- Email: [email protected]
Construction of a type of Biorthogonal Binary Wave Functions by Multiresolution Analysis Approach
Available online at www.sciencedirect.com
Physics Procedia 33 (2012) 1346 – 1353
2012 International Conference on Medical Physics and Biomedical Engineering
Construction of a type of Biorthogonal Binary Wave Functions by Multiresolution Analysis Approach Delin Hua School of Education Nanyang Institute of Technology Nanyang 473000, China [email protected]
Abstract Wavelet analysis is nowadays a widely used tool in applied mathematics. The concept of vector-valued binary wavelets with two-scale dilation factor associated with an orthogonal vector-valued scaling function is introduced. The existence of orthogonal vector-valued wavelets with two-scale is discussed. A necessary and sufficient condition is provided by means of vector-valued multiresolution analysis and paraunitary vector filter bank theory. An algorithm for constructing a sort of orthogonal vector-valued wavelets with compact support is proposed, and their orthogonal properties are investigated.
©2012 2011Published Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name Committee. organizer] © by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Index Terms—Dilation equation; vector-valued multiresol-ution analysis; binary scaling functions; binary wavelets; matrix theory; orthogonal; frame wavelets
Introduction Construction of wavelet bases is an important aspect of wavelet analysis, and there are many kinds of scalar scaling functions and scalar wavelet functions. The main advantage of wavelets is their timefrequency localization property. Already they have led to exciting applications in signal analysis [1], fractals [2], image processing [3] and so on. Sampling theorems play a basic role in digital signal processing. They ensure that continuous signals can be processed by their discrete samples. Vector-valued wavelets are a class of generalized multiwavelets [4]. Chen and Cheng [5] introduced the notion of vectorvalued wavelets and showed that multiwavelets can begenerated from the component functions in vectorvalued wavelets. Vector-valued wavelets and multi-wavelets are different in the following sense. In real life, Video images are vector-valued signals. Vector-valued wavelet transforms have been recently studied
1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Committee. doi:10.1016/j.phpro.2012.05.221
1347
Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
for image coding by W. Li. Chen and Cheng studied orthogonal compactly supported vector-valued wavelets with 2-scale. Inspired by [5-7], we are about to investigate the construction of a class of orthogonal compactly supported vector-valued wavelets with three-scale. Similar to uni-wavelets, it is more complicated and meaningful to investigate vector-valued wav-elets with 4-scale. Based on an observation in [5,6], another purpose of this article is to introduce the notion of orthogonal vector-valued wavelet packets with two-scale and investigate their properties. Notations and multiresolution analysis Set s be a constant and 2 s Z , where Z and Z den--ote all non-negative integers and all integers, respectively. By L2 ( R 2 , C v ) , we denote the aggregate of arbitrary vector-valued functions F (t ) ,i.e, L2 ( R 2 , C s ) : { F (t ) ( f1 (t ), f 2 (t ), , f s (t ) )T : f (t ) L2 ( R 2 ), 1, 2, , s }, where T means the transpose of a vector. For example, video images an -d digital films are examples of vectorth column at time t. For F (t ) L2 ( R 2 , C v ), valued functions where g (t ) denotes the pixel on the u 2 1/ 2 F denotes the norm of vector-valued function F (t ), i.e., F : ( | f (t ) | dt ) , and its 1 R integration is defined as R F (t ) dt : ( R2 f1 (t ) dt , f 2 (t ) dt , , R T f ( t ) dt ) .The Fourier transform of F (t ) is defined by s 2 2
2
2
R
F( ) :
R
2
F (t ) e
it
dt .
For two vector-valued functions F , G F ( ), G ( ) : F (t ) G (t )* dt , 2
L2 ( R 2 , C v ) , their symbol inner product is defined by
R
(1)
where * means the transpose and the complex conjugate, and I u denotes the v v identity matrix. A sequence {Fl (t )}l Z U L2 ( R 2 , C u ) is called an orthonormal set of the subspace U , if the following condition is satisfied
Fj ( ), Fk ( )
I , j, k
j ,k v
Z,
(2)
where j , k is the Kronecker symbol, i.e., j , k 1 as j k and j , k 0 otherwise. Definition 1. We say that F (t ) Y L2 ( R 2 , C s ) is an orthogonal vector-valued function of Y if its translates { F (t v)}v Z 2 is an orthonormal set of the subspace Y , i.e.,
F(
n), F (
v)
I , n, v Z 2 .
n ,v u
(3)
Definition 2[5]. A sequence {Fv (t )}v Z 2 U L2 ( R 2 , C s ) is called an orthonormal basis of Y , if it satisfies (2), and for any G (t ) U , there exists a unique sequence of s s constant matrices { M k }k Z 2 such that
G (t ) v Z2
M v Fv (t ).
