construction of trivariate biorthogonal compactly supported wavelets

Chaos, Solitons and Fractals 34 (2007) 1412–1420

www.elsevier.com/locate/chaos

Construction of trivariate biorthogonal compactlysupported wavelets

Lei Sun *, Zhengxing Cheng, Yongdong Huang

School of Sciences, Xi’an Jiaotong University, Xi’an 710049, PR China

Accepted 9 October 2006

Communicated by Prof. L. Marek-Crnjac

Abstract

In this paper, first we introduce trivariate multiresolution analysis and trivariate biorthogonal wavelets. A sufficientcondition on the existence of a pair of trivariate biorthogonal scaling functions is derived. Then, the pair of nonsepa-rable or separable trivariate biorthogonal wavelets can be achieved from the pair of trivariate biorthogonal scalingfunctions.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that nature is clearly not continuous, not periodic, but self-similar. Mohamed EI Naschie has intro-duced a mathematical formulation to describe phenomena that is resolution dependent. His E-infinity view appears tobe clearly a new framework for understanding and describing nature. As reported in [1–3], space–time is an infinitedimensional fractal that happens to have D = 4 as the expectation value for topological dimension. The topologicalvalue 3 + 1 means that, in our low energy resolution, the world appears to us if it were four-dimensional. In [4–6], Iov-ane shows that the dimension changes if we consider different energies, corresponding to different lengths-scale in uni-verse. The Fourier’s transform is a mathematical tool to consider the motion either in the frequency domain or in thetime domain. It cannot be simultaneously described in the frequency domain as well as in the time domain. Because ofthis, we need a transform that takes simultaneously account into the two aspects. Fortunately, the wavelet transform,which permits a multiresolution analysis of data with different behavior on different scales, can make up for thedisadvantage.

The wavelet transform is a simple and practical mathematical tool that cuts up data or functions or operations intodifferent frequency components and then studies each component with a resolution matched to its scale. The main fea-ture of the wavelet transform, which has been noticed by many physicians and engineers, is to hierarchically decomposea general function, as a signal or a process, into a set of approximations functions with different scales. Wavelets are a

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.10.027

* Corresponding author.E-mail address: [email protected] (L. Sun).

mailto:[email protected]

L. Sun et al. / Chaos, Solitons and Fractals 34 (2007) 1412–1420 1413

fairly simple mathematical tool with a variety of possible applications. Already they have led to exciting application infractals [7], signal analysis [8] and image processing [9] in the last two decades. In order to implement the wavelet trans-form, we need all kinds of wavelets possessing good properties such as orthogonality, compact support, symmetry andantisymmetry .

Multivariate wavelet analysis is a powerful tool for processing and analyzing multi-dimensional signals. A majorityof real-world signals are multi-dimensional signals, for example, graphic information and video signal. Multi-dimen-sional signal processing has been an important branch of signal processing science. At the present time, since tensorproduct wavelets have merit of simple algorithm and easy implementation, people usually use tensor product waveletsto deal with multi-dimensional signals. Numerical experiments with decomposition and reconstruction procedure usingnonseparable wavelets reveal more feature in the high-frequency band than does by a separable wavelet [9], therefore,nonseparable wavelets were used in pattern recognition and texture analysis and edge detection. The design of nonsep-arable orthogonal wavelets is still a challenging problem and only a few references have dealt with this subject [9–18]. Sois nonseparable biorthogonal wavelets.

In this paper, inspired by [15–18], we shall investigate the design of a class of trivariate biorthogonal wavelets.The paper is organized as follows. In Section 2, we recall trivariate multiresolution analysis and trivariate bior-thogonal wavelets. In Section 3, starting from the given real-coefficient functions satisfying certain conditions,we obtain a sufficient condition on the existence of trivariate biorthogonal scaling functions. Then, by virtue oftrivariate biorthogonal scaling functions, we construct the corresponding trivariate biorthogonal wavelets bymatrix extension.

2. Trivariate multiresolution analysis of L2(R3)

We begin with some basic theory and notations to be used throughout this paper. Let Z and R be the set of all inte-gers and real numbers, respectively. Denote

L2ðR3Þ ¼ f :

ZR3

jf ðxÞj2 dx <1� �

and

l2ðR3Þ ¼ sm :Xm2Z3

jsmj2 <1( )

:

If let m0 = (0,0,0), m1 = (1,0,0), m2 = (0,1,0), m3 = (0,0,1), m4 = (1,1,0), m5 = (1,0,1), m6 = (0,1,1), m7 = (1,1,1), then

