Skip to main content
Springer Nature Link
Log in

Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In several applications, data are collected in the frequency (Fourier) domain non-uniformly, either by design or as a consequence of inexact measurements. The two major bottlenecks for image reconstruction from non-uniform Fourier data are (i) there is no obvious way to perform the numerical approximation, as the non-uniform Fourier data is not amenable to fast transform techniques and resampling the data first to uniform spacing is often neither accurate or robust; and (ii) the Gibbs phenomenon is apparent when the underlying function (image) is piecewise smooth, an occurrence in nearly every application. Recent investigations suggest that it may be useful to view the non-uniform Fourier samples as Fourier frame coefficients when designing reconstruction algorithms that attempt to mitigate either of these fundamental problems. The inverse polynomial reconstruction method (IPRM) was developed to resolve the Gibbs phenomenon in the reconstruction of piecewise analytic functions from spectral data, notably Fourier data. This paper demonstrates that the IPRM is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted \(L_2\) space of orthogonal polynomials. Moreover, the IPRM can also be used to remove the Gibbs phenomenon from the Fourier frame approximation when the underlying function is piecewise smooth. The one-dimensional numerical results presented here demonstrate that using the IPRM in this way yields a robust, stable, and accurate approximation from non-uniform Fourier data.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

Notes

  1. We note that iterative procedures using \(l^1\) regularization are also commonly employed, see e.g. [30]. While these methods are often very effective, the purpose of our paper is to develop an analytical framework for a direct solver. Indeed, since the IPRM is a linear construction, it might serve to produce an objective funtion to which a regularization term can be applied.

  2. The original theorem is written for the interval \([-\pi ,\pi ]\).

  3. Since \(e_k = e^{i\lambda _k \pi x}\) and \(\lambda _k\) are sampling the conditioning and invertibility of T depends on the sampling pattern.

References

  1. Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55. National Bureau of Standards, New York (1964)

    MATH  Google Scholar 

  2. Adcock, B., Hansen, A.C.: Stable reconstruction in Hilbert spaces and the resolution of the Gibbs phenomenon. Appl. Comput. Harmon. Anal. 32, 357–388 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adcock, B., Gataric, M., Hansen, A.C.: On stable reconstructions from nonuniform Fourier measurements. SIAM J. Imaging Sci. 7(3), 1690–1723 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Adcock, B., Gataric, M., Hansen, A.C.: Recovering piecewise smooth functions from nonuniform Fourier measurements. arXiv:1410.0088 (2014)

  5. Barkhudaryan, A., Barkhudaryan, R., Poghosyan, A.: Asymptotic behavior of Eckhoff’s method for Fourier series convergence acceleration. Anal. Theory Appl. 23(3), 228–242 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Batenkov, D.: Complete algebraic reconstruction of piecewise-smooth functions from Fourier data. Math. Comput. 84(295), 2329–2350 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benedetto, J.J., Wu, H.C.: Non-uniform sampling and spiral MRI reconstruction. Proc. SPIE 4119(1), 130–141 (2000)

    Article  Google Scholar 

  8. Cerda, J.O., Seip, K.: Fourier frames. Ann. Math. 155(3), 789–806 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, H., Shizgal, B.D.: A spectral solution of the Sturm–Liouville equation; comparison of classical and nonclassical basis sets. J. Comput. Appl. Math. 136, 17–35 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Christensen, O., Lindner, A.M.: Frames of exponentials: lower frame bounds for finite subfamilies and approximation of the inverse frame operator. Linear Algebra Appl. 323(1–3), 117–130 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, New York (2002)

    MATH  Google Scholar 

  12. Christensen, O.: Finite-dimensional approximation of the inverse frame operator. J. Fourier Anal. Appl. 6(1), 79–91 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Christensen, O., Strohmer, T.: The finite section method and problems in frame theory. J. Approx. Theory 133, 221–237 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Duffin, R.J., Schaeffer, A.C.: class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72(2), 341–366 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eckhoff, K.S.: Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions. Math. Comput. 64(210), 671–690 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Feichtinger, H.G., Gröchenig, K., Strohmer, T.: Efficient numerical methods in non-uniform sampling theory. Numer. Math. 69, 423–440 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gelb, A., Hines, T.: Recovering exponential accuracy from nonharmonic Fourier data through spectral reprojection. J. Sci. Comput. 51, 158–182 (2011)

