Detection of edges in spectral data

Anne Gelb, Eitan Tadmor

Research output: Contribution to journalArticlepeer-review

140 Scopus citations

Abstract

We are interested in the detection of jump discontinuities in piecewise smooth functions which are realized by their spectral data. Specifically, given the Fourier coefficients, {f̂k = ak + ibk}Nk=1, we form the generalized conjugate partial sum N S̃σN[f](x) = ∑ σ(k/N)(aksin kx - bkcos kx). The classical conjugate partial sum, k=1 the jump function [f](x) := f(x+) - f(x-); thus, -π/log NS̃N[f](x) tends to "concentrate" near the edges of f. The convergence, however, is at the unacceptably slow rate of order script O sign(1/log N). To accelerate the convergence, thereby creating an effective edge detector, we introduce the so-called "concentration factors," σk,N = σ(k/N). Our main result shows that an arbitrary C2[0, 1] nondecreasing σ(•) satisfying ∫11/Nσ(x)/x dx→-π leads N→∞ to the summability kernel which admits the desired concentration property, S̃σN[f](x)→[f](x), with convergence rate, \S̃σN[f](x)\ ≤ Const(log N/N + |σ(1/N)|) N→∞ for x's away from the jump discontinuities. To improve over the slowly convergent conjugate Dirichlet kernel (corresponding to the admissible σN(x) ≡ -π/log N), we demonstrate the examples of two families of concentration functions (depending on free parameters p and α): the so-called Fourier factors, σFα(x) = -π/Si(α) sin αx, and polynomial factors, σp(x) = -pπxp. These yield effective detectors of (one or more) edges, where both the location and the amplitude of the discontinuities are recovered.

Original languageEnglish (US)
Pages (from-to)101-135
Number of pages35
JournalApplied and Computational Harmonic Analysis
Volume7
Issue number1
DOIs
StatePublished - Jul 1999

Keywords

  • Concentration factors
  • Conjugate partial sums
  • Fourier expansion
  • Piecewise smoothness

ASJC Scopus subject areas

  • Applied Mathematics

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