Z3 ¼[7j¼0

ðmj þ 2Z3Þ; ðmi þ 2Z3Þ \ ðmj þ 2Z3Þ ¼ ;; i 6¼ j;

where 0 6 i, j 6 7, 2S = {2x: x 2 S}, S1 ± S2 = {x1 ± x2: x1 2 S1, x2 2 S2}. In other words, Z3 can be split into eightdisjoint subsets:

fð2i; 2j; 2kÞ : i; j; k 2 Zg; fð2i; 2jþ 1; 2kÞ : i; j; k 2 Zg; fð2i; 2j; 2k þ 1Þ : i; j; k 2 Zg;fð2i; 2jþ 1; 2k þ 1Þ : i; j; k 2 Zg; fð2iþ 1; 2j; 2kÞ : i; j; k 2 Zg; fð2iþ 1; 2jþ 1; 2kÞ : i; j; k 2 Zg;fð2iþ 1; 2j; 2k þ 1Þ : i; j; k 2 Zg; fð2iþ 1; 2jþ 1; 2k þ 1Þ : i; j; k 2 Zg:

Definition 1. For an arbitrary sequence flmgm2Z3 , let D = {m: m 2 Z3, lm 5 0}. If D is finite, then flmgm2Z3 is called afinite sequence.

It is well known that multiresolution analysis provides a nature framework for the understanding of wavelet bases,and for the construction of wavelets. Now we recall briefly trivariate multiresolution analysis. Define a close subspaceVj � L2(R3) by

Vj ¼ closL2ðR3Þðspanfuð2jx� aÞgÞ; a 2 Z3: ð1Þ

Definition 2 [19]. We say that the function u(x) generates a trivariate multiresolution analysis {Vj}j2Z of L2(R3), if thesequence {Vj}j2Z defined in (1) satisfies the following:

1414 L. Sun et al. / Chaos, Solitons and Fractals 34 (2007) 1412–1420

(1) � � �� V�1 � V0 � V1 � � � � ;(2) \j2ZVj = {0}; [j2ZVj is dense in L2(R3);(3) f(x) 2 V0 if and only if f(2jx) 2 Vj, j 2 Z;(4) there exists u(x) 2 V0 such that the sequence {u(x � a), a 2 Z3} forms a Riesz basis of V0.

And the function u(x) is called the scaling function of the trivariate multiresolution analysis.By the theory of trivariate multiresolution analysis, there exist functions wl(x) (1 6 l 6 7) such that

fwlðx� aÞ : 1 6 l 6 7; a 2 Z3g

forms a Riesz basis of W0, the complement subspace of V0 in V1, then we say wl(x) (1 6 l 6 7) are wavelet functionsassociated with the scaling function u(x).

Let uðxÞ; euðxÞ be a pair of scaling functions. If it satisfies

huðx� aÞ; euðx� bÞi ¼ dab; a; b 2 Z3; ð2Þ

where dab is the Kronecker symbol such that dab = 1 when a = b and dab = 0 when a 5 b. Then we refer it to as a pair ofbiorthogonal scaling functions.

Let wðxÞ; ewðxÞ be a pair of wavelet functions. If it satisfies

hwðxÞ; euðxÞi ¼ 0;

hewðxÞ;uðxÞi ¼ 0;

hwðx� aÞ; ewðx� bÞi ¼ dab:

Then we refer it to as a pair of biorthogonal wavelet functions, respectively, associated with uðxÞ; euðxÞ.Suppose uðxÞ; euðxÞ satisfy the following equations:

uðxÞ ¼Xa2Z3

pauð2x� aÞ; ð3Þ

euðxÞ ¼Xa2Z3

epabeuð2x� aÞ; ð4Þ

whereP

a2Z3 pa ¼ 8 andP

a2Z3epa ¼ 8.Let PðxÞ; eP ðxÞ, respectively, denote the mask functions of fpaga2Z3 ; fepaga2Z3 :

P ðxÞ ¼ 1

8

Xa2Z3

pae�ix�a; eP ðxÞ ¼ 1

8

Xa2Z3

epae�ix�a; ð5Þ

where x Æ a denotes dot product of two vectors x and a.If uðxÞ; euðxÞ which are defined, respectively, in (3) and (4) are a pair of biorthogonal scaling functions, then
X7
j¼0

P ðxþ pmjÞeP ðxþ pmjÞ ¼ 1: ð6Þ

Conversely, if the mask functions P ðxÞ; eP ðxÞ satisfy (6) and Cohen’s condition [20], that is, there exists a constantC > 0 such that

fjP ð2�jxÞjjeP ð2�jxÞj; j P 1; x 2 TgP C;

where T = [�p,p]3, then uðxÞ; euðxÞ are a pair of biorthogonal scaling functions.

