in terms of artifacts removal and our method potentially has the ability to reconstruct high number of shot of DWI.

KEYWORDS: Diffusion weighted imaging, Hankel matrix, image reconstruction, low-rankness, magnetic resonance imaging.


1.     Background

Diffusion weighted magnetic resonance imaging (MRI) is a unique examining method noninvasively detecting the Brownian motion of water molecules in the tissues in biomedical imaging. It is widely used in tumor diagnosis, acute stroke evaluation and neuroscience research. As a conventional method of diffusion weighted imaging (DWI) acquisition, single-shot echo-planar imaging (EPI) has the advantages of motion immunity and short acquisition time, but suffers from image distortion, blurring and low spatial resolution. some methods were proposed to overcome the distortion in DWI, such as spatiotemporal encoding and multi-shot EPI.

The multi-shot interleaved EPI fully acquires the k-space data by sampling different segment in each shot, as shown in Figure 1. Multi-shot EPI provides higher spatial resolution than single-shot EPI. However, multi-shot is sensitive to physiological motions, which will induce phase variations from shot to shot. Directly interleaving multi-shot data together into fully sampled k-space will lead to severe artifacts in image.

Figure 1. A schematic diagram of 3-shot interleaved DWI. Note: The solid lines represent the collecting lines in the k-space (Fourier space) of images, and the dotted lines denote the lines where signals are not sampled.


2.     Phase-constrained Low Rank Hankel Matrix reconstruction (PLRHM)

We exploit the partial sum of singular values to constrain low-rankness, the proposed rank minimization model is:

where  denotes concatenated matrices of k-space data of shots, ,  is an operator that converts  into so called  matrix in LORAKS  the Fourier transform operator,  the inverse Fourier transform operator,  the  channel coil sensitivity map,  an operator that under-samples k-space data and zero-fills the non-sampled data points,  the  channel sampled k-space data, where  is the rank of ,  the matrix size of ,  a regularization parameter that balances the data consistency and low-rankness constraint.

3.     Main results

The proposed method was compared with two state-of-the-art navigator-free DWI image reconstruction methods, including the POCS-ICE and MUSSELS. Figure 2 show two slices reconstructions of 8-shot head DWI. Figure 2(d) exhibit the references reconstructed by IRIS. Slight artifacts still remain in the reconstructions of POCS-ICE (Figure 2(a)), as marked by red arrows. POCS-MUSSELS reconstructions (Figure 2(b)) show no obvious artifacts but they look dark in the center of images, as marked by yellow arrows. While our results (Figure 2(c)) can effectively reconstruct the image with minimal artifacts.

Figure 2. Reconstructions of slice 9 of 8-shot in vivo head DWI using different reconstruction methods. (a) POCS-ICE, (b) POCS-MUSSELS, (c) the proposed method, (d) reference reconstructed by IRIS. The residual artifact is marked by the red arrow and the dark region is marked by yellow arrow.


Figure 3 shows the reconstructions of 3 slices of 12-shot head DWI. Directly inverse Fourier transformation induces severe aliasing artifacts (Figure 3(a)). POCS-ICE fails to remove the severe aliasing artifacts (Figure 3(b)). POCS-MUSSELS removes the artifacts to some extent but slight artifacts still remain in the image (Figure 3(c)). While the proposed method can effectively reconstruct the image with minimal artifacts and shaper edges than POCS-MUSSELS, as shown in Figure 3(d). In this case, the shot number is up to 12, which is an aggressive high shot number for reconstruction. POCS-ICE and POCS-MUSSELS have difficulty to recover the artifact-free image, while the proposed method has the potential to handle the case with high number of shots.

Figure 3. Reconstructions of 3 slices of 12-shot in vivo head DWI using different reconstruction methods. (a) direct reconstruction without correction, (b) POCS-ICE, (c) POCS-MUSSELS, (d) the proposed method.




This work was supported in part by National Key R&D Program of China (2017YFC0108700), National Natural Science Foundation of China (61971361, 61871341, 61811530021 and 61672335), Natural Science Foundation of Fujian Province of China (2018J06018), Fundamental Research Funds for the Central Universities (20720180056), Science and Technology Program of Xiamen (3502Z20183053), and China Scholarship Council.

