A class of bidimensional nonseparable wavelet packets

2002.22B( 1): 131-137

..Atat~cta,spcientia

1~~Jm~lIl A CLASS OF BIDIMENSIONAL NONSEPARABLE WAVELET PACKETS 1

4=-** )

Tia,n Xiong/ei ( ~ .. ~) Li Yunzhang ( Sch.ool of Applied Math.ematics and Physics, Beijing Polytechnic Unit'ersity, Beijing 100022, China Abstract

2-band wavelct packets in L 2(R') were constructed in [3]. In this note, a way

to construct bidimensional orthonormal wavelet packets related to the dilation matrix M ( 11

=

1 ) is obtained, M-wavelets are used in quincunx subsampling in two dimensions

-1

for image processing. What is more, the approach of this paper can be generalized to construct wavelet packets in L 2 (R') related to a general diltion matrix. Key words

Scaling function, wavelet, wavelet packet

1991 MR Subject Classification

1

42C15, 41A58, 41A63

Introduction In [1], Coifman, Meyer, and Wickerhauser introduced orthogonal wavelet packets for L 2(R)

and, by using tensor products, constructed orthogonal wavelet packets in L 2(R2 ) . In [2], Chui and Li studied nonorthogonal wavelet packets and their dual wavelet packets in the univariate case, thus generalizing the orthorgonal wavelet packets of [1]. In [3], Shen constructed nontensor wavelet packets in L 2 ( R S ) . Applications of wavelet packets in signal processing and compression can be found in [4] and [5]. Since signals, as well as images, are multidimensional, most applications are multivariate. Unfortunately, all the above results correspond to the dilation matrix M denotes unit matrix. In practice, this is not always the case. Let sEN. For 1 ::; p < R' for which

00"

= 21,

where I

we denote by LP(R') the Banach space of all functions / on

II/lip =

(1 dxl/(x)IP); R'

< 00.

For / E L 1 (R'), we define the Fourier transform of / by

Let M be an s x s dilation matrix, i.e., M is an s x s integer matrix with the eigenvalues

larger than 1 in absolute value. A ladder of closed subspaces {l0 LEZ of L 2(R') is called a multiresolution analysis related to M if the following conditions hold:

1 Received October 29,1999, revised September 25,2000, Supported by the National Natural Science FOtUldation (198010U5). the Youth Foundation of Beijing, the N atural Science Foundation of Beijing (1013005), and

the Edueational Committee Foundation of Beijing(01KJ-019),

132

ACTA MATHEMATICA SCIENTIA

Voi C Voi - 1 for j E Z; 2(R'); (2) njEz Voi = {OJ, UjEz Voi = L (3) f(·) E Voi if and only if f(Mj.) E Vo for

Vol.22 Ser.B

( 1)

j' E Z:

(4) there exists a function ¢(.) in Vo such that the set {¢(. - k) hEZ' is an orthonormal basis for Vo. Here ¢(.) is called a scaling function of the multiresolution analysis. Since Vo C V_I- ¢(.)

has to satisfy the equation

¢O = Idet MI~

:L hj¢(M. -j), jEZ'

where {hj}jEZ' is called the mask, and Ho(O = IdetMI-~ of the refinement equation. It is well known that Idet MI

L: j 'EZ " hje-ij~ -

is called the symbol 1 basic wavelets are required to

characterize the orthogonal complement of Vo in V-I. Especially, when s basic wavelet

',p

= 1 and M = 2, the

can be defined by

't/J(o)

= J2:L(-1)jhl-j¢(2

0

-j).

jEZ

(~ ~1),

For the case M =

it is easy to check that, if

{Voi

}jEZ

is a multiresolution

analysis related to M, and ¢ is the corresponding scaling function with Ho(O being its symbol, then the basic wavelet

4'

can be defined by

where

In what follows, M is always referred to be the matrix

(~ ~

1)

(1.1) without specification.

