The Mallat Algorithm of Biorthogonal Two direction Bivariate Wavelets and Application in Communication Engineering

2017 2nd International Conference on Software, Multimedia and Communication Engineering (SMCE 2017) ISBN: 978-1-60595-458-5

The Mallat Algorithm of Biorthogonal Two-direction Bivariate Wavelets

and Application in Communication Engineering

Wei-jun YAN

College of Mathematics and Statistics, Yulin University, Yulin, China

Keywords: Biorthogonal two-direction refinement function, Biorthogonal two-direction wavelet, Decomposition and Reconstruction, Communication Engineering.

Abstract. In this paper we investigate the Mallat algorithms of biorthogonal bivariate two-direction wavelets. We introduce the concepts of biorthogonal bivariate two-direction refinement function and biorthogonal two-direction bivariate wavelet. The Mallat algorithms of biorthogonal two-direction bivariate wavelets was investigated. Moreover, a necessary and sufficient condition of complete reconstruction with decomposition is deduced. This algorithm is important for the decomposition and reconstruction of discrete signal with finite energy.

Introduction

Wavelet analysis has intently attracted many researchers’ attention in the last twenty years. It have been widely used in signal processing, image processing, voice recognition and synthesis, music, radar, machine fault diagnosis and monitoring, and so on [1–6]. Multiwavelets, which can offer properties as symmetry, orthogonality, short support at the same time, have been an active research field of wavelets analysis. To obtain some beautiful features, Yang and Li [7] introduce the concepts of two-direction scaling function and two-direction wavelets, give the definition of orthogonal two-direction scaling function and construct the orthogonal two-direction wavelets. Moreover, they investigate the approximation order of two-direction scaling functions. After the notion of two-direction wavelets is introduced, lots of researchers pay attention to the research of two-direction wavelets, such as the works [8, 9]. Two-direction scaling functions  and two-direction wavelets  associated with  , which are more general setting than the one-direction function and wavelets. The two-direction setting is more flexible than the one-direction setting. In the multiwavelet theory, the Mallat algorithms for two-direction bivariate wavelets is important. So, the notion of two-direction bi-orthogonal wavelets is introduced and to derive the decomposition and reconstruction algorithms biorthogonal two-direction wavelets is given in this paper.

Notations and Fundamentals on Two-direction Wavelets

We start from the following notations. Z and R denote all integers and all real numbers. We use

2 2

( )

L R denote the space of all square integrable functions on R2 .

2 1 2

x( ,x x )R ,k( ,k k_{1 2}), n( ,n n_{1 2}), l=( , )l l_{1 2} Z2. Let E is a 2 2 matrix whose all entries are integers and all eigenvalues is larger than one in modulus. The absolute value of the determinant of matrix E is denote m , i.e., | detE|m .For two functions f(x), (x)g L R2( 2) , by

(x), (x)

f g

 , their symbol inner product is defined (x), (x) ₃ (x) (x) x

R

f g f g d

 



, where the

(x)

g means the complex conjugate.

In the following, two-direction scaling function is introduced. (x)is called two-direction

scaling function, if (x)L R2( 2) satisfies the following two-direction scaling equation

3 3

(x) _k ( x k) _k (k x)

k Z k Z

p E p E

  

 

Definition 1. We say that the two-direction scaling bivariate functions (x)and (x)L R2( 2) is biorthogonal, if they satisfy:

0,k

(x), (x k) , (x), (k x) 0

    

      (2) where _0,k is the Kronecker symbol. Define a sequence of subspace { }V_{j j Z}

2 2

2

2 2

( ){ ( x k), (l x) : k,l ,}

j j

j j

j L R

V Clos m  E  m  E Z , jZ (3) Definition 2. We say that trivariable function (x) in Eq.(1) generates a two-direction multiresolution analysis { }V_{j j Z} , if the sequence { }V_{j j Z} defined in (3) satisfies:

1) V_j V_j1; 2) _{j Z} V_j {0}, _{j Z} V_j L R2( 2); 3) (x)V_j ( x)E V_j1; 4) The family

2

{ (x k), (n x) : k,n    Z } is a Riesz basis of subspace V₀.