The authors would like to thank Dr. Guobin Li in United Imaging Company for providing the 12-shot head DWI data in this paper.



[1]      D. Le Bihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis, and M. Laval-Jeantet, "MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders," Radiology, vol. 161, no. 2, pp. 401-407, 1986.

[2]      D. Le Bihan, J. F. Mangin, C. Poupon, C. A. Clark, S. Pappata, N. Molko, and H. Chabriat, "Diffusion tensor imaging: concepts and applications," Journal of Magnetic Resonance Imaging, vol. 13, no. 4, pp. 534-546, 2001.

[3]      S. Mori and J. Zhang, "Principles of diffusion tensor imaging and its applications to basic neuroscience research," Neuron, vol. 51, no. 5, pp. 527-539, 2006.

[4]      A. G. Sorensen, F. S. Buonanno, R. G. Gonzalez, L. H. Schwamm, M. H. Lev, F. R. Huang-Hellinger, T. G. Reese, R. M. Weisskoff, T. L. Davis, and N. Suwanwela, "Hyperacute stroke: evaluation with combined multisection diffusion-weighted and hemodynamically weighted echo-planar MR imaging," Radiology, vol. 199, no. 2, pp. 391-401, 1996.

[5]      M. D. Budde and N. P. Skinner, "Diffusion MRI in acute nervous system injury," Journal of Magnetic Resonance, vol. 292, pp. 137-148, 2018.

[6]      F. Farzaneh, S. J. Riederer, and N. J. Pelc, "Analysis of T2 limitations and off-resonance effects on spatial resolution and artifacts in echo-planar imaging," Magnetic Resonance in Medicine, vol. 14, no. 1, pp. 123-139, 1990.

[7]      E. Solomon, N. Shemesh, and L. Frydman, "Diffusion weighted MRI by spatiotemporal encoding: analytical description and in vivo validations," Journal of Magnetic Resonance, vol. 232, pp. 76-86, 2013.

[8]      H. K. Jeong, J. C. Gore, and A. W. Anderson, "High-resolution human diffusion tensor imaging using 2-D navigated multishot SENSE EPI at 7 T," Magnetic Resonance in Medicine, vol. 69, no. 3, pp. 793-802, 2013.

[9]      X. Ma, Z. Zhang, E. Dai, and H. Guo, "Improved multi-shot diffusion imaging using GRAPPA with a compact kernel," Neuroimage, vol. 138, pp. 88-99, 2016.

[10]    N. K. Chen, A. Guidon, H. C. Chang, and A. W. Song, "A robust multi-shot scan strategy for high-resolution diffusion weighted MRI enabled by multiplexed sensitivity-encoding (MUSE)," Neuroimage, vol. 72, no. 2, pp. 41-47, 2013.

[11]    H. Guo, X. Ma, Z. Zhang, B. Zhang, C. Yuan, and F. Huang, "POCS‐enhanced inherent correction of motion‐induced phase errors (POCS‐ICE) for high‐resolution multishot diffusion MRI," Magnetic Resonance in Medicine, vol. 75, no. 1, pp. 169-180, 2016.

[12]    M. Mani, M. Jacob, D. Kelley, and V. Magnotta, "Multi‐shot sensitivity‐encoded diffusion data recovery using structured low‐rank matrix completion (MUSSELS)," Magnetic Resonance in Medicine, vol. 78, no. 2, pp. 494-507, 2017.

[13]    Y. Hu, E. G. Levine, Q. Tian, C. J. Moran, X. Wang, V. Taviani, S. S. Vasanawala, J. A. McNab, B. A. Daniel, and B. L. Hargreaves, "Motion‐robust reconstruction of multishot diffusion‐weighted images without phase estimation through locally low‐rank regularization," Magnetic Resonance in Medicine, vol. 81, no. 2, pp. 1181-1190, 2019.

[14]    L. Guo, F. Huang, Z. Xu, Y. Mei, W. Fang, X. Ma, E. Dai, H. Guo, Q. Feng, and W. Chen, "eIRIS: Eigen-analysis approach for improved spine multi-shot diffusion MRI," Magnetic Resonance Imaging, vol. 50, pp. 134-140, 2018.