Suppose a pair of biorthogonal wavelet functions wlðxÞ;fwlðxÞ ð1 6 l 6 7Þ associated with uðxÞ; euðxÞ satisfies thefollowing equations:

wlðxÞ ¼Xb2Z3

qlbuð2x� bÞ; ð7Þ

fwlðxÞ ¼Xb2Z3

eqlb euð2x� bÞ: ð8Þ

Let QlðxÞ;fQlðxÞ, respectively, denote the mask functions of fqlbgb2Z3 ; f eql

bgb2Z3 :


QlðxÞ ¼ 1

8

Xb2Z3

qlbe�ix�b; fQlðxÞ ¼ 1

8

Xb2Z3

eqlbe�ix�b: ð9Þ

Then
X7
j¼0

P ðxþ pmjÞfQlðxþ pmjÞ ¼ 0; ð10Þ

X7

j¼0

eP ðxþ pmjÞQlðxþ pmjÞ ¼ 0; ð11Þ

X7

j¼0

fQlðxþ pmjÞQmðxþ pmjÞ ¼ dlm; ð12Þ

or equivalently,

MðxÞfM�ðxÞ ¼ I ;

where the matrices MðxÞ ¼ ðQiðxþ mjpÞÞ;fMðxÞ ¼ ðeQiðxþ mjpÞÞ ð0 6 i; j 6 7Þ with Q0ðxÞ ¼ P ðxÞ; eQ0ðxÞ ¼ eP ðxÞand * denotes the complex and the transpose.

As usual, for a given pair of trivariate biorthogonal scaling functions, the construction of trivariate biorthogonalwavelets is reduced to the construction of MðxÞ and fMðxÞ such that

MðxÞfM�ðxÞ ¼ I :

The method for constructing trivariate biorthogonal wavelets is called matrix extension [19].

3. Construction of trivariate biorthogonal wavelets

In this section, we consider constructing trivariate biorthogonal wavelets. Without the special mention made, thesequence under consideration is finite and real-valued.

For the complex number n with jnj = 1 and the positive integer r, let the real-coefficient functions L(n), S(n),eLðnÞ; eSðnÞ satisfy the following:

Sðn2ÞeSðn2Þ ¼ 1� 1� n2

4

� �r1� n�2

4

� �r

Q2; ð13Þ

LðnÞeLðnÞ ¼Xr�1

j¼0

r þ j� 1

j

� �1� u

2

� �j

; u ¼ 1

2ðnþ n�1Þ; ð14Þ

Sð1Þ ¼ eSð1Þ ¼ Lð1Þ ¼ eLð1Þ ¼ 1; Q ¼ ð�1ÞrLð�1Þ ¼ ð�1ÞreLð�1Þ; Lð0ÞQ 6¼ 0; eLð0ÞQ 6¼ 0: ð15Þ

Define the functions AðnÞ;BðnÞ; eAðnÞ; eBðnÞ as follows:

AðnÞ ¼ n4r�1 1þ n�1

2

� �r

Lðn�1ÞSðn2Þ; ð16Þ

BðnÞ ¼ 1þ n2

� �reLð�nÞ 1� n2

� �2r

Q; ð17Þ

eAðnÞ ¼ n4r�1 1þ n�1

2

� �reLðn�1ÞeSðn2Þ; ð18Þ

eBðnÞ ¼ 1þ n2

� �r

Lð�nÞð1� n2Þ2rQ: ð19Þ

Lemma 1. Let AðnÞ;BðnÞ; eAðnÞ; eBðnÞ be as defined in (16)–(19). Then
eAðnÞBðnÞ þ eAð�nÞBð�nÞ ¼ 0;
AðnÞeBðnÞ þ Að�nÞeBð�nÞ ¼ 0:


Proof. By (17) and (18), we have

eAðnÞBðnÞ þ eAð�nÞBð�nÞ ¼ n1�4r 1þ n2

� �2reLðnÞeSðn2ÞeLð�nÞQ 1� n2

� �2r

� n1�4r 1þ n2

� �2reLð�nÞeSðn2ÞeLðnÞQ 1� n2

� �2r

¼ 0:

Similarly, according to (16) and (19),

AðnÞeBðnÞ þ Að�nÞeBð�nÞ ¼ n1�4r 1þ n2

� �2r

LðnÞSðn2ÞLð�nÞQ 1� n2

� �2r

� n1�4r 1þ n2

� �2r

Lð�nÞSðn2ÞLðnÞQ 1� n2

� �2r

¼ 0

as required. h

The next result, in fact, comes from [10].

Lemma 2. LðnÞ; eLðnÞ be as defined in (14) and n ¼ e�ix0 . Then, for r P 1,

1� n2

� �r1� n

2

� �r

Lð�1ÞeLð�1Þ 6Xr�1

j¼0

r þ j� 1

j

� �sin2 x0

2

� �j: ð20Þ

Moreover, the equality holds if and only if x0 = (2k + 1)p, k 2 Z.

