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Wavelet Based Approach to Fractals and Fractal Signal Denoising Carlo Cattani DiFarma, University of Salerno, Via Ponte Don Melillo I-84084 Fisciano (SA) [email protected]

Abstract. In this paper localized fractals are studied by using harmonic wavelets. It will be shown that, harmonic wavelets are orthogonal to the Fourier basis. Starting from this, a method is defined for the decomposition of a suitable signal into the periodic and localized parts. For a given signal, the denoising will be done by simply performing a projection into the wavelet space of approximation. It is also shown that due to their self similarity property, a good approximation of fractals can be obtained by a very few instances of the wavelet series. Moreover, the reconstruction is independent on scale as it should be according to the scale invariance of fractals. Keywords: Harmonic Wavelets, Weierstrass Function, Scale Invariance, Fractals, Denoising.

1

Introduction

Wavelets are some special functions ψkn (x) [6,11,12,17,22] which depend on two discrete valued parameters: n is the scale (refinement, compression, or dilation) parameter and k is the localization (translation) parameter. These functions fulfill the fundamental axioms of multiresolution analysis so that by a suitable choice of the scale and translation parameter one is able to easily and quickly approximate any localized functions (even tabular) with decay to zero. Due to this multiscale approach the approximation method by wavelets has some advantages due to the minimum set of coefficients for representing the phenomenon and the direct projection into a given scale, thus giving a direct physical interpretation of the phenomenon. In each scale the wavelet coefficients and, in particular, the detail coefficients βkn describe “local” oscillations. Therefore wavelets seems to be the more expedient tool for studying those problems which are localized (in time or in frequency) and/or have some discontinuities. Among the many different families of wavelets a special attention was paid to the complex Littlewood Paley bases, also called harmonic wavelets [4,6,7,8,9,10, 18, 19]. They seems to be one of the most expedient tool for studying processes which are localized in Fourier domain or have high frequency oscillation in time. 

Preliminary results presented at the International Conference on Computational Science and Applications (ICCSA 2008), June 30-July 3, 2008 Perugia (It) [7].

M.L. Gavrilova and C.J.K. Tan (Eds.): Trans. on Comput. Sci. VI, LNCS 5730, pp. 143–162, 2009. c Springer-Verlag Berlin Heidelberg 2009 

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It has been recently observed [7, 9, 10] that also some kind of highly oscillating functions and random-like or fractal-like functions can be easily reconstructed by using the harmonic wavelets. For this reason they will be used, in the following, to study the regularity of a slightly modified Weierstrass function [2, 3, 21], which has a fractal behavior. Indeed the Weierstrass function is based on periodic functions ranging from −∞ to ∞, therefore a slight modification is needed in order to let its reconstruction, by harmonic wavelet series, converges in L2 (R). Therefore we will focus on the so-called deterministically self-similar signals (see e.g. [22]) which are deterministic self-similar functions with compact support. The self-similarity property of fractals will be analyzed in terms of wavelet coefficients (see also [22]) and it will be shown that to reconstruct a fractal only a small set of coefficients, at the lower scale, are needed. In the last part of this paper a signal made by a 2π–periodic function and a localized self-similar function will by analyzed by using harmonic wavelets. A fundamental theorem, which characterizes harmonic wavelets with respect to the periodic functions, will be given. It will be shown that harmonic wavelets are orthogonal to the Fourier basis, for this reason the wavelet reconstruction of a 2π–periodic function will trivially vanish. Based on this, the denoising of the signal will be obtained by a projection into a wavelet space with no error (contrary to what usually happens with ordinary techniques of hard/soft denoising). In this case the only error is due to the scale of wavelet approximation and it can be shrunk to zero by simply increasing the scale.

