The Research of Affine Bivariate Dual Frames Associated with a Generalized Multiresolution Analysis and Filter Banks

Available online at www.sciencedirect.com

Physics Procedia 33 (2012) 1224 – 1231

2012 International Conference on Medical Physics and Biomedical Engineering

The Research of Affine Bivariate Dual Frames Associated with a Generalized Multiresolution Analysis and Filter Banks Han Ke-zhong College of Information and Statistics,Zhengzhou Institute of Aeronautical Industry management ,Zhengzhou 450011, China [email protected]

Abstract The rise of frame theory in applied mathematics is due to the flexibility and redundancy of frames. In the work, the notion of bivariate affine pseudoframes is introduced and the no-tion of a bivariate generalized multiresolution analysis is introduced. A novel approach for designing one GMRA of Paley Wiener subspaces 2 2 (GMRA) of L ( R ) is proposed. The sufficient condition for the existence of a sort of affine pseudoframes with fi-filter banks is obtained by virtue of a generalized multiresolution analysis. The pyramid decomposition scheme is established based on such a generalized multiresolution analysis. An approach for designing a sort of affine biariate dual frames in two-dimensional space is presented. ©©2012 by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International 2011Published Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [nameCommittee. organizer] Keywords : bivariate, the pyramid decomposition sche-me, pseudofames, dual fames, oblique frames, filter banks

1.Introduction Wavelet transform is a signal processing technique and has been widely applied in the field of image processing,data compression, signal processing, noise removal and time series analysis. In general, multiresolution analysis is the most application of wavelet transform[1]. According to MRA theory, wavelet transform can be processed iteratively. The wavelet fra- me is one of the most widely applied wavelet transform. The structured frames are much easier to be constructed than str- uciured orthonormal bases. The notion of frames was introduced by Duffin and Schaeffer [2] and popularized greatly by the work of Daubechies and her co-authors[3]. After this ground breaki ng work, the theory of frames began to be more widely studied both in theory and in applications[4-7], such as data

1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Committee. doi:10.1016/j.phpro.2012.05.203

1225

Han Ke-zhong / Physics Procedia 33 (2012) 1224 – 1231

compression and sampling theory, and so on. Every frame(or Bessel sequence) determines an analysis operator, the range of which is important for a number of applications. The notion of Frame multiresolution analysis as described by [5] generalizes the no tion of multiresolution analysis by allowing non-exact affine frames. However, subspaces at different res- -olutions in a FMRA are still generated by a frame formed by translates and dilates of a single function. This article is motivated from the observation that standard methods in sampling theory provide examples of multirsolution structure which are not FMRAs. Inspired by [5] and [7], we introduce the notion of a generalized 2 2 multiresolution str-ucture (GMRA) of L ( R ) generated by several functions of integer grid translations 2 2 in space L ( R ) . , we denote a separable Hilbert space. We recall that a sequence {

By

:

Z

} is a frame

2

, if there exist positive real numbers C1 , C 2 such that

for

, C1 A sequence

2

,

C2

Z2

,

.

1

is a Bessel sequence if (only) the upper inequality of (1) follows. If only

for all

, the upper inequality of (1) holds, the sequence

with respect to (w.r.t.) the subspace frame {

,

. If

is a Bessel sequence

is a frame of a Hilbert space

, there exist a dual

} such that

h

,h

h,

h,

Z2

.

2

Z2

2.The generalized multiresolution analysis In what follows, we consider the case of generators, which yield affine pseudoframes of integer 2 2 2 2 2 be two sequences in L ( R ) . grid translates for subspaces of L ( R ) . Let u h and { u h} v Z 2 2 Let be a closed subspace of L ( R ) . We say that u h forms a dual bivariate pseudoframe for the subspace with respect to (w.r.t.) 2 { u h} (u Z ) if

, g s

g s

u Z2

where we define a translate operator, v function s L2 (R2 ) . It is important to note that contained in .

Vn

Vn if

Vn

Dg s

1

n

Vn

Z ; (b) 1

,

2

n Z n Z

n Z ,

2

of L ( R )

Vn

{0} ;

and n Z

where

u

h s .

