Scaling, Fractals and Wavelets

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243 Local regularity The local regularity properties of process sample paths have been introduced in section 224 Their wavelet counterparts most often derive from the orthogonal discrete wavelet transform, given that they could be extended, normally quite easily, to the continuous (surfaces) varieties (see 3) 0 in t0 and a THEOREM 21 Let X be a signal with H lder regularity h h ) Hence, there exists a constant c > 0 such suf ciently regular wavelet (n that for any j, k Z Z: |dX (j, k)| c 2 ( 2 +h)j 1 + |2j t0 k|h

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Conversely, if for any j, k Z Z: |dX (j, k)| c 2 ( 2 +h)j 1 + |2j t0 k|h

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for h < h, thus X has H lder regularity h in t0 The proof of the theorem was established independently by Jaffard [JAF 89] and Holschneider and Tchamitchian [HOL 90] In the light of this result, we note once again that it is the decrease of wavelet coef cients through scales which characterizes the local regularity of the sample path of X Furthermore, this result is not surprising since the H lder regularity of a function is a particular cause for the 1/f spectral behavior at high frequencies The second part of Theorem 21 also shows that knowledge of the coef cients located vertically to the singular point (|2j t0 k| = 0) is itself not suf cient to determine the local regularity of X in t0 Strictly speaking, it would be necessary to consider the decomposition in its entirety, thus implying that an isolated singularity can affect all the coef cients dX (j, k) inside a cone, called an in uence cone For a wavelet whose temporal support is nite, this cone is also limited at each scale From the estimation point of view, the direct implication of Theorem 21 is to highlight the practical limits of (discrete) orthogonal wavelet transforms, because it is quite unlikely that the abscissa t0 of the singularity coincides with the coef cients line on the dyadic grid Hence, in practice, it is more often a continuous analysis diagram which is preferred, for which we possess a less precise and incomplete version (direct implication) of Theorem 21 (see the following proposition) PROPOSITION 25 If X is of H lder regularity n < h < n + 1 in t0 , for a wavelet analysis possessing n h vanishing moments, then we have the following asymptotic behavior: TX (t0 , a) O(ah+ 2 ),

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Scale Invariance and Wavelets

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Proof Let the continuous wavelet transform (221) constructed with n > n be: TX (t0 , a) = = a a (u) X(t0 + au) du

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(u) X(t0 + au)

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cr ar ur

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where cr represent the Taylor expansion coef cients of X in the vicinity of t0 The signal X is of regularity n < h < n + 1 in t0 and is a localized time function Thus, in the limit of in nitely ne resolutions (a 0+ ): lim+ TX (t0 , a) C a ah+ 2

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(u)|au|h du |u|h (u) du = C ah+ 2

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It is important to underline that if the wavelet is not of suf cient regularity, it is the term of degree n in the Taylor polynomial which dominates at nite scales and it is thus the regularity of the wavelet which imposes the decrease of the coef cients through scales However, one should not be misled by the interpretation of Proposition 25 It is only because we focus on the limited case of in nite resolution that the in uence of the singularity seems to be perfectly localized in t = t0 In reality, it is shown in [MAL 92] that, in the case of non-oscillating singularities (see 3), it is necessary and suf cient to consider the maximum local lines of the wavelet coef cients situated inside the in uence cone, {TX (a, t) : |t t0 | < c a}, to be able to characterize the local regularity of the process In addition, the practical use of this property is made more dif cult by the necessarily nite resolution imposed by the sampling of the data, which does not permit detailed scrutiny of the data beyond a minimum scale, which is noted by convention a = 1 Furthermore, the different aspects of the study of the local regularity of a function constitutes an important object in other chapters of this work This is of true for 3, which tackles the issue of the characterization of functional regularity spaces, of 6 and 5 which expose the case of multifractional processes and their sample path regularity, and nally of 1, which presents the multifractal spectra as statistical and geometric measures of the distribution of pointwise singularities of a process Finally, let us note that, as previously indicated in section 224, the increments of stochastic stationary processes with stationary increments for which the H lder

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