A REMARK ON THE ORTHOGONALITY OF A CLASS BIDIMENSIONAL NONSEPARABLE WAVELETS

2004,24B( 4) :569-576

A REMARK ON THE ORTHOGONALITY OF A CLASS BIDIMENSIONAL NONSEPARABLE WAVELETS 1 Li Yunzhang (

~-;..*

)

Department of Applied Mathematics, Beijing University of Technology, Beijing 100022, China E-mail: [email protected]

Abstract

Let M

= (11

1 ) In this paper, a necessary condition and an optimal -1 .

sufficient condition on the orthogonality of M-wavelets are obtained by the introduction of cycle rei at to M. Key words

Cycle, nonseparable wavelet

2000 MR Subject Classification

1

42C40

Introduction Throughout this paper, M is always referred to be the matrix

transform of a function

,clat:

I:::1tt

(1) Vj C Vj -

n

1

f E L 1 (R 2 ) is defined by , f(·)

1 = -2 7r

j

R2

(

11

1 ). The Fourier -1

.

dxf(x)e- 1x , .

~:b):':h: :~:~m: ::~:::o:':~:: amultiresolution analysis (MRA)

for j E Zj

(2) j E Z ~ = {OJ, UjEz Vj = L 2(R2 ) j (3) fO E Vj if and only if f(Mj·) E Vo for j E Z; (4) there exists a function ¢>(.) in Vo such that the set {¢>(. - k) basis for Vo. Since Vo C V-I, ¢>(.) has to satisfy some M-refinement equation

¢>(.) =

Vi

L

hEZ2

is an orthonormal

hn¢>(M· -n).

nEZ 2 1 Received

April 15, 2002; revised May 28, 2003. Supported by Natural Science Foundation of Beijing.

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It is well-known that the associated wavelet 'l/J is then given by

Jz

where H1W = e-i~lHo(E, + err, 7f)T), Ho(E,) = L:"EZ2 hl1e-i"~ for'; = (6, 6)T, Ho(E,) is called the symbol of the MRA. Analogous to the arguments in [10], we have

for a.e. E, E R 2 . In recent years, wavelets in higher dimensions, especially bidimensional wavelets, have attracted the interest of many mathematicians. The details caJ.! be seen in [1]-[9], [11] and [13]. Analogous to that of [10], a compact set K C R 2 is called congruent to [-7f, 7fj2 modulo 27f Z2 if IKI = 47f 2, and, for any E, E [-7f, 7f]2, there exists l E Z2 such that E, + 27ft E K. In this paper, using the symbol as a starting point as in [10], we investigate the orthogonality of M-wavelets. Definition 1.1 A mapping 7 is called a cycle mapping related to M if 7E, = ME, (mod 2) 27fZ for E, E [-7f, 7f]2, and 7 maps [-7f, n]2 into [-n, 7fj2. It is easy to verify that Proposition 1.1 Assume that 7 is a cycle mapping related to M, and that 0:, (3 E [-7f, 7f]2 satisfy (1.1) 0: = 7(3. Then either (3 = M-10: or (3 = M-10: + nc for some f. E { (x, y)T : x, y E {I, -I} }. Definition 1.2 Let 7 be a cycle mapping related to M. A finite set

is called a cycle related to M if E,i = 7E,i+l' E,,, = 76. The cycle is called non-trivial if it is also different from {a}. Examples Both

and

are cycles related to 1\1. Our main results can be stated as follows.

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Li: REMARK ON ORTHOGONALITY OF WAVELETS

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Theorem 1.1

Assume that H o is a trigonometric polynomial satisfying that

Define ¢ by

Suppose {¢(. - n) : n E Z2} are orthonormal, then there exists no non-trivial cycle {6, 6, ... ,~n} C [-7f,7f]2 related to M such that IHo(~i)1 = 1 for 1::; i::; n. Remark 1.1 It is unresolved that whether {¢(. - n) : n E Z2} are orthonormal if there exists no non-trivial cycle {~l, 6, ... ,~n } C [-7f, 7fF related to M satisfying that IHo(~i)1 = 1 for 1 :::; i :::; n. Theorem 1.2 Assume that H o is a trigonometric polynomial satisfying that

Define ¢ by

D denotes the region {~: ~

= (6, 6) T ,16 +61::; 7f, 16 -61::; n , I~il::; 327f

for i

= 1, 2}.

Suppose H o has no zeros in D, then {¢(. - n) : n Eo Z2} are orthonormal. Remark 1.2 Theorem 1.2 is optimal in the following sense: it is impossible to find a < ~ such that the absence of zeros of H o on D Q; = {~ : ~ = (6, 6 f, 16 + 61 ::; 7f, 16 - 61 :::; n , I~i I :::; a7f for i = 1, 2 }

guarantees the orthonormality of {¢(. - n) : n E Z2}.

