Detection of edges in spectral data II. Nonlinear enhancement

Anne Gelb, Eitan Tadmor

Research output: Contribution to journalArticle

96 Scopus citations

Abstract

We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where [f](x) := f(x+) - f(x-) ≠ 0. Our approach is based on two main aspects - localization using appropriate concentration kernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration kernels, Kε(·), depending on the small scale ε. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small W-1,∞-moments of order script O sign(ε)) satisfy Kε * f(x) = [f](x) + script O sign(ε), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form KσN(t) = Σ σ(k/N) sin kt to detect edges from the first 1/ε = N spectral modes of piecewise smooth f's. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of exponential factors, σexp(·), with superior localization properties. The other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges, where Kε * f(x) ∼ [f](x) ≠ 0, and the smooth regions where Kε * f = script O sign(ε) ∼ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.

Original languageEnglish (US)
Pages (from-to)1389-1408
Number of pages20
JournalSIAM Journal on Numerical Analysis
Volume38
Issue number4
DOIs
StatePublished - Dec 1 2001

Keywords

  • Concentration kernels
  • Piecewise smoothness
  • Spectral expansions

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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