(4)
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Delin Hua / Physics Procedia 33 (2012) 1346 – 1353 [5]
Definition 3 . A vector-valued multiresolution analysis of L2 ( R 2 , C s ) is a nested sequence of closed 2 2 s subspaces {Vl }l z such that (i) Vl Vl 1 , l Z ; (ii) l z Vl {0}; l z Vl is dense in L ( R , C ) , 2 2 s l where 0 is the zero vector of L ( R , C ) ; (iii) F (t ) V0 if and only if F (2 t ) Vl , ; (iv) there is (t ) V0 such that the sequence { (t v ), v Z 2 } is an orthonormal basis of subspace V0 . On the basis of Definition 2 and Definition 3, we obtain (t ) satisfies the following equation
(t ) u Z2
Pv (2t u ),
(5)
constant matrices, i.e., {Pk }k z has only Where {Pk }k Z 2 , is a finite supported sequence of finite non-zero terms, and the others are zero matrices. By taking the Fouries transform for the both sides of (5), and assuming ( ) is continuous at zero, we have
(2 )
R2 ,
( ) ( ),
4 ( ) u Z2
(6)
Pu exp{ iu }.
(7)
Let W j ( j Z ) denote the orthocomplement subspace of V j in V j 1 and there exist three vectorvalued functions Gs (t ) L2 ( R 2 , C s ), r 1, 2,3 such that their translations and dilations form a Riesz basis of W j , i.e., j 2 W j clos (span{Gs (2 t u ) : s 1, 2, 3; u Z }), j Z . ( 8) L ( R ,C ) Since r (t ) W0 V1 , r 1, 2,3 there exist three finitely supported sequences {Bk( s ) }k z of v v constant matrices such that 2
r
u
(t ) u Z2
4
(s)
Bu( r ) (2t u ), r 1, 2,3.
( ) u Z
2
(9)
Bu( s ) exp{ iu }.
(10)
Then, the refinement equation (10) becomes thefollowing
(2 )
r
(r )
( ) ( ), s 1, 2,3,
R2 .
(11)
If (t ) L2 ( R 2 , C s ) is an orthogonal vector-valued scaling function, then it follows from (3) that
( ),
(
v)
v Z 2.
I ,
0,u v
(12)
We say that r (t ) L2 ( R 2 , C s ), r onal vector-valued wavelet functions associated with the vector-valued scaling function (t ) , if they satisfy
(
n),
and the family { r
( ),
(
r r
(
n)
Lemma 1 . Let F (t )
F( k Z
O, r 1, 2,3, n, v Z ,
(13)
(t v), r 1, 2,3, v Z } is an orthonomal basis of W0 . Thus we have
[6]
2
v)
1, 2,3 are orthog
2k ) F (
r,
I , r,
0, n u 2
2
1, 2,3; n Z .
(14)
L ( R , C ) . Then F (t ) is an orthogonal vector-valued function if and only if
2k )*
s
Is ,
R2 .
(15)
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Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
Lemma 2. If (t ) L2 ( R 2 , C s ) , defined by (5), is an orthogonal vector-valued scaling function, then for v Z , we have the following equalities,
P (P
4v
)*
4
I.
(16)
0, v s
Z 3
(
) (
)
*
(17)
2
Is ,
R .
0
Proof. By substituting equation (5) into the relation (12), for
(
I
0, k s
k ), ( ) 2
l Z u Z
1 4
2
l Z u Z
2
(
Pl
2k l ), (
R
2
k
Z , we obtain that
Pl (2t 2k l ) (2t u )* ( Pu )* dt
u ) ( Pu )*
1 4u
Z
2
Pu ( Pu
4k
)* .
