A Review of Wavelets and its Basic Properties

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_______________________________________________________________________________________________

1Meenakshi, ²Viney Kumar Popli

1Department of Mathematics, Lajpat Rai DAV College, Jagraon, India.

2Department of Mathematics, FCET, Ferozeshah E-mail: [email protected], [email protected]

Abstract: Wavelet theory is a refinement of Fourier analysis. The replacement of Fourier analysis by this theory very often yields much better results. Applications of wavelets have been studied in diverse fields of science and technology. We have studied the mathematical aspects of this theory. Recently we have studied the concept of Haar-Vilenkin wavelet in [10] which is a generalization of Haar wavelet. In this paper we have present a basic review of wavelets and its properties. We have also introduced the concept of Vector-valued non-uniform wavelet packets.

I. INTRODUCTION

Wavelets are a versatile tool with rich mathematical tool and great potential for applications. Wavelet analysis is a refinement of Fourier analysis and has been developed by practical considerations in last three decades.

Wavelet theory is the outcome of multi-disciplinary endeavour that has brought together mathematicians, physicists and engineers. This relationship has created a flow of ideas that goes well beyond the construction of new bases or transforms.

Fourier has laid the foundations with his theories of frequency analysis. It has been used in diverse fields of science and technology, however scientists started realizing its shortcomings around 1940’s. The big disadvantage of Fourier expansion was that it had only frequency resolution and no time resolution. The wavelet and Gabor analysis are probably the most recent tools to overcome the shortcomings of the Fourier transform. Gabor analysis was introduced by Noble Laureate of Physics Dennis Gabor. He introduced the Window-Fourier transforms by using a Gaussian distribution function as a window function. But there was one major drawback. Gabor transform used the same window for the analysis of entire signal.

Haar[7] is credited with the first use of wavelet.

Wavelets in their present theoretical form were introduced at the beginning of the eighties by Morlet et al. as a tool for signal analysis in view of applications for the analysis of seismic data. Grossman and Morlet[5]

led to detailed mathematical study of the continuous wavelet transform and their various applications.

Daubechies[3] made a remarkable contribution to wavelet theory by constructed the families of compactly supported orthonormal wavelets with some degree of smoothness. Mallat[9] introduced the concept of multiresolution analysis(MRA) provided a major role in Mallat's algorithm for the decomposition and reconstruction of an image in his work.

There have been significant applications of wavelet analysis to a wide variety of problems in many diverse fields including mathematics, physics, medicine, computer science and engineering. expansions in L²(R) with good time-frequency and regularity approximation properties. The wavelet transform is rapidly gaining popularity and recognition with applications ranging from pure mathematics to virtually every field of engineering, from astrology to economics and from oceanography to seismology. Applications of wavelets have been studied in diverse fields such as numerical simulation of partial differential equations, modelling real world problems, signal and image processing specially more accurate understanding of medical signals such as EEG and ECG. This theory has also been used in oil exploration, analyzing meteorological data and prediction of financial time series.

II. BASIC INGREDIENTS OF WAVELET ANALYSIS

The name wavelet means small waves, and in brief, a wavelet is an oscillation that decays quickly.

Mathematically, a function



is a wavelet if it satisfies the following conditions:

(i) A wavelet must have a finite energy, i.e. E =











 dt t)²



( . (ii) The following condition must hold:

 ^









 





 d

C

)2

( _.

This condition is known as the admissibility condition and C_ is called the admissibility constant.

(iii) The function



has a zero mean, i.e.

. 0 ) 0 ˆ ( 



Examples of Wavelet

Example 2.1: (Haar Wavelet) The first wavelet ever constructed was Haar wavelet, see Figure 1. It is thesimplest example of an orthonormal wavelet. The Haar wavelet is defined as:

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Figure1

Example 2.2 (Mexican Hat Wavelet) The function

defined by the equation ²

2

) 1 ( )

( x   x

²

e

^^x



^is

known as Mexican Hat Wavelet. This wavelet is the negative of second derivative of Gaussian Distribution function.

A wavelet transform(WT) differs from Fourier methods in that they allow localization of a signal in both time and frequency. A WT of a signal outperforms FT when the signal under consideration contains singularities and sharp spikes.

In wavelet theory, a function is represented by infinite series expansion in terms of dilated and translated version of a basis function



called mother wavelet satisfying the above conditions.