(3)

s v , v Z 2 , for an any Z 2 need not be u h( s) and { u h( s)} v

s

Definition 1. A bivariate generalized multiresolution analysis closed linear subspaces

g, uh

elements

Vn is dense in

Dg s

{Vn , (s), (s)} is a sequence of s , (s) 2

L (R

2

L (R 2

2

) : (c) g s

2g 2s , for

g s

) ,such that (a)

Vn if and only 2

L (R

2

) ; (d)

1226

Han Ke-zhong / Physics Procedia 33 (2012) 1224 – 1231

g ( s) V0 implies

v

g ( s ) V0 , for all, v Z 2 ; (e) {

for V0 with respect to {

v

Z 2 } constitutes a pseupseudoframe

( s ), v

(s), v Z 2} .

v

3.A gmra of paley-wiener subspaces A necessary and sufficient condition for the construction of pseudoframe of translates for PaleyWiener subspaces is presented as follows. Theorem 1. Let

2

( s)

2

L (R

) be such that | ( ) | 0 a.e. on a connected neighbourhood of 0 in

2

[ 1/ 2,1/ 2) and { R 2 | ( ) | c 0} , and | ( ) | 0 a.e. otherwise. Define 2 2 2 2 L ( R ) { Z 2 } is a , V0 PW {g (s) L ( R ) : sup p( g ) } . Then, for v , v pseudoframe of translates for V0 w. r. t. { v v Z 2} if and only if ( ) where

is the characteristic function on

a. e. ,

(4)

f (s)

, and the Fourier transform of an integrable function

is defined by

( Ff )( )

f( )

R2

2 is

f ( s )e 2

R2 ,

ds ,

(5)

2

which, as usual, can be naturally extended to functions in spce L ( R ) . For a sequence

c {c(v)}

2

2

( Z 2 ) , we define the discrete-time Fourier transform as the function in L2 0,1 given

by

Fc

C

v Z2

2

then

{

n

g s

s , v Z 2} and { ,

g s

g,

v

| 0 a.e. othe--rwise, that

f, v Z2

= v Z2

u Z

[0,1]

f v Z

2

v

v

h)

f, v Z2

e2

f

R2

2

2

g,

PW , consider

F(

v

| 0

(6)

holds,

, i. e. ,

s

v

v Z2

f s

(6)

.

(s), v Z 2 } are a pair of commutative pseudoframes for

n

v Z2

Proof of Theorem 1. For all

2 iv

is also such that |

2

Note that the discrete-time Fourier transform is Z -periodic.Moreover, if a.e. on a connected neighbourhood of 0 in [ 1/ 2,1/ 2) , and |

c v e

iv

v

v

d

e2

F

e i v

d

v

2 iv

e

2 iu

v

v

s .

(7)

1227

Han Ke-zhong / Physics Procedia 33 (2012) 1224 – 1231

f v Z

v

v

f

2

| 0 only on [ 1/ 2,1/ 2) 2 ,and that

where we use the fact that |

v Z2

f

v is Z 2 -

v

periodic. Therefore

( ) ( ) is a necessary and sufficient condition for {

{

v

_

a. e. ,

(8)

2

,v

Z } to be a pseudoframe for V0 with respect to

2

v

,v Z } .

Direct calculation also shows that (7) is satisfied if in Theorem 1. Thereof,

v

,v Z2

and

. Let

( s ), ( s )

,v Z2

v

h and h satisfy supported conditions specified

are a pair of multiple pseudoframes for

, where

1, 2,3 . Theorem 2

7

L2 ( R 2 ) have the properties specified in Theorem 1 such that the

condition (6) is satisfied. Assume that Vn is defined by (8). Then

{Vn , (s), (s)} forms a GMRA.