Since M -refinable functions and M -wavelets were used in quincunx subsampling in two dimensions for image processing, and the basic wavelet has the precise expression, they attract interest of many mathematicians ( See [6]- [8] ). In [6]. a class examples of mult iresolut ion analysis related to the M were given. In [9], the regularity of M -refinable functions was discussed. In this paper, the construction of M- wavelet packets is obtained, and our approach can be used to the case of a general dilation matrix. The main result of this paper is as follows. Theorem 1.1 Assume that {Voi }jEZ is a multiresolution analysis related to M

(1 1), ¢

is the corresponding scaling function with Ho(O being its symbol, 1 -1 responding wavelet:

H1(O is defined as

4'.,," .... ,'i

by

111

(1.1), and Wj is the orthogonal complement of Vj

4'

=

is the cor-

III Voi-l. Define

No.1

Tian & Li: A CLASS OF BIDIMENSIONAL NONSEPARABLE WAVELET PACKETS

where "I E {O. 1}. Z+ denotes the set of all strictly positive integers. Then { 4'" n E Z2 } is an orthonormal basis in W _j for j E Z+. In particular,

,'2"",',; (. -

133 n) :

is an orthonormal basis for £2 (R 2 ) .

2

Proof of Theorem 1.1

Lemma 2.1 Let H o(' ) and H 1 (-) be as in Theorem 1.1. Assume that in L 2 (R 2 ) so that the functions f (. - n), n E Z2 are orthonormal. Define F1 ( · )

=

L "EZ

F2 ( · )

=

f

is any function

h"f(' - n), 2

L

g"f(' - n),

"EZ2

HdO = 2-~ L:"EZ 2 g"e-i,,~. Mk), F2(· - Mk) : k E Z2 } is an orthonormal basis for

where g" is such that Then {F1 ( ·

-

E

= Span{ f(· -

n) : n E Z2 }.

M k), F2(. - M k) : k E Z2 } is an orthonormal system. Let Q be the quadrilateral ABCD, where A = (0,0), B = (71", -71"), C = (271",0), D = (71",71"). It is easy to check that R 2 = U, E Z 2 ( Q + 7I"MI), where Q + 7I"MI and Q + 7I"MI' share no common interior point for I i- I'. Therefore, Proof First we show that {F1 (.

r =r

-

dxFdx)F1(x - Mk)

JR2

JU ,EZ2(Q+1rMI)

d~IF1(OI2eim'e

1d~( L IF1(~ + = 1d~( L IF1(~ + =

Q

IEZ 2

Q

IEZ 2

7I"MIW)e iAH e 7I"M2IW)eiMke

+ 1 d~( L IF1(~ + 7I"M(MI + (1,

1d~( L IF1(~ + + 1d~( L IF1(~ + Q

=

0)T))12)eiMke

IEZ 2

271"1)12)eiAHe

IEZ 2

Q

7I"M(1, O)T

+ 271"lW)e Lla e

IEZ 2

Q

= 21 d~IHo(OI2(L I!(~ + 271"1)12)eiMke IEZ 2

Q

+21 d~IHo(~ + (71", 7I")T)1 2( L I!(~ + (71". Q

IEZ

2

7I")T

+ 271"IW)e iAa e.

134

ACTA MATHEMATICA SCIENTIA

Vo1.22 Ser.B

Since fl' - n). n E Z2 are orthonormal. :L1EZ21j(~ + 2?rl)1 2 = 4~2 for a. e. ~ E R 2. And by the same procedure as that of (10), it is easy to check that the matrix

is a unitary matrix for a. e. ~ E R 2 • Hence,

L2

dxF1(x)Fdx - Mk)

1') f ?r- lQ

= 2

d~(IHo(~W + IHo(~ + (?r, ?r)TW)ei.\H·{

= 2~Z ~ d~eiMk{ = -.;.

f

.d~eiZk{

?r- lro,ll']>

= tik,o Similarly,

L,

dxF1(x)Fz(x - Mk)

= 21 d~Ho(OH1(~)(2: Ij(~ + 2?rIW)e iM k { lEZ'

Q

+21 d~Ho(~ + (?r, ?r?)Hl(~ + (?r, ?r)T)( 2: Ij(~ + (?r, ?r? + 2?rIW)e iM k {

hd~[Ho(OH1(O

lEZ 2

Q

= 2~Z

+ Ho(~ + (?r, ?r)T)Hl(~ + (?r, ?r)T)]eiAH{

= 0, where in the last equality, we used the fact that the matrix

Ho(O HdO) ( Ho(~ + (?r, ?r)T) Hl(~ + (?r, ?r)T) is a unitary matrix for a. e. ~ E R Z• Hence, {F1 ( · orthonormal system. Finally, wc show that span{ F 1 ( ·

