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.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Bovik, A. C. (2000). Handbook of image and video processing (1st ed.). San Diego: Academic Press.
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.
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.
Candes, E. J., & Wakin, M. B. (2008). An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2), 21–30.
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.
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.
Donoho, D. L. (2006). Compressed sensing. IEEE Transactions on Information Theory, 52(4), 1289–1306.
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.
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.
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.
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.
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.
Luenberger, D. G., & Ye, Y. (2008). Linear and nonlinear programming (3rd ed.). New York: Springer Science + Business Media, LCC.
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.
Lustig, M., Donoho, D. L., Santos, J. M., & Pauly, J. M. (2008). Compressed sensing MRI. IEEE Signal Processing Magazine, 25(2), 72–82.
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.
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.
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.
Vogel, C. R. (2002). Computational methods for inverse problems, frontiers in applied mathematics (Vol. 23). Philadelphia: Society for industrial mathematics.
Webb, A. G. (2002). Introduction to biomedical imaging. New York: Wiley-IEEE press.
Wolfe, P. (1969). Convergence conditions for ascent methods. Siam Review, 11(2), 226–235.
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.
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.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Work by the second author was supported by CNPq grant 301064/2009-1.
Rights and permissions
About this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11045-016-0383-6