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Journal of Computational Mathematics ›› 2022, Vol. 40 ›› Issue (6): 913-935.DOI: 10.4208/jcm.2103-m2019-0256

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IMPROVED HARMONIC INCOMPATIBILITY REMOVAL FOR SUSCEPTIBILITY MAPPING VIA REDUCTION OF BASIS MISMATCH

Chenglong Bao1, Jianfeng Cai2, Jae Kyu Choi3, Bin Dong4, Ke Wei5   

  1. 1. Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China;
    2. Department of Mathematics, The Hong Kong University of Science and Technology, Hongkong, China;
    3. School of Mathematical Sciences, Tongji University, Shanghai 200092, China;
    4. Beijing International Center for Mathematical Research and Laboratory for Biomedical Image Analysis Beijing Institute of Big Data Research, Peking University, Beijing 100871, China;
    5. School of Data Science, Fudan University, Shanghai 200433, China
  • Received:2019-11-08 Revised:2020-04-18 Online:2022-11-15 Published:2022-11-18
  • Contact: Jae Kyu Choi, Email: jaycjk@tongji.edu.cn
  • Supported by:
    The research of the first author is supported in part by the NSFC Youth Program 11901338. The research of the second author is supported by the Hong Kong Research Grant Council (HKRGC) GRF 16306317 and 16309219. The research of the third author is supported by the NSFC Youth Program 11901436 and the Fundamental Research Program of Science and Technology Commission of Shanghai Municipality (20JC1413500). The research of the fourth author is supported by the NSFC grant 11831002. The research of the fifth author is supported by the National Natural Science Foundation of China Youth Program grant 11801088 and the Shanghai Sailing Program (18YF1401600).
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Chenglong Bao, Jianfeng Cai, Jae Kyu Choi, Bin Dong, Ke Wei. IMPROVED HARMONIC INCOMPATIBILITY REMOVAL FOR SUSCEPTIBILITY MAPPING VIA REDUCTION OF BASIS MISMATCH[J]. Journal of Computational Mathematics, 2022, 40(6): 913-935.

In quantitative susceptibility mapping (QSM), the background field removal is an essential data acquisition step because it has a significant effect on the restoration quality by generating a harmonic incompatibility in the measured local field data. Even though the sparsity based first generation harmonic incompatibility removal (1GHIRE) model has achieved the performance gain over the traditional approaches, the 1GHIRE model has to be further improved as there is a basis mismatch underlying in numerically solving Poisson’s equation for the background removal. In this paper, we propose the second generation harmonic incompatibility removal (2GHIRE) model to reduce a basis mismatch, inspired by the balanced approach in the tight frame based image restoration. Experimental results shows the superiority of the proposed 2GHIRE model both in the restoration qualities and the computational efficiency.
Key words: Quantitative susceptibility mapping, Magnetic resonance imaging, Deconvolution, Partial differential equation, Harmonic incompatibility removal, (Tight) wavelet frames, Sparse approximation

CLC Number: 