-

MI.:), Fz(· - Mk) : k E ZZ} is an

Mk), F 2(· - MI.:) : k E Z2 } = E. By the definition of

-

F 1 and F 2 , it suffices to show that

E C span] F l

Since Z2

= {n: n = Ml +

E,

1 E Z2,

(· E

Mk), F 2(· - MI.:) : I.: E Z2}.

= (0, O)T or(l, O)T}, and

is invariant under M Z2- shifts. it reduces to show that

f(·),

fl' - (I, O)T) E span{ F l

{· -

Ml;;). F2(· - MI.~): k E Z2}.

No.1

Tirol & Li: A CLASS OF BIDIMENSIONAL NONSEPARABLE WAVELET PACKETS

f(f.)

135

= f(f.)[IH o(OI2 + !Hl(f.W + Ho(f.)Ho(f. + (1r, 1rf) +H1(f.)Hdf. + (1r, 1r)T)]

= f(f.)Ho(f.)[Ho(O

+ Ho(f. + (1r,

1r)T)]

+f(OH1(C)[H1(O + H1(c + (1r. 1r)T)] 1 = J2F1(f.)[Ho(O + Ho(f. + (1r, 1r)T)] A

- -

1 + J2F2(c)[H I (C) + H1(f. + (1r, 1r)T)] A

=

~FI(f.)

- -

L [1 + (_l)n l+n']hnein{ + ~F2(O L [1 + (_l)n l+n']Ynein{. nEZ'

nEZ'

It is easy to check that, for n = (nl, n2f E Z2, nl + n2 is even if and only if M-1n E Z2. Therefore,

Hence,

fO

=

L

hMnF1(· + Mn) +

nEZ'

L

YMn F2(' + Mn),

nEZ'

and consequently, f(·) E span] F 1 ( ·

Putting y(.)

= f(· -

-

Mk), F2(· - Mk) : k E Z2}.

(1, O)T). Then

g(e) = e-i{l j(c)

= e-i{l j(c) [IHo(cW + IH1(eW -Ho(f.)Ho(c + (1r, 1I')T) - H1(c)H1(f. + (11', 1I')T)]

= e-i{l j(c)Ho(f.)[Ho(e) - Ho(e + (11', 1I')T)] +e-i(l j(e)H1(e)[H1(f.) - H1(e + (11', 1I')T)]

=

~e-i{l F1(e)[Ho(c) -

v2 .

Ho(c + (11', 1r)T)]

+ ~e-i{IF2(e)[HI(e) - H1(e + (1r, 1r)T)]

= ~FI(e)

L [1- (_1t l+

n']h

ne

i[n-(I.O)T](

nEZ'

+~F2(C)

L

[1- (_1)"1+n']Yn ei[n-(I.O)T]{.

nEZ'

It is easy to check that, for n = (nll n2)T E Z2, nl + n2 is odd if and only if M-I[n - (1, O)T] E Z2. Therefore,

g(c) = F1(f.)

L

hMn+(I.O)TeiMn{ + F2(f.)

nEZ'

Hence, y(.)

=L nEZ'

hMn+(I,O)TF1(· + Mn) +

L

YMn+(I.O)Te iMn{.

nEZ'

L nEZ'

YMn+(I.O)TF2 ( · + Mn),

136

ACTA MATHEMATICA SCIENTIA

Vol.22 Ser.B

and consequently,

The proof is completed. Proof of Theorem 1.1

Since cc

L 2(R2 )

= Vo EB Wo EB W_ j , j=1

it suffices to show that

w.,

{V' el h, ... ,ej(.

for j E Z+. By induction. Define

- n) : ei E {O, I}, n E Z2 } is an orthonormal basis in

'0'0 by

Since {V,(·-n) : n E Z2} is an orthonormal basis for W o , by Lemma 2.1, we have that {,J"o('M n) : fO E {O, 1 }, n E Z2 } is an orthonormal basis for Woo Hence, { I2v',o (M . - M n) : eo E

{O, 1 },

E Z2 } is an orthonormal basis for W -1. It is easy to check that the Fourier transform of 120'0(M . -Mn) is exactlyJ,'o(~)e-i"( Therefore, {V'el(. - n) : fl E {O, I}, n E Z2} is an orthonormal basis for W -1. ti

Assume that {,p'1 ,'2,...,'j (. - n) : f/ E {O, 1 }, n E Z2 } is an orthonormal basis in W _ j for j E Z+.

