Skip to main content
SpringerLink
Log in

Improving compressive sensing in MRI with separate magnitude and phase priors

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

Compressive sampling/compressed sensing (CS) has shown that it is possible to perfectly reconstruct non-bandlimited signals sampled well below the Nyquist rate. Magnetic Resonance Imaging (MRI) is one of the applications that has benefited from this theory. Sparsifying operators that are effective for real-valued images, such as finite difference and wavelet transform, also work well for complex-valued MRI when phase variations are small. As phase variations increase, even if the phase is smooth, the sparsifying ability of these operators for complex-valued images is reduced. If the phase is known, it is possible to remove it from the complex-valued image before applying the sparsifying operator. Another alternative is to use the sparsifying operator on the magnitude of the image, and use a different operator for the phase, i.e., one related to a smoothness enforcing prior. The proposed method separates the priors for the magnitude and for the phase, in order to improve the applicability of CS to MRI. An improved version of previous approaches, by ourselves and other authors, is proposed to reduce computational cost and enhance the quality of the reconstructed complex-valued MR images with smooth phase. The proposed method utilizes \(\ell _1\) penalty for the transformed magnitude, and a modified \(\ell _2\) penalty for phase, together with a non-linear conjugated gradient optimization. Also, this paper provides an extensive set of experiments to understand the behavior of previous methods and the new approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Notes

  1. The implementation in Zhao et al. (2012) is based on iterative soft thresholding for wavelet transforms, which cannot be used for TV penalty. In order to make the comparison as fair as possible we follow the implementation in (6)

References

  • Bovik, A. C. (2000). Handbook of image and video processing (1st ed.). San Diego: Academic Press.

    MATH  Google Scholar 

  • Candes, E. J., & Tao, T. (2005). Decoding by linear programming. IEEE Transactions on Information Theory, 51(12), 4203–4215. doi:10.1109/TIT.2005.858979.

    Article  MathSciNet  MATH  Google Scholar 

  • Candes, E. J., & Tao, T. (2006). Near optimal signal recovery from random projections: Universal encoding strategies? IEEE Transactions on Information Theory, 52(12), 5406–5425.

    Article  MathSciNet  MATH  Google Scholar 

  • Candes, E. J., & Wakin, M. B. (2008). An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2), 21–30.

    Article  Google Scholar 

  • Cetin, M., & Karl, W. C. (2001). Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization. IEEE Transactions on Image Processing, 10(4), 623–631. doi:10.1109/83.913596.

    Article  MATH  Google Scholar 

  • De Poorter, J., De Wagter, C., De Deene, Y., Thomsen, C., Ståhlberg, F., & Achten, E. (1995). Noninvasive MRI thermometry with the proton resonance frequency (PRF) method. In vivo results in human muscle. Magnetic Resonance in Medicine, 33(1), 74–81.

    Article  Google Scholar 

  • Donoho, D. L. (2006). Compressed sensing. IEEE Transactions on Information Theory, 52(4), 1289–1306.

    Article  MathSciNet  MATH  Google Scholar 

  • Duarte, M. F., Danvenport, M. A., Takhar, D., Laska, J. N., Sung, T., Kelly, K. F., et al. (2008). Single-pixel imaging via compressive sampling. IEEE Signal Processing Magazine, 31(2), 83–91.

    Article  Google Scholar 

  • Farsiu, S., Robinson, M. D., Elad, M., & Milanfar, P. (2004). Fast and robust multiframe super resolution. IEEE Transactions on Image Processing, 13(10), 1327–1344.

    Article  Google Scholar 

  • Fessler, J., & Noll, D. (2004). Iterative image reconstruction in MRI with separate magnitude and phase regularization. In IEEE International symposium on biomedical imaging, vol. 1 (pp. 209–212). IEEE.

  • Fessler, J. A. (2010). Model-based image reconstruction for MRI. IEEE Signal Processing Magazine, 27(4), 81–89.

    Article  Google Scholar 

  • Funai, A., Fessler, J. A., Yeo, D., Olafsson, V. T., & Noll, D. C. (2008). Regularized field map estimation in MRI. IEEE Transactions on Medical Imaging, 27(10), 1484–1494.

    Article  Google Scholar 

  • IEEE Signal Processing Magazine: Special issue on sensing, sampling, and compression, vol. 25 (2008). http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=4472102.

  • Jonsson, E., Huang, S. C., & Chan, T. (1998). Total variation regularization in positron emission tomography. Technical report, mathematics, UCLA 98(48).