[1] L.de Rochefort, T. Liu, B. Kressler, J. Liu, P. Spincemaille, V. Lebon, J. Wu and Y. Wang, Quantitative susceptibility map reconstruction from MR phase data using Bayesian regularization: validation and application to brain imaging, Magn. Reson. Med., 63:1(2010), 194-206.
[2] J.K. Seo, E.J. Woo, U. Katscher and Y. Wang, Electro-Magnetic Tissue Properties MRI, Imperial College Press, London, 1st edition, 2014.
[3] Y. Kee, Z. Liu, L. Zhou, A. Dimov, J. Cho, L.de Rochefort, J.K. Seo and Y. Wang, Quantitative susceptibility mapping (QSM) algorithms: mathematical rationale and computational implementations, IEEE Trans. Biomed. Eng., 64:11(2017), 2531-2545.
[4] W. Chen, S.A. Gauthier, A. Gupta, J. Comunale, T. Liu, S. Wang, M. Pei, D. Pitt and Y. Wang, Quantitative susceptibility mapping of multiple sclerosis lesions at various ages, Radiology, 271:1(2014), 183-192, PMID: 24475808.
[5] J. Acosta-Cabronero, G.B. Williams, A. Cardenas-Blanco, R.J. Arnold, V. Lupson and P.J. Nestor, In vivo quantitative susceptibility mapping (QSM) in Alzheimer’s disease, PloS one, 8:11(2013), e81093.
[6] J.M.G.van Bergen, J. Hua, P.G. Unschuld, I.A.L. Lim, C.K. Jones, R.L. Margolis, C.A. Ross, P.C.M.van Zijl and X. Li, Quantitative susceptibility mapping suggests altered brain iron in premanifest Huntington disease, Amer. J. Neuroradiol, 37:5(2016), 789-796.
[7] E.M. Haacke, J. Tang, J. Neelavalli and Y.C.N. Cheng, Susceptibility mapping as a means to visualize veins and quantify oxygen saturation, J. Magn. Reson. Imag., 32:3(2010), 663-676.
[8] J. Klohs, A. Deistung, F. Schweser, J. Grandjean, M. Dominietto, C. Waschkies, R.M. Nitsch, I. Knuesel, J.R. Reichenbach and M. Rudin, Detection of cerebral microbleeds with quantitative susceptibility mapping in the arcabeta mouse model of cerebral amyloidosis, J. Cerebr. Blood F. Met., 31:12(2011), 2282-2292, PMID: 21847134.
[9] A.V. Dimov, Z. Liu, P. Spincemaille, M.R. Prince, J. Du and Y. Wang, Bone quantitative susceptibility mapping using a chemical species? Specific r2* signal model with ultrashort and conventional echo data, Magn. Reson. Med., 79:1(2018), 121-128.
[10] T. Liu, P. Spincemaille, L.de Rochefort, R. Wong, M. Prince and Y. Wang, Unambiguous identification of superparamagnetic iron oxide particles through quantitative susceptibility mapping of the nonlinear response to magnetic fields, Magn. Reson. Imaging, 28:9(2010), 1383-1389.
[11] C. Milovic, B. Bilgic, B. Zhao, C. Langkammer, C. Tejos and J. Acosta-Cabronero, Weak-harmonic regularization for quantitative susceptibility mapping, Magn. Reson. Med., 81:2(2019), 1399-1411.
[12] Y. Wang and T. Liu, Quantitative susceptibility mapping (QSM): decoding MRI data for a tissue magnetic biomarker, Magn. Reson. Med., 73:1(2015), 82-101.
[13] T. Liu, I. Khalidov, L.de Rochefort, P. Spincemaille, J. Liu, A.J. Tsiouris and Y. Wang, A novel background field removal method for MRI using projection onto dipole fields (PDF), NMR Biomed, 24:9(2011), 1129-1136.
[14] T. Liu, J. Liu, L.de Rochefort, P. Spincemaille, I. Khalidov, J.R. Ledoux and Y. Wang, Morphology enabled dipole inversion (MEDI) from a single-angle acquisition: comparison with COSMOS in human brain imaging, Magn. Reson. Med., 66:3(2011), 777-783.
[15] T. Liu, P. Spincemaille, L.de Rochefort, B. Kressler and Y. Wang, Calculation of susceptibility through multiple orientation sampling (COSMOS): a method for conditioning the inverse problem from measured magnetic field map to susceptibility source image in MRI, Magn. Reson. Med., 61:1(2009), 196-204.
[16] E.M. Stein and R. Shakarchi, Functional Analysis. Introduction to Further Topics in Analysis, volume 4 of Princeton Lect. Anal., Princeton University Press, Princeton, NJ, 2011.
[17] E.M. Haacke, S. Liu, S. Buch, W. Zheng, D. Wu and Y. Ye, Quantitative susceptibility mapping: current status and future directions, Magn. Reson. Imaging, 33:1(2015), 1-25.
[18] J.K. Choi, H.S. Park, S. Wang, Y. Wang and J.K. Seo, Inverse problem in quantitative susceptibility mapping, SIAM J. Imaging Sci., 7:3(2014), 1669-1689.
[19] T. Liu, C. Wisnieff, M. Lou, W. Chen, P. Spincemaille and Y. Wang, Nonlinear formulation of the magnetic field to source relationship for robust quantitative susceptibility mapping, Magn. Reson. Med., 69:2(2013), 467-476.
[20] S. Wang, T. Liu, W. Chen, P. Spincemaille, C. Wisnieff, A.J. Tsiouris, W. Zhu, C. Pan, L. Zhao and Y. Wang, Noise effects in various quantitative susceptibility mapping methods, IEEE Trans. Biomed. Eng., 60:12(2013), 3441-3448.
[21] C. Bao, J. Choi and B. Dong, Whole brain susceptibility mapping using harmonic incompatibility removal, SIAM J. Imaging Sci., 12:1(2019), 492-520.
[22] F. Trèves, Basic Linear Partial Differential Equations, Dover Publications, Inc., Mineola, NY, 2006, Reprint of the 1975 original.
[23] H. Sun and A.H. Wilman, Background field removal using spherical mean value filtering and tikhonov regularization, Magn. Reson. Med., 71:3(2014), 1151-1157.
