Optimization and Engineering

, Volume 5, Issue 4, pp 485–502 | Cite as

Rapid, Embeddable Design Method for Spiral Magnetic Resonance Image Reconstruction Resampling Kernels

  • Christopher Kumar Anand
  • Tamás Terlaky
  • Bixiang Wang


After formulating the design problem for Resampling Kernels used in Magnetic Resonance Spiral Image Reconstruction, we show that an iterative Gauss-Seidel-type interior-point optimization method is suitable (fast and light-weight) for embedded uses. In contrast to previous practice, we directly optimize a computationally efficient, piecewise-linear kernel rather than an analytic function (Kaiser-Bessel). We also optimize our kernels for worst-case (infinity-norm) signal aliasing, rather than the usual proxy energy function (2-norm) minimization. In numerical simulations of undesirable near-frequency systematic noise the new kernel significantly outperforms a conventional Kaiser-Bessel-based solution.

magnetic resonance imaging spiral imaging regridding non-uniform Fourier transform interior point method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. A. Aldroubi and K. Grüchenig, “Nonuniform sampling and reconstruction in shift-invariant spaces,” SIAM Review vol. 4, pp. 585–620, 2001.Google Scholar
  2. E. D. Andersen, J. Gondzio, C. Meszaros, and X. Xu, “Implementation of interior-point methods for large scale linear programs,” in Interior Point Methods of Mathematical Programming,T. Terlaky (Ed.), Kluwer, 1996.Google Scholar
  3. M. S. Bazzaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming,Wiley-Interscience, 1993.Google Scholar
  4. P. Boernert, Morning Course in Image Reconstruction, ISMRM 2001, Glasgow.Google Scholar
  5. A. S. Fahmy, B. S. Tawfik, and Y. M. Kadah, “Image reconstruction using shift-variant resampling kernel for magnetic resonance imaging,” in Medical Imaging 2002: Image Processing, M. Sonka and J. M. Fitzpatrick (Eds.), SPIE, 2002, pp. 825–833.Google Scholar
  6. J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Tr. Sig. Proc. vol. 51, pp. 560–574, 2003.Google Scholar
  7. E. M. Haacke, R. W. Brown, M. R. Thompson, and R. Venkatesan, Magnetic Resonance Imaging: Physical Principles and Sequence Design, John Wiley & Sons, 1999.Google Scholar
  8. R. W. Hamming, Digital Filters, Prentice-Hall, New Jersey, 1983.Google Scholar
  9. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE vol. 4, pp. 51–83, 1978.Google Scholar
  10. J. Jackson, C. H. Meyer, and D. G. Nishimura, “Selection of a convolution function for Fourier inversion using gridding,” IEEE Trans. Med. Imaging vol. 10, pp. 473–478, 1991.Google Scholar
  11. Z.-P. Liang and P. C. Lauterbur, Principles of Magnetic Resonance Imaging: A Signal Processing Perspective, Wiley-IEEE, 2000.Google Scholar
  12. F. Marvasti (Ed.), Nonuniform Sampling: Theory and Practice (Information Technology: Transmission, Processing, and Storage), Plenum, 2001.Google Scholar
  13. J. O'Sullivan, “A fast sinc function gridding algorithm for Fourier inversion in computer tomography,” IEEE Trans. Med. Imaging vol. 4, pp. 200–207, 1985.Google Scholar
  14. J. G. Pipe and P. Menon, “Sampling density compensation in MRI: Rationale and an iterative numerical solution,” Magn. Reson. Med. vol. 41, pp. 179–186, 1999.Google Scholar
  15. V. Rasche et al., “Resampling of data between arbitrary grids using convolution interpolation,” IEEE Trans. Medical Imaging vol. 18, pp. 385–392, 1999.Google Scholar
  16. D. Rosenfeld, “An optimal and efficient new gridding algorithm using singular value decomposition,” Magn. Reson. Med. vol. 40, pp. 14–23, 1998.Google Scholar
  17. H. Sedarat and D. G. Nishimura, “On the optimality of the gridding reconstruction algorithm,” IEEE Trans. Medical Imaging vol. 19, pp. 306–317, 2000.Google Scholar
  18. R. J. Vanderbei, “LOQO user's manual—version 3.10. Interior point methods,” Optimization Methods and Software, vols. 11/12, pp. 485–514, 1999.Google Scholar
  19. F. T. A. W. Wajer et al., “Interpolation from arbitrary to Cartesian sample positions: Gridding,” Proc. ProRISC/IEEE Workshop, 2000.Google Scholar
  20. F. T. A. W Wajer et al., “Simple formula for the accuracy of gridding,” Proc. Intern. Soc. Mag. Resonance Med., 2001a.Google Scholar
  21. F. T. A. W. Wajer et al., “Interpolation from arbitrary to Cartesian sample positions: Gridding,” Proc. ProRISC/IEEE Workshop, 2001b.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Christopher Kumar Anand
    • 1
  • Tamás Terlaky
    • 1
  • Bixiang Wang
    • 1
  1. 1.Department of Computer ScienceMcMaster UniversityHamilton, OntarioCanada

Personalised recommendations