Thus, both Theorem 1 and formulas (16), (23) and (24) provide an approach to design a class of compactly supported orthogonal vector-valued wavelets. Design of vector-valued wavelets
In the following, we begin with considering the existence of a class of orthogonal vector- valued wavelets.
compactly
supported
(t ) L2 ( R 2 , C s ) defined by (5), be an orthogonal vector-valued scaling function. Assume r (t ) L2 ( R 2 , C s ), r 1, 2,3, and ( ) and (r ) ( ) are defined by (7) and (10), respectively. Then r (t ) are orthogonal vector-valued wavelet functions associated with (t ) if and only if
Theorem 1. Let
3
(
(r )
)
)*
(
O, r 1, 2,3
(18)
0 3
(r )
(
( )
)
)*
(
r,
Is ,
(19)
0
where r , s 1, 2, l Z
l Z
2
2
Pl ( Bl( r 2) u )* Bl( r ) ( Bl(
R . or equivalently,
O, r 1, 2,3, u Z 2 ;
) * 2u
)
4
I r , s 1, 2, 3, u Z 2 .
r,
0,u v ,
(20) (21)
Proof. Firstly, we prove the necessity. By Lemma 1 and (6), (11) and (13), we have
(2
O u Z
u )
s
(2
u )*
2
(
u ) (
u )
u Z2
u )*
( 3
( 0
(r )
/ 2)
u )*
( (r )
(
/ 2)* .
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Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
It follows from formula (14) and Lemma 1 that
Iv
r,
r u Z
(2
2u )
2u )*
(2
2
(r )
(
u ) (
u )
2
u Z
u )*
( 3
(r )
( )
(
u )*
( ( )
)
)* .
(
0
Next, the sufficiency of the theorem will be proven. From the above calculation, we have
(2 u Z
u )
r
(2
u )*
2
3
(
/ 2)
(r )
/ 2)*
(
I .
0, r u
0
Furthermore
( ), r
r
r
(
1
k)
2
2u )* e 4ik d 1 ( k) 2
(2
( ),
[0, / 2]2
(2
2u )
u Z2
Z2
O, r 1, 2,3, k [0, / 2]2
r
(2
2u )
u Z2
2u )* e 4ik d Z2 0, k r , I s , k 1, 2,3} are a family of Thus, (t ) and r (t ), r 1, 2,3 are mutually orthogonal, and { (t ), orthogonal vector-valued functions. This shows the orthogonality of { s ( v ), r 1, 2,3}v z . Similar to [7,Proposition 1], we can prove its completeness in W0 . Theorem 2. Let (t ) L2 ( R 2 , C s ) be a 5-coefficient compactly supported orthogonal vector-valued (2
scaling functions satisfying the following refinement equation:
(t )
P0 (2t ) P1 (2t 1)
P4 (2t 4).
Assume there exists an integer 4 , such that (4 I v P ( P )* ) 1 P ( P )* is a positive definite matrix. Define Qr ( r 1, 2,3) to be two essentially distinct Hermitian matrice, which are all invertible and satisfy
(Qr ) 2
P ( P )* ] 1 P ( P )* .
[4 I s
(22)
Define B (j r ) B
Then r
Qs Pj ,
(r) j
(t )
j
1
(Qs ) Pj , r
j
r 1, 2,3, j {0,1, 2,3, 4}.
(t ) ( r 1, 2,3) , defined by (24), are orthogonal vector-valued wavelets associated with (r)
B0
(2t )
(r)
B1
(2t 1)
(r)
B4
(2t
4)
(23)
(t ) : (24)
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Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
1 . By Lemma 2, (20) and (21), it suffices to show that {B0( r ) , B1( r ) ,
Proof. For convenience, let
B2( r ) , B3( r ) , B4( r ) , r 1, 2,3} satisfy the following equations: P0 ( B4( r ) )*
O,
(r ) * 4 0 (r ) * 0 0 (r ) ( ) * 0 4 (r ) (r ) * 0 0
P (B ) P (B )
B (B )
B (B )
r 1, 2,3,
O,
(25)
r 1, 2,3, (r ) * 1
P1 ( B ) O,
P4 ( B )
r,
(r ) 1
(26) (r ) * 4
O,
(27)
{1, 2,3} ,
(s) * 1
(28) (s) 4
B (B )
(s) * 4
B (B )
4I s .
(29)
If {B0( r ) , B1( r ) , B2( r ) , B3( r ) , r 1, 2} are given by (23), then equations (26), (27) and (29) follow from (16). For the proof of (28) and (30), it follows from (16) and (27) that
P0 ( B0( r ) )*
P1 ( B1( r ) )*
P2 ( B2( r ) )*
[ P0 ( P0 )*
P2 ( P2 )*
( P1 ( P1 )*
P1 ( P1 )* )( Bs )
B0r ( B0r )*
P3 ( P3 )*
B1r ( B1r )*
1
P3 ( B3( r ) )*
P4 ( P4 )* ] Qs
P4 ( B4( r ) )*
P1 ( P1 )* (Qs )
1
O.