The familiar scaling relationships associated with MRAs between dilates of the function

s still hold in GMRAs. Define filter functions B( ) and B ( ) , respectively by

well as that of

B( )

s as

v Z2

b(v)e

2 iv

and B(

)

n Z

2

p(n)e

2 in

of the two sequences B

{b(n)} and

B {b(n)} , respectively, wherever the sum is defined. Proposition 1

7

Assume also that

Let sequ.

B( )

b(v)

be such that B(0)

v Z2

2 and B( ) 0 in a neighborhoood of 0.

2 . Then there exist ( s ) L2 ( R 2 ) such that s

2

b(v) (2s v) .

(9)

v Z2

t under the same conditions as that of p for a

Similarly, there exist one scaling relationship for seq. p , i.e.,

s

2

b n

2s v .

(10)

v Z2

B 2

4.Affine dual frames of space Definition 2. Let Vn , 2

2

4

2

a. e..

(11)

2

L (R )

s ,

s

2

functions in L ( R ) . We say {

B

n

be a given GMRA, and let

( s),

n

( s),

( s) and

( s ) be 2 band-pass

1, 2, 3} forms a pseudoframe of translates for V1 w.

1228

Han Ke-zhong / Physics Procedia 33 (2012) 1224 – 1231

r. t.

{

v

(s),

(s),

v

f s

f, v Z

Proposition

f s

1, 2,3} if for v

s

v

v Z

8 2 . Let {Vn , (s), (s)} be a given

the family of binary functions {

{

f,

2

pseudoframe of translates for V1 with respect to {

respect to

V1 ,

n,k

,

:n , k

,

n ,k

,

f,

n ,k

To characterize the condition for which {

{

n

(s),

n

{

GMRA, and let

( s),

n

( s),

n

1, 2, 3} be a

(s), n (s), 1, 2,3} . Then, for every n Z , } forms a pseudoframe of translates for Vn 1 with

n

,

f,

n,k

k Z2

for V1 with respect to

( s).

v

f ( s) Vn 1 ,

} , i.e.,

f s

:n , k

v

2

n

( s),

( s) :

:n , k

.

}forms a pseudoframe of translates

( s) :

n

s

} , we begin with developing the

( s)

-

:n , k

k Z2

(s)

and

( s) 2

based on a GMRA, namely,

v Z2

( s) 2

2

2

q (v) (2s v) in

L (R

q (v) (2s v) in

L (R

v Z2

2

),

2

).

(13) (14)

Similar to Proposition 3, we have the following fact. Proposition

4

7

. Let {q (k ),

2

T

0,1 . Let

s

2

} be such that Q (0)

2

L ( R ) and be defined by (10). Assume that

( s)

in Proposition 3. Then there exist Let

0 and Q

2

L (R

2

),

L T , where

{b(k )} satisfies the conditions

generated from (14).

be the characteristic function of the interval

defined in Proposition 1.

k . k Z2

Theorem 3 k k

(s), (s),

k

8

. Let

(s),

be the bandwidth of the subspace V0 defined in Theorem 1. forms a pseudoframe of translates for V1 with respect to {

} if and only if there exist functions P

and

D

,

(s),

k

in

L2 0,1

2

such

that

P

B

P

B

D 1 2

Q D

4 Q

1 2 2

2

;

(16.1)

0.

(16.2)

be functions in L R defined by (14), (15), Theorem 4. Let s , s , ( s) and ( s ) , (18) and (19), respectively. Assume that conditions in Theorem 3 are satisfied. Then, for any function ( s) L2 ( R 2 ) , and any integer n,

1229

Han Ke-zhong / Physics Procedia 33 (2012) 1224 – 1231 3

,

n,k

n,k

k Z2

(s)

Furthermore, for any

n 1

s

,

:v , k

s .

:v , k

k Z2

1 v

(18)

L2 ( R 2 ) , 3

s

,

Consequently, if {

:v , k

} and { 2

}, (

:v , k

:v , k

s

:v , k

k Z2

1 v

.

(19)

Z 2 ) are also Bessel sequences, they are a

, v Z, k

2

pair of affine frames for L ( R ) . Proof. (i)

E

0 ,

Consider, for

s

s

,

2

L (R

Z , the operator E :

,k

2

)

L (R

2

) such that

s .

