An Intriguing Property of Scaling Function in Wavelet Theory and its Verification Using

Daubechies-Lagarias Algorithm

Soman K P Computational Engineering and Networking

Amrita Vishwa Vidyapeetham Coimbatore, India.

kp_soman@ettimadai.amrita.edu

Arathi T Computational Engineering and Networking

Amrita Vishwa Vidyapeetham Coimbatore, India.

arathiviswam09@yahoo.co.in

Abstract—The advent of wavelet in itself is a revolution in the field of signal processing. The simultaneous localization of signal in both its time and frequency domain was what attracted the engineers the most. However, most of them still fail to appreciate the contribution of Ingrid Daubechies, whose scaling and wavelet functions have several surprising features. Here, we try to throw light into the astonishing features of the Daubechies scaling and wavelet functions. Understanding of these features appears to be very important for mathematicians for exploring and exploiting new function spaces. The main purpose of this article is to convince ourselves (readers) the exotic properties of scaling and wavelet functions through computational experiments. Keywords—Scaling function, Fractals, Orthogonality

I. INTRODUCTION Though wavelet have been around for many years

and is popular because of extensive applications of wavelets in various Scientific and engineering applications, most practicing engineers are confined with the use of standard wavelet filters and are not much bothered about interesting properties of scaling and wavelet functions. This is partly because wavelet and scaling function explicitly does not come into picture in most engineering applications.

It is quoted in [1] that - “The Daubechies wavelets have surprising features—such as intimate connections with the theory of fractals. If their graph is viewed under magnification, characteristic jagged wiggles can be seen, no matter how strong the magnification is. This exquisite complexity of detail means that there is no simple formula for these wavelets. They are ungainly and asymmetric; nineteenth-century mathematicians would have recoiled from them in horror. But like the Model-T Ford, they are beautiful, because they work. The Daubechies wavelets turn the theory into a practical tool that can be easily programmed and used by any scientist with a minimum of mathematical training”.

This article discusses the properties of the wavelet and scaling function that would have made nineteenth-century mathematicians recoil from them in horror. The first and foremost property that we discuss is called “partition of unity” property of scaling function. That is scaling function ( )xφ

satisfies the relation ( ) 1n

x nφ + =∑ , where n is an integer.

For a 4 tap Daubechies scaling function whose support is (0, 3), this means that,

(0.1) (1.1) (2.1) 1φ φ φ+ + = (1)

Also, (0.2) (1.2) (2.2) 1φ φ φ+ + = (2)

Or, in general, ( ) (1 ) (2 ) 1, 0 1 x x x xφ φ φ+ + + + = ≤ ≤ (3)

Figure 1: Daubechies 4-tap scaling function

Till 1980s, Mathematicians did not think of such

functions and would not have believed such functions exist. The greatness of Daubechies lies in proving that such a function exists and secondly in showing that integer translates of such functions are orthogonal and are thus useful for signal representation. Unfortunately there is no explicit representation of such functions (except Haar) and like fractals, ‘bigger’ function is expressed in terms of scaled and translated version of itself (smaller functions). i.e. ( ) 2 ( ) (2 )

kx h k x kφ φ= −∑ (4)

Proving the above relation requires Poisson summation formula. We discuss it below.

II. POISSON SUMMATION FORMULA The Poisson Summation Formula (PSF) expresses the

infinite sum of periodic samples of any function f(x), at the integer points, to be equal to a similar sum, when taken in its Fourier Transform domain.

2009 International Conference on Advances in Recent Technologies in Communication and Computing

978-0-7695-3845-7/09 $25.00 © 2009 IEEE

DOI 10.1109/ARTCom.2009.189

228

2009 International Conference on Advances in Recent Technologies in Communication and Computing

978-0-7695-3845-7/09 $26.00 © 2009 IEEE

DOI 10.1109/ARTCom.2009.189

228

2009 International Conference on Advances in Recent Technologies in Communication and Computing

978-0-7695-3845-7/09 $26.00 © 2009 IEEE

DOI 10.1109/ARTCom.2009.189

228

Consider a continuous function f(x), in the time interval (0, 1).