For any fixed fll f2, .... fj E {O, I}, define ·0'1, '2, ''','';+1 by fj+l

= 0, 1,

then, by Lemma 2.1, {~"1' '2.... ,'.;+1 (. - Mn) : f/ E {O. I}, n E Z2} is an orthonormal basis for span{ V"I, '2, ....,j(. - n) : f/ E {O, I}, n E Z2}. Since

W_ j

EB

=

span{V"I"2,·",'j(·-n):f/E{0,1},nEZ 2},

('I "2, ... ,'j)E {O, 1 }j

{ '0'1. '2, "','';+1 (. - M n) : f/ E {O, I}, n E Z2} is an orthonormal basis for W l : Therefore. r

{12·0'1' '2,""'.i+1 (M· -Mn) : f/ E {O, I}, n E Z2} is an orthonormal basis of W-j-l. It is

easy to check that the Fourier transform of y'2;j,'I"2'''','j+l(M. -Mn) is exactly

Therefore. {4"1.'2, ....'j+l (. - n) : f/ E {O. I}, n E Z2} is an orthonormal basis for W_ j proof is completed. Example 2.1

where

For any N EN, let

1•

The

No.1

Tian & Li: A CLASS OF BIDIMENSIONAL NONSEPARABLE WAVELET PACKETS

Define

137

= II Ho(M-j '), 00

¢(.)

j=1

then it follows from [6, Theorem 4.1] that ¢ generates a multiresolution analysis related to M. According to Theorem 1.1, we can obtain a wavelet packet decomposition for L 2(R2 ) . For a given s x s dilation matrix M and a multiresolution analysis related to M, there

exist Idet M I- 1 wavelets and Idet M I filters, we can define the new 1.1. and obtain the wavelet packet decomposition.

V'

analogous to Theorem

References 1 Coifman R R. Meyer Y, Wickerhauser M V. Wavelet analysis and signal processing. In:Ruskai M B et al, eds. Wavelets and their Applications. Jones and Bartlett, 1992. 153-178 2 Clmi C K, Li C. Nonorthogonal wavelet packets. SIAM J Math Anal, 1993,24(3): 712-738 3 Zuowei Shen. Nontensor product wavelet packets in L 2(R'). SIAM J Math Anal, 1995, 26(4): 1061-1074 4 Coifman R R, Meyer Y, Quake S, Wickerhauser M V. Signal processing and compression with wave packets. In: Meyer Y, Roques S, eds. Progress in Wavelet Analysis and Application. France: Editions Frontieres, Gif-sur-Yvette, 1993. 77-93 5 Wickerhauser M V. Acoustic signal compression with wavelet packets. In: Chui C Ked. Wavelets: A tutorial in theory and applications. New York: Academic Press, 1992. 679-700 6 Albert Cohen, Ingrid Daubechies. Non-separable bidimensionale wavelet bases. Revista Mat Iberoamericana, 1993, 9(1): 51-137 7 Kovacevic J, Vet terli M. Perfect reconstruction filter banks for HDTV representation and coding. Image Comm, 1990, 2(3): 349-364 8 Kovacevic J, Vetterli M. Nonseparable mnltidimensional perfect reconstruction filter banks and wavelet bases for R". IEEE Trans Inf Theory, 1992, 38(2): 533-555 9 Li Yunzhang. The regularity of a class bidimensional nonseparable refinable functions. Acta Math Sinica, 1999,42(6): 1053-1064 (Chinese edition) 10 Ingrid Daubechies. Ten Lectures on Wavelets. SIAM Philadelphia, 1992