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## Edge detection using Fourier coefficients

Venue: | Amer. Math. Monthly |

Citations: | 3 - 1 self |

### Citations

142 | On the Gibbs phenomenon and its resolution,
- Gottlieb, Shu
- 1997
(Show Context)
Citation Context ...non [11] appears and truncating the series after any finite number of terms always leads to O(1) oscillations in the reconstructed signal. (For a nice, detailed treatment of the Gibbs phenomenon, see =-=[6]-=-.) Considering the question again, however, one realizes that if a discontinuity is characterized by a “phenomenon,” then the existence of the discontinuity is indeed encoded in the coefficients. The ... |

64 | Detection of edges in spectral data”,
- Gelb, Tadmor
- 1999
(Show Context)
Citation Context ...done. We restrict ourselves to periodic (or compactly supported) functions and only consider Fourier series. (Those interested in seeing a more general theory of concentration factors are referred to =-=[3, 4]-=-.) Much of the information in this article is well known [3, 4]. The use of the EulerMascheroni constant to improve the performance of the concentration factor in Section 4 is, to the best of our know... |

35 |
Discourse on Fourier Series
- Lanczos
- 1966
(Show Context)
Citation Context ...es. 500 c○ THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 115Now suppose that f (t) is periodic with period T and is square integrable in each period—that f (t) ∈ L2 [0, T ]. Then (as described in =-=[8]-=-) the Fourier series converges to the function in L2 [0, T ] (and this is a weaker form of convergence than uniform convergence). Additionally, for square integrable functions Parseval’s equation stat... |

24 | capturing by the spectral viscosity method
- Tadmor, Shock
- 1989
(Show Context)
Citation Context ... interest rather than directly estimating the solution. The spectral viscosity method, a numerical method used to solve nonlinear partial differential equations (PDEs), is an example of such a method =-=[12]-=-. The method approximates the Fourier coefficients of the solution of a PDE. The Fourier coefficients are then used to calculate an approximation to the solution. The accurate reconstruction of the so... |

9 |
Landau-Ramanujan constant. From MathWorld – a Wolfram web resource. http://mathworld.wolfram.com/Landau-RamanujanConstant.html
- Weisstein
(Show Context)
Citation Context ...tuted ln(N) for the partial sum. It is well known 508 c○ THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 115that lim N→∞ (( N∑ k=1 ) ) 1 − ln(N) ≡ γ = 0.577215 .... k Furthermore, it has been shown =-=[13]-=- that 1 2(N + 1) < ( ) N∑ 1 − ln(N) − γ< k 1 2N . k=1 The constant γ is known as the Euler-Mascheroni constant. Rather than dividing the sum by ln(N), wedivideitbyln(N) + γ . This defines a second, im... |

7 | Reducing the effects of noise in image reconstruction
- Archibald, Gelb
- 2002
(Show Context)
Citation Context ... the Fourier series or transform associated with the signal is not uniform; in such cases the Gibbs phenomenon [11] appears and truncating the series after any finite number of terms always leads to O=-=(1)-=- oscillations in the reconstructed signal. (For a nice, detailed treatment of the Gibbs phenomenon, see [6].) Considering the question again, however, one realizes that if a discontinuity is character... |

1 |
spectral viscosity approximations for conservation laws
- Enhanced
(Show Context)
Citation Context ...urier coefficients are then used to calculate an approximation to the solution. The accurate reconstruction of the solution requires that the positions of the discontinuities of the solution be known =-=[5]-=-. In this paper we discuss techniques for using a function’s Fourier coefficients to determine the locations and sizes of the jump discontinuities of the function. At first glance the spectral represe... |

1 |
Gibbs Phenomenon, available at http://www.sosmath.com/fourier/ fourier3/gibbs.html
- MATHematics
(Show Context)
Citation Context ...on about discontinuities in the signal. When a signal is discontinuous the convergence of the Fourier series or transform associated with the signal is not uniform; in such cases the Gibbs phenomenon =-=[11]-=- appears and truncating the series after any finite number of terms always leads to O(1) oscillations in the reconstructed signal. (For a nice, detailed treatment of the Gibbs phenomenon, see [6].) Co... |

1 |
Wikipedia: The Free Encyclopedia, Dirichlet kernel, http://www.wikipedia.org/wiki/Dirichlet_ kernel. SHLOMO ENGELBERG received his
- E, his
- 2006
(Show Context)
Citation Context ...lly to ln(2)/π). To proceed with our analysis we must analyze the partial sums gN (ξ) = N∑ n=1 cos(2πnξ) , ξ = t − τ nπ more carefully. To this end, we consider the properties of the Dirichlet kernel =-=[14]-=- defined by DN (ξ) ≡ N∑ n=−N e i2πnξ = 1 + 2 N∑ cos(2πnξ). (3) June–July 2008] EDGE DETECTION USING FOURIER COEFFICIENTS 505 n=1This is a finite geometric series whose sum is −i2π Nξ DN (ξ) = e 2N∑ e... |