B2r ( B2r )*
B3r ( B3r )*
B4r ( B4r )*
Qr {P1 (P1 )* [P1 (P1 )* ] 1[4Is P1 (P1 )* ]P1 (P1 )*}(Qr ) 1 Qr {P1 ( P1 )*
4I s
P1 ( P1 )* }(Qr )
1
4I s .
So, (28), (30) follow. This completes the proof of Thm 2. Example1. Let
(t )
L2 ( R 2 , C 3 ) be 5-coefficient orthogonalvector valued scaling function satisfy the
following equation:
(t )
P0 (2t ) P1 (2t 1)
Where P3
P4
O, P0 ( P4 )*
P4 (2t 4). O,
P0 ( P0 )*
P0
P1 ( P1 )* 2 2 1 2
2 2 2 3
0
0
P2 ( P2 )*
P3 ( P3 )*
0 1 2 3 3
, P1
P4 ( P4 )*
4I3 .
1
0
0
0
2 6
0 ,
0
0
3 3
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Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
P2
2 Q1
2 2 2 3
0
0
0
2 3 3 2
.
0
53
0
4
0
53
0
, Q2
2
0
0
2
0
53
0
1 ,
0
53
0
0
1 . By using (22), we can choose
Suppose 2
2 2 1 2
2
0
4
By applying formula (24), we get that 1
1
2
2 53
(1)
B0
106 0
0
106
53
159
53
2 (1)
, B1
106
0
0
0
6
B2(1)
1 2 53 106 0
B0(2)
0
0 2
6 3
1 2 106 159 0
1 2 53 106
0
6
6
0
0
1 2 106 159 0
0 53 , 53 6 6 0 53 . 53 6 6
Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
2 B1(2)
B2(2)
0
0
0
106 6
0
0
0
1 2 53 106
1 2 106 159
0
0
,
2 6 3 0 53 . 53 6 6 ( )
(2t ) Applying Theorem 2, we obtain that G (t ) B0 B1( ) (2t 1) B4( ) (2t 4), 1, 2,3 are orthogonal vector-valued wavelet functions associated with the orthogonal vector-valued scaling function Conclusion
A necessary and sufficient condition on the existence of a class of orthogonal vector-valued wavelets is presented. An algorithm for constructing a class of compactly supported orthogonal vectorvalued wavelets is proposed. References [1]
N. Zhang X.Wu,``Lossless of color masaic images". IEEE Trans Image Delivery}, 2006, 15(6), pp. 1379-1388
[2]
S. Efromovich, J. akey , M. Pereyia, N. Tymes, ``Data-Diven and Optimal Denoising of a Signal and Recovery of its
Derivation Using Multiwavelets", IEEE Trans Signal Processing, 2,004, 52(3), pp.628-635 [3]
Z. Shen , Nontensor product wavelet packets in\ L2 ( R s ) [J]. SIAM Math Anal.1995, 26(4): 1061-1074.
[4]
X. G. Xia, B. W. Suter, `Vector-valued wavelets and vector filter banks", IEEE Trans Signal Processing}, 1996, 44(3),
pp. 508-518. [5] Solitons
Q. Chen, Z..Cheng, ``A study on compactly supported orthogonal vector-valued wavelets andwavelet packets”, Chaos,
& Fractals}, 2007, 31(4), pp.1024-1034.
[6]
C.K.Chui, J.Lian, ``A study of orthonormal multiwavelets". Appli Numer Math, 1996, 20(1), pp. 273-298
[7]
