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Detection of Edges in Spectral Data III—Refinement of the Concentration Method

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Abstract

Edge detection from Fourier spectral data is important in many applications including image processing and the post-processing of solutions to numerical partial differential equations. The concentration method, introduced by Gelb and Tadmor in 1999, locates jump discontinuities in piecewise smooth functions from their Fourier spectral data. However, as is true for all global techniques, the method yields strong oscillations near the jump discontinuities, which makes it difficult to distinguish true discontinuities from artificial oscillations. This paper introduces refinements to the concentration method to reduce the oscillations. These refinements also improve the results in noisy environments. One technique adds filtering to the concentration method. Another uses convolution to determine the strongest correlations between the waveform produced by the concentration method and the one produced by the jump function approximation of an indicator function. A zero crossing based concentration factor, which creates a more localized formulation of the jump function approximation, is also introduced. Finally, the effects of zero-mean white Gaussian noise on the refined concentration method are analyzed. The investigation confirms that by applying the refined techniques, the variance of the concentration method is significantly reduced in the presence of noise.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287, USA

    Anne Gelb & Dennis Cates

Authors
  1. Anne Gelb
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  2. Dennis Cates
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Correspondence to Anne Gelb.

Additional information

This work was partially supported by NSF grants CNS 0324957, DMS 0510813, DMS 0652833, and NIH grant EB 025533-01 (AG).

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Gelb, A., Cates, D. Detection of Edges in Spectral Data III—Refinement of the Concentration Method. J Sci Comput 36, 1–43 (2008). https://doi.org/10.1007/s10915-007-9170-8

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  • DOI: https://doi.org/10.1007/s10915-007-9170-8

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