  • Liang, Z. P., & Lauterbur, P. C. (2000). Principles of magnetic ressonance imaging: A signal processing perspective. New York: IEEE Press.

    Google Scholar 

  • Luenberger, D. G., & Ye, Y. (2008). Linear and nonlinear programming (3rd ed.). New York: Springer Science + Business Media, LCC.

    MATH  Google Scholar 

  • Lustig, M., Donoho, D. L., & Pauly, J. M. (2007). Sparse MRI: The application of compressed sensing for rapid MR imaging. Magnetic Resonance in Medicine, 58(6), 1182–1195. doi:10.1002/mrm.21391.

    Article  Google Scholar 

  • Lustig, M., Donoho, D. L., Santos, J. M., & Pauly, J. M. (2008). Compressed sensing MRI. IEEE Signal Processing Magazine, 25(2), 72–82.

    Article  Google Scholar 

  • Nielsen, J. F., & Nayak, K. S. (2009). Referenceless phase velocity mapping using balanced SSFP. Magnetic Resonance in Medicine, 61(5), 1096–1102. doi:10.1002/mrm.21884.

    Article  Google Scholar 

  • Romberg, J. K., Candes, E. J., & Tao, T. (2006). Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 52(2), 489–509.

    Article  MathSciNet  MATH  Google Scholar 

  • Sutton, B., Noll, D., & Fessler, J. (2001). Simultaneous Estimation of Image and Inhomogeneity Field Map. In Proceedings of the ISMRM minimum data acquisition workshop, vol. 2 (pp. 15–18). Citeseer.

  • Trzasko, J., & Manduca, A. (2009). Highly undersampled magnetic resonance image reconstruction via homotopic \(\ell _{0}\)-minimization. IEEE Transactions on Medical Imaging, 28(1), 106–121. doi:10.1109/TMI.2008.927346.

    Article  Google Scholar 

  • Vogel, C. R. (2002). Computational methods for inverse problems, frontiers in applied mathematics (Vol. 23). Philadelphia: Society for industrial mathematics.

    Book  Google Scholar 

  • Webb, A. G. (2002). Introduction to biomedical imaging. New York: Wiley-IEEE press.

    Book  Google Scholar 

  • Wolfe, P. (1969). Convergence conditions for ascent methods. Siam Review, 11(2), 226–235.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, F., Fessler, J. A., Nielsen, J. F., & Noll, D. C. (2011). Separate magnitude and phase regularization via compressed sensing. Proceedings of the International Society for Magnetic Resonance in Medicine, 19, 2841.

    Google Scholar 

  • Zhao, F., Noll, D. C., Nielsen, J. F., & Fessler, J. A. (2012). Separate magnitude and phase regularization via compressed sensing. IEEE Transactions on Medical Imaging, 31(9), 1713–1723. doi:10.1109/TMI.2012.2196707.

    Article  Google Scholar 

  • Zibetti, M. V. W., & De Pierro, A. R. (2009). A new distortion model for strong inhomogeneity problems in Echo-Planar MRI. IEEE Transactions on Medical Imaging, 28(11), 1736–1753. doi:10.1109/TMI.2009.2022622.

    Article  Google Scholar 

  • Zibetti, M. V. W., & De Pierro, A. R. (2010). Separate magnitude and phase regularization in MRI with incomplete data: Preliminary results. In IEEE international symposium on biomedical imaging (pp. 736–739). IEEE. doi:10.1109/ISBI.2010.5490069.

  • Zibetti, M. V. W., & Mayer, J. (2007). A robust and computationally efficient simultaneous super-resolution scheme for image sequences. IEEE Transactions on Circuits and Systems for Video Technology, 17(10), 1288–1300. doi:10.1109/TCSVT.2007.903801.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DMAEC-CPGEI, Federal University of Technology - Paraná (UTFPR), Curitiba, Brazil

    Marcelo V. W. Zibetti

  2. CNPq, Campinas, Brazil

    Alvaro R. De Pierro

Authors
  1. Marcelo V. W. Zibetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alvaro R. De Pierro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcelo V. W. Zibetti.

Additional information

Work by the second author was supported by CNPq grant 301064/2009-1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zibetti, M.V.W., De Pierro, A.R. Improving compressive sensing in MRI with separate magnitude and phase priors. Multidim Syst Sign Process 28, 1109–1131 (2017). https://doi.org/10.1007/s11045-016-0383-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-016-0383-6

Keywords

Search

Navigation