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## Singularity Detection And Processing With Wavelets (1992)

Venue: | IEEE Transactions on Information Theory |

Citations: | 589 - 13 self |

### Citations

4593 | A Computational Approach to Edge Detection
- Canny
- 1986
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Citation Context ... to characterize the different singularities but no effective method has been derived yet. In computer vision, it is extremely important to detect the edges that appear in images and many researchers =-=[6, 16, 17, 21, 22]-=- have developed techniques based on multiscale transforms. These multiscale transforms are equivalent to a wavelet transform but have been studied before the development of the wavelet formalism. Let ... |

2892 |
The Fractal Geometry of Nature
- Mandelbrot
- 1982
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Citation Context ...em 2 Let f (x)sL 2 (R). If f (x) is Lipschitz a at x 0 , 0sasn, then there exists a constant A such that for all point x in a neighborhood of x 0 and any scale s | Wf (s,x) |sA (s a + | x-x 0 | a ) . =-=(15)-=- Conversely, f (x) is Lipschitz a at x 0 ,0sasn, if the two following conditions holds. # There exists e > 0 and a constant A such that for all points x in a neighborhood of x 0 and any scale s | Wf (... |

1270 | Theory of edge detection
- Marr, Hildreth
- 1980
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Citation Context ... to characterize the different singularities but no effective method has been derived yet. In computer vision, it is extremely important to detect the edges that appear in images and many researchers =-=[6, 16, 17, 21, 22]-=- have developed techniques based on multiscale transforms. These multiscale transforms are equivalent to a wavelet transform but have been studied before the development of the wavelet formalism. Let ... |

1204 |
Probability, random variables, and stochastic processes, volume 196
- Papoulis, Pillai, et al.
- 1965
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Citation Context ... that both equations (73) and (78) are valid for xs]x 0 -e,x 0 +e[ and ss2 with e = 4 K-C ##### s 0 and s 1 = 4K K-C ##### s 0 . -- -- Page 60 Appendix 4 White noise Wavelet Transform It is well know =-=[20]-=- that the density of zero-crossings of a differentiable Gaussian process whose autocorrelation is R (t) is # ### p 2 R (0) -R (2) (0) ######## , (79) where R (n) (t) is the n th derivative of R (t). T... |

794 |
The wavelet transform, time-frequency localization, and signal analysis
- Daubechies
- 1990
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Citation Context ...ation (3), the Fourier transform of Wf (2 j , x) is given by W f (2 j , w) = y (2 j w) f (w) . (38) The function f (x) can be reconstructed from its wavelet transform and the reconstruction is stable =-=[7, 14]-=- if and only there exists two constants A > 0 and B > 0 such that Asj =- S + | y (2 j w) | 2sB . (39) Let us denote by || Wf (2 j , x) || the L 2 (R) norm of the function Wf (2 j , x) along the variab... |

403 |
Trigonometric Series
- Zygmund
- 1968
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Citation Context ...ocally f (x) so the oscillations are attenuated and the Lipschitz exponent in x 0 increases by more than 1. Singularities with such an oscillatory behavior have been thoroughly studied in mathematics =-=[24]-=-. A classical example is the function f (x) = sin ( x 1 ## ) in the neighborhood of x = 0. This function is not continuous in 0 but is bounded in the neighborhood of 0 so it is Lipschitz 0 in x = 0. T... |

343 |
Ondelettes et opérateurs
- Meyer
- 1990
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Citation Context ...but it is not well adapted for finding the location and the spatial distribution of singularities. This was a major motivation in mathematics and in applied domains for studying the wavelet transform =-=[10, 18]-=-. By decomposing signals into elementary building blocks that are well localized both in space and frequency, the wavelet transform can characterize the local regularity of signals. The wavelet transf... |