2

Proofs of Theorems Proof of Theorem 1.1

By contradiction, suppose that there exists a non-trivial cycle

{~i h:'S;i:'S;n such that IHo(~i)1 = 1 for 1 ::; i ::; n, it suffices to show that

J(~i + 27fk) = 0 for 1 :::; i :::; nand k E Z2. First, we claim that (2.1) holds for ~i+l t

E { (x,

(2.1)

= M-l~i + tt«

with

yf : x, y E {1, -1} }

and k satisfying M':' k E Z2. In fact, without loss of generality, let t

6

= M- 1 6

+ tt« for some

E { (x, y)T : x, y E {1, -1} }.

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Then M-l(~1 + 21rk) =

Since IH o(6 )1= 1, H o(M - 1(6

6

+ 1rMk - 1rE.

+ 21rk)) = H o(6 + n) = 0, and consequently,

¢(6 + 21rk)

= H o(6 + 1rE)¢(M- 1(6 + 21rk)) =

o.

Secondly, we show that

¢(~i + 21rk) = 0 for ~i+l = M-l~i or k with M- 1k cj. Z2. By induction on k = h/kr + kn where k = (k 1, k 2)T, [x] denotes the integer part of x for x E R. (I) (2.2) holds for k = O. Without loss of generality, we only show that ¢(6) = O. Since { 6, 6, ... , ~n } is a cycle, we can choose the smallest io such that ~ia+I = M- 1 ~ia + 1rEia for some Eia E {(x, y)T : x, y E {I, -I}}. So, M- ia6 = M-l~ia = ~ia+I - 1rEia' and consequently, H o(M- ia6 ) = O. Therefore,


(il

H,lM- j 6 ) )
(II) (2.2) holds for k = 1, 2, 3, 4. We only show that (2.2) holds for k = 1. The others can be showed analogously. Without loss of generality, we only show that (2.2) holds fori = 1 and k = 1. (111) M- 1k cj. Z2, and k = 1. At this time, k E {(O, ±If, (±1, of}. Since

6

=

76, a and (3 being replaced by 6 and 6 in Proposition 1.1, 6

6 = M- 16 + 1rE for some E E { (x, If 6 = M- 1 6 , then ¢(6 ± 21r(O, l)T) Since

IHo(6 )1= 1, H o(6

= M- 1 6 or

y)T : x, y E {I, -I} }.

16 ± (1r, -1r)T)¢(M-l~1 ± (1r, _1r)T) TAT = H o (6 ± (1r, -1r) )¢(6 ± (1r, -1r) ). = H o(M -

± (1r, _1r)T) = 0, and consequently,

¢(6 ± 21r(0, l)T)

=

o.

Analogously, ¢(6 ± 21r(1, O)T) = o. If 6 = M- 16 +(1r, _1r)T E [-1r, 1rj2, 6 E [-1r, 1rj2, then ¢(6 +21r(0, l)T) o by (I), and

¢(6 + 21r(0, _l)T)

= Ho(6)¢(6) =

= H o(6)¢(6 + 21r( -1, l)T)

= Ho(6)Ho(1\1-16)¢(1\1-16 + 21r(O, 1\1-16 E {(x, yf : y - 1r ::; X

::;

_l)T),

1r - y, ~ ::; y ::; 1r}.

(2.3)

(2.4)

Li: REMARK ON ORTHOGONALITY OF WAVELETS

No.4

573

It follows from (2.4) that

When 6 = M- 16+(-1r, _1r)T, or M- 16+(1r, _1r)T, we have Ho(M- 16) = Ho(6+(1r, 1r)T) = 0, or H o(M - 1 6 ) = H o(6 + (-1r, 1r)T) = 0. This together with (2.3) leads to that ¢(6 +

21r(0, _l)T) = 0.

When

6 = M- 1 6 , we have

¢(M- 16

+ 21r(0, _l)T) = ¢(6 + 21r(0, _l)T) = H o (M - 1 6 + (-1r, 1r)T)¢(M- 1 6 + (-1r, 1r)T).

(2.5)

By (2.4),

M -1 c3 E

(x y) T : - -1r < Y < x - -1r x < -1r } , 2 2' - 2 '

{

"

which implies that

H o (M - 16

+ (-1r, 1r)T) = H O(€4 + (-1r,

which together with (2.3) and (2.5) leads to that ¢(6 When M- 16 + (-1r, 1r)T = €4, by (I), we have

which together with (2.3) and (2.5) leads to that have ¢(6 ± 21r(1, O)T) = 0.

