Wavelets and Fractals

Transcription

1 Wavelets and Fractals Bikramjit Singh Walia Samir Kagadkar Shreyash Gupta 1 Self-similar sets The best way to study any physical problem with known symmetry is to build a functional basis with symmetry as close to that of the problem under consideration as possible. For this reason a system of spherical functions is the best one to fit the problem with spherical symmetry, while Fourier decomposition is apt for problems invariant under translations. The matter turns out to be even more conspicuous when one studies fractals - non-differentiable self-similar objects. On the other hand, the invariance under scale transformations (or self-symmetry) is the very symmetry group wavelet transformation is based upon; on the other, with no requirements of differentiability, the wavelet transform seems to be an ultimate tool for studying singular objects. According to B. Mandelbrot, a fractal is an object a part of which in some way similar to the whole. Fractal objects are present everywhere in Nature. The fractal behaviour is common to hydrodynamic turbulence, the Feynman paths in quantum mechanics, financial time series, terrestrial landscapes and thin film surfaces, etc. The most remarkable mathematical examples of fractals are: triadic Cantor sets, Sierpinski gasket, Koch triadic curve, etc. In contrast to the classical geometrical objects, the dimension of a fractal set is not an integer number. For a classical object, if covered with balls (or boxes) of size δ, the total no. of balls required for the minimal covering increases as N(δ) δ D, for δ 0 (1) For regular geometrical objects the dimension D is always an integer: D=1 for line, D=2 for surface etc. In general D may or may not be an integer. For the Triadic Cantor set obtained by sequentially erasing the central 1/3 part of the unit interval(see Fig. 1) at the ith stage of the construction 1

2 process there are N i = 2 i parts of size l i = 3 i ; thus N(l) l D F, D F = ln 2/ ln 3 which is less than 1 (the dimension of a differentiable curve). The dimension of the Koch curve, shown in Fig. 2, in contrast is greater than that of a differential curve D F = ln 4/ ln 3. However fractal sets are self-similar by construction. Figure 1: The construction of the triadic Cantor set from the unit length rod 2 Hausdorff-Bezikovich dimension It is convenient to characterize fractal sets by the power behaviour of corresponding measures. Let us consider a set A and a function M d (A) = i δd, which is a measure defined on the coverage of the set A by δ-balls, δ 0. If there exists such d R that the measure M d (A) has a discontinuity at d = D F lim M d (A) = δx d δ 0 i N(δ)δ d (2) x i A The set A is said to have Hausdorff-Bezikovich dimension D F. For example, the fractal dimension of a triadic Cantor set is easily derived from its construction procedure. In the ith generation (see Fig. 1), we have N = 2 i equal parts of length l = 3 i. Thus N = 2 i = 2 ln l ln 3 = l ln 2 ln 3 DF = ln (3) ln 3 2

3 3 Fractals and multifractals There are two types of fractals: regular (or geometric) fractals, as those mentioned above, and random fractals - self-similar objects without any regular geometric structure, e.g. the path of a Brownian particle. Both of them are characterized by singularity strength (or Lipschitz-Holder exponent) and the fractal dimensions f(α) of all subsets I α with the singularity strength equal to α. Any subset of a regular geometric fractal is similar to the whole fractal, and hence has the same fractal dimension. In contrast different subsets of real fractals present in Nature may have different dimensions. Therefore for simple geometrical fractals such as the Cantor set, there is no dependence of the singularity strength of the subsets on their fractal dimension both are constants (α = f(α) =const). For real fractals appearing in nature such dependence is often present, sometimes with intrinsic behaviour. Figure 2: The iterative construction of the Koch curve: on each stage of the construction the central one-third part of unit interval is substituted by polygon with two equal sides The theory of multifractals fractal objects with the dimensions of sub- 3

4 sets dependent on the singularity strength stems from the Mandelbrot papers and was developed by several authors in mid-1980s. It is known as the multifractal approach or multifractal formalism. The simplest example of a multifractal set is a binomial multiplicative process. Let us consider a population of arbitrary objects initially distributed homogenously on a unit interval, and a process which redistributes the population with the probability p to the left half of the interval, and with probability q = 1 p to the right half. After the first iteration we will have the probability measure (p, q) for the whole interval; after the second iteration (p 2, pq, qp, q 2 ), etc. After the nth iteration the unit interval is divided into N = 2 n cells which can be labeled by binary coordinates n x = 2 ν x ν, x ν = 0, 1. (4) ν=1 The population of a cell x containing k digit zeros in binary coordinates is p k q N k ; the sum of population s of all cells is equal to the initial population (p + q) n = 1 n = 1. The population measure of the unit interval after the 8th stage of binomial multiplicative process with p = 0.25 is shown in Fig. 3. Let ζ = k/n be the relative number of zeros in the binary representation (4). For a given ζ there exists cells with the measure N n (ζ) = n! (ζn)!((1 ζ)n)! (5) µ ζ = n (ζ), where (ζ) = p ζ (1 p) 1 ζ. (6) According to definition of fractal dimension (2), the population of a given cell (4) should scale as N n (ζ) δ f(ζ), where f(ζ) is the fractal dimension of the subset containing exactly k = ζn zeros in the binary representation. Substituting the Stirling approximation for the factorial into the equation (5) we get N n (ζ) 1 2πnζ(1 ζ) exp( n(ζ ln ζ + (1 ζ) ln(1 ζ)), (7) and taking into account that n = ln δ/ln2 we arrive at the explicit equation for the fractal dimension of the subsets generated by binomial multiplicative process as a function of ζ f(ζ) = ζ ln ζ + (1 ζ) ln(1 ζ). (8) ln 2 4