S. Yang. Z.Cheng, H. Wang, ``Construction of biorthogonal multiwavelets", Math Anal Appl}, 2002, 276(1), pp.1-12.
[8]
S. Li and H. Ogawa., “Pseudoframes for Subspaces with Applications,” J. Fourier Anal. Appl.., Vol. 10,No 4, pp.409-
431, 2004. [9]
S. Yang , Biorthogonal two-direction refinable function and two-direction wavelet. Appl Math & Comput . 2006 , 1822
(3) : 1717-1724 [10]
Q. Chen , A . Huo , “The research of a class of biorthogonal compactly supported vector-valued wavelets , Chaos ,
Solitons & Fractals , 41 (2 ), pp . 951-961, August 2009”
1353
Physics Procedia 33 (2012) 1346 – 1353
2012 International Conference on Medical Physics and Biomedical Engineering
Construction of a type of Biorthogonal Binary Wave Functions by Multiresolution Analysis Approach Delin Hua School of Education Nanyang Institute of Technology Nanyang 473000, China [email protected]
Abstract Wavelet analysis is nowadays a widely used tool in applied mathematics. The concept of vector-valued binary wavelets with two-scale dilation factor associated with an orthogonal vector-valued scaling function is introduced. The existence of orthogonal vector-valued wavelets with two-scale is discussed. A necessary and sufficient condition is provided by means of vector-valued multiresolution analysis and paraunitary vector filter bank theory. An algorithm for constructing a sort of orthogonal vector-valued wavelets with compact support is proposed, and their orthogonal properties are investigated.
©2012 2011Published Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name Committee. organizer] © by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Index Terms—Dilation equation; vector-valued multiresol-ution analysis; binary scaling functions; binary wavelets; matrix theory; orthogonal; frame wavelets
Introduction Construction of wavelet bases is an important aspect of wavelet analysis, and there are many kinds of scalar scaling functions and scalar wavelet functions. The main advantage of wavelets is their timefrequency localization property. Already they have led to exciting applications in signal analysis [1], fractals [2], image processing [3] and so on. Sampling theorems play a basic role in digital signal processing. They ensure that continuous signals can be processed by their discrete samples. Vector-valued wavelets are a class of generalized multiwavelets [4]. Chen and Cheng [5] introduced the notion of vectorvalued wavelets and showed that multiwavelets can begenerated from the component functions in vectorvalued wavelets. Vector-valued wavelets and multi-wavelets are different in the following sense. In real life, Video images are vector-valued signals. Vector-valued wavelet transforms have been recently studied
1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Committee. doi:10.1016/j.phpro.2012.05.221
1347
Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
for image coding by W. Li. Chen and Cheng studied orthogonal compactly supported vector-valued wavelets with 2-scale. Inspired by [5-7], we are about to investigate the construction of a class of orthogonal compactly supported vector-valued wavelets with three-scale. Similar to uni-wavelets, it is more complicated and meaningful to investigate vector-valued wav-elets with 4-scale. Based on an observation in [5,6], another purpose of this article is to introduce the notion of orthogonal vector-valued wavelet packets with two-scale and investigate their properties. Notations and multiresolution analysis Set s be a constant and 2 s Z , where Z and Z den--ote all non-negative integers and all integers, respectively. By L2 ( R 2 , C v ) , we denote the aggregate of arbitrary vector-valued functions F (t ) ,i.e, L2 ( R 2 , C s ) : { F (t ) ( f1 (t ), f 2 (t ), , f s (t ) )T : f (t ) L2 ( R 2 ), 1, 2, , s }, where T means the transpose of a vector. For example, video images an -d digital films are examples of vectorth column at time t. For F (t ) L2 ( R 2 , C v ), valued functions where g (t ) denotes the pixel on the u 2 1/ 2 F denotes the norm of vector-valued function F (t ), i.e., F : ( | f (t ) | dt ) , and its 1 R integration is defined as R F (t ) dt : ( R2 f1 (t ) dt , f 2 (t ) dt , , R T f ( t ) dt ) .The Fourier transform of F (t ) is defined by s 2 2
2
2
R
F( ) :
R
2
F (t ) e
it
dt .
For two vector-valued functions F , G F ( ), G ( ) : F (t ) G (t )* dt , 2
L2 ( R 2 , C v ) , their symbol inner product is defined by
R
(1)
where * means the transpose and the complex conjugate, and I u denotes the v v identity matrix. A sequence {Fl (t )}l Z U L2 ( R 2 , C u ) is called an orthonormal set of the subspace U , if the following condition is satisfied
Fj ( ), Fk ( )
I , j, k
j ,k v
Z,
(2)
where j , k is the Kronecker symbol, i.e., j , k 1 as j k and j , k 0 otherwise. Definition 1. We say that F (t ) Y L2 ( R 2 , C s ) is an orthogonal vector-valued function of Y if its translates { F (t v)}v Z 2 is an orthonormal set of the subspace Y , i.e.,
F(
n), F (
v)
I , n, v Z 2 .
n ,v u
(3)
Definition 2[5]. A sequence {Fv (t )}v Z 2 U L2 ( R 2 , C s ) is called an orthonormal basis of Y , if it satisfies (2), and for any G (t ) U , there exists a unique sequence of s s constant matrices { M k }k Z 2 such that
G (t ) v Z2
M v Fv (t ).