    Article  MATH  Google Scholar 

  19. Gelb, A., Song, G.: A frame theoretic approach to the nonuniform fast Fourier transform. SIAM J. Numer. Anal. 52(3), 1222–1242 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gottlieb, D., Shu, C.-W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39(4), 644–668 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gottlieb, D., Shu, C.-W., Solomonoff, A., Vandeven, H.: On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function. J. Comput. Appl. Math. 43, 81–92 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gröchenig, K.: Acceleration of the frame algorithm. Trans. Signal Process. 41(12), 3331–3340 (1993)

    Article  MATH  Google Scholar 

  23. Gröchenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10(2), 105–132 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hrycak, T., Gröchenig, K.: Pseudospectral Fourier reconstruction with the modified inverse polynomial reconstruction method. J. Comput. Phys. 229(3), 933–946 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  26. Jung, J.-H.: A hybrid method for the resolution of the Gibbs phenomenon. In: Hesthaven, J.S., Rønquist, E. M. (eds.) Lecture Notes in Computational Science and Engineering 76, New York, pp. 219–227 (2011)

  27. Jung, J.-H., Shizgal, B.D.: Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomena. J. Comput. Appl. Math. 172(1), 131–151 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jung, J.-H., Shizgal, B.D.: Short note: on the numerical convergence with the inverse polynomial reconstruction method for the resolution of the Gibbs phenomenon. J. Comput. Phys. 224, 477–488 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kvernadze, G.: Approximation of the discontinuities of a function by its classical orthogonal polynomial Fourier coefficients. Math. Comput. 79, 2265–2285 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lustig, M., Donoho, D., Pauly, J.: Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn. Reson. Med. 58(6), 1182–1195 (2007)

    Article  Google Scholar 

  31. 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 

  32. Shizgal, B.D.: Spectral methods based on nonclassical basis functions; the advection diffusion equation. Comput. Fluids 31, 825–843 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shizgal, B.D., Jung, J.-H.: Towards the resolution of the Gibbs phenomena. J. Comput. Appl. Math. 161(1), 41–65 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Solomonoff, A.: Reconstruction of a discontinuous function from a few Fourier coefficients using Bayesian estimation. J. Sci. Comput. 10(1), 29–80 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Song, G., Gelb, A.: Approximating the inverse frame operator from localized frames. Appl. Comput. Harmon. Anal. 35, 94–110 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Viswanathan, A.: Spectral sampling and discontinuity detection methods with application to magnetic resonance imaging. Master’s thesis, Arizona State University, Tempe, Arizona (2008)

  37. Viswanathan, A., Gelb, A., Cochran, D., Renaut, R.: On reconstruction from non-uniform spectral data. J. Sci. Comput. 45(1–3), 487–513 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980)

    MATH  Google Scholar 

Download references

Acknowledgements

Anne Gelb was partially supported by Grants NSF-DMS 1216559, NSF-DMS 1521600, NSF 1502640, and AFOSR FA9550-15-1-0152.

Author information

Authors and Affiliations

  1. Department of Mathematics, Jimei University, Xiamen, Fujian Province, China

    Xinjuan Chen

  2. Department of Mathematics, University at Buffalo, The State University of New York, Buffalo, NY, 14260-2900, USA

    Jae-Hun Jung

  3. Department of Data Science, Ajou University, Yeongtong-gu, Suwon, 16499, Korea

    Jae-Hun Jung

  4. Department of Mathematics, Dartmouth College, Hanover, NH, 03755-3551, USA

    Anne Gelb

Authors
  1. Xinjuan Chen
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jae-Hun Jung
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Anne Gelb
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jae-Hun Jung.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, X., Jung, JH. & Gelb, A. Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method. J Sci Comput 76, 1127–1147 (2018). https://doi.org/10.1007/s10915-018-0655-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0655-4

Keywords