Edge Detection Free Postprocessing for Pseudospectral Approximations

  • Scott A. Sarra


Pseudospectral Methods based on global polynomial approximation yield exponential accuracy when the underlying function is analytic. The presence of discontinuities destroys the extreme accuracy of the methods and the well-known Gibbs phenomenon appears. Several types of postprocessing methods have been developed to lessen the effects of the Gibbs phenomenon or even to restore spectral accuracy. The most powerful of the methods require that the locations of the discontinuities be precisely known. In this work we discuss postprocessing algorithms that are applicable when it is impractical, or difficult, or undesirable to pinpoint all discontinuity locations.


Fourier Chebyshev Pseudospectral Gibbs Spectral filtering Digital total variation filtering 


  1. 1.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2000) Google Scholar
  2. 2.
    Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006) zbMATHGoogle Scholar
  3. 3.
    Chan, T., Osher, S., Shen, J.: The digital TV filter and nonlinear denoising. IEEE Trans. Image Process. 10(2), 231–241 (2001) zbMATHCrossRefGoogle Scholar
  4. 4.
    Gelb, A.: A hybrid approach to spectral reconstruction of piecewise smooth functions. J. Sci. Comput. 15, 293–322 (2001) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Gelb, A., Cates, D.: Detection of edges in spectral data III: Refinement of the concentration method. J. Sci. Comput. 36(1), 1–43 (2008) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gelb, A., Tadmor, E.: Detection of edges in spectral data. Appl. Comput. Harmon. Anal. 7, 101–135 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gelb, A., Tadmor, E.: Detection of edges in spectral data II: Nonlinear enhancement. SIAM J.  Numer. Anal. 38(4), 1389–1408 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007) zbMATHGoogle Scholar
  9. 9.
    Hesthaven, J.S., Kirby, R.M.: Filtering in Legendre spectral methods. Math. Comput. 77, 1425–1452 (2008) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Jung, J.-H., Shizgal, B.: Inverse polynomial reconstruction of two dimensional Fourier images. J. Sci. Comput. 25, 367–399 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kosloff, R., Tal-Ezer, H.: A modified Chebyshev pseudospectral method with an O(1/n) time step restriction. J. Comput. Phys. 104, 457–469 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ma, H.: Chebyshev-Legendre super spectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 35, 893–908 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Osher, S., Shen, J.: Digitized PDE method for data restoration. In: Anastassiou, G. (ed.) Analytic-Computational Methods in Applied Mathematics, Chap. 16, pp. 751–771. Chapman and Hall/CRC (2000) Google Scholar
  14. 14.
    Sarra, S.A.: Digital Total Variation filtering as postprocessing for Chebyshev pseudospectral methods for conservation laws. Numer. Algorithms 41, 17–33 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Sarra, S.A.: Digital Total Variation filtering as postprocessing for Radial Basis Function Approximation Methods. Comput. Math. Appl. 52, 1119–1130 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sarra, S.A.: The Matlab postprocessing toolkit. Submitted to ACM Trans. Math. Softw. (2009) Google Scholar
  17. 17.
    Sod, G.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000) zbMATHGoogle Scholar
  19. 19.
    Vandeven, H.: Family of spectral filters for discontinuous problems. SIAM J. Sci. Comput. 6, 159–192 (1991) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsMarshall UniversityHuntingtonUSA

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