(4)
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Delin Hua / Physics Procedia 33 (2012) 1346 – 1353 [5]
Definition 3 . A vector-valued multiresolution analysis of L2 ( R 2 , C s ) is a nested sequence of closed 2 2 s subspaces {Vl }l z such that (i) Vl Vl 1 , l Z ; (ii) l z Vl {0}; l z Vl is dense in L ( R , C ) , 2 2 s l where 0 is the zero vector of L ( R , C ) ; (iii) F (t ) V0 if and only if F (2 t ) Vl , ; (iv) there is (t ) V0 such that the sequence { (t v ), v Z 2 } is an orthonormal basis of subspace V0 . On the basis of Definition 2 and Definition 3, we obtain (t ) satisfies the following equation
(t ) u Z2
Pv (2t u ),
(5)
constant matrices, i.e., {Pk }k z has only Where {Pk }k Z 2 , is a finite supported sequence of finite non-zero terms, and the others are zero matrices. By taking the Fouries transform for the both sides of (5), and assuming ( ) is continuous at zero, we have
(2 )
R2 ,
( ) ( ),
4 ( ) u Z2
(6)
Pu exp{ iu }.
(7)
Let W j ( j Z ) denote the orthocomplement subspace of V j in V j 1 and there exist three vectorvalued functions Gs (t ) L2 ( R 2 , C s ), r 1, 2,3 such that their translations and dilations form a Riesz basis of W j , i.e., j 2 W j clos (span{Gs (2 t u ) : s 1, 2, 3; u Z }), j Z . ( 8) L ( R ,C ) Since r (t ) W0 V1 , r 1, 2,3 there exist three finitely supported sequences {Bk( s ) }k z of v v constant matrices such that 2
r
u
(t ) u Z2
4
(s)
Bu( r ) (2t u ), r 1, 2,3.
( ) u Z
2
(9)
Bu( s ) exp{ iu }.
(10)
Then, the refinement equation (10) becomes thefollowing
(2 )
r
(r )
( ) ( ), s 1, 2,3,
R2 .
(11)
If (t ) L2 ( R 2 , C s ) is an orthogonal vector-valued scaling function, then it follows from (3) that
( ),
(
v)
v Z 2.
I ,
0,u v
(12)
We say that r (t ) L2 ( R 2 , C s ), r onal vector-valued wavelet functions associated with the vector-valued scaling function (t ) , if they satisfy
(
n),
and the family { r
( ),
(
r r
(
n)
Lemma 1 . Let F (t )
F( k Z
O, r 1, 2,3, n, v Z ,
(13)
(t v), r 1, 2,3, v Z } is an orthonomal basis of W0 . Thus we have
[6]
2
v)
1, 2,3 are orthog
2k ) F (
r,
I , r,
0, n u 2
2
1, 2,3; n Z .
(14)
L ( R , C ) . Then F (t ) is an orthogonal vector-valued function if and only if
2k )*
s
Is ,
R2 .
(15)
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Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
Lemma 2. If (t ) L2 ( R 2 , C s ) , defined by (5), is an orthogonal vector-valued scaling function, then for v Z , we have the following equalities,
P (P
4v
)*
4
I.
(16)
0, v s
Z 3
(
) (
)
*
(17)
2
Is ,
R .
0
Proof. By substituting equation (5) into the relation (12), for
(
I
0, k s
k ), ( ) 2
l Z u Z
1 4
2
l Z u Z
2
(
Pl
2k l ), (
R
2
k
Z , we obtain that
Pl (2t 2k l ) (2t u )* ( Pu )* dt
u ) ( Pu )*
1 4u
Z
2
Pu ( Pu
4k
)* .
