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with two other methods which make it possible to obtain satisfactory estimations The rst method is based on the wavelet transform and is called the wavelet transform maxima modules (WTMM) (cf 3 for a detailed description of this method) The second method [GON 92b] uses Wigner-Ville distributions 951 Principles of the method For the sake of simplicity, our study is limited to continuous functions on [0; 1] The calculation algorithm of the H lder exponent is based on Proposition 94 To apply this proposition to the calculation of the H lder exponents for a real continuous signal f , we have to begin by calculating the coef cients cj of a GIFS k whose attractor is f This amounts to solving the inverse problem for GIFS, which is a generalization of the ordinary inverse problem for IFS The latter problem was studied by many authors, either from a theoretical point of view [ABE 92, BAR 88, BAR 85a, BAR 85b, BAR 86, CEN 93, DUV 92, FOR 94, FOR 95, VRS 90] or in

DEFINITION 12 Let f : R R be a function, s > 0 and x0 a real number Then f C s (x0 ), if and only if there is a real number > 0, a polynomial P of degree s and a constant C, such that x B(x0 , ), |f (x) P (x x0 )| C|x x0 |s (116)

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success of any business firm The larger (and harder) question is how to create this degree of separation from the competition What are the steps our organization can take to give us the operational leg-up on our competitors Many companies experiment with various formulas: cost cutting, down-sizing and corporate reorganizations, total quality focus, and so forth While all of these ideas may be useful and appropriate in their particular circumstance, this book has offered a different premise For the projectbased organizations that will increasingly dominate the world stage, doing it right is not a sufficient substitute for getting it right; that is, being internally efficient is not the same (or as useful) as offering a product that fits customer needs We made the argument that it is only in creating and maintaining a customer-based outlook that project organizations are best able to reap the benefits of a new, chaotic, but highly lucrative marketplace Creating this customer focus is not easy, despite the obvious surface merits of such a philosophy Indeed, we suspect that every firm we have ever consulted with, studied, or worked for would quickly assert that they have such a customer focus to their

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sup s/f C s x0 If 0 < s < 1, then the polynomial P is simply the constant f (x0 ) and the increments of f over [x0 , x0 + ] are indeed compared to Considering P allows us to take into account the higher order singularities, ie in the derivatives of f We remove the regular part of f to exhibit the singular behavior Consider for example f (x) = x + |x|3/2 Using P allows us to nd p = 3 , whereas the simple increment 2 would give p = 1, a non-signi cant value in this case The pointwise exponent, whose de nition is natural, has a geometric interpretation To begin with, let us remove the signal s regular part, thus performing the difference f (x) P (x x0 ) Around x0 , the signal thus obtained is entirely contained in a hull of the form C|x x0 | p + for any > 0, and this hull is optimal, ie any smaller hull C|x x0 | p , does not contain an in nite number of points of the signal We observe that the smaller the p is, the more irregular f is in the neighborhood of x0 and vice versa In addition, p > 1 implies that f is derivable at x0 , and a discontinuous function at x0 is such that p = 0 In many applications, the regularity of a signal is as important as, or more important, than its amplitude For example, the edges of an image do not vary if an af ne transformation of the gray-levels is carried out This will modify the amplitude, but not the exponents The pointwise H lder exponent will thus be one of the main ingredients for the fractal analysis of a signal or image Generally speaking, measuring and studying the pointwise exponent is particularly useful in the processing of strongly irregular signals whose irregularity is expected to contain relevant information Examples includes biomedical signals and images (ECG, EEG, echography, scintigraphy), Internet traf c logs, radar images, etc The pointwise exponent, however, presents some drawbacks For example, it is neither stable by integral-differentiation nor, more generally, under the action of pseudo-differential operators This means that it is not possible to predict the f exponent at x of a primitive F of f knowing p (x) It is only guaranteed that F f p (x) p (x) + 1 In the same way, the exponent of the analytical signal associated with f is not necessarily the same as f This is a problem in signal processing, since.

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