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Detection of Edges from Nonuniform Fourier Data

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Abstract

Edge detection is important in a variety of applications. While there are many algorithms available for detecting edges from pixelated images or equispaced Fourier data, much less attention has been given to determining edges from nonuniform Fourier data. There are applications in sensing (e.g. MRI) where the data is given in this way, however. This paper introduces a method for determining the locations of jump discontinuities, or edges, in a one-dimensional periodic piecewise-smooth function from nonuniform Fourier coefficients. The technique employs the use of Fourier frames. Numerical examples are provided.

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References

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

    Article  Google Scholar 

  2. Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, Mineola (2001)

    MATH  Google Scholar 

  3. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Scientific Computation. Springer, Berlin (2006). Fundamentals in single domains

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Christensen, O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2003)

    Google Scholar 

  6. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Dutt, A., Rokhlin, V.: Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Comput. 14(6), 1368–1393 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Engelberg, S.: Edge detection using Fourier coefficients. Am. Math. Mon. 115(6), 499–513 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Engelberg, S., Tadmor, E.: Recovery of edges from spectral data with noise—a new perspective. SIAM J. Numer. Anal. 46(5), 2620–2635 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Gelb, A., Cates, D.: Segmentation of images from Fourier spectral data. Commun. Comput. Phys. 5(2–4), 326–349 (2009)

    MathSciNet  Google Scholar 

  13. Gelb, A., Hines, T.: Recovering exponential accuracy from non-harmonic Fourier data through spectral reprojection (2010)

  14. Gelb, A., Tadmor, E.: Detection of edges in spectral data. Appl. Comput. Harmon. Anal. 7(1), 101–135 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Gelb, A., Tadmor, E.: Adaptive edge detectors for piecewise smooth data based on the minmod limiter. J. Sci. Comput. 28(2–3), 279–306 (2006) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 26. Society for Industrial and Applied Mathematics, Philadelphia (1977)

    Book  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

  20. 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 

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

    Article  Google Scholar 

  22. 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 

  23. Solomonoff, A.: Locating a discontinuity in a piecewise-smooth periodic function using Bayes estimation (2006)

  24. Stefan, W., Viswanathan, A., Gelb, A., Renaut, R.: A high order edge detection method for blurred and noisy Fourier data (2010)

  25. Tadmor, E., Zou, J.: Three novel edge detection methods for incomplete and noisy spectral data. J. Fourier Anal. Appl. 14(5–6), 744–763 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

  27. 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 

  28. Ziou, D., Tabbone, S.: Edge detection techniques—an overview. Int. J. Pattern Recognit. Image Anal. 8, 537–559 (1998)

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, AZ, 85287-1804, USA

    Anne Gelb & Taylor Hines

Authors
  1. Anne Gelb
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  2. Taylor Hines
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Correspondence to Anne Gelb.

Additional information

Communicated by Yang Wang.

Research supported in part by NSF-DMS-FRG award 0652833.

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Gelb, A., Hines, T. Detection of Edges from Nonuniform Fourier Data. J Fourier Anal Appl 17, 1152–1179 (2011). https://doi.org/10.1007/s00041-011-9172-7

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  • DOI: https://doi.org/10.1007/s00041-011-9172-7

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