Geodesic
Spectral reconstruction of piecewise smooth functions from their discrete data
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 36 (2002) no. 2, pp. 155-175.

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This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coefficients and physical space interpolants have been discussed extensively in the literature, and it is clear that an a priori knowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurate results with high resolution near the discontinuities. Hence detection of the jump discontinuities is critical for all methods. Here we formulate a new localized reconstruction method adapted from the method developed in Gottlieb and Tadmor (1985) and recently revisited in Tadmor and Tanner (in press). Our procedure incorporates the detection of edges into the reconstruction technique. The method is robust and highly accurate, yielding spectral accuracy up to a small neighborhood of the jump discontinuities. Results are shown in one and two dimensions.

DOI : 10.1051/m2an:2002008
Classification : 42A10, 42A50, 65T40
Keywords: edge detection, nonlinear enhancement, concentration method, piecewise smoothness, localized reconstruction 
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     title = {Spectral reconstruction of piecewise smooth functions from their discrete data},
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     pages = {155--175},
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Gelb, Anne; Tadmor, Eitan. Spectral reconstruction of piecewise smooth functions from their discrete data. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 36 (2002) no. 2, pp. 155-175. doi : 10.1051/m2an:2002008. http://geodesic.mathdoc.fr/articles/10.1051/m2an:2002008/

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