Thus, both Theorem 1 and formulas (16), (23) and (24) provide an approach to design a class of compactly supported orthogonal vector-valued wavelets. Design of vector-valued wavelets
In the following, we begin with considering the existence of a class of orthogonal vector- valued wavelets.
compactly
supported
(t ) L2 ( R 2 , C s ) defined by (5), be an orthogonal vector-valued scaling function. Assume r (t ) L2 ( R 2 , C s ), r 1, 2,3, and ( ) and (r ) ( ) are defined by (7) and (10), respectively. Then r (t ) are orthogonal vector-valued wavelet functions associated with (t ) if and only if
Theorem 1. Let
3
(
(r )
)
)*
(
O, r 1, 2,3
(18)
0 3
(r )
(
( )
)
)*
(
r,
Is ,
(19)
0
where r , s 1, 2, l Z
l Z
2
2
Pl ( Bl( r 2) u )* Bl( r ) ( Bl(
R . or equivalently,
O, r 1, 2,3, u Z 2 ;
) * 2u
)
4
I r , s 1, 2, 3, u Z 2 .
r,
0,u v ,
(20) (21)
Proof. Firstly, we prove the necessity. By Lemma 1 and (6), (11) and (13), we have
(2
O u Z
u )
s
(2
u )*
2
(
u ) (
u )
u Z2
u )*
( 3
( 0
(r )
/ 2)
u )*
( (r )
(
/ 2)* .
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Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
It follows from formula (14) and Lemma 1 that
Iv
r,
r u Z
(2
2u )
2u )*
(2
2
(r )
(
u ) (
u )
2
u Z
u )*
( 3
(r )
( )
(
u )*
( ( )
)
)* .
(
0
Next, the sufficiency of the theorem will be proven. From the above calculation, we have
(2 u Z
u )
r
(2
u )*
2
3
(
/ 2)
(r )
/ 2)*
(
I .
0, r u
0
Furthermore
( ), r
r
r
(
1
k)
2
2u )* e 4ik d 1 ( k) 2
(2
( ),
[0, / 2]2
(2
2u )
u Z2
Z2
O, r 1, 2,3, k [0, / 2]2
r
(2
2u )
u Z2
2u )* e 4ik d Z2 0, k r , I s , k 1, 2,3} are a family of Thus, (t ) and r (t ), r 1, 2,3 are mutually orthogonal, and { (t ), orthogonal vector-valued functions. This shows the orthogonality of { s ( v ), r 1, 2,3}v z . Similar to [7,Proposition 1], we can prove its completeness in W0 . Theorem 2. Let (t ) L2 ( R 2 , C s ) be a 5-coefficient compactly supported orthogonal vector-valued (2
scaling functions satisfying the following refinement equation:
(t )
P0 (2t ) P1 (2t 1)
P4 (2t 4).
Assume there exists an integer 4 , such that (4 I v P ( P )* ) 1 P ( P )* is a positive definite matrix. Define Qr ( r 1, 2,3) to be two essentially distinct Hermitian matrice, which are all invertible and satisfy
(Qr ) 2
P ( P )* ] 1 P ( P )* .
[4 I s
(22)
Define B (j r ) B
Then r
Qs Pj ,
(r) j
(t )
j
1
(Qs ) Pj , r
j
r 1, 2,3, j {0,1, 2,3, 4}.
(t ) ( r 1, 2,3) , defined by (24), are orthogonal vector-valued wavelets associated with (r)
B0
(2t )
(r)
B1
(2t 1)
(r)
B4
(2t
4)
(23)
(t ) : (24)
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Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
1 . By Lemma 2, (20) and (21), it suffices to show that {B0( r ) , B1( r ) ,
Proof. For convenience, let
B2( r ) , B3( r ) , B4( r ) , r 1, 2,3} satisfy the following equations: P0 ( B4( r ) )*
O,
(r ) * 4 0 (r ) * 0 0 (r ) ( ) * 0 4 (r ) (r ) * 0 0
P (B ) P (B )
B (B )
B (B )
r 1, 2,3,
O,
(25)
r 1, 2,3, (r ) * 1
P1 ( B ) O,
P4 ( B )
r,
(r ) 1
(26) (r ) * 4
O,
(27)
{1, 2,3} ,
(s) * 1
(28) (s) 4
B (B )
(s) * 4
B (B )
4I s .
(29)
If {B0( r ) , B1( r ) , B2( r ) , B3( r ) , r 1, 2} are given by (23), then equations (26), (27) and (29) follow from (16). For the proof of (28) and (30), it follows from (16) and (27) that
P0 ( B0( r ) )*
P1 ( B1( r ) )*
P2 ( B2( r ) )*
[ P0 ( P0 )*
P2 ( P2 )*
( P1 ( P1 )*
P1 ( P1 )* )( Bs )
B0r ( B0r )*
P3 ( P3 )*
B1r ( B1r )*
1
P3 ( B3( r ) )*
P4 ( P4 )* ] Qs
P4 ( B4( r ) )*
P1 ( P1 )* (Qs )
1
O.