256 |
Decomposition of Hardy functions into square integrable wavelets of constant shape
- GROSSMANN, MORLET
- 1984
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Citation Context ...) is uniformly Lipschitz a over intervals ]a +e , b -e[ for any e > 0, if and only if for any e > 0, there exists a constant A e such that for any xs]a+e,b -e[ and any scale s, | Wf (s,x) |sA e s a . =-=(10)-=- If f (x)sL 2 (R), for any scale s 0 > 0, by applying the Schwartz inequality, we can easily prove that the function | Wf (s,x) | is bounded over the domain s > s 0 . Hence, equation (10) is really a ... |

183 |
Scale space filtering
- Witkin
- 1983
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Citation Context ...y s (u -x) du s ds ### . (4) The wavelet transform Wf (s,x) is a function of the scale s and the spatial position x. The plane defined by the couple of variables (s,x) is called the scale-space plane =-=[22]-=-. Any function F (s,x) is not a priori the wavelet transform of some function f (x). One can prove that F (s,x) is a wavelet transform if and only if it satisfies the reproducing kernel equation F (s ... |

122 |
Edge and Curve Detection for Visual Scene Analysis
- Rosenfeld, Thurston
- 1971
(Show Context)
Citation Context ... to characterize the different singularities but no effective method has been derived yet. In computer vision, it is extremely important to detect the edges that appear in images and many researchers =-=[6, 16, 17, 21, 22]-=- have developed techniques based on multiscale transforms. These multiscale transforms are equivalent to a wavelet transform but have been studied before the development of the wavelet formalism. Let ... |

38 |
Exposants de Hölder en des points donnés et coefficients d’ondelettes
- Jaffard
- 1989
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Citation Context .... Theorem 1 gives a characterization of the Lipschitz regularity over intervals but not precisely at a point. The second theorem proved independently by Holschneider and Tchamitchian [12] and Jaffard =-=[13]-=- shows that one can also estimate the Lipschitz regularity of f (x) precisely at a point x 0 . The theorem gives a necessary condition and a sufficient condition but not a necessary and sufficient con... |

21 |
Complete Signal Representation With Multiscale Edges
- Zhong
- 1989
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Citation Context ... the oscillations can be measured from the wavelet transform local maxima. It has been shown that one and two-dimensional signals can be reconstructed from the local maxima of their wavelet transform =-=[14]-=-. As an application, we develop an algorithm that removes white noises by discriminating the noise and the signal singularities through an analysis of their wavelet transform maxima. In two-dimensions... |

18 |
Wavelet analysis of fully developed turbulence data and measurement of scaling exponents, in: Turbulence and coherent structures, edited by
- Bacry, Arneodo, et al.
- 1989
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Citation Context ...y being studied [2]. A well known example is the turbulence for high Reynold numbers where there are still no comprehensive theory to understand the nature and repartition of the irregular structures =-=[4]-=-. In signal processing, singularities often carry most of the signal information. This is well illustrated in image processing where edges provide reliable features for recognition purposes. The detec... |

14 |
Wavelet transform of multifractals
- Arneodo, G, et al.
- 1988
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Citation Context ...gnals, the interesting information is given by transient phenomena such as peaks. In physics, it is important to study irregular structures to infer properties about the underlined physical phenomena =-=[1, 2, 15]-=-. Until recently, the Fourier transform was the main mathematical tool for analyzing singularities. The Fourier transform is global and provides a description of the overall regularity of signals but ... |

14 |
Regularite locale de la fonction "nondifferentiable" de Riemann
- HOLSCHNEIDER, PH
- 1990
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Citation Context ...elet y(x) is continuously differentiable and that it has a compact support although this last condition is not strictly necessary. The first theorem is a well known result and a proof can be found in =-=[12]-=-. Theorem 1 Let f (x)sL 2 (R). The function f (x) is uniformly Lipschitz a over intervals ]a +e , b -e[ for any e > 0, if and only if for any e > 0, there exists a constant A e such that for any xs]a+... |

