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On Reconstruction from Non-uniform Spectral Data

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Abstract

This paper addresses the reconstruction of compactly supported functions from non-uniform samples of their Fourier transform. We briefly investigate the consequences of acquiring non-uniform spectral data. We summarize two often applied reconstruction methods, convolutional gridding and uniform re-sampling, and investigate the reconstruction accuracy as it relates to sampling density. Finally, we provide preliminary results from employing spectral re-projection methods in the reconstruction.

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References

  1. Ahn, C., Kim, J., Cho, Z.: High-speed spiral-scan echo planar NMR imaging. IEEE Trans. Med. Imag. 5(1), 2–7 (1986)

    Article  Google Scholar 

  2. Archibald, R., Chen, K., Gelb, A., Renaut, R.: Improving tissue segmentation of human brain MRI through preprocessing by the Gegenbauer reconstruction method. NeuroImage 20(1), 489–502 (2003)

    Article  Google Scholar 

  3. Archibald, R., Gelb, A.: A method to reduce the Gibbs ringing artifact in MRI scans while keeping tissue boundary integrity. IEEE Trans. Med. Imag. 21, 305–319 (2002)

    Article  Google Scholar 

  4. Atkinson, K.: An Introduction to Numerical Analysis. Wiley, New York (1978)

    MATH  Google Scholar 

  5. Bateman, H., Erdelyi, A.: Higher Transcendental Functions. McGraw-Hill, New York (1953)

    Google Scholar 

  6. Cocosco, C., Kollokian, V., Kwan, R., Evans, A.: Brainweb: online interface to a 3D MRI simulated brain database. NeuroImage 5(4), 425 (1997)

    Google Scholar 

  7. Collins, D., Zijdenbos, A., Kollokian, V., Sled, J., Kabani, N., Holmes, C., Evans, A.: Design and construction of a realistic digital brain phantom. IEEE Trans. Med. Imag. 17(3), 463–468 (1998)

    Article  Google Scholar 

  8. Dym, H., McKean, H.: Fourier Series and Integrals. Academic Press, New York (1972)

    MATH  Google Scholar 

  9. Fessler, J.A., Sutton, B.P.: Nonuniform fast Fourier transforms using min-max interpolation. IEEE Trans. Signal Process. 51(2), 560–574 (2003)

    Article  MathSciNet  Google Scholar 

  10. Fourmont, K.: Non-equispaced fast Fourier transforms with applications to tomography. J. Fourier Anal. Appl. 9(5), 431–450 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gelb, A., Tadmor, E.: Detection of edges in spectral data. Appl. Comput. Harmonic Anal. 7, 101–135 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gelb, A., Tadmor, E.: Detection of edges in spectral data II: nonlinear enhancement. SIAM J. Numer. Anal. 38(4), 1389–1408 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gottlieb, D., Orszag, S.: Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia (1977)

    MATH  Google Scholar 

  14. Gottlieb, D., Shu, C.: On the Gibbs phenomenon and its resolution. Siam Rev. 39(4), 644–668 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hansen, P.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. Society for Industrial Mathematics, Philadelphia (1998)

    Google Scholar 

  16. Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-dependent Problems. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  17. Hoge, R.D., Kwan, R.K., Pike, G.B.: Density compensation functions for spiral MRI. Magn. Reson. Med. 38(1), 117–128 (1997)

    Article  Google Scholar 

  18. Jackson, J.I., Meyer, C.H., Nishimura, D.G., Macovski, A.: Selection of a convolution function for Fourier inversion using gridding. IEEE Trans. Med. Imag. 10(3), 473–478 (1991)

    Article  Google Scholar 

  19. Kwan, R., Evans, A., Pike, G.: MRI simulation-based evaluation of image-processing and classification methods. IEEE Trans. Med. Imag. 18(11), 1085–1097 (1999)

    Article  Google Scholar 

  20. McConnell Brain Imaging Center: BrainWeb: Simulated Brain Database. http://www.bic.mni.mcgill.ca/brainweb/

  21. O’Sullivan, J.D.: Fast sinc function gridding algorithm for Fourier inversion in computer tomography. IEEE Trans. Med. Imag. 4(4), 111 (1985)

    Google Scholar 

  22. Pipe, J.G., Menon, P.: Sampling density compensation in MRI: rationale and an iterative numerical solution. Magn. Reson. Med. 41(1), 179–186 (1999)

    Article  Google Scholar 

  23. Remi, K., Evans, A., Pike, G.: An extensible MRI simulator for post-processing evaluation. In: Proceedings of the 4th International Conference on Visualization in Biomedical Computing, pp. 135–140. Springer, Berlin (1996)

    Google Scholar 

  24. Rosenfeld, D.: An optimal and efficient new gridding algorithm using singular value decomposition. Magn. Reson. Med. 40(1), 14–23 (1998)

    Article  Google Scholar 

  25. Rosenfeld, D.: New approach to gridding using regularization and estimation theory. Magn. Reson. Med. 48(1), 193–202 (2002)

    Article  Google Scholar 

  26. Sammis, I., Strain, J.: A geometric nonuniform fast Fourier transform. J. Comput. Phys. 228(18), 7086–7108 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sedarat, H., Nishimura, D.G.: On the optimality of the gridding reconstruction algorithm. IEEE Trans. Med. Imag. 19(4), 306–317 (2000)

    Article  Google Scholar 

  28. Steidl, G.: A note on fast Fourier transforms for nonequispaced grids. Adv. Comput. Math. 9(3), 337–352 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  29. Tadmor, E.: Filters, mollifiers and the computation of the Gibbs phenomenon. Acta Numer. 16, 305–378 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tadmor, E., Tanner, J.: Adaptive mollifiers for high resolution recovery of piecewise smooth data from its spectral information. Found. Comput. Math. 2(2), 155–189 (2002)

    MATH  MathSciNet  Google Scholar 

  31. Tadmor, E., Tanner, J.: Adaptive filters for piecewise smooth spectral data. IMA J. Numer. Anal. 25(4), 635–647 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Tadmor, E., Zou, J.: Three novel edge detection methods for incomplete and noisy spectral data. J. Fourier Anal. Appl. 14(5), 744–763 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ, 85287-5706, USA

    Adityavikram Viswanathan & Douglas Cochran

  2. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287-1804, USA

    Anne Gelb & Rosemary Renaut

Authors
  1. Adityavikram Viswanathan
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  2. Anne Gelb
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  4. Rosemary Renaut
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Correspondence to Adityavikram Viswanathan.

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In memory of David Gottlieb, beloved teacher, scholar, and friend. His pioneering work in spectral methods and in particular in resolving the Gibbs phenomenon has had a profound influence on this research.

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Viswanathan, A., Gelb, A., Cochran, D. et al. On Reconstruction from Non-uniform Spectral Data. J Sci Comput 45, 487–513 (2010). https://doi.org/10.1007/s10915-010-9364-3

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  • DOI: https://doi.org/10.1007/s10915-010-9364-3

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