Where a>0, Then the continuous wavelet transform

T

 of

f  L

²

( R )

with respect to wavelet



is defined as

  dt

t f a b a f T

R



t^ab





( )

^*

)

,

(

¹²



where a,b lies in R, a>0.

Here a denotes the scale or frequency parameter and b denotes the translation parameter. The inverse wavelet transform is defined as

The inverse transform allows the original signal to be recovered from its wavelet transform by integrating over all scales and locations a and b. For detailed study see [1,8,11].

. Remark 2.1

(i) It is clear that the wavelet transform is a function of the scale a or frequency a(variable



of the Fourier transform) and the spatial position or time b (In case of Fourier transform one of them is missing). The plane defined by the variable (a,b) is called the scale-space or time frequency plane

T

_

f ( a , b )

measures the variation of f in a neighborhood of b.

(ii) For compactly supported wavelet, that is, wavelets vanishing outside a closed interval of real line the value of the wavelet transform T_ f(a,b) depends upon the value of f in a neighborhood of b of size proportional to the scale a. At small scale, provides localized information such as localized regularity of f(t). The local regularity of a function (or signal) is often measured with Lipschitz exponents.

III. MULTIRESOLUTION ANALYSIS

The concept of multiresolution analysis is a heart of wavelet theory which provides a method to construct wavelets. Multiresolution means that we want to use different degrees of resolution in order to approximate a given function f, ranging from coarse to fine structures, like a camera zooming in on an object. These different degrees of resolution are described mathematically by a nested sequence of vector spaces of functions,

_

 V

_₂

 V

_₁

 V

₀

 V

₁

 V

₂



_ (3.1) Definition 3.1[15,16]: A one dimensional Multiresolution analysis (abbreviated MRA) introduced by Mallat under the guidance of Meyer in 1989, is an increasing sequence of closed subspaces

  V

j _j__Z^of L²(R) such that

(i)

(ii) is dense in L²(R).

(iii)

f ( x )  V

_jiff f(2x)V_j_₁ and _f₍_x₎__V_j iff Vj

m x

f(  ) ^for

j  Z

.

(iv) There exists a function

 (x )

in L²(R) called the scaling function such that the collection

  ( x  n ) 

for

n  Z

is an orthonormal system of translates and

 ( ) 

0

span x n

V   

.

Remark 3.1: A MRA is defined by first identifying the subspace 0

V

, defining Vj by taking

 f ( x ) : f ( x ) D

2

g ( x ), g ( x ) V

0

 ,

V

j

 

j



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_______________________________________________________________________________________________

So that the first part of condition (iii) of Definition 3.1 is satisfied and then proving that the Definition 3.1(i), (ii), (iv) and equation 3.1 hold. V0 can be defined by first identifying the function

 (x )

such that

 

(xn)



is an orthonormal system of translates and then defining

 ( ) 

0

span x n

V   

.

Example 3.1 (Haar MRA)

Let V0 is the space of all functions which are square integrable on R and constant on all intervals [k, k+1) for integer k. According to condition (iv) of Definition 3.1, Vj is taken as the space of all square integrable functions which are constant on all intervals of the form [k.2^-j, (k+1).2^-j). For



, we choose

 

  



 otherwise

x x

x 0 ,

1 0 , ) 1 ( )

( 

_[₀_,₁₎



^.

One can show that {Vj} is a multiresolution analysis with scaling function



. The Haar wavelet

is related to



by

 ( t )   ( 2 t )   ( 2 t  1 )

. Example 3.2 (Shannon MRA)

Let V0 is the space of all square integrable functions having bandwidth not exceeding



, that is

f ˆ (  )  0

whenever

|  |  

. According to condition (iii) of Definition 3.1, Vj is taken to be the space of all square integrable functions having bandwidth not exceeding

j



2

. The Shannon scaling function is given by

t t t



 ( )  sin(  )

^.

I can be verified that the system of translates

T

_k



forms an orthonormal basis for V0.

IV. WAVELET DECOMPOSITION AND RECONSTRUCTION ALGORITHM

Consider the general structure of MRA where

  V

j is generated by some scaling function

  L

²

( R )

and

 

Wj is generated by some wavelet

  L

²

( R )

. B y Definition 3.1(ii), every function fL²(R) can be approximated as closely as desired by

f

N

 V

N for some

N  Z

. Since

V

_j

 V

_j_₁

 W

_j_₁

 j  Z

.

