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IFS: Local Regularity Analysis and Multifractal Modeling of Signals
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Let us consider the operator W : H H de ned by:
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We call any set A H which is a xed point of W an attractor of the IFS {K, Sn : n = 1, 2, , m}, ie, it veri es: W (A) = A An IFS always possesses at least one attractor Indeed, given any set G H, the closure of the accumulation set of points {W (m) (G)} , with W (m) (G) = m=1 W (W (m 1) (G)), is a xed point of W If all Sn functions are contractions, then the IFS is said to be hyperbolic In this case, W is also a contraction for the Hausdorff metric; thus, it possesses a single xed point which is the single attractor of the IFS When the IFS is hyperbolic, the attractor can be obtained in the following manner [BAR 85a]: let p = (p1 , , pm ) be a probability vector with pn > 0 and n pn = 1 From the xed point x0 of S1 , let us de ne the sequence xi by successively choosing xi {S1 (xi 1 ), , Sm (xi 1 )}, where the probability pn is linked to the occurrence xi = Sn (xi 1 ) Then, the attractor is the closure of the trajectory {xi }i N In this chapter, we focus on IFS which make it possible to generate continuous function graphs [BAR 85a] Given a set of points {(xn , yn ) [0; 1] [u; v], n = 0, 1, , m}, with (u, v) 2 , let us consider the IFS given by m contractions Sn (n = 1, , m) which are de ned on [0; 1] [u; v] by: Sn (x, y) = Ln (x); Fn (x, y) where Ln is a contraction which transforms [0; 1] into [xn 1 ; xn ] and where Fn : [0; 1] [u; v] [u; v] is a contraction with respect to the second variable, which satis es: Fn (x0 , y0 ) = yn 1 ; Fn (xm , ym ) = yn (91)
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Then, the attractor of this IFS is the continuous function graph which interpolates the points (xn , yn ) In general, this type of function is called a fractal interpolation function [BAR 85a] The most studied class of IFS is that of af ne iterated function systems, ie, IFS for which Ln and Fn are af ne functions We will study this class later We also assume that the interpolation points are equally spaced Then, Sn (0 n < m) can be written in a matrix form as: Sn t x = 1/m 0 an cn t n/m + x bn
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Let f be the function whose graph is the attractor of the corresponding IFS Let us note that once cn is xed, an and bn are uniquely determined by (91) so as to ensure the continuity of f We are now going to calculate the H lder function of f and see if we can control the local regularity with these af ne iterated function systems PROPOSITION 91 ([DAO 98]) Let t [0; 1) and 0 i1 ik be its base m decomposition (when t possesses two decompositions, we select the one with a nite number of digits) Then: f (t) = min lim inf log(ci1 cik ) log(cj1 cjk ) log(cl1 clk ) , lim inf , lim inf k ) k ) k + k + log(m log(m log(m k )
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where, for any integer k, if we note tk = m k [mk t], the k-tuples (j1 , , jk ) and (l1 , , lk ) are given by: t+ = tk + m k = k t = tk m k = k
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