B2r ( B2r )*
B3r ( B3r )*
B4r ( B4r )*
Qr {P1 (P1 )* [P1 (P1 )* ] 1[4Is P1 (P1 )* ]P1 (P1 )*}(Qr ) 1 Qr {P1 ( P1 )*
4I s
P1 ( P1 )* }(Qr )
1
4I s .
So, (28), (30) follow. This completes the proof of Thm 2. Example1. Let
(t )
L2 ( R 2 , C 3 ) be 5-coefficient orthogonalvector valued scaling function satisfy the
following equation:
(t )
P0 (2t ) P1 (2t 1)
Where P3
P4
O, P0 ( P4 )*
P4 (2t 4). O,
P0 ( P0 )*
P0
P1 ( P1 )* 2 2 1 2
2 2 2 3
0
0
P2 ( P2 )*
P3 ( P3 )*
0 1 2 3 3
, P1
P4 ( P4 )*
4I3 .
1
0
0
0
2 6
0 ,
0
0
3 3
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Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
P2
2 Q1
2 2 2 3
0
0
0
2 3 3 2
.
0
53
0
4
0
53
0
, Q2
2
0
0
2
0
53
0
1 ,
0
53
0
0
1 . By using (22), we can choose
Suppose 2
2 2 1 2
2
0
4
By applying formula (24), we get that 1
1
2
2 53
(1)
B0
106 0
0
106
53
159
53
2 (1)
, B1
106
0
0
0
6
B2(1)
1 2 53 106 0
B0(2)
0
0 2
6 3
1 2 106 159 0
1 2 53 106
0
6
6
0
0
1 2 106 159 0
0 53 , 53 6 6 0 53 . 53 6 6
Delin Hua / Physics Procedia 33 (2012) 1346 – 1353
2 B1(2)
B2(2)
0
0
0
106 6
0
0
0
1 2 53 106
1 2 106 159
0
0
,
2 6 3 0 53 . 53 6 6 ( )
(2t ) Applying Theorem 2, we obtain that G (t ) B0 B1( ) (2t 1) B4( ) (2t 4), 1, 2,3 are orthogonal vector-valued wavelet functions associated with the orthogonal vector-valued scaling function Conclusion
A necessary and sufficient condition on the existence of a class of orthogonal vector-valued wavelets is presented. An algorithm for constructing a class of compactly supported orthogonal vectorvalued wavelets is proposed. References [1]
N. Zhang X.Wu,``Lossless of color masaic images". IEEE Trans Image Delivery}, 2006, 15(6), pp. 1379-1388
[2]
S. Efromovich, J. akey , M. Pereyia, N. Tymes, ``Data-Diven and Optimal Denoising of a Signal and Recovery of its
Derivation Using Multiwavelets", IEEE Trans Signal Processing, 2,004, 52(3), pp.628-635 [3]
Z. Shen , Nontensor product wavelet packets in\ L2 ( R s ) [J]. SIAM Math Anal.1995, 26(4): 1061-1074.
[4]
X. G. Xia, B. W. Suter, `Vector-valued wavelets and vector filter banks", IEEE Trans Signal Processing}, 1996, 44(3),
pp. 508-518. [5] Solitons
Q. Chen, Z..Cheng, ``A study on compactly supported orthogonal vector-valued wavelets andwavelet packets”, Chaos,
& Fractals}, 2007, 31(4), pp.1024-1034.
[6]
C.K.Chui, J.Lian, ``A study of orthonormal multiwavelets". Appli Numer Math, 1996, 20(1), pp. 273-298
[7]
S. Yang. Z.Cheng, H. Wang, ``Construction of biorthogonal multiwavelets", Math Anal Appl}, 2002, 276(1), pp.1-12.
[8]
S. Li and H. Ogawa., “Pseudoframes for Subspaces with Applications,” J. Fourier Anal. Appl.., Vol. 10,No 4, pp.409-
431, 2004. [9]
S. Yang , Biorthogonal two-direction refinable function and two-direction wavelet. Appl Math & Comput . 2006 , 1822
(3) : 1717-1724 [10]
Q. Chen , A . Huo , “The research of a class of biorthogonal compactly supported vector-valued wavelets , Chaos ,
Solitons & Fractals , 41 (2 ), pp . 951-961, August 2009”
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