Therefore fN has a unique decomposition

1

1 

 

 _N _N

N f g

f ^where

f

_N_₁

 V

_N_₁ and

1

1 





_N

N

W

g

. By repeating this process we have

M N M N N

N

g g g f

f 

_₁



_₂

  

_



_ , (4.1) Where

f

_j

 V

_j and

g

_j

 W

_j for all

j  Z

.

The decomposition in (4.1) is called a wavelet decomposition. The stopping criteria is to require

M

f

N_ to be smaller than some threshold.

Since both the scaling function

  V

0 and

  W

0 are in V1 and since V1 is generated by

Z k k x

k(x)2¹² (2  ), 

,

1



^, ^there ^exist ^two

sequences

 

p_k ^and

 

g_k l² such that

R x k x g x k x p x

k k   k k   





⁽² ^), ⁽ ⁾



⁽² ⁾

)

(

  



^(4.2)

On the other hand, since

 ( 2 x )

and

 ( 2 x  1 )

are in V1 and

V

₁

 V

₀

 W

₀, there are four

l

2sequences which are denoted by {a-2k}, {b-2k}, {a1-2k} and {b1-2k},

Z

k 

such that

 ( ) ( )  ,

) 2

(  

₂

 

₂



k

a

k

x k b

k

x k

x  



 



^ ^ ^



 _ _

k a k x k b k x k

x 1) ( ) ( )

2

( ₁ ₂



₁ ₂





,

 x  R

. (4.3)

By the formula (4.3), we have

 

   





_ _

k

a

l k

x k b

l k

x k

l

x ) ( ) ( )

2

(

₂



₂





^,

Z

l 

, (4.4)

which is called a decomposition relation of



and



. Now, we have two pair of sequences

     p

k

, q

k



^and

   

 a

_k

, b

_k



all of which are unique due to direct sum relationship

V

₁

 V

₀

 W

₀. These sequences are used to formulate the decomposition and reconstruction algorithms. Hence

  p

_k and

 

q_k are called the reconstruction sequences, while

  a

_k ^and

  b

_k ^are

called the decomposition sequences. Every f_jV_j^and

j

j W

g  have unique series representations:

     



k j j jk

k j

j

x c x k c c l

f ( )

_,

 ( 2 ),

_, ²^.

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V. GENERALIZATION OF HAAR WAVELETS

Haar function is the simplest example of wavelet and it generates classical multiresolution analysis, see for example [15]. In the recent past various extensions of Haar have been studied [2,4,6,12,13]. The notion of Haar function was generalized by Vilenkin [14] and is often called Haar-Vilenkin function. Recently we have studied the concept of Haar-Vilenkin wavelet and Haar- Vilenkin scaling function in[10]. Haar-Vilenkin system is defined as follows:

Let P denote the set of positive integers and let

k  P

can be written as

1 ) 1

(   



 M r m s

k

_n _n (5.1)

Where

n  N

,

r  0 , 1 ,..., M

_n

 1

and 1

,..., 2 ,

1 

 m_n

s , N be the set of non-negative integers and

{ m

k

}

k_N be a sequence of natural numbers for

 2

m

k . Let M0 =1 and Mk = mk-1Mk-1,

k  P

^.^An

arbitrary element

t  [ 0 , 1 )

can be written as



^

 







0 1

) 0

( ,

k k k

k

t m

M

t t

. (5.2)

Vilenkin [14] introduced the following function known as the generalized Haar function or Haar type Vilenkin function [11].



 

   



otherwise M t r M

r m

M ist t

h

_k ⁿ _nⁿ _n _n

0 2 1 ) exp

(



(5.2)

The function hk(t) is a mother wavelet for

k  P

and for

t  [ 0 , 1 ).

The function hk(t) is called a Haar- Vilenkin Wavelet and the system

{ 

a_,b

( t )}

a_,b_Z is referred as the Haar-Vilenkin system where

).

( )

(

^/²

,_b

t m

_n^a

h

_k

m

_n^a

t b

a

 



For

k  P

and

t  [ 0 , 1 )

as defined in (5.1) and (5.2) the Haar-Vilenkin scaling function is defined as:









 



n nM

r M n r

k t M

p ₁

,

)

(







   



otherwise M t r M M r

n n

n

0

, 1 (5.3)

The collection

{ 

a_,b

( t )}

a_,b_Z is referred to as the system of Haar Vilenkin scaling functions where

).

