THE GIBBS PHENOMENON

A collection of selected references on

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




  • D. Gottlieb and E. Tadmor
    Recovering pointwise values of discontinuous data within spectral accuracy
    in "Progress and Supercomputing in Computational Fluid Dynamics", Proc. of US-Israel workshop held in Jerusalem in 1984, Vol. 6 (E. M. Murman and S. S. Abarbanel, eds.), Birkhauser, Boston (1985) 357-375..

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

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

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

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

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

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

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  • R. Archibald and A. Gelb
    A method to reduce the Gibbs ringing artifact in MRI scans while keeping tissue boundary integrity
    IEEE Transactions of Medical Imaging 21(4) (2002) 100-114.

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  • A. Gelb and E. Tadmor
    Spectral reconstruction of one- and two-dimensional piecewise smooth functions from their discrete data
    Mathematical Modeling and Numerical Analysis 36(2) (2002) 155-175.

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  • R. Archibald and A. Gelb
    Reducing the effects of noise in image reconstruction
    Journal of Scientific Computing 17(1-4) (2002) 167-180.

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  • E. Tadmor and J. Tanner
    An adaptive order Godunov type central scheme
    in "Hyperbolic Problems: Theory, Numerics, Applications", Proc. 9th International Hyp Conference held in CalTech, Pasadena in 2002 (T. Hou & E. Tadmor eds.), Springer (2003) 871-880..

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

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  • G. Kvernadze
    Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients
    Mathematics of Computation 73(246) (2003) 731-751.

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

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  • J. Boyd
    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.

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  • Anne Gelb and Jared Tanner
    Robust reprojection methods for the resolution of the Gibbs phenomenon
    Applied Computational Harmonic Analysis 20 (2006) 3-25.

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

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  • L. Hu and X. L. Shi
    Concentration factors for functions with harmonic bounded mean variation
    Acta Mathematica Hungaria (2007).

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

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

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

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

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

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  • D. Batenkov, N. Sarig and Y. Yomdin
    An “algebraic” reconstruction of piecewise-smooth functions from integral measurements
    Functional Differential Equations 19(1-2) (2012) 13-30.

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  • D. Batenkov and Y. Yomdin
    Algebraic Fourier reconstruction of piecewise smooth functions
    Mathematics of Computation 81(277) (2012) 277-318.

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

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  • D. Batenkov
    Complete algebraic reconstruction of piecewise-smooth functions from Fourier data
    Mathematics of Computation 84(295) (2015) 2329–2350.

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  • Beong In Yun
    Enhanced edge detection method based on a threshold function for discrete data
    Applied Mathematical Sciences 10(58) (2016) 2881-2893.

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