2

Preliminary Remarks on Self-similar Functions

Self-similarity, or scale invariance, is a characteristic feature of many natural phenomena [1, 3, 14, 13, 15, 16, 20, 21, 22], biological signals, geological deformations, natural phenomena as clouds, tree branches evolution, leaves formation, stocks and financial data etc. All these phenomena can be represented by some kind of singular (nowhere differentiable) functions which are fractals. They are mathematically known through tabular and recursive-analytical formulas both deterministic and stochastic thus being represented by a very large family of functions. Most of them are characterized by the scale invariance (or self-similarity). This, however, is a feature of many functions, including fractals, and behaves as if the function is identical to any of its rescaled instances up to a suitable renormalization of the amplitude. In particular, the following definition holds: Definition 1: The function f (x) , x ∈ A ⊂ R is scale invariant with scaling exponent H, (0 < H < 1) (Hausdorff dimension) if, for any μ ∈ R, it is f (μx) = μH f (x) .

(1)

μ = 2C ⇒ C = log2 μ ,

(2)

It is assumed μ > 0 so that

Wavelet Based Approach to Fractals and Fractal Signal Denoising

and the scale invariance reads as   f 2C x = 2CH f (x) .

145

(3)

Indeed the condition (1) is quite strong and can’t be easily fulfilled, in the sense that there are not so many functions for which (1) holds true. For example the Cantor set function is invariant only for μ = 3k , k ∈ N. For this reason the invariance it is assumed to be satisfied by only few values of μ and the corresponding weakened version of scale invariance is called discrete scale invariance [1, 2]. Some examples of such a kind of functions are: – The Weierstrass function [2, 21] f (x) =

∞ 

ak cos bk x (a > 1, b > 1) .

(4)

k=1

– The Mandelbrot-Weierstrass function [2] f (x) =

∞ 

  a−kH 1 − cos ak x (a > 1, 0 < H < 1) .

(5)

k=−∞

– The Cantor function [13, 15, 16] ⎧ f0 (x) = x ⎪ ⎪ ⎪ ⎪ ⎪ ⎧1 ⎪ ⎪ ⎪ ⎪ fn (3x) ⎪ , 0 ≤ x ≤ 13 ⎪ ⎪ ⎪ ⎨ 2 ⎪ ⎪ ⎪ ⎨ 1 2 1 ⎪ ⎪ f (x) = , 1) .

(6)

A characteristic feature of the above functions is that they show also some high frequency oscillations, thus suggesting us that somehow it should be possible to study fractals with high oscillating functions. 2.1

Self-similar Functions with Finite Energy

Our investigation will be restricted to the family of self-similar functions f (x), such that f (x) ∈ L2 (R). ⎡ ∞ ⎤1/2  2 Definition 2. Let f (x) : R → C, f (x) ∈ L2 (R) if ⎣ |f (x)| dx⎦ < ∞. −∞

Therefore some additional hypotheses should be done, on the self-similar function, so that it would belong to L2 (R). However it can be easily checked that

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Theorem 1. If f (x) is self-similar then f (x) ∈ L2 (R). Proof: In fact, by taking into account definition 2, it is ⎤1/2 ⎡ ∞  2 ⎣ |f (x)| dx⎦ ≤M 2n 1, m ≤ 2n

and, in particular, 

 ϕ0k (x) , e2πmix = 0, m > 1 .   For the second product ψkn (x) , e2πmix we have, according to (12), 

ψkn

(x) , e

2πmix



∞ = 2π −∞

2−n/2 −iωk/2n e χ (ω/2n ) δ (ω − 2mπ) dω 2π

from where 

4

ψkn

(x) , e

2πmix



 =

0, m > 2n+1 1, m ≤ 2n+1 .

Harmonic Wavelet Reconstruction

Let f (x) ∈ L2 (R), its reconstruction in terms of harmonic wavelets can be obtained by the formula (see e.g. [4, 7, 8, 9, 10, 19])   ∞ ∞  ∞   0 n n αk ϕk (x) + βk ψk (x) f (x) = n=0 k=−∞

k=−∞

 +

∞  k=−∞

α∗k ϕ0k

(x) +

∞  ∞ 

 n βk∗n ψ k

(17)

(x)

n=0 k=−∞

which involves (8) and the corresponding conjugate functions. The series at the r.h.s. represents the projection of f (x) into the ∞-dimensional subspace of ∞ L2 (R), done by the operator W∞ ; so that formally we can write ∞ f (x) = W∞ [f (x)]

and since f (x) ∈ L2 (R) there follows its convergence.