5 Figure 3: Measure generated by the binomial multiplicative process with p=0.25. The x-axix is in x = i2 8 units. As seen from Fig. 4, the sets with equal number of 1 and 0 in binary representation (4) have maximal fractal dimension. 4 Multifractals: Thermodynamic formalism and wavelets The multifractal approach accounts for the statistical scaling properties of singular measures by means of singularity spectrum which determines the behaviour of f(h), just as the poles of a complex variable function defines its behaviour. If a fractal subset B x0 = {x ρ(x, x 0 ) l} (9) of the set A, where ρ is a metric on A and l is the diameter of B, is covered by δ-balls centered at x i. For a mono-fractal set B the power behaviour of µ(b) in δ 0 limit is described by the singularity strength of the measure. lim µ(b) δ 0 lh, (10) where h does not depend on x 0. In general multifractal case the power behaviour of the measure can be dependent on the point. The fractal dimension of the sets of all points x i A, such that µ(b xi ) δ h, where B xi is 5

6 sufficiently small neighborhood of x i, is denoted by f(h). The simplest measure is incapable of accounting for the variety of all multifractal properties. In multifractal analysis the properties of the singular objects are described Figure 4: Fractal dimensions of substes generated by binomial multiplicative process as a function of relative number of zeros in binary representation. in the terms of the measure M d (q, δ), more general than M d (δ): M d (q, δ) x i A µ q i δd = Z(q, δ) δ d τ(q). (11) The practical reason for introducing a weighted measure is clear enough. If one considers a set A covered with the cells of size δ, it may happen that some cells contain only a few points of A, while some other cells contain plenty of them. The Hausdorff measure accounts for all non-empty cells with the same weight. This is unfair either to low populated cells or to densely populated cells. What one needs is a method separately sensitive to both extreme cases. This method, called weighted curdling, is based on measure, with µ i = N i /N being the relative population of the ith δ-cell, where N = #A. In the case of negative q the measure is more sensitive to low populated cells, and the positive q measures are more sensitive densely populated cells. The multifractal formalism has been established and developed to account separately for the subsets of different singularity strength and to reveal the scaling properties of singular measures arising in different physical situations, first of all in hydrodynamic turbulence. 6

7 The multi-fractal formalism associates the fractal dimension f(h) with the fractal subset of a given singularity strength h of the considered set. By thermodynamic analogy it is possible to consider Z(q, δ) = µ q i δ τ(q) (12) as a partition function, with q regarded as a counter part of the inverse temperature. The power behaviour of the partition function in δ 0 limit is expressed in terms of the mass exponent τ(q), an analog of free energy in thermodynamics. The scaling exponent τ(q) and the fractal dimension f(h) are related by means of the Legendre transform τ(q) = min[f(h) qh], f(h) = min[qh + τ(q)]. (13) h q The so called generalized fractal dimension D q also called a Renyi dimension, is expressed by means of τ(q) D q = lim δ 0 ( 1 q 1 ln Z(q, δ) ) = τ(q) ln δ 1 q, (14) where D 0 is the fractal dimension of the support of the measure D 1 (information dimension), and D 2 the two point correlation dimension. For a monofractal, as it was already mentioned, α = f(α) = D 0 = D q q. For multifractals D q is nothing but the dimension of the subset for which the singularity strength is equal to q: D q = d(α = q).it usually decreases with q. The multi fractal formalism was first successfully applied in physics to the description of cascade processes in hydro dynamic turbulence, for example, the so called p-model. To recall it briefly, the p-model describes a non-equal sharing of the energy flux from a large eddy of size l to 2 d small ones of size l/2 where d is space dimension. The simplest hypothesis is that a p 1 fraction of the energy goes to one half of them and a fraction p 2 = 1 p 1 goes to another half. The qth moment of the energy dissipated by the eddies of the given size l can be used as a measure E q l = E q L( l L) (q 1)Dq, (15) where L is the maximal size of the eddies the process starts with. For the nth stage of the process there will be all possible eddies: E l = p n m 1 p m 2 E l, m n, l = L/2 n, (16) 7

8 and hence the Renyi dimension can be written in the form Two limiting cases D q = log 2 [p q 1 + pq 2 ] 1 1 q, 1/2 p 1 1. (17) D = log 2 p 1 1 and D = log 2 p 1 2 (18) are of mostly of experimental interest. They correspond to the domination of the most strong and the most weak domains of the cascade. The typical dependence D q vs. q is shown in Fig. 5. For the case of p-model D(q) is a decreasing function of q. Figure 5: The dependencies of the Renyl dimensions D(q) versus q in the p-model for the cases: a)p 1 = 0.6, p 2 = 0.4, b)p 1 = 0.9, p 2 = 0.1 The partition function Z(q, a) can be directly evaluated using the calculated set of wavelet coefficients W ψ (a, x): Z(q, a 0 ) = W ψ (a, x) q. (19) overall maxima(x,a a 0 ) In this construction, to calculate the partition function for a given scale a 0 we have to sum up over all maxima lines l : (x, a a 0 ) starting from (x, a 0 ) and going to smaller scales a a 0. In practice, this often means that it is sufficient to take a section W ψ (a = a 0, x) and sum up over all maxima in x. The wavelet coefficients in the partition function are taken in L 1 norm W ψ (a, b)[f] = 1 ψ ( ) x b a a f(x)dx. 8