Abstract This article consists of basics of wavelet analysis required for understanding of and use of wavelet

ISSN (Online): 2320-9364, ISSN (Print): 2320-9356

www.ijres.org Volume 4 Issue 2 ǁ February. 2016 ǁ PP.31-35

A Short Report on Different Wavelets and Their Structures.

Sumana R. Shesha* and Achala L. Nargund*

*Post Graduate Department of Mathematics and Research Centre in Applied Mathematics MES College, Malleswaram 15th cross, Bangalore-03

Abstract

: This article consists of basics of wavelet analysis required for understanding of and use of wavelet theory. In this article we briefly discuss about HAAR wavelet transform their space and structures.

Keywords

: HAAR wavelet transform, Multiresolution analysis (MRA), Daubechies wavelet transforms, Biorthogonal wavelets.

Mathematical Classification: 42C40

I. Introduction to wavelets [1,2,6,7]

Wavelet analysis is used to decomposes sounds and images into component waves of varying durations, called wavelets which are localised vibrations of a sound signal or localized detail in an image. This analysis can be used in signal processing for removing noise. Jean Baptist Joseph Fourier (1807) has introduced frequency analysis leading Fourier analysis. He also explained the fact that functions can be represented as the sum of sine and cosine which led to Fourier Transform.

Alfred Haar in the year 1909 introduced wavelets. Haar’s contribution to wavelets is very evident so the entire wavelet family named after him. The Haar wavelets are the simplest of the wavelet families.

II.

Haar wavelets

[2,3,4]

A Haar wavelet is the simplest type of wavelet. In discrete form, Haar wavelets are related to a mathematical operation called the Haar transform. The Haar transform serves as a prototype for all other wavelet transforms.

III.

Haar spaces

Let 

 

t be the box function defined by,

 

   



  



otherwise t t

0

1 0 1

 . (1)

The equation,

 

t 



 

2t 





2t1





₍₂₎

is called the dilation equation. The Haar function 

 

t is typically called a scaling function. The space







t k



L

 

R span

V k Z

2

0      is called the Haar space

V

0 generated by the Haar function



 

t . The set







 tk



kZ forms an orthonormal basis for V0 . The set







tk





kZ also forms a Riesz basis for V0.

[Riesz basis: Suppose that 

 

t L2

 

R and 0 A B are constants such that

 

_





 



 

Z n

n Z

n n n Z

n

n c t B c

c

A 2



2 (3)

for any square-summable sequence

c



. Then we say that the translates





_n

 

t



_nZ form a Riesz basis of

 



n

t



n Z

span

V

0





_ .] The projection of the functions

f

 

t

L

 

R

2



into

V

₀ is given by,

 



f t



f

  

t t k

 

t k



P

Z k

 







 

,

(4)

The vector space V span





jt k





_{k Z} L

 

R j

2

2   

  is called the Haar space V_j.V_j L2

 

R is a vector space of piecewise constant functions with possible breakpoints at ₂jZ _{. For each}

Z

k , define the function

   

 



 t k

j j

k

j, 22 22

 . (5)

The set



 



Z k k j, t 

 is an orthonormal basis for j

V . The compact support for the functions



j,k

 

t is given

 

_ 

 

 k_j k _j

2 1 , 2

supp



j ,k . (6)

The dyadic intervalI_j,k, j,kZ, dilation equation and projection function are defined by

   



 _{  }  

   

 

 _{j j j j}

k j

k t k R t k

k I

2 1 2

2 1 , 2

, . (7)

 

t j k

 

t j k

 

t

k

j, 1,2 1,2 1

2 2 2

2

 

 









. (8)

 



f t



 

t f

 

t

 

t

P jk

Z k

k j

j



 , ,  ,



 , (9)

where

j

P is called the approximation operator and the space Vj is also known as approximation space. The

Haar spaces

 

Z j j

V _ satisfy

 V_1 V0 V1 V2   The function

 

j

V t

f  if and only if f

 

2t Vj1.



 0 2

1 0

1  

        

 

    



V V V V V

Z j

j

. (10)



_{ }

R L V

V V V V

Z j

j

2 2

1 0

1 

       

 



    



IV.

Haar wavelet spaces

The wavelet function 

 

t is given by,

 

 





  

   

 

  

   

 

  

 

   

otherwise t t t

t t

0

1 2

1 1

2 1 0

1

1 2

2 

 

. (11)

The equation,

 

t 

 

2t 



2t1



 _, is called the dilation equation. The space W span





t k





k Z L

 

R

2

0      is

called the Haar wavelet space W₀ generated by the Haar wavelet function

 

t . The set







tk





_k_Z forms

an orthonormal basis forW₀ . The set







tk





kZ also forms a Riesz basis forW0 . The Haar wavelet space

0

W satisfies the following properties.

 If f

 

t V₀ and g

 

t W₀, then f

   

t , gt 0. 

0

V and W0 are perpendicular to each other. (12)



0 0 1 V W

V   .

The vector space _{W span}





_{t k}





_L

 

_R Z

k j

j

2

2  _ 

  is called the Haar wavelet space

j

W .