Figure 3: 1 f(x), 2 ej2Π0x, 3 ej2Π1x, 4 ej2Π2x

( ) ( ). ( ) ( )k k

k

f x f x e x e x=∑ , 0 1x≤ ≤ (5)

where, ( ), ( )kf x e x is the inner product of the function f(x)

with the kth complex base 2j kxe π . Since each of the complex bases 2j kxe π is orthogonal

in the unit interval and are periodic with periodicity 1, the same bases can be used for representing a part of the function f(x) in any unit interval.

Figure 4: The function f(x) in any unit interval

i.e. for the function ( )f x shown in Fig.4, part of the function in the interval (n-1/2) to (n+1/2), where n is an integer and can be represented as

( ) ( ), ( ) ( )k kk

f x f x e x e x=∑ , 0.5 0.5n x n− ≤ ≤ + (6)

Replacing ‘x’ with ‘n’, where n is an integer ( ) , k

k

f n f e=∑ , (7)

since, ( ) 1ke x = ( 2 2 2 1 0 1j kx i kn j ne e e jπ π π= = = + = ) where,

, kf e = 1/ 2

2

1/ 2

( )n

j kx

n

f x e dxπ+

−∫

(8)

Now,

2 ( )( ) , , 0 1j k n yk

k

f n y f e e yπ ++ = ≤ ≤∑ (9)

Note that the inner product is unaffected by change of variable.

2( ) , , 0 1j kyk

kf n y f e e yπ+ = ≤ ≤∑ , (10)

‘y’ is an arbitrary shift from integer positions Summing over ‘n’ we obtain,

2( ) , , 0 1j kyk

n n k

f n y f e e yπ+ = ≤ ≤∑ ∑∑ (11)

On interchanging the summation, 2( ) , , 0 1j ky

kn k n

f n y f e e yπ+ = ≤ ≤∑ ∑∑ (12)

2 2( ) ( ) , 0 1j kx j ky

n kf n y f x e dx e yπ π

∞

−∞

⎛ ⎞+ = ≤ ≤⎜ ⎟

⎝ ⎠∑ ∑ ∫

(13)

2ˆ( ) ( 2 ) , 0 1j k y

n kf n y f k e yππ+ = ≤ ≤∑ ∑

(14)

This is the general expression for Poisson summation formula. If y = 0, we get a special case of Poisson summation formula: ˆ( ) ( 2 )

n kf n f kπ=∑ ∑

(15)

III PROOF FOR “PARTITION OF UNITY’ OF SCALING FUNCTION

The Fourier transform of the refinement relation in wavelets is:

( ) ( / 2). ( / 2)Hφ ω ω φ ω∧ ∧

= (16)

where, φ∧

is the wavelet scaling function in the transform domain and H the transform domain representation of the coefficients of the scaling function. For ω = 0, we have:

(0) (0). (0)Hφ φ∧ ∧

= (17)

Since, we assume that (0) 0,φ∧

≠ it is evident that H(0) must be equal to 1. The scaling function is assumed to be normalized. i.e. the area under the scaling function φ(x) is equal to 1.

i.e. ( ). 1.x dxφ =∫ By periodicity, we have, (2 ) 1H kπ = for any kεZ. Then,

(4 ) (2 ). (2 ) (2 )k H k k kφ π π φ π φ π∧ ∧ ∧

= = (18) In general, (18) can be written as:

(2 ) (2 ),n k kφ π φ π∧ ∧

= n≥1 (19)

Since, | ( ) | 0φ ω∧

→ as | |ω → ∞ ,

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(2 ) 0kφ π∧

= for kεZ, k ≠ 0. (20) From Poisson Summation formula,

1 12

0 0( ) (2 ) (0) ( ). 1.

N Njkx

k kx k e k x dxπφ φ π ϕ φ

− − ∧ ∧

= =

+ = = = =∑ ∑ ∫

where, ( )x kφ + represents the integer translated versions of the scaling function.

IV. DAUBECHIES-LAGARIAS ALGORITHM There is no explicit expression for the scaling function φ(x) and wavelet function ψ(x), as they are related by means of the refinement relation that expresses φ(x) and ψ(x) in terms of scaled and translated versions of itself. Hence, most algorithms which need the scaling and wavelet functions are formulated based on the filter coefficients, h(k) and g(k). There are certain methods which allow us to evaluate the values of the scaling and wavelet functions at dyadic (powers of 2) points. These methods include the Cascade algorithm (Direct evaluation at Dyadic Rational Points), Successive Approximation Method, Subdivision scheme [1], etc. However, there is only one method known till date, which enables the calculation of the scaling and wavelet function value at the point, to a desired accuracy. These evaluations are useful in wavelet density estimators, wavelet based classification, etc. This method is the Daubechies-Lagarias algorithm [2, 3, 4] A. Algorithm: Let φ be the scaling function of a wavelet system with a compact support and let the support be (0, N-1).