[24] L.C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010.
[25] H. Ji and K. Wang, Robust image deblurring with an inaccurate blur kernel, IEEE Trans. Image Process., 21:4(2012), 1624-1634.
[26] J. Li, C. Miao, Z. Shen, G. Wang and H. Yu, Robust frame based x-ray CT reconstruction, J. Comput. Math., 34:6(2016), 683-704.
[27] J.F. Cai, R.H. Chan, L. Shen and Z. Shen, Convergence analysis of tight framelet approach for missing data recovery, Adv. Comput. Math., 31:1-3(2009), 87-113.
[28] J.F. Cai, R.H. Chan and Z. Shen, Simultaneous cartoon and texture inpainting, Inverse Probl. Imaging, 4:3(2010), 379-395.
[29] R.H. Chan, T.F. Chan, L. Shen and Z. Shen, Wavelet algorithms for high-resolution image reconstruction, SIAM J. Sci. Comput., 24:4(2003), 1408-1432.
[30] J.F. Cai, T. Wang and K. Wei, Fast and provable algorithms for spectrally sparse signal reconstruction via low-rank Hankel matrix completion, Appl. Comput. Harmon. Anal., 46:1(2019), 94-121.
[31] G. Ongie and M. Jacob, Off-the-grid recovery of piecewise constant images from few Fourier samples, SIAM J. Imaging Sci., 9:3(2016), 1004-1041.
[32] C. Milovic, B. Bilgic, B. Zhao, J. Acosta-Cabronero and C. Tejos, Fast nonlinear susceptibility inversion with variational regularization, Magn. Reson. Med., 80:2(2018), 814-821.
[33] J.F. Cai, B. Dong, S. Osher and Z. Shen, Image restoration: total variation, wavelet frames, and beyond, J. Amer. Math. Soc., 25:4(2012), 1033-1089.
[34] J.F. Cai, B. Dong and Z. Shen, Image restoration: a wavelet frame based model for piecewise smooth functions and beyond, Appl. Comput. Harmon. Anal., 41:1(2016), 94-138.
[35] L.I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60:1-4(1992), 259-268, Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991).
[36] K. Bredies, K. Kunisch and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3:3(2010), 492-526.
[37] Y. Kee, J. Cho, K. Deh, Z. Liu, P. Spincemaille and Y. Wang, Coherence enhancement in quantitative susceptibility mapping by means of anisotropic weighting in morphology enabled dipole inversion, Magn. Reson. Med., 79:2(2018), 1172-1180.
[38] Z. Liu, P. Spincemaille, Y. Yao, Y. Zhang and Y. Wang, MEDI+0: morphology enabled dipole inversion with automatic uniform cerebrospinal fluid zero reference for quantitative susceptibility mapping, Magn. Reson. Med., 79:5(2017), 2795-2803.
[39] J. Eckstein and D.P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55:3, Ser. A (1992), 293-318.
[40] J.F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Model. Simul., 8:2(2009/10), 337-369.
[41] T. Goldstein and S.J. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2:2(2009), 323-343.
[42] Y.R. Fan, T.Z. Huang, T.H. Ma and X.L. Zhao, Cartoon-texture image decomposition via nonconvex low-rank texture regularization, J. Franklin Inst., 354:7(2017), 3170-3187.
[43] B. He, M. Tao and X. Yuan, Alternating direction method with Gaussian back substitution for separable convex programming, SIAM J. Optim., 22:2(2012), 313-340.
[44] C. Wisnieff, T. Liu, P. Spincemaille, S. Wang, D. Zhou and Y. Wang, Magnetic susceptibility anisotropy: cylindrical symmetry from macroscopically ordered anisotropic molecules and accuracy of MRI measurements using few orientations, NeuroImage, 70(2013), 363-376.
[45] C. Langkammer, F. Schweser, K. Shmueli, C. Kames, X. Li, L. Guo, C. Milovic, J. Kim, H. Wei, K. Bredies, S. Buch, Y. Guo, Z. Liu, J. Meineke, A. Rauscher, J.P. Marques and B. Bilgic, Quantitative susceptibility mapping: Report from the 2016 reconstruction challenge, Magn. Reson. Med., 79:3(2018), 1661-1673.
[46] K. Shmueli, J.A.de Zwart, P.van Gelderen, T. Li, S.J. Dodd and J.H. Duyn, Magnetic susceptibility mapping of brain tissue in vivo using MRI phase data, Magn. Reson. Med., 62:6(2009), 1510-1522.
[47] B. Kressler, L.de Rochefort, T. Liu, P. Spincemaille, J. Quan and Y. Wang, Nonlinear regularization for per voxel estimation of magnetic susceptibility distributions from MRI field maps, IEEE Trans. Med. Imag., 29:2(2010), 273-281.
[48] W. Li, A.V. Avram, B. Wu, X. Xiao and C. Liu, Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping, NMR in Biomedicine, 27:2(2014), 219-227, NBM-13-0182.R2.
[49] Z. Wang, A.C. Bovik, H.R. Sheikh and E.P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13:4(2004), 600-612.
[50] L.de Rochefort, R. Brown, M.R. Prince and Y. Wang, Quantitative MR susceptibility mapping using piece-wise constant regularized inversion of the magnetic field, Magn. Reson. Med., 60:4(2008), 1003-1009.
[51] D.C. Ghiglia and M.D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software, Wiley-Interscience publication, Wiley, 1998.
[52] D. Zhou, T. Liu, P. Spincemaille and Y. Wang, Background field removal by solving the Laplacian boundary value problem, NMR in Biomedicine, 27:3(2014), 312-319, NBM-13-0115.R3.
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