Corresponding to a two-direction scaling bivariate function (x), a family trivariable functions (x), {1, 2, ,m 1}



    is called a two-direction wavelets if

2

{_(xk),_(nx) : k, nZ ,} forms a Riesz basis of subspace W₀, so that V₁  V₀ W₀,

1

j j j

V_ V W ._{ 2 _{( x k),} 2 _{(l x) : k, l} 2_}

j j

j j

m_ E  m_ E Z forms a Riese basis of L R2( 2).

Hence, a two-direction wavelets _(x) associated with a two-direction refinable function (x) must satisfies the following equation

2 2

,k ,k

(x) ( x k) (k x)

k Z k Z

q E q E

  

    

 





 



 (4) Definition 3. Let 2 2

(x), (x) L R( )

   be an biorthogonal two-direction scaling functions, and the two-direction wavelets _(x),_(x), satisfy the Eq. (4), where {q,k}_{k Z}2



 and

2

,k _k

{q }_Z are finitely supported sequences. If (x), (x) L R2( 2) and (x),(x), satisfy the following conditions

0,k ,

(x), (x k) (x), (k x) 0

(x), (x k) ( ), (k x) 0

(x), (x k)

(x), (k x) 0

x

 

 

   

 

   

   

   

 

    



    



  



  



(5)

Like Eq. (1) and Eq. (4), there are following equations

2 2

k k

k

( ) ( x k) (k x)

k Z Z

x p E p E

  

 





 



 (6)

2 2

,k ,k

k k

(x) ( x k) (k x),

Z Z

q E q E

  

     

 





 



  (7) Where {p_k}_{k Z}2



 ,{p_k}_{k Z}2



 ,{q_,k}_{k Z}2



 and {q_,k}_{k Z}2



 are finitely supported sequences.

Denote

/ 2 / 2

, , (x) ( x n), , , (x) (n x)

j j j j

j n m E j n m E

   

 _  _  _  _

,

/ 2 / 2

, (x) ( x n), , (x) (n x)

j j j j

j n m E j n m E

 _  _  _  _

/ 2 / 2

, ,n(x) ( x n), , ,n(x) (n x)

j j j j

j m E j m E

   

 _  _  _  _

/ 2 / 2

, (x) ( x n), , (x) (n x)

j j j j

j n m E j n m E

 _  _  _  _

The Main Results

In this part, we will proceed to study the decomposition and reconstruction Mallat algorithm of biorthogonal two-direction trivariable wavelets and a present a necessary and sufficient condition of complete reconstruction.

For energy finite signal 2 2

(x) ( )

f L R , the approximation f_j1 of signal f at the scale j 1

m 

will respectively be piecewise constant and piecewise affine on V_j . Because of

(1) (2) ( 1)

1 ,

m

j j j j j j j

V_ V W V W W  W  jZ , here the spaces

2 3

( ) 2

, , , ,

( ){ , : n }, ,

j L R j n j n

W  clos _ _ Z  j Z. So f_j1 can represent as following

1(x) k 1, 1,k(x) k 1,k 1,k(x)

j j k j j j

f _ 



c_ _ 



c_ _ (8) Furthermore, the right side of Eq.(8) can be expressed as following

,k ,k ,k ,k , ,k , ,k , ,k ; ,k

kcj j (x) kcj j (x)  { kd j  j (x) kd j  j (x)}

       



  





 



(9)

In the following, we give the decomposition algorithms of biorthogonal two-direction wavelets. Theorem 1. Let f_j1V_j1, then we have following formula

2

1/ 2

,l _{k Z} ( 1,k k l 1,k k l)

j j E j E

c m



_ c_ p_ c_ p_ (10)

2

1/ 2

,l _{k Z} ( 1,k l k 1, l k)

j j E j k E

c m



_ c_ p_ c_ p_ (11)

2

1/ 2

, ,lj _{k Z} ( j 1,k ,k El j 1,k ,k El)

d_ m



_ c_ q_ _ c_ q_ _ (12)

2

1/ 2

, ,lj _{k Z} ( j 1,k , l kE j 1,k , l kE )

d_ m



_ c_ q_ _ c_ q_ _ (13) Proof. To make an inner product on both ends of Eq. (8) by j l,



, we have

, 1, 1, (x), , 1, 1, (x), ,

j l _k j k j k j l _k j k j k j l

c 



c_ _   



c_ _   (14) Inspired by Eq.(1),we have

1/ 2

1,k(x), ,l(x) l

j j m pk E

   

 

  , j 1,k(x),j,l m 1/ 2pk El

   

 

  (15) Substitute the Eq.(15) into Eq.(14) , we obtain the Eq. (9). Similarly, the Eq. (10) can be obtained. The following we prove the Eq. (12) and Eq. (13).

