Detection of edges in spectral data

Anne Gelb, Eitan Tadmor

Research output: Contribution to journalArticle

131 Citations (Scopus)

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}N k=1, we form the generalized conjugate partial sum N S̃σ N[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 ∫1 1/Nσ(x)/x dx→-π leads N→∞ to the summability kernel which admits the desired concentration property, S̃σ N[f](x)→[f](x), with convergence rate, \S̃σ N[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, σF α(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
StatePublished - Jul 1999

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Discontinuity
Jump
Partial Sums
Dirichlet Kernel
Detector
Concentration Function
Piecewise Smooth Functions
sin(-x)
Detectors
Summability
Fourier coefficients
Accelerate
Convergence Rate
Polynomials
Tend
kernel
Polynomial
Arbitrary
Demonstrate
Family

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Detection of edges in spectral data. / Gelb, Anne; Tadmor, Eitan.

In: Applied and Computational Harmonic Analysis, Vol. 7, No. 1, 07.1999, p. 101-135.

Research output: Contribution to journalArticle

Gelb, A & Tadmor, E 1999, 'Detection of edges in spectral data', Applied and Computational Harmonic Analysis, vol. 7, no. 1, pp. 101-135.
Gelb, Anne ; Tadmor, Eitan. / Detection of edges in spectral data. In: Applied and Computational Harmonic Analysis. 1999 ; Vol. 7, No. 1. pp. 101-135.
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N2 - 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}N k=1, we form the generalized conjugate partial sum N S̃σ N[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 ∫1 1/Nσ(x)/x dx→-π leads N→∞ to the summability kernel which admits the desired concentration property, S̃σ N[f](x)→[f](x), with convergence rate, \S̃σ N[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, σF α(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.

AB - 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}N k=1, we form the generalized conjugate partial sum N S̃σ N[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 ∫1 1/Nσ(x)/x dx→-π leads N→∞ to the summability kernel which admits the desired concentration property, S̃σ N[f](x)→[f](x), with convergence rate, \S̃σ N[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, σF α(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.

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