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151

Thanks to the orthonormality of the basis, the wavelet coefficients can be computed by ⎧ ∞ ⎪   ⎪ 0 ⎪ α = f (x) , ϕ (x) = f (x) ϕ0k (x) dx ⎪ k k ⎪ ⎪ ⎪ ⎪ −∞ ⎪ ⎪ ⎪ ∞ ⎪ ⎪   ⎪ ∗ 0 ⎪ ⎪ αk = f (x) , ϕk (x) = f (x) ϕ0k (x) dx ⎪ ⎪ ⎨ −∞

(18)

∞ ⎪ ⎪ n ⎪ n n ⎪ βk = f (x) , ψk (x) = f (x) ψ k (x) dx ⎪ ⎪ ⎪ ⎪ ⎪ −∞ ⎪ ⎪ ⎪ ∞ ⎪   ⎪ ⎪ n ∗n ⎪ ⎪ β = f (x) , ψ (x) = f (x) ψkn (x) dx ⎪ k ⎩ k −∞

or, according to (14), with the equivalent corresponding equations in the Fourier domain [4, 7, 8, 9, 10], ⎧  ∞  2π ⎪ 0 0   ⎪ αk = 2π f (x), ϕk (x) = k (ω)dω = f (ω)ϕ f(ω)eiωk dω ⎪ ⎪ ⎪ −∞ 0 ⎪ ⎪ ⎪ ⎪  2π ⎪ ⎪ ⎪ 0  ∗ ⎪ f(ω)e−iωk dω = 2π

f (x), ϕ (x) = . . . = α ⎪ k ⎪ ⎨ k 0 (19)

 2n+2 π ⎪ ⎪ n ⎪ −n/2 n n  ⎪ βk = 2π f (x), ψk (x) = . . . = 2 f(ω)eiωk/2 dω ⎪ ⎪ ⎪ 2n+1 π ⎪ ⎪ ⎪ ⎪ n+2  ⎪ 2 π ⎪ ⎪ n n  ⎪ −n/2 ⎩ β ∗nk = f (x), ψ f(ω)e−iωk/2 dω , k (x) = . . . = 2 2n+1 π

being f (x) = f(−ω). The wavelet approximation, at the scale N , is obtained by fixing an upper limit in the series expansion (17), so that N < ∞, M < ∞ . There follows that N [f (x)] defined as the function f (x) is approximated by the projection WM  N f (x) ∼ [f (x)] = = WM

def

M  k=0

n=0 k=−M

M 

N M  

 +

αk ϕ0k (x) +

N M  

k=0

α∗k ϕ0k (x) +

n=0 k=−M

 βkn ψkn (x)  n βk∗n ψ k

(x) .

(20)

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As a consequence of (15),(17),(20), we have the following   Corollary 1. The harmonic wavelet projection of the Fourier basis e2πmix , with m ∈ N, is identically zero:   N 2πmix WM e =0

5

∀M, N ≤ ∞ .

,

(21)

Scale Invariance and Wavelet Coefficients

Let us compute the wavelet coefficients (16) for a self-similar function (3). It can be shown that (see also [22]) Theorem 3. The scaling coefficients of a self-similar function (3) are not independent on the lower scale (n = 0) scaling coefficients since it is −C(H+1/2) αk αC k = 2

,

∀k ∈ Z

(22)

with C defined by (2). ∞ f (x) ϕ0k (x) dx , by the substitution x = 2C X it is

Proof : From (16), αk = −∞

∞ αk = 2

C

    f 2C X ϕ0k 2C X dX

−∞

and, by taking into account the definition (8),   ϕ0k 2C X = 2−C/2 ϕC k (X) . Thus, we have ∞ αk = 2

C

  (AA) C/2 f 2C X 2−C/2 ϕC k (X) dX = 2

−∞

2CH f (X) ϕC k (X) dX

−∞

∞

= 2C(H+1/2)

∞

f (X) ϕC k (X) dX

−∞

that is αk = 2C(H+1/2) αC k

(23)

so that (22) follows. Conditions (22),(23) can be also used to define a self-similar function, as well as the following:

