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Area Mathematics - Functions / Integral transformations

IEV ref103-04-11

en
wavelet
small localized wave, represented by a function having a zero mean value and a practically finite duration

Note 1 to entry: From a mother wavelet ψ(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaaaa@3972@ , daughter wavelets are obtained through shifting and scaling (expansion or compression): ψ a,b (t)= 1 a ψ( tb a ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5naaBaaaleaacaWGHbGaaiilaiaadkgaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0ZaaSaaaeaajugabiaaigdaaOqaamaakaaabaGaamyyaaWcbeaaaaGccqaHipqEdaqadaqaamaalaaabaGaamiDaiabgkHiTiaadkgaaeaacaWGHbaaaaGaayjkaiaawMcaaaaa@4691@ , where a is a scale parameter and b a position parameter.

Note 2 to entry: Examples (see Figures 3 and 4):

  • Haar wavelet: ψ(t)=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaiabg2da9iabgkHiTKqzaeGaaGymaaaa@3C8F@ for −1/2 < t < 0, ψ(t)=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaiabg2da9KqzaeGaaGymaaaa@3BA2@ for 0 < t < 1/2, ψ(t)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaiabg2da9KqzaeGaaGimaaaa@3BA1@ outside;
  • Morlet wavelet: ψ(t)= e t 2 /2 e jωt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaKqzaeGaeqiYdKNaai ikaiaadshacaGGPaGaeyypa0JaciyzaOWaaWbaaSqabeaacqGHsisl caWG0bWaaWbaaWqabeaajugOaiaaikdaaaWccaGGVaqcLbmacaaIYa aaaKqzaeGaciyzaOWaaWbaaSqabeaacqGHsisljugWaiGacQgaliab eM8a3jaayIW7caWG0baaaaaa@4B8D@ (example of exponential damping; Figure 4 gives the real part).

Figure 3 – Haar wavelet

Figure 3 – Ondelette de Haar

Figure 4 – Morlet wavelet

Figure 4 – Ondelette de Morlet


fr
ondelette, f
petite onde localisée, représentée par une fonction ayant une valeur moyenne nulle et une durée pratiquement finie

Note 1 à l'article: À partir d'une ondelette mère ψ(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaaaa@3972@ , des ondelettes filles sont obtenues par décalage et changement d'échelle (dilatation et compression): ψ a,b (t)= 1 a ψ( tb a ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5naaBaaaleaacaWGHbGaaiilaiaadkgaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0ZaaSaaaeaajugabiaaigdaaOqaamaakaaabaGaamyyaaWcbeaaaaGccqaHipqEdaqadaqaamaalaaabaGaamiDaiabgkHiTiaadkgaaeaacaWGHbaaaaGaayjkaiaawMcaaaaa@4691@ , où a est un paramètre d'échelle et b un paramètre de position.

Note 2 à l'article: Exemples (voir les Figures 3 et 4):

  • ondelette de Haar: ψ(t)=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaiabg2da9iabgkHiTKqzaeGaaGymaaaa@3C8F@ pour −1/2 < t < 0, ψ(t)=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaiabg2da9KqzaeGaaGymaaaa@3BA2@ pour 0 < t < 1/2, ψ(t)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaiabg2da9KqzaeGaaGimaaaa@3BA1@ ailleurs;
  • ondelette de Morlet: ψ(t)= e t 2 /2 e jωt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaKqzaeGaeqiYdKNaai ikaiaadshacaGGPaGaeyypa0JaciyzaOWaaWbaaSqabeaacqGHsisl caWG0bWaaWbaaWqabeaajugOaiaaikdaaaWccaGGVaqcLbmacaaIYa aaaKqzaeGaciyzaOWaaWbaaSqabeaacqGHsisljugWaiGacQgaliab eM8a3jaayIW7caWG0baaaaaa@4B8D@ (exemple d'amortissement exponentiel; Figure 4 représente la partie réelle).

ar
الموجة

de
Wavelet, n

es
ondícula

it
wavelet
onda elementare (unitaria)

ja
ウェーブレット

pl
falka

pt
ôndula

sr
таласна трансформација, ж јд

sv
vågpaket

zh
小波

Publication date: 2009-12
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