[15]    W. Liu, X. Zhao, Y. Ma, X. Tang, and J.-H. Gao, "DWI using navigated interleaved multishot EPI with realigned GRAPPA reconstruction," Magnetic Resonance in Medicine, vol. 75, no. 1, pp. 280-286, 2016.

[16]    E. Dai, Z. Zhang, X. Ma, Z. Dong, X. Li, Y. Xiong, C. Yuan, and H. Guo, "The effects of navigator distortion and noise level on interleaved EPI DWI reconstruction: a comparison between image‐and k‐space‐based method," Magnetic Resonance in Medicine, vol. 80, no. 5, pp. 2024-2032, 2018.

[17]    K. P. Pruessmann, M. Weiger, M. B. Scheidegger, and P. Boesiger, "SENSE: sensitivity encoding for fast MRI," Magnetic Resonance in Medicine, vol. 42, no. 5, pp. 952-962, 1999.

[18]    X. Zhang, D. Guo, Y. Huang, Y. Chen, L. Wang, F. Huang, and X. Qu, "Image reconstruction with low-rankness and self-consistency of k-space data in parallel MRI," arXiv preprint arXiv:1909.02846, 2019.

[19]    J. P. Haldar, "Low-rank modeling of local k-space neighborhoods (LORAKS) for constrained MRI," IEEE Transactions on Medical Imaging, vol. 33, no. 3, pp. 668-681, 2013.

[20]    K. H. Jin, D. Lee, and J. C. Ye, "A general framework for compressed sensing and parallel MRI using annihilating filter based low-rank Hankel matrix," IEEE Transactions on Computational Imaging, vol. 2, no. 4, pp. 480-495, 2016.

[21]    G. Ongie and M. Jacob, "Off-the-grid recovery of piecewise constant images from few Fourier samples," SIAM Journal on Imaging Sciences, vol. 9, no. 3, pp. 1004-1041, 2016.

[22]    F. Lam, C. Ma, B. Clifford, C. L. Johnson, and Z. P. Liang, "High‐resolution 1H‐MRSI of the brain using SPICE: data acquisition and image reconstruction," Magnetic Resonance in Medicine, vol. 76, no. 4, pp. 1059-1070, 2016.

[23]    X. Qu, M. Mayzel, J. F. Cai, Z. Chen, and V. Orekhov, "Accelerated NMR spectroscopy with low‐rank reconstruction," Angewandte Chemie International Edition, vol. 54, no. 3, pp. 852-854, 2015.

[24]    X. Qu, Y. Huang, H. Lu, T. Qiu, D. Guo, T. Agback, V. Orekhov, and Z. Chen, "Accelerated nuclear magnetic resonance spectroscopy with deep learning," Angewandte Chemie International Edition, DOI: 10.1002/anie.201908162, 2019.

[25]    H. Lu, X. Zhang, T. Qiu, J. Yang, J. Ying, D. Guo, Z. Chen, and X. Qu, "Low rank enhanced matrix recovery of hybrid time and frequency data in fast magnetic resonance spectroscopy," IEEE Transactions on Biomedical Engineering, vol. 65, no. 4, pp. 809-820, 2017.

[26]    J. Ying, H. Lu, Q. Wei, J.-F. Cai, D. Guo, J. Wu, Z. Chen, and X. Qu, "Hankel matrix nuclear norm regularized tensor completion for N-dimensional exponential signals," IEEE Transactions on Signal Processing, vol. 65, no. 14, pp. 3702-3717, 2017.

[27]    J. Ying, J.-F. Cai, D. Guo, G. Tang, Z. Chen, and X. Qu, "Vandermonde factorization of Hankel matrix for complex exponential signal recovery—Application in fast NMR spectroscopy," IEEE Transactions on Signal Processing, vol. 66, no. 21, pp. 5520-5533, 2018.

[28]    G. Ongie and M. Jacob, "Recovery of Piecewise Smooth Images from Few Fourier Samples," In 2015 International Conference on Sampling Theory and Applications (SampTA), pp. 543-547, 2015.