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153

Corollary 2. A localized function is self-similar if there exists an integer n which is solution of the equation   α  1 log2  nk  . n= H + 1/2 αk

(24)

Equation (24) shows a relation between the values of the scaling coefficients and the scale parameter n. As can been seen by a direct computation, a simple solution of (24) is αnk = a−(H+1/2)n g(k) , a > 1 . Analogously, for the detail coefficients, it can be shown from (16) that Theorem 4. The wavelet coefficients of a localized self-similar function are not independent on the lower scale (n = 0) wavelet coefficients, since it is βkn+C = 2−C(H+1/2) βkn

,

∀n ∈ N, k ∈ Z

(25)

with C defined by (2). ∞ Proof : From

βkn

n

f (x) ψ k (x) dx , by the substitution x = 2C X, it is

= −∞

∞ βkn

=2

C

  n  f 2C X ψ k 2C X dX

−∞

and, by taking into account (8),  n n+C ψ k 2C X = 2−C/2 ψ k (X) , there follows ∞ ∞  C  −C/2 n+C (AA) C/2 n+C C n βk = 2 f 2 X 2 ψk (X) dX = 2 2CH f (X) ψ k (X) dX −∞

∞

= 2C(H+1/2)

−∞ n+C

f (X) ψ k

(X) dX ,

−∞

so that βkn = 2C(H+1/2) βkn+C . In other words for a self-similar function, characterized by a factor C, there are only ∞C+1 independent scaling-wavelet coefficients: ∞  ∞  ∞  ∞ {αk }k=−∞ , βk0 k=−∞ , βk1 k=−∞ , ...., βkC−1 k=−∞ . (26) According to this, the description of self-similar functions requires a number of coefficients which is less than those needed for regular functions.

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As a consequence of Eq. (25), we also have that Theorem 5. The wavelet coefficients of a self-similar function (3) are selfsimilar functions of the form: βkn = F (2n , k)

(27)

with F (2n , k) self-similar function of the scale F (μ2n , k) = μS F (2n , k) , being μ=2 so that

C

! " 1 , S=− H+ 2

(28)

,

  F 2n+C , k = 2−C(H+1/2) F (2n , k) .

(29)

Proof : From equation (25) let us write βkn = F (an , k) with the base a to be determined. It is     F an+C , k = F aC an , k and, according to (25),

  F aC an , k = 2−C(H+1/2) F (an , k) .

Let us assume that F (an , k) is a self-similar function so that F (λan , k) = λS F (an , k) . Comparing with the previous equation it is aC = λ , λS = 2−C(H+1/2) from where λ = 2C , S = − (H + 1/2) . For example, according to (29), we can take F (2n , k) = 2−n(H+1/2) k q , q ∈ R

(30)

which fulfills (29). The wavelet coefficients are βkn = 2−n(H+1/2) k q , q ∈ R

(31)

and the equation (25) becomes βkn+C = 2−(n+C)(H+1/2) k q , q ∈ R More in general we can write F (2n , k) = 2−n(H+1/2) g (k) and

βkn = 2−n(H+1/2) g (k) .

.

Wavelet Based Approach to Fractals and Fractal Signal Denoising

6

155

Wavelet Coefficients of the Weierstrass Function

This section deals with the harmonic wavelet reconstruction of a slightly modified version of the Weierstrass function (4). In fact, in order to give a good representation of these kind of functions and to study their properties by using wavelets one should localized them in space. Therefore we consider the following Weierstrass function ∞ 

f (x) =

a−kH cos ak x

,

(a > 1 , 0 ≤ H ≤ 1)

(32)

k=−∞

with a further condition to be limited in space, i.e. with compact support, as should be any physical signal. This implies that the sum will be taken up to finite fixed values. The Fourier transform of (32) is F [f (x)] =