,k

k Z2

2

2

2

Then the operator E are well defined and uniformly bounded in the norm on L ( R ) . To show that the 0 asm limit E , it is sufficient to show that, for all g s in any dedense subspace of band2 2 limited functions in L ( R ) ,

g,

k Z2

,k

0 as

,k

In particular, we may choose the dense set of functions g

.

t , whose Fourier transform have compact

support, is continuous, and vanishes in a neighborhood of 0. 1

g,

,k

B

,k

1

g,

2

k

where B is the Bessel bound of

,k k

,k

|2

1

B

4

2

(20)

. Taking standard calculation of the right-hand side of (20),

g

k

2

,k

k

1

| g,

2

4 u

2

2

4

v

4

g

d

where

B is the Bessel

u

bound of

,k

term 4

k

g

. Following the lead of [8] and since 4 n

2

1

2

C

2

f

is continuous with compact support, the

, being a Riemann sum to the finite integral

g

t dt .

n

Moreover, since

g

vanishes in a neighborhood of 0 for all

g,

2 ,k

B

1

2

C 4

g

4

, we get that

g

d

k

B

1

2

1

2

C g

2

4

2

d

g

Note that the last integral at the right-hand side tends to 0 as theorem since, by using (18) recursively, we have

4

t

. This proves the first part of the

1230

Han Ke-zhong / Physics Procedia 33 (2012) 1224 – 1231 3

,

n,k

n ,k

k Z2

V

(ii) Since

g

L2 R 2 , for any

1 C

2

1

,

L2 R 2 and any Vn s.t. g s

k Z

. Now, by (18), for all n 3

} and {

0 there exists n0 g,

2

v,k

k Z

C

g

:v , k

t .Moreover, for C

BB ,

2

g

2

2

1 C

} are Bessel sequences, then equation

:v , k

{

} will be affine frames. In fact, the lower frame bound of {

:v , k

0 , and for

:v , k

2

If {

:v , k

v,k

n0

n0 , we have ,

2

t .

:v , k

n 1

1 v

g

:v , k

k Z2

1 v

n0 there exists g Vn0

any n

n 1

s

.

(19) implies that both :v , k

} and {

:v , k

{

:v , k

} and

} is implied by the

upper Bessel bound of the other. Therefore, this completes the proof of the second part of Theorem 4. 9

Z n be a family of multivariate vector-valued wavelet packets with Z n , we have respect to the orthogonal vector-valued scaling func tion ( x) , then , for ,

Theorem 5 . If

( x),

( ),

(

k)

,

0, k s

,k

Z n.

(21)

5.Conclusion 2

2

A sort of pseudoframes for the subspaces of L ( R ) are characterized, and a method for constructing 2

2

a GMRA of Paley-Wiener subspaces of L ( R ) is presented. The pyramid decomposition scheme is derived based on such a GMRA. References [1] Q. Chen, Z. Cheng,.

-valued wavelets and wav

. Chaos, Solitons

Fractals, Vol.31, No. 4, PP. 1024-1034, 2007 -366, 1952.

[2] -frequency

[3] 36, pp.961-1005, 1990..

, of mathe-matics, Vol. 4, No. 2, pp. 129-201, 2000. ory of multiresolution

comput.

Harmon. Anal., Vol. 5, pp. 389-427, 1998. L2(Rd). (II) Dual -

-637, 1997. comput. Harmon. Anal., Vol. 11, No. 2, pp. 289-304, 2001. and pseudofram--

[8]

7, No.

1, pp. 23-40, 2001. -valued multivariate wavelet packets associated with a dilation matrix, Chaos, Solitons & Fractals, Vol.35, No. 3, pp. 323-332, 2008.

Han Ke-zhong / Physics Procedia 33 (2012) 1224 – 1231 Acade-mic Press, New York, 1980. [11] S. Li and H. Ogawa., Pseudoframes for Subspaces with Applications 2004.

Fourier Anal. Appl.., Vol. 10,No 4, pp.409-431,

1231