°

1rf) = 0,

+ 21r(0, -l)T)

¢(6 + 21r(0, -l)T)

= 0.

= 0. Analogously, we also

Therefore, ¢(6 + 21rk) = when 6 = M- 1 6 + (-1r, 1r)T). By the same procedure as above, we can show that ¢(6 + 21rk) = under other cases. (II2) 6 = M- 16 , k = 1, and M- 1k E Z2. At this time,

°

k E {(±1,

¢(6

±l)T, (±1, =j=l)T}.

± 21r(1, t)") = H o(M- 16 ± 21r(1, 0)T)¢(M- 1 6 ± 21r(1, o)T) T

= H o(6 )¢(6 ± 21r(1, 0) ). A

Analogous to (III),

¢(6

± 21r(1, O)T) = 0. Hence,

¢(6 Analogously,

6

¢(6

(2.6)

± 21r(1, -l)T)

± 21r(1, l)T) = 0.

= 0.

°:s

(III) Suppose (2.2) holds for k :S m, m 2: 4, and 1 :S i :S n. Without loss of generality, we only show that ¢(6 + 21rk) = for k = m + 1. Since 6 = 76, applying Proposition 1.1 to 6 and 6, we obtain that 6 = M- 1 6 or = 1'1- 1 6 + 1rE for some E

E { (x, yf : x, y E {I, -I} } .

°

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(IIIl) 6 = M- 1 6 · If k = Mt for some t E Z2, then i :::; [m~2]

< m + 1, and by inductive assumption, = O. Hence, ¢(6 + 27fk) = H o(M- l (6 + 27fk))¢(M- l (6 +

+ 27fk)) = ¢(6 + 27ft) = O. If k = Mt + (1, O)T for some t l

¢(M- (6 27fk))

E

Z2, then H o(M- l (6 + 27fk))

and consequently, ¢(6 + 27fk) = O. (III2) 6 = M- 1 6 + (7f, _7f)T, and M- lk ~ Z2. Suppose k = Mt + (0, I)T for some t E Z2, then

= H o(6 + (7f, 7f)T) = 0,

i:::; [mJi3] :::; m. Hence,

¢(M- l (6

+

= ¢(6 + 27ft) = 0 by inductive assumption, and consequently, ¢(6 + 27fk) = O. The proofs of the others are analogous. By induction, the proof is completed. Proof of Theorem 1.2 By the same procedure as in [10], we have the following Lemma: Lemma 2.1 Assume that H o is a trigonometric polynomial satisfying

27fk))

Define ¢ by

Then { ¢(. - n), n E Z2 } are orthonormal if and only if there exists a compact set K congruent to [-7f, 7fj2 (modulo 27rZ 2 ) and containing a neighborhood of 0 such that inf inf IHo(M-k~)1

k>O€EK

> O.

By Lemma 2.1, it suffices to construct a compact set K congruent to [-7f, 7fj2 (modulo 27f Z2) and containing a neighborhood of 0 such that inf inf IHo(M-k~)1

k>O€EK

> O.

= (6, 6)T, 16 + 61 :::;

7f, 16 - 61 :::; 7f}, taking > O. = (6, 6)T, 16 + 61 :::; 7f, 16 - 61 :::; 7f}, [-7f, 7f]2 is no

When H o has no zeros in {~ : ~

K = [-7f, 7fj2, then we have infk>oinf€EK IHo(M-k~)1

When H o has zeros in {~ : ~ longer a satisfactory choice for K. Define D, = {~ : ~ = (6, 6f, 16 D2 =

H : ~ = (6, 6f, 16 + 61:::; 7f, 16 - 61 :::;

D 3 = {~ : ~ = (6, 6f, 16 D4 =

+ 61 :::; 7f, 16 - 61 :::; 7f, 6 2:: 0, 6 2:: O}, tt ,

6 :::; 0, 6 2:: O},

+ 61:::; n , 16 - 61 :::; n , 6 :::; 0, 6 :::; O},

H : ~ = (6, 6 f, 16 + 61 :::; 7f, 16 - 61 :::; 7f, 6 2:: 0, 6 :::; 0 }; 51 = [-71", 7f]2 n {~: ~ = (6, 6f, 62:: 0, 62:: O}, 52 = [-7f, 7f]2 n {~: ~ = (6, 6f, 6:::; 0, 62:: O}, 53 = [-7f, 7f]2 n {~: ~ = (6, 6f, 6

< 0,6:::; O},

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Li: REMARK ON ORTHOGONALITY OF WAVELETS