∞  k=−∞

∞       1  a−kH F cos ak x = a−kH δ ω − ak + δ ω + ak , 2 k=−∞

so that according to (19) it is 2π αk = 0

=

f(ω) e−iωk dω =

∞ 

a−hH

h=−∞

1# 2

2π  ∞

a−hH

0 h=−∞ h

eia

k

h

+ e−ia

k

$ =

   1  δ ω − ah + δ ω + ah e−iωk dω 2 ∞ 

  a−hH cos kah ,

h=−∞

that is αk =

∞ 

 (32)  a−hH cos kah = f (k) .

h=−∞

Analogously we get ⎧ ∞  (32) ⎪ ∗ ⎪ α ⎪ = − a−hH cos(kah ) = − f (k) k ⎪ ⎪ ⎪ ⎪ h=−∞ ⎪ ⎪ ! ! " " ⎪ ∞ ⎨  k h (32) k n −n/2 −hH βk = 2 a cos a = f n n 2 2 ⎪ ⎪ h=−∞ ⎪ ⎪ ⎪ ! ! " " ∞ ⎪  ⎪ k h (32) k ⎪ −n/2 −hH ∗n ⎪ β = −2 a cos a = − f , ⎪ ⎩ k 2n 2n h=−∞

(33)

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and the reconstruction formula (17) for the Weierstrass function (32) gives ! " ∞  ∞     k n n ψ f (k) ϕ0k (x) + ϕ0k (x) + f (x) + ψ (x) k k 2n n=0 k=−∞ k=−∞ ! " ∞ ∞  ∞  k (11)  0 = f (k)Φk (x) + f Ψkn (x) n 2 n=0 k=−∞ k=−∞ %  ∞ ∞     = a−hH cos kah Φ0k (x) ∞ 

f (x) =

k=−∞

+

h=−∞



∞  ∞  n=0 k=−∞

∞ 

a

−hH

! cos

h=−∞

k h a 2n

"% Ψkn (x) .

Thus for the Weierstrass function the scaling parameter n increases the frequency of oscillations of the wavelet coefficients. Let us see, now, if the self similarity conditions can be fulfilled by the function (32), and then by the wavelet coefficients (22) that is let us show if there exists a factor C such that condition (25) holds. Theorem 6. The Weirstrass function (32) is self-similar, and fulfills (25), if ah = 2h+C . Proof : From definition (22), it is ∞ 

βkn+C = 2−(n+C)/2

a−hH cos

!

k

"

ah n+C

2 ! " ∞  k ah −hH a cos . 2n 2C

h=−∞



= 2−C/2 2−n/2

h=−∞ h

h+C

By assuming a = 2  βkn+C

we have

" k 2h+C =2 2 cos 2 2n 2C h=−∞  ! " ∞  k h −C(h+1/2) −n/2 −hH 2 cos 2 2 = 2−C(h+1/2) βkn , =2 2n −C/2

−n/2

∞ 

−(h+C)H

!

h=−∞

with

!

" k h+C 2 2n h=−∞ ! " ∞  k h −n/2 −(h+C)H =2 2 cos 2 . 2n

βkn = 2−n/2

∞ 

h=−∞

2−(h+C)H cos

Wavelet Based Approach to Fractals and Fractal Signal Denoising

157

Therefore the scale invariance is fulfilled for the Weierstrass series (32) when the series is taken in the form f (x) =

∞ 

2−kH cos 2k x

, (0 ≤ H ≤ 1) .

(34)

k=−∞

7

Denoising Fractal Signals

As application of the above theory let us consider a few examples of wavelet projection of signals made by two components: a 2π–periodic function and a fractal function (Weierstrass–like (34)). We show that, given the signal, where the two components are mixed together, it is possible to separate the periodic function from the other by a suitable projection. Let us consider, for example, the artificial signal (Fig. 1,a) f (x) =

5 1  −0.4k 1 cos 4πx + 2 cos 2k x 30 20

(35)

k=−5

where the amplitude of the periodic function is small comparing with the localized fractal pulse, and assume that the r.h.s. is unknown, so that we know only that there is a 2π–periodic component . The unknown functions, the periodic and the fractal pulse, are drawn in Fig. 1,b,d, respectively. After projection by harmonic wavelets, taking into account Eq. (21), it is   10 1 cos 4πx = 0 , W50 30 and we get (Fig. 1,c)  10 W50 [f

