0 k<2n 1 n Pk cn 2k+i

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(97)

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for any n > 1 where, for any k {0, , 2n 1 1}:

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n Pk

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cn 1 k

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n j n 1 j j=1 ij (k) n 1 j 2 j 2 j=1 | 0 | + | 1 |

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Scaling, Fractals and Wavelets

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where the sequence (i1 (k), , in 1 (k)) is the single sequence of {0, 1}n 1 n 1 n j 1 such that we have k = Moreover, for any n > 1, we have j=1 ij (k)2 n n 1 Pk = 1 0 k<2 Unfortunately, such a procedure does not provide us with appropriate representations in practice and the reasons for this can be analyzed as follows: each cj is de ned as the ratio of two wavelet coef cients While it is assumed that k j all wk are non-zero, some arbitrarily small values are likely to exist for most real applications This yields both very large and small values of cj Obviously, such a k large variation range is an issue for the proposed modeling, since all the coef cients {cj , k = 0, , 2j 1} are replaced by only two values and thus a control on the k dispersion of the cj is needed However, what draws our interest is the representation k of irregular signals (otherwise, a fractal approach would not make sense) As far as irregular signals are concerned, energy is present at most scales, and consequently the majority of the cj should vary within an intermediate range Moreover, from k a fractal analysis viewpoint, large cj are not interesting as they do not contribute k to the regularity of f In addition, if we assume that f is nowhere differentiable, then the H lder exponent in each point is less than 1; thus, many cj , including k those which control the multifractal properties of f , will belong to [ 1 , 1] (see 2 [DAO 98] for details) Thus, it appears reasonable to ignore, in our representation, the large cj , and consider only those that are less than 1 More precisely, we keep k unchanged the cj when they are greater than 1 and we calculate ( j ) that yield the i k best L2 -approximation by considering only the remaining cj Evidently, for this k strategy to make sense, it is necessary that the cardinal of {cj : cj 1} is negligible k k as compared with that of {cj : cj < 1} This depends on the nature of the signal and k k on the choice of g = These constraints lead us to the following criteria for the sub-optimal choice of the wavelet analysis:

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j (C1) {wk = 0} =

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(C2)

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the cardinal of {cj : cj < 1} is maximal k k

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In practice, because of edge artifacts, wavelet decomposition is often limited to a certain scale j0 > 0 As a consequence, the (cj ) are de ned only for j > j0 In k practice, if we write the signal f as:

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f (x) =

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0 k<2j0

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aj0 (2j0 x k) + k

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n=j0 0 k<2n

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n wk (2n x k)

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then the problem at hand is to nd, for any j > j0 , two positive scalars j and j such 0 1 ) satisfying that, when we replace all (cj ) satisfying cj < 1 (respectively (cj 2k 2k 2k+1

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j j 2 cj 2k+1 < 1) by 0 (respectively 1 ), we obtain the best L -approximation of the original signal f The resulting approximate signal f is hence a WSA function de ned by:

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f (x) =

0 k<2j0 J 1

aj0 (2j0 x k) + k

0 k<2j0

j wk0 (2j0 x k)

n=j0 +1 (i1 ,,in ) {0,1}n

wj0 j0

j0 p p=1 ip 2

sgn wn

n n p p=1 ip 2

j=j0 +1

cj

j j p p=1 ip 2

1 1 Sin Si1 (x)

where: c

j j p p=1 ip 2

cj = j ij

j j p p=1 ip 2

if cj

j j p p=1 ip 2

1 (98)

otherwise

wk n

Let us observe that, for n > j0 and k {0, , 2n 1}, the wavelet coef cient of f is given by:

n n wk = sgn(wk ) w[j0 n

k 2n j0

] j=j0 +1

cj [

k 2n j

Since the study is restricted to orthogonal wavelet transforms that preserve energy the goal is to nd two positive sequences ( n )n=j0 +1,,J 1 and 0 ( n )n=j0 +1,,J 1 that satisfy: 1