[29]    P. J. Shin, P. E. Z. Larson, M. A. Ohliger, M. Elad, J. M. Pauly, D. B. Vigneron, and M. Lustig, "Calibrationless parallel imaging reconstruction based on structured low-rank matrix completion," Magnetic Resonance in Medicine, vol. 72, no. 4, pp. 959-970, 2014.

[30]    J. P. Haldar and J. Zhuo, "P-LORAKS: Low-rank modeling of local k-space neighborhoods with parallel imaging data," Magnetic Resonance in Medicine, vol. 75, no. 4, pp. 1499-1514.

[31]    T. H. Kim, K. Setsompop, and J. P. Haldar, "LORAKS makes better SENSE: Phase-constrained partial fourier SENSE reconstruction without phase calibration," Magnetic Resonance in Medicine, vol. 77, no. 3, pp. 1021-1035, 2017.

[32]    Z.-P. Liang, "Spatiotemporal imaging with partially separable functions," in 2007 4th IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2007, pp. 988-991: IEEE.

[33]    B. Zhao, J. P. Haldar, A. G. Christodoulou, and Z.-P. Liang, "Image reconstruction from highly undersampled (k, t)-space data with joint partial separability and sparsity constraints," IEEE Transactions on Medical Imaging, vol. 31, no. 9, pp. 1809-1820, 2012.

[34]    S. G. Lingala, Y. Hu, E. DiBella, and M. Jacob, "Accelerated dynamic MRI exploiting sparsity and low-rank structure: kt SLR," IEEE Transactions on Medical Imaging, vol. 30, no. 5, pp. 1042-1054, 2011.

[35]    R. A. Lobos, T. H. Kim, W. S. Hoge, and J. P. Haldar, "Navigator-free EPI ghost correction with structured low-rank matrix models: New theory and methods," IEEE Transactions on Medical Imaging, vol. 37, no. 11, pp. 2390-2402, 2018.

[36]    Z. Hu, X. Ma, T.-K. Truong, A. W. Song, and H. Guo, "Phase-updated regularized SENSE for navigator-free multishot diffusion imaging," Magnetic Resonance in Medicine, vol. 78, no. 1, pp. 172-181, 2017.

[37]    B. Recht, M. Fazel, and P. A. Parrilo, "Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization," SIAM Review, vol. 52, no. 3, pp. 471-501, 2010.

[38]    Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, "Fast and accurate matrix completion via truncated nuclear norm regularization," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 9, pp. 2117-2130, 2012.

[39]    T.-H. Oh, H. Kim, Y.-W. Tai, J.-C. Bazin, and I. So Kweon, "Partial sum minimization of singular values in RPCA for low-level vision," in Proceedings of the IEEE International Conference on Computer Vision, 2013, pp. 145-152.

[40]    S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, "Distributed optimization and statistical learning via the alternating direction method of multipliers," Foundations and Trends® in Machine learning, vol. 3, no. 1, pp. 1-122, 2011.

[41]    R. Chandra, S. Eisenstat, and M. Schultz, "Conjugate gradient methods for partial differential equations," Yale University New Haven, CT, 1978.

[42]    T.-H. Oh, Y. Matsushita, Y.-W. Tai, and I. So Kweon, "Fast randomized singular value thresholding for nuclear norm minimization," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2015, pp. 4484-4493.

[43]    G. H. Golub and C. F. Van Loan, "Matrix computations," The Johns Hopkins University Press, Baltimore, USA, 1989.

[44]    M. Uecker, P. Lai, M. J. Murphy, P. Virtue, M. Elad, J. M. Pauly, S. S. Vasanawala, and M. Lustig, "ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: Where SENSE meets GRAPPA," Magnetic Resonance in Medicine, vol. 71, no. 3, pp. 990-1001, 2014.

[45]    B. Bilgic, I. Chatnuntawech, M. K. Manhard, Q. Tian, C. Liao, S. S. Iyer, S. F. Cauley, S. Y. Huang, J. R. Polimeni, and L. L. Wald, "Highly accelerated multishot echo planar imaging through synergistic machine learning and joint reconstruction," Magnetic Resonance in Medicine, vol. 82, pp. 1343-1358, 2019.