( )

(

^/²

,_b

t m

_n^a

p

_k

m

_n^a

t b

a

 



Remark 5.1: Haar wavelet is a particular case of Haar- Vilenkin wavelet for mn=2. The process is shown in the example 5.1

Example 5.1: This example shows the process of making wavelets

If

m

_n

 2 ( n  N )

the system

( h

_n

, n  N )

is the Haar system.

Then m= (2, 2, 2,…) i.e.

m

₀

 2 , m

₁

 2 , m

₂

 2 ,...

Here

,...

2 ,...,

2 ,

1 ₁ ₂ ² ₃ ³

0

n

M n

M M

M

M     

Thus 



^    

  3

2 2 1 0

0 1 2 2 2

t t t M t t

k k

k

Here

n n n

n k

t r ist r

M t

h 2

1 , 2

exp )

( 





 

0 , otherwise

_k _n _n

r

_n

r t ist t

h

n

2 1 , 2

exp 2 )

( 

²

   

0 , otherwise When n=0

r=0, s=1

k  M

₀

 r ( m

₀

 1 )  s  1

= r+s

When r=0, s=1 then k=1 When n=1

r=0, 1, s=1, k=r+s+1 When r=0, s=1 then k=2 When r=1, s=1 then k=3 Here

1 0 , exp )

(

₀

1

t  it  t 

h 

0, otherwise Thus

21 1

( t )  1 , 0  t  h

1 ,

1

₂¹

 



 t

= 0, otherwise

As

t

₀

 0

when

0  t 

¹₂ and

t

₀

 1

when ₂¹

 t  1

and

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_______________________________________________________________________________________________

21 1

2

( t )  2 exp it , 0  t 

h 

0 , otherwise Thus

4 2

( t )  2 , 0  t 

1

h

2 1 4

,1

2  

 t

0, otherwise

As

t

₁

 0

for

0  t 

¹₄ and

t

₁

 1

for₄¹

 t 

¹₂. and

1 ,

exp 2 )

(

₁ ₂¹

3

t  it  t 

h 

0 , otherwise 43 21 3

( t )  2 ,  t 

h

1 ,

2

₄³

 

 t

0, otherwise

As

t

₁

 0

for ¹₂

 t 

₄³ and

t

₁

 1

for ₄³

 t  1

And

41 2

4

( t )  2 exp it , 0  t 

h 

0, otherwise Thus

8 4

( t )  2 , 0  t 

1

h

41 81

,

2  

 t

0, otherwise

As

t

₂

 0

for

0  t 

₈¹ and

t

₂

 1

for ₈¹

 t 

₄¹. Thus we have

n k n

t r t r

h

n

2 , 2

2 )

(

²



²¹







n n

t r

n

r

2 1 , 2

2

²



¹²

  



0, otherwise

We have studied the basic properties of Haar-Vilenkin system in [10]. We have proved the orthogonality of Haar-Vilenkin wavelet, convergence of Haar-Vilenkin wavelet series an properties of Haar-Vilenkin wavelet coefficients. Our main goal in this paper is to examine whether Haar-Vilenkin mother wavelet is associated with any multiresolution analysis. We have introduced a

multiresolution analysis where translation and dilation are taken by (b Z)

M b

n

 and mn respectively.

A special type of Multiresolution Analysis

Definition 5.1 For k as in (5.1),a special type of multiresolution analysis is a sequence

{ V

j

}

j_Z of closed subspaces of L²(R) such that

1.

V

_j

 V

_j1 for all

j  Z

. 2.



j_Z

V

j is dense in L²(R).

3.



_j__Z

V

_j

 { 0 }

.

4.

f ( x )  V

j iff

f ( m

_n^^j

x )  V

0 for all

j  Z

. 5. There exists a function gk(x) in L²(R) , called the scaling function such that the collection

 

Mb b Z

k

t

_n

g  }

_

{

is an orthonormal system of

translates and

. )

0

(

 



 

 span  T g x

V

_k

Mbn

Remark 2.1 A special type of multiresolution analysis is defined by first identifying the space V0, defining Vj

by letting

} ) ( ), ( )

( : ) (

{ f x f x D g x g x V

₀

V

j

mn

j

  

so that the Definition 2.1(4) is satisfied and then proving that Definition 2.1(1), (2), (3) and (5) hold. V0 can be defined by just identifying the function gk(x) such that

Z b k

x g T

Mbn

( )}



{

is an orthonormal system of

translates and then defining

. )

0

(

 



 

 span  T g x

V

_k

Mbn

Definition 5.2 For each

a , b  Z

define



_a_,_b

( x )

by

).

