Scaling, Fractals and Wavelets

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Moreover, this result is optimal, ie there exists a function of global regularity > 0 such that p (x) = l (x) for all x outside a set of zero Hausdorff dimension THEOREM 16 Let 0 < < 1 and f : [0, 1] [ , 1] be a lower limit of continuous functions Let g : [0, 1] [ , 1] be a lower semi-continuous function Assume that for all t [0, 1], f (t) g(t) Then, there exists a continuous function F : [0, 1] R such that: for all x, l (x) = g(x); for all x outside a set of zero Hausdorff dimension, p (x) = f (x) This theorem shows that, when the compatibility condition f (t) g(t) is satis ed, we can simultaneously and independently prescribe the local and pointwise regularity of a function outside a small set These two measures of irregularity are thus to some extent independent and provide complementary information 134 Signal dimension theorem Let us investigate the relationships between the dimension of a signal and its H lder exponents There is no general result concerning the Hausdorff dimension, apart from obvious upper bounds resulting from the inequalities dim( ) Dim( ) ( ) Here is a result for Dim( ) [TRI 86a] THEOREM 17 If is the graph of a continuous function f , then: 2 sup (x)

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Dim( )

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The same inequalities are true if we use the grid of the dyadic intervals: 2 sup (x)

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Dim( )

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We do not provide the demonstration of these results, which requires an evaluation of the packing measure of the graph In the same context, we could show that if the local H lder exponents (un (x)) tend uniformly to a real , then this number is also equal to ( ) However, a much more interesting equality may be given for Minkowski-Bouligand dimension of the graph which is both simple and general THEOREM 18 Let f be a continuous function de ned on an interval J, and non-constant on J For any > 0, let us call -variation of f on J the arithmetic mean of -oscillations: 1 f, [x , x + ] J dt var (f ) = |J| J

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Fractal and Multifractal Analysis in Signal Processing

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then: ( ) = lim sup 2

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log var (f ) log

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The assumption that f is not constant is necessary, as otherwise the oscillations are all zero and var (f ) = 0 In this case, the graph is a horizontal segment and the value of its dimension is 1 Proof A proof [TRI 99] using geometric arguments consists of estimating the area of the Minkowski -sausage ( ) We show that this is equivalent to that of the union of the horizontal segments t J [t , t + ] {z(t)} centered on the graph This is equal to the variation var (z) EXAMPLE 17 The graph of a self-af ne function de ned on J = [a, b] may be obtained as the attractor of an iterated functions system, like the self-similar curves of Example 12 (see also s 9 and 10) For this, it is suf cient to de ne: an integer N 2; N + 1 points in the plane A1 = (x1 , y1 ), , AN +1 = (xN +1 , yN +1 ), such that x1 = a < < xi < < xN +1 = b; N af ne triangular applications of the plane T1 , ,TN , such that, for each Ti , the image of the segment A1 AN +1 is the segment Ai Ai+1 These may be written as: Ti = i hi 0 i +

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where 0 < i = (xi+1 xi )/(b a) < 1 and | i | < 1 The ve parameters of Ti are related by the relations Ti (A1 ) = Ai and Ti (AN +1 ) = Ai+1 In the particular case where i = 1/N for any i and i | i | 1, we may verify that if = N k , the quantity var (f ) is of the order of (( i | i |)/N )k We then obtain: ( ) = 2 |log( log i | i | i | i |)/N | =1+ log N log N

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(see the classical example of Figure 13 where N = 4 and i = 1 for any i) The 2 H lder exponent is calculated using 4-adic intervals Its uniform value is 1 Therefore 2 ( ) = Dim( ) = 3 Let us note that the Hausdorff dimension, strictly lower than 2 3 2 , is much more dif cult to estimate [MCM 84]

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