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Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

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Abstract

We propose three novel methods for recovering edges in piecewise smooth functions from their possibly incomplete and noisy spectral information. The proposed methods utilize three different approaches: #1. The randomly-based sparse Inverse Fast Fourier Transform (sIFT); #2. The Total Variation-based (TV) compressed sensing; and #3. The modified zero crossing. The different approaches share a common feature: edges are identified through separation of scales. To this end, we advocate here the use of concentration kernels (Tadmor, Acta Numer. 16:305–378, 2007), to convert the global spectral data into an approximate jump function which is localized in the immediate neighborhoods of the edges. Building on these concentration kernels, we show that the sIFT method, the TV-based compressed sensing and the zero crossing yield effective edge detectors, where finitely many jump discontinuities are accurately recovered. One- and two-dimensional numerical results are presented.

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Authors and Affiliations

  1. Department of Mathematics, Institute for Physical Science and Technology and Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD, 20742, USA

    Eitan Tadmor

  2. Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD, 20742, USA

    Jing Zou

Authors
  1. Eitan Tadmor
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  2. Jing Zou
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Corresponding author

Correspondence to Eitan Tadmor.

Additional information

Communicated by Ronald A. DeVore.

Research was supported in part by NSF grants DMS04-07704, DMS07-07949 and ONR grant N00014-91-J-1076.

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Tadmor, E., Zou, J. Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data. J Fourier Anal Appl 14, 744–763 (2008). https://doi.org/10.1007/s00041-008-9038-9

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  • DOI: https://doi.org/10.1007/s00041-008-9038-9

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