( )

(

^/²

,_b

x m

_n^a

g

_k

m

_n^a

x b D

_m

T

_b

g

_k

x

a ^a



n



 

For each

a  Z

, define the approximation operator

P

a on the functions

f ( x )  L

²

( R )

by . ,

)

( _, _,

Mbn Mbn a

b a

a f x f

P ^

  

Lemma 5.1 For all continuous function f(x) having a compact support on R

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1.

lim

_a__

P

_a

f  f

₂

 0

, and 2.

lim

_a__

P

_a

f

₂

 0 .

Proof of Lemma is a special case of Proposition 1 in [8].

VI. WAVELET PACKETS ASSOCIATED WITH VECTOR- VALUED NONUNIFORM WAVELETS

We have introduced vector-valued nonuniform multiresolution analysis(VNUMRA) where the associated subspace V0 of L²(R;C^s) has, an orthonormal basis, a collection of translates of a vector valued function over the set

  { 0 , r / N }  2 Z

, where r and N are relatively prime and

1  r  2 N  1

and the corresponding dilation factor is 2N. The notion of vector-valued nonuniform wavelet packets is introduced in this paper and their various properties are investigated.

Let s be a constant and

2  s  Z

. By L²(R;C^s) we denote the set of all vector-valued functions

f (t )

i.e.

 



^f(t) ⁽^), ⁽ ^),..., ⁽⁾ ^: ^, ⁽⁾ ⁽ ⁾



) C (R;

L² ^s   f₁t f₂ t f_s t ^T tR f_k t L² R (6.1)

Where ^T denotes the transpose and C^s denotes the s- dimensional complex Eucledian space.

Definition 6. 1: Given integers

N  1

and r odd with

1 2

1  r  N 

such that r and N are relatively prime and

  L

²

( R , C

^s

)

^generates ^a ^VNUMRA

  V

j _j__Z^{of L}²^(R;C^s), if the sequence

  V

j _j__Z satisfies:

a)

...  V

_₁

 V

₀

 V

₁

 ...

b)



j_Z

V

j is dense in L²(R;C^s).

c)



_j__Z

V

_j

 { 0 }

, where 0 is the zero vector of L²(R;C^s).

d)

 ( t )  V

_j if and only if

. )

2

( Nt  V

_j_₁

 j  Z



e) There exists

 ( t )  V

0 such that the sequence

{  ( t   ) :    }

is an orthonormal basis of V0 where

  { 0 , r / N }  2 Z

. The vector valued function

 (t )

is called a scaling function of the VNUMRA.

Define a closed subspace

V

_j

 L

²

( R , C

^s

)

^by

Z j t

N span clos

V_j  _L2₍_R_,_Cs₎( {((2 )^j ),}),  . (6.2)

Given a VNUMRA let Wm denotes the orthogonal complement of Vm in Vm+1, for any integer m. It is clear from the conditions (a), (b) and (c) of Definition 1 that

m Z

s m

W

C R

L

²

( , )  

_

The main purpose of VNUMRA is to construct orthonormal basis of L²(R;C^s) given by appropriate translates and dilates of finite collection of functions called the associated wavelets.

Definition 6.2: A collection

{ 

_k

}

_k_₁_,₂_,...,₂_N_₁ of functions in V1 is called a set of wavelets associated with given VNUMRA if the family of functions







 

 _

 ( )}

₁_,₂_,...,₂ ₁_,

{

_k

x

_k _N is an orthonormal

system of W0. Let

1 2 ,..., 2 , 1 ), ( ) ( ), ( )

0

( t   t 

^k

t  

_k

t k  N 



The family of vector-valued nonuniform functions ,...

1 , 0 , ), (

{²^Nn^^k t nZ_ k …, 2N-1} is called a vector-valued nonuniform wavelet packet w.r.t the orthogonal vector-valued scaling function



⁰

( t )

, where

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

( ⁽ ⁾

2 



  





^k t Q ^k ⁿ Nt k N

Nn

 

 



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