10 |
Edge representation from wavelet transform maxima
- Zhong
- 1990
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Citation Context ... of y(x) is -- -- Page 17 y (w) = (iw/2) # # # w/2 sin(w/2) ######## # # # 3 . (26) This wavelet belongs to the class of wavelets for which the wavelet transform can be computed with a fast algorithm =-=[23]-=-. Fig. 2: (a): Graph a wavelet y(x) with compact support and one vanishing moment. It is a quadratic spline. (b): Graph of the primitive q(x). -- -- Page 18 (a) (b) (c) (d) Fig. 3: see the caption nex... |

9 |
Detection of abrupt changes in sound signals with the help of wavelet transforms, Inverse problems: an interdisciplinary study (Montpellier
- Grossmann, Holschneider, et al.
- 1986
(Show Context)
Citation Context ...only in mathematics but also in signal processing. In section 4, we explain the relation between the multiscale edge detection algorithms used in computer vision and the approach of Grossmann et. al. =-=[9]-=- which finds singularities by following the lines of constant phase in a wavelet transform. The detection of the wavelet transform local maxima is strongly motivated by these techniques. Section 5 is ... |

7 |
Propagation et interaction des singularites pour les solutions des equations aux derivees partielles non-lineaires
- Bony
- 1983
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Citation Context ...not satisfy this property. Since local Lipschitz exponents do not behave well with respect to differentiation, one can not extend directly this notion to tempered distributions. For this purpose Bony =-=[5]-=- extended the concept of Lipschitz exponents through the 2-microlocalization which is closely related to the wavelet transform as shown by Meyer [19]. We do not take this approach and use a simpler ex... |

2 |
Wavelet analysis of fractal growth process
- Argoul, Ameodo, et al.
- 1988
(Show Context)
Citation Context ...gnals, the interesting information is given by transient phenomena such as peaks. In physics, it is important to study irregular structures to infer properties about the underlined physical phenomena =-=[1, 2, 15]-=-. Until recently, the Fourier transform was the main mathematical tool for analyzing singularities. The Fourier transform is global and provides a description of the overall regularity of signals but ... |

2 |
Transformation en ondelettes et Turbulence pleinement developpee,” Rapport de Magistere
- Bacry
- 1989
(Show Context)
Citation Context .... It is therefore sufficient to use a wavelet with only one vanishing -- -- Page 22 moment. In signals obtained from turbulent fluids, interesting structures have a Lipschitz exponent between 0 and 2 =-=[3]-=-. We thus need a wavelet with two vanishing moments to analyze the turbulent structures. In the following, we suppose that the wavelet y(x) is the n th derivative of a positive functionsq(x) that has ... |

2 |
Wavelet representation and time-scaled matched receiver for asymptotic signals
- Escudie, Torresani
- 1990
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Citation Context ...long its scale and spatial variables. These general maxima points estimate locally the main frequency component of a signal. This approach is closely related to the algorithm of Escudie and Torresani =-=[8]-=- for measuring the modulation law of asymptotic signals. -- -- Page 3 An algorithm that reconstructs one and two-dimensional signals from the wavelet transform local maxima has been implemented by Zho... |

2 |
A real-time algorithm for singal analysis with the help of the wavelet transform,” preprint from CPT
- Holschneider, Kronland-Martinet, et al.
- 1988
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Citation Context ...hat the decay of the wavelet transform at scales larger than 1 characterize the Lipschitz exponent of the function up to the scale 1. Fast algorithms to compute the wavelet transform are described in =-=[11, 14]-=-. We shall not worry anymore about the opposition between asymptotic measurements and finite resolutions. The local maxima of the wavelet transform of fig. 3(b) are shown in fig. 3(c). The black lines... |

1 |
La 2-microlocalisation
- Meyer
- 1989
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Citation Context ...sly differentiable but its n th derivative is singular in x 0 and a 0 characterizes this singularity. -- -- Page 7 One can prove that if f (x) is Lipschitz a then its primitive g (x) is Lipschitz a+1 =-=[19]-=-. However, it is not true that if a function is Lipschitz a at a point x 0 , then its derivative is Lipschitz a-1 at the same point. Section 5.3 gives an example of function that does not satisfy this... |