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

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Annual Reviewswww.annualreviews.org/aronline406 FARGEIn this integral we use the invariant measure d#(O) of the rotation groupSO(n) defined as:withn--1kdid(O) = A. I-[ YI sin J- ’~dO~. Oj(10)k--lj--Ik= ! 27~k/2and the Euler angles50_
Annual Reviewswww.annualreviews.org/aronline3.2 Elementary PropertiesWAVELET TRANSFORMS 407We will now list some of the main properties of the continuous wavelettransform. For the sake of simplicity, from here on we will consider theone-dimensional wavelet transform, the generalization to n dimensionsbeing straightforward, and discard the prime in x’. We will denote thecontinuous wavelet transform of a functionf(x) by the operator notationW[f](x), and the resulting wavelet coefficients by f(l, x).LINEARITY The wavelet transform is linear because it is an inner productbetween the signal f and the wavelet ~O. Likewise, the continuous wavelettransform of a vector function is a vector whose components are thecontinuous wavelet transform of the different components.COVARIANCE BY TRANSLATION AND DILATION The continuoustransform is covariant under any translation x0:waveletW[f] (x-- Xo) = f(1, x-- Xo), (12)which in particular implies that differentiation commutes with continuouswavelet transform; namely we havec?~ W ( f ) = W ~xV[W(f)] = W(Vf),V.[W(f)] = W(V.f). (13)A consequence of the translation covariance is the fact that the frequencyof a monochromatic signal can be read off from the phase of the waveletcoefficients (Escudi6 & Torrtsani 1989, Guillemain 1991, TchamitchianTorrtsani 1991, Delprat et al 1991). The number of zeros of the phase onlines with l = constant gives the frequency of the signal (Figure 2b). Thisproperty is independent of the wavelet chosen.The continuous wavelet transform is also covariant under any dilationby/0:IV[f] (lox) = IF 1.y(/0/, lox). (14)A consequence of the dilation covariance is the fact that the wavelettransform of a power-law function is fully determined by its restriction toany line l = constant. The lines of constant phase point out the possiblesingularities of the function (Figure 2a). This property is also independentof the wavelet choice.
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