To make an inner product on both ends of Eq. (8) and Eq. (9)by __{, ,j l}(x), we have

, .j l _k j 1,k j 1,k(x), , ,j l(x) _k j 1,k j 1,k(x), ; ,j l(x)

d_ 



c_ _ _  



c_ _ _  (16) Inspired by Eq.(1), we have

1/ 2

1,k(x), , ,l(x) ,k l

j  j m q E

   

 

  , _j1,k(x),__{, ,lj} (x)m1/ 2q__{,k E l} (17) Thus, we obtain the Eq. (12). Similarly, the Eq. (13) can be obtained.

In the following, we give the reconstruction algorithms of biorthogonal two-direction bivariate wavelets.

1/2 1/2

1, _k( ,k ,k ) _k( , ,k ; k , ,k , k)

j l j l Ek j Ek l j l E j l E

c_ m



c p_ c p _ m

 

_ d_ q_ _ d_ q_ _ (18)

1/2 1/2

1, _k( ,k k ,k k ) _k( , ,k , k , ,k , k)

j l j l E j E l j l E j l E

c_ m



c p_ c p _ m

 

_ d_ q_ _ d_ q_ _ (19) Proof. Firstly, we prove the Eq. (18). To make an inner product on both ends of Eq. (8) and Eq.(9) by _j_1,l, we have

1,l _k ,k ,k 1,l _k ,k ,k 1,l

, ,k , ,k 1,l , ,k , ,k 1,l

k k

(x), (x) (x), (x)

[ (x), (x) (x), (x) ]

j j j j j j j

j j j j j j

c c c

d_{ } d_{ }

                                       





 



Inspired by Eq.(1),we have

1/ 2 ,k(x), φ 1,l(x) l k

j j m p E

   

 

  , j,k(x),j 1,l(x) m 1/ 2pEk l

   

 

 

1/ 2 , ,kj (x), j 1,l(x) m q ,l Ek

 

   

 

  , , ,kj (x), φj 1,l(x) m 1/ 2q, k lE

   

 

 

Thus, we obtain the Eq. (18). Similarly, the Eq. (19) can be obtained.

In following, we discuss the conditions of complete reconstruction after signal decomposition according to the Mallat algorithm .

Theorem 3. Let (x), (x) be an biorthogonal two-direction scaling functions, and the two-direction wavelets _(x),_(x)associated with (x), (x) , then the sufficient and necessary condition of complete reconstruction as follows

2 2

2 2

2

k-En k n k 0,n , ; ,k n ,k 0,n ,

k Z k Z

n k k n k k ,k n k λ,k n k

k Z k Z

, n k , k k k ,k n

k Z

( ) , ( )

( ) 0 , ( ) 0

( ) 0 , (

k E k En k E

E E E E

E k En E k

p p p p m q q q q m

p p p p q p q p

q p q p p q p q

                                                             











2 2 2 ,k n k Z

k , n k k , n k ,k , n k ,k , n k

k Z k Z

) 0

( ) 0 , ( ) 0

E

E E E E

p q p q q q q q

                                  _{   }  







(20)

Proof. Fromthe definition of biorthogonality, for all nZ2, we have

0,n k

k k

l l k l

l l k l

k l k l

k l k l

k l k l

(x n), (x) [ (x n) k] [k (x n)],

( x l) (l x) ( x n k), ( x l)

( x n k), (l x) (k n x), ( x l)

(

k

p E p E

p E p E p p E E E

p p E E E p p E E E

p p                                                         

















k k k k

k

1

k En Ex), (l Ex) (p Enp p Enp ) m

    

 

   





so the identity _{k Z}2(pk Enpk pk Enpk) m0,n

   

 

  



is holds. Similarly, the other can be obtained. Thus, Theorem 3 is proved.

Summary

The notion of two-direction biorthogonal bivariate two-direction wavelets is introduced. Mattlat algorithm of two-direction biorthogonal bivariate two-direction wavelets has been proposed. A necessary and sufficient condition of complete reconstruction with decomposition are established.

Acknowledgement

This research was financially supported by the National Science Foundation of China (Grant No:11641001) and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No:2014JQ2-1003).

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