1 2 3( ) { , , ,.... ,....}ndyad x d d d d= is the set of numbers between the digits 0 and 1 in dyadic representation of x:

1

2 jj

j

x d∞

−

=

=∑ (21)

dyad (x, n) denotes the first n digits from the dyad of x. Let { (0), (1),.... ( 1)}h h h h N= − be the vector of scaling

function coefficients. Then, we define two (N–1)x(N–1) matrices as:

0 2 (2 1)T h i j= − − , for 1 , 1i j N≤ ≤ − and

1 2( (2 )T h i j= − , for 1 , 1i j N≤ ≤ − Then the theorem can be expressed as:

1 2

( ) ( ) ... ( )( 1) ( 1) ... ( 1)

lim . ....... ... ... ...

( 2 _ ( 2) ... ( 2)

d d dnn

x x xx x x

T T T

x N x N x N

φ φ φφ φ φ

φ φ φ→∞

⎡ ⎤⎢ ⎥+ + +⎢ ⎥=⎢ ⎥⎢ ⎥+ − + − + −⎣ ⎦

The average value of the first row is taken as φ(x), average of second and third rows as φ(x+1) and φ(x+2)

respectively. The sum of these values will be equal to 1, as proved in the previous section. This will be true for the value of φ, at any given point. This has been experimentally verified in Matlab for Daub-4 and Daub-6 wavelet systems and the results tabulated below.

TABLE I. RESULTS FOR DAUB-4 SYSTEM Shift (over Time)

Φ(0 + Shift) Φ(1 + Shift) Φ(2 + Shift) Sum

0 0 1.3657 -0.3657 1 0.1 0.3713 0.7234 -0.0947 1 0.2 0.5436 0.4788 -0.0224 1 0.3 0.6626 0.3408 -0.0034 1 0.4 0.7959 0.1742 0.0299 1 0.5 0.9330 0.0000 0.0670 1 0.6 0.9701 0.0258 0.0041 1 0.7 1.0606 -0.0551 -0.0055 1 0.8 1.1653 -0.1645 -0.0008 1 0.9 1.2662 -0.2663 0.0001 1

TABLE II.RESULTS FOR DAUB-6 SYSTEM Shift (over Time)

Φ(0 + Shift)

Φ(1 + Shift)

Φ(2 + Shift)

Φ(3 + Shift)

Φ(4 + Shift)

Sum

0 0 1.2863 -0.3858 0.0953 0.0042 1 0.1 0.1013 1.1574 -0.3338 0.0727 0.0024 1 0.2 0.2154 1.0008 -0.2650 0.0487 0.0001 1 0.3 0.3371 0.8299 -0.1895 0.0235 -0.0010 1 0.4 0.4578 0.6701 -0.1310 0.0030 0.0000 1 0.5 0.6052 0.4411 -0.0150 -0.0315 0.0002 1 0.6 0.7166 0.3303 -0.0278 -0.0191 0.0000 1 0.7 0.8377 0.2003 -0.0311 -0.0069 0.0000 1 0.8 0.9731 0.0373 -0.0113 0.0009 0.0000 1 0.9 1.1280 -0.1740 0.0467 -0.0007 0.0000 1

V. CONCLUSION One of the intriguing properties of the wavelet

scaling function is the property of partition of unity. This property is demonstrated using Daubechies-Lagarias algorithm. It is also shown how to demonstrate the orthogonal property of integer translates of the scaling and wavelet functions.

REFERENCES

[1] Wavelets: Seeing the forest and the trees, http://www.beyonddiscovery.org/content/view.txt.asp?a=1952.

[2] Daubechies I, and Lagarias J. (1991). Two-scale difference equations I. Existence abd global regularity of solutions, SIAM J Math Anal, 22(5), 1388-1410.

[3] Daubechies I and Lagarias J. (1992). Two-scale difference equations II. Local regularity, infinite products of matrices and fractals, SIAM J.Math Anal,23 (4), 1031-1079.

[4] Vidakovic, B. (1999), Statistical Modeling by Wavelets. Wiley, NY.

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