(x)] =

10 W50

 5 5 1  −0.4k 1  −0.4k k 2 cos 2 x ∼ 2 cos 2k x , = 20 20 k=−5

(36)

k=−5

so that the periodic function follows from the difference (Fig. 1, e) 1 10 cos 4πx ∼ [f (x)] . = f (x) − W50 30 As a second example, let us take the artificial signal (Fig. 2,a) f (x) = 5 sin 2πx +

5 1  −0.4k 2 cos 2k x 10

(37)

k=−5

where the fractal noise is neglectable with respect to the periodic function 5 sin 2πx, so that signal looks like a smooth function (Fig. 2,a). Let us assume

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Fig. 1. Extraction of a periodic function from a noised signal (the amplitude of the periodic function is small): a) the given noised signal f , Eq. (35); b) the fractal compo10 f , Eq. (36), giving nent of the noised signal; c) the projection of the noised signal W50 the approximation of the fractal component; d) the periodic component of the noised 10 f , giving the approximation of the periodic component signal; e) the difference f − W50

Wavelet Based Approach to Fractals and Fractal Signal Denoising

159

Fig. 2. Extraction of a fractal noise from a noised signal (the amplitude of the periodic function is large): a) the given noised signal f , Eq. (37); b) the fractal component of the noised signal; c) the projection of the noised signal W00 f , giving a very rough approximation of the fractal component; d) the periodic component of the noised signal; e) the difference f − W00 f , giving the coarse approximation of the periodic component

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Fig. 3. Extraction of a fractal noise from a noised signal (the amplitude of the periodic function is large): a) the given noised signal f , Eq. (37); b) the fractal component of 10 f , giving a good approxthe noised signal; c) the projection of the noised signal W50 imation of the fractal component; d) the periodic component of the noised signal; e) 10 f , giving a good approximation of the periodic component the difference f − W50

Wavelet Based Approach to Fractals and Fractal Signal Denoising

161

only that one component of the given signal (Fig. 2,a) is 2π–periodic. The unknown fractal and periodic components are drawn in Fig. 2,b,d, respectively. After projection by harmonic wavelets, taking into account Eq. (21), it is at the coarse level W00 [5 sin 2πx] = 0 , and we get (Fig. 2,c)  W00 [f

(x)] =

W00

 5 5 1  −0.4k 1  −0.4k k 2 cos 2 x ∼ 2 cos 2k x . = 10 10 k=−5

k=−5

The scale of approximation is very low and, as a consequence, the resulting periodic function from the difference (Fig. 2,e) 5 sin 2πx ∼ = f (x) − W00 [f (x)] shows a quite large error of approximation. If we increase instead the scale of 10 [5 sin 2πx] = 0. We get (Fig. 3,c) approximation (Fig. 3) it is still W50   5 5  1  −0.4k 10 10 1 −0.4k k W50 [f (x)] = W50 2 cos 2 x ∼ 2 cos 2k x = 10 10 k=−5

k=−5

with a better approximation. The periodic function follows from the difference (Fig. 3,e) 10 [f (x)] . 5 sin 2πx ∼ = f (x) − W50

8

Conclusion

In this paper harmonic wavelets were applied to the reconstruction of localized self-similar functions. The fundamental property of these wavelets of being orthonormal to the Fourier basis was used for the decomposition of a noised signal into a 2π-periodic and a localized fractal component. This method could be easily generalized to different periodicity components by a suitable definition of the harmonic wavelets (see also [9]).

References 1. Abry, P., Goncalves, P., L´evy-V´ehel, J.: Lois d’´eschelle, Fractales et ondelettes, Hermes (2002) 2. Borgnat, P., Flandrin, P.: On the chirp decomposition of Weierstrass-Mandelbrot functions, and their time-frequency interpretation. Applied and Computational Harmonic Analysis 15, 134–146 (2003) 3. Bunde, A., Havlin, S. (eds.): Fractals in Science. Springer, Berlin (1995) 4. Cattani, C.: Harmonic Wavelets towards Solution of Nonlinear PDE. Computers and Mathematics with Applications 50(8-9), 1191–1210 (2005)

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