wavelet transforms and their applications to turbulence - Wavelets ...

Annual Reviewswww.annualreviews.org/aronline(Paul 1984, 1985a,b, 1986,(Figure 3b):WAVELET TRANSFORMS 4131989), is the Paul wavelet which is analytic~/m = F(m+l) (%~l)m(1 ’+m’ -,fS x)(25)or{~ m(k) kme k fork > 0,~,,,(k) = for k _< 0.The higher the order rn, the better the cancellations (vanishing moments)the wavelet. The complex-valued wavelets are called progressive wavelets iftheir Fourier coefficients are zero for negative wavenumbers. They are welladapted to analyze causal signals, i.e. signals for which there is some actionof causality. This is because progressive wavelets preserve the direction oftime and do not create parasitic interference between the past and future.Progressive wavelets are used in particular to analyze musical sounds(Kronland-Martinet ct al 1987, Kronland-Martinct 1988).EXAMPLES OF REAL-VALUED WAVELETS Some commonly used real-valuedwavelets are the ruth derivatives of the Gaussian (Figures 3c and 3d):orI[Im(X )= (-- 1) m ~(d--lx12/2), (26)ax~ffm(k)m(~-- lk) m e -Ikl2/2.The higher order derivatives imply more cancellations (vanishingmoments) of the wavelet. Among these derivatives of the Gaussian, themost widely used is the Marr wavelet, also called the "Mexican hat"(Figure 3d), which is the Laplacian of a Gaussian (m ~ 2). Thewavelet in its generalization to n dimensions is isotropic and thereforecannot discriminate different directions in the signal. Jaffard (1991 b) hasproposed to use, instead of a Laplacian of a Gaussian, a Gaussian differentiatedin only one direction, which will then be nonisotropie with a goodFigure 3 Examples of wavelets ~b commonly used for the continuous wavelet transform(continuous line for the real part and broken line for the imaginary part). We visualize ~b(x)(left) and ~(k) (right).(a) Morlet wavelet, for k~ = 6, (b) Paul’s wavelet for m = 4, (c) Firstderivative of a Gaussian, (d) Marr wavelet (second derivative of a Gaussian), (e)(Difference of two Gaussians).