No.4

S4 = [-1r, 1r]2 n {~: ~ = (6,6f, 62:: 0, 6 ~ O}. We denote by H i- I (0) the closure of the set of zeros of H o in D, for 1 ~ i ~ 4. For each ~i E H i- I (0), choose J, to be the intersection with S, of a small cube neighborhood of ~i, small enough so that J, does not overlap with the lines 6 = ± 2311" , 6 = ± 2311" , and so that IH a (1]) I ~ ~ for 1] E J i . By the finite covering theorem, there exist a finite number of JiI' 1 ~ 1 ~ L, such that H i- I (0) C Uf~1 i«. Define

for 2

~

1~

t.; and K

= {[-1r,

1r]2 \

(Ut=1 Uf~1

MliI ) }

U{Uf~IMIll n [-1r, 1r]2 - (21r, Ov}

u{Uf~1 M 121 n [-1r, 1r]2 + (0,

21rV}

u{Uf~1 M 131 n [-1r, 1r]2 + (21r,

OV}

u{Uf~1 M 141 n [-1r, 1r]2 - (0, 21rV} = K o UKI UK 2 UK 3 UK 4 . It is obvious that K is congruent to [-1r, 1rF (modulo 21rZ 2) and contains a neighborhood of O. In the next, we show that H o has no zeros in M:" K for k 2:: 1. Since

M-IKo c {(:

(=

(6, 6f, 16 +61

< 1r, 16 -61 ~ 1r} \ (Ut=1 utI Iii)'

we obtain that Ho(M-IO ::j:. 0 for ~ E K o, and

for k 2: 2, which leads to that Ho(M-k~) ::j:. 0 for ( E K o and k u-» K o for k 2:: 1.

2:: 2. Hence, H o has no zeros in

M-IKI C {Uf~IIll- (1r, 1r)T}, and IHo(1/)! ~ ~ for 1] E III which leads to that (1r, 1r)T)1 = l-IHo(1]) 12 2:: ~ for 1] E Ill, Ho has no zeros in M- I K I • It is easy to check

Since

IHo(1] that M- 2 K I CD, and

for k 2:: 3. Hence, H o has no zeros in M-kK I for k 2:: 1.

u-» K I

for k

2:: 2, and consequently,

H o has no zeros in

Analogously, n; has no zeros in M-k(K2 U K 3 u K 4 ) for k 2: 1. Therefore, H o has no zeros in M:" K for k 2:: 1. The proof is completed. The optimality of Theorem 2.2 will be explained in the following example. Example 2.1 Assume that Ho(~) = ~(1 + e-3i~1) for ~ = (6, 6)T. It is easy to check that

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Define
n}

{(+). en, (~:;). (-

is a cycle related to M, and that

Ho(~) =

1 for

Hence, by Theorem 2.1, {
2 4 5 6 7 8 9 10 11 12 13

Daubechies I. Orthonormal bases of compactly supported wavelets. Comm Pure Appl Math, 1988, 41: 909-996 Cohen A, Daubechies I. Non-separable bidirnensionale wavelet bases. Rev Mat Iberoarnericana, 1993, 9: 51-137 Kovacevic J, Vetterli M. Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for H": IEEE Trans Inf Theory, 1992, 38: 533-555 Strang G, Nguyen T. Wavelets and filter banks. Wellesley, MA: Wellesley-Cambridge Press, 1996 Kovacevic J, Vetterli M. Perfect reconstruction filter banks for HDTV representation and coding. Image Comm, 1990, 2: 349-364 Belogay Eugene, Wang Yang. Arbitrarily smooth orthogonal nonseparable wavelets in R2. SIAM J Math Anal, 1999,30: 678-697 Grochenig K, Madych W. Multiresolution analysis, Haar bases, and self-similar tilings. IEEE Trans Infom Theory, 1992, 38: 558-568 Huang Daren, Li Yunzhang, Sun Qiyu. Refinable function and refinement mask with polynomial and exponential decay. Chin Ann Math, 1999, 20A: 483-488 (Chinese Edition) Li Yunzhang. An estimate of the regularity of a class bidimensional nonseparable refinable functions. Acta Math Sinica, 1999, 42: 1053-1064 (Chinese Edition) Daubechies I. Ten Lectures on Wavelets. SIAM, Philadelphia, 1992 Long Ruilin. Wavelet analysis in higher dimensions. Beijing: World Book Pulication Coorporation, 1995 (Chinese Edition) Tian Xiongfei, Li Yunzhang. A class of bidimensional nonseparable wavelet packets. Acta Math Sci, 2002, 22B(I): 131-137