For eachkZ, define the function 2



/ 2



, 2 2

j j

j k t k

    . The set



 



Z k k

j, t 

 is an orthonormal

basis for

W

_j. The compact support for the functions



j,k

 

t

is given by,





_



 

 k_j k _j

2 1 , 2

supp_{j ,k} . (13)

 

t j k

 

t j k

 

t

k

j, 1,2 1,2 1

2 2 2

2

 

 

  

 . (14)

For each jZ, the detail operator Qjon functions f

 

t L

 

R

2

 is defined by,

 

f t  P f t  P f t 

Qj  j1  j . (15)

The Haar wavelet spaces

 

Z j j

W _ satisfy the following properties.

If j and l are integers withl  j, then V_j and W_j are perpendicular to each other. If j and l are integers withl  j, then _Wj and W_l are perpendicular to each other.

j j

j V W

V 1   _.

Let f_j1

 

t V_j1 be defined as

 



 



 

Z m

m j m

j t a t

f ₁  _, and suppose that h 

is the vector given by





T T

h h

h _

  

   

2 2 , 2

2 , ₁

0



, then the projection of fj1

 

t into Vj can be written as

 

_

 

_

 

 

 

  

 

Z k

k j Z

k

k j j

k j k

j t b t f t t t

f



, 1 ,



,



, (16)

where

_

_

k

k k

k a a h a

b  2  2 1  

2

2 _{for kZ and}





T

k k k

a a

a  2 , 2 1

 _.

Let f_j1

 

t V_j1 be defined as

 

_

 

 

 

Z m

m j m

j t a t

f ₁  _1, . (17)

Suppose that f_j

 

t is the projection of _fj1

 

_t into

j

V and

g



is the vector given by





T T

g g

g _

  

 

 



2 2 , 2

2 , 1

0

 _{. (18)}

If gj

 

t  fj1

 

t  fj

 

t is the residual function in Vj1, then gj

 

t Wj and is thus given by

 

_

 





Z k

k j k

j t c t

g  , (19)

where





k

k k

k a a g a

c  ₂  _{2 1}   2

2 _for

_k

_

_Z

_and

_

_

T

k k k

a a

a  2 , 2 1

 _.

Thus, a piecewise constant function fj

 

t _, j₀ with possible breakpoints at points in Z

j

2 can be

decomposed into a piecewise constant approximation function _fj1

 

_t with possible breakpoints on the coarser grid j 1Z

2  and a piecewise constant detail function g_j₁

 

t with possible breakpoints in 2jZ. This process can be iterated and finally fj

 

t can be written as the sum of an approximation function f0

 

t with possible

breakpoints at the integers and detail functions _g0

 

_{t , g1}

 

_{t ,}_{, gj1}

 

_t with possible breakpoints at

Z Z

Z _{, ,2}j

4 1 , 2 1

   

 respectively. The next step is to model the decomposition in terms of linear

transformations (matrices). The technique to process discrete data via a discrete version of the decomposition process is called the discrete Haar wavelet transformation.

Suppose that

N

is an even positive integer. The discrete Haar wavelet transformation is defined as,                                                                                                                                                      2 2 2 2 0 0 0 0 0 0 2 2 2 2 0 0 0 0 0 0 2 2 2 2 2 2 2 2 0 0 0 0 0 0 2 2 2 2 0 0 0 0 0 0 2 2 2 2 2 2 N N N G H W (20) The N N  2 block 2 N

H is called the averages block and the N _N 2

block G_N2 is called the details block.

We have, _{H a b} N

  _

2 and GN a c

  _

2 . The inverse discrete Haar wavelet transformation is given by,

       c b W a T N  

 _{. (21)}

The two-dimensional discrete Haar wavelet transformation B of a matrix

A

is defines as,

T N

M AW

W

B  .

The inverse of two-dimensional discrete Haar wavelet transformation is given by,

N T

MBW

W

A .

VI.

Advantages of Haar Wavelet Transform

a) It is very simple.

b) It is fast.

c) It can be calculated without any need of temporary array so it needs less memory. d) It can be reversed exactly without the edge effects.

VII.

GRAPHS

[5]

Haar Wavelet

Haar Wavelet for

f



sin

x

VIII.

Acknowledgement

We are very much thankful to Prof. N. M. Bujurke, UGC Emeritus professor and INSA Senior Scientist for introducing us to such a deep subject and helping us in preparing this report. The first author would like to thank the three academies, namely, the Indian Academy of Sciences, Bangalore, the Indian National Science Academy, New Delhi and the National Academy of Sciences India, Allahabad for providing an opportunity to study this subject under Prof. N. M. Bujurke through Summer Research Fellowship Programme.

Refernces

[1.] A brief history of Wavelets – Brendon Keift - Internet

[2.] A Primer on WAVELETS and their Scientific Applications – James S. Walker –Chapman & Hall/CRC.

[3.] WAVELET THEORY: An Elementary Approach with Applications – David K. Ruch & Patrick J. Van Fleet – Wiley.

[4.] An Introduction to Wavelet Analysis – David F. Walnut – Birkhäuser. [5.] Wavelets and Multiwavelets – Fritz Keinert – Chapman & Hall/CRC.

[6.] Wavelets: Tools for Science and Technology – Stephane Jaffard, Yves Meyer, Robert D. Ryan – Siam. [7.] THE WORLD According to WAVELETS: The Story of a Mathematical Technique in the Making –