COMPUTATION of the GIBBS PHENOMENON

A collection of selected references on

Detection of edges and reconstruction of discontinuous data
from its (pseudo-)spectral information


  • D. Gottlieb & E. Tadmor (1985)
    Recovering pointwise values of discontinuous data within spectral accuracy
    ``Progress and Supercomputing in Computational Fluid Dynamics", Proc. of a 1984 U.S.-Israel Workshop (E. M. Murman and S. S. Abarbanel, eds.), Birkhauser, Boston v. 6 (1985) 357-375.

  • S. Abarbanel, D. Gottlieb & E. Tadmor (1986)
    Spectral methods for discontinuous problems
    ``Numerical Methods for Fluid Dynamics II", Proc. 1985 Conference on Numerical Methods for Fluid Dynamics (K. W. Morton and M. J. Baines, eds.), Clarendon Press, Oxford(1986) 129-153.

  • S. Mallat & W. L. Hwang (1992)
    Singularity detection and processing with wavelets
    IEEE Transactions on Information Theory 38(2) (1992) 617-643.

  • A. Gelb & E. Tadmor (1999)
    Detection of edges in spectral data
    Applied and Computational Harmonic Analysis 7 (1999) 101-135.

  • A. Gelb & E. Tadmor (2000)
    Detection of edges in spectral data II. Nonlinear enhancement
    SIAM Journal on Numerical Analysis 38 (2000), 1389-1408.

  • A. Gelb (2000)
    A hybrid approach to spectral reconstruction of piecewise smooth functions
    Journal of Scientific Computing 15(3) (2000), 293-322.

  • G. Kvernadze (2001)
    Approximation of the singularities of a bounded function by the partial sums of its differentiated Fourier series
    Applied Comput. Harmonic Analysis 11(3) (2001), 439-454.

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  • E. Tadmor & J. Tanner (2002)
    Adaptive mollifiers -- high resolution recovery of piecewise smooth data from its spectral information
    Foundations of Computational Mathematics 2(2) (2002) 155-189.

  • R. Archibald & A. Gelb (2002)
    A method to reduce the Gibbs ringing artifact in MRI scans while keeping tissue boundary integrity
    IEEE Transactions of Medical Imaging (2002) 100-114.

  • R. Archibald & A. Gelb (2002)
    Reducing the effects of noise in image reconstruction
    Journal of Scientific Computing 17(1-4) (2002) 100-114.

  • A. Gelb & E. Tadmor (2002)
    Spectral reconstruction of one- and two-dimensional piecewise smooth functions from their discrete data
    Mathematical Modeling and Numerical Analysis 36 (2002) 167-180.

  • E. Tadmor & J. Tanner (2003)
    An adaptive order Godunov type central scheme
    ``Hyperbolic Problems: Theory, Numerics, Applications'', Proc. 9th International Conference held in CalTech Pasadena, (T. Hou and E. Tadmor, eds.), Springer (2003) 871-880.

  • S. Sarra (2003)
    The spectral signal processing suite
    ACM Transactions on Mathematical Software 29(2) (2003), 195-217.

  • G. Kvernadze (2004)
    Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients
    Mathematics of Computation 73(246) (2004), 731-751.

  • E. Tadmor & J. Tanner (2005)
    Adaptive filters for piecewise smooth spectral data
    IMA Journal of Numerical Analysis 25(4) (2005) 635-647.

  • J. Boyd (2005)
    Trouble with Gegenbauer reconstruction for defeating Gibbs' phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations
    Journal o Computational Physics 204 (2005) 253-264.

  • A. Gelb & E. Tadmor (2006)
    Adaptive edge detectors for piecewise smooth data based on the minmod limiter
    Journal of Scientific Computing 28(2-3) (2006) 279-306.

  • A. Gelb & J. Tanner (2006)
    Robust reprojection methods for the resolution of the Gibbs phenomenon
    Applied Computtaional Harmonic Analysis 20 (2006) 3-25.

  • J. Tanner (2006)
    Optimal filter and mollifier for piecewise smooth spectral data
    Mathematics of Computations 75(254) (2006) 767-790.

  • Q. Shi & X. Shi (2006)
    Determination of jumps in terms of spectral data
    Acat Math. Hungar. 110(3) (2006) 193-206.

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  • E. Tadmor (2007)
    Filters, mollifiers and the computation of the Gibbs phenomenon
    Acta Numerica 16 (2007) 305-378.

  • L. Hu & X. L. Shi (2007)
    Concentration factors for functions with harmonic bounded mean variation
    Acta Math. Hungar. (2007).

  • S. Engelberg (2008)
    Edge detection using Fourier coefficients
    American Mathematical Monthly (2008) 499-513.

  • S. Engelberg & E. Tadmor (2008)
    Recovery of edges from spectral data with noise---a new perspective
    SIAM Journal on Numerical Analysis 46(5) (2008) 2620-2635.

  • E. Tadmor & J. Zou (2008)
    Novel edge detection methods for incomplete and noisy spectral data
    Journal of Fourier Analysis and Applications 14(5) (2008) 744-763.

  • A. Gelb & D. Cates (2009)
    Segmentation of images from Fourier spectral data
    Communications in Computational Physics 5(2-4) (2009) 326-349.

  • B. I. Yun, K. S. Rim (2013)
    Local edge detectors using a sigmoidal transformation for piecewise smooth data
    Applied Mathematics Letters 26(2) (2013) 270-276.