Let x ¼ ðx1;x2;x3Þ; z1 ¼ e�ix1 ; z2 ¼ e�ix2 ; z3 ¼ e�ix3 ; PðxÞ ¼ Pðz1; z2; z3Þ, we have

Theorem 1. Let

P ðz1; z2; z3Þ ¼ Aðz1Þ þ z2k12 z2k2

3 Bðz1Þ�

Aðz2Þ þ z2k31 z2k4

3 Bðz2Þ�


2 Bðz3Þ�

; ð21ÞeP ðz1; z2; z3Þ ¼ eAðz1Þ þ z2k12 z2k2

3eBðz1Þ

� � eAðz2Þ þ z2k31 z2k4

3eBðz2Þ


2eBðz3Þ

� �; ð22Þ

where ki (1 6 i 6 6) is a non-negative integer. Then there exist the solutions uðxÞ; euðxÞ to Eqs. (3) and (4) determined,

respectively, by fpaga2Z3 ; fepaga2Z3 and

(1) uðxÞ; euðxÞ is a pair of biorthogonal scaling functions;

(2) If ki = 0 (1 6 i 6 6), then uðxÞ; euðxÞ is a pair of biorthogonal tensor product scaling functions ;

(3) If there exists at least ki 5 0 for some i 2 {1,2,3,4,5,6}, then uðxÞ; euðxÞ are nonseparable.

Proof.

(1) According to (21) and (22), we have

P ðxÞeP ðxÞ ¼ Pðz1; z2; z3ÞeP ðz1; z2; z3Þ ¼ Aðz1Þ þ z2k12 z2k2

3 Bðz1Þ� eAðz1Þ þ z2k1

2 z2k23eBðz1Þ

� �Aðz2Þ þ z2k3

1 z2k43 Bðz2Þ

� eAðz2Þ þ z2k3

1 z2k43eBðz2Þ

� �Aðz3Þ þ z2k5

1 z2k62 Bðz3Þ

� eAðz3Þ þ z2k51 z2k6

2eBðz3Þ

� �¼ Aðz1ÞeAðz1Þ þ Bðz1ÞeBðz1Þ þ z2k1

2 z2k23 Bðz1ÞeAðz1Þ þ z�2k1

2 z�2k23 Aðz1ÞeBðz1Þ

n oAðz2ÞeAðz2Þ þ Bðz2ÞeBðz2Þ þ z2k3



n oAðz3ÞeAðz3Þ þ Bðz3ÞeBðz3Þ þ z2k5



n o:

Thus,


X7

j¼0

P ðxþ pmjÞeP ðxþ pmjÞ ¼ Aðz1ÞeAðz1Þ þ Bðz1ÞeBðz1Þ þ z2k12 z2k2

3 Bðz1ÞeAðz1Þ þ z�2k12 z�2k2

3 Aðz1ÞeBðz1ÞnþAð�z1ÞeAð�z1Þ þ Bð�z1ÞeBð�z1Þ þ z2k1

2 z2k23 Bð�z1ÞeAð�z1Þ

þz�2k12 z�2k2

3 Að�z1ÞeBð�z1Þo

Aðz2ÞeAðz2Þ þ Bðz2ÞeBðz2Þ þ z2k32 z2k4

3 Bðz2ÞeAðz2Þn

þz�2k32 z�2k4

3 Aðz2ÞeBðz2Þ þ Að�z2ÞeAð�z2Þ þ Bð�z2ÞeBð�z2Þ þ z2k32 z2k4

3 Bð�z2ÞeAð�z2Þ

þz�2k32 z�2k4

3 Að�z2ÞeBð�z2Þo


3 Bðz3ÞeAðz3Þn

þz�2k52 z�2k6

3 Aðz3ÞeBðz3Þ þ Að�z3ÞeAð�z3Þ þ Bð�z3ÞeBð�z3Þ þ z2k52 z2k6

3 Bð�z3ÞeAð�z3Þ

þz�2k52 z�2k6

3 Að�z3ÞeBð�z3Þo:

If jnj = 1, then

1þ n2

� �r 1þ n2

� �r

LðnÞeLðnÞ þ 1� n2

� �r 1� n2

� �r

Lð�nÞeLð�nÞ ¼ 1 ð23Þ

and

Aðz1ÞeAðz1Þ þ Bðz1ÞeBðz1Þ þ Að�z1ÞeAð�z1Þ þ Bð�z1ÞeBð�z1Þ

¼ 1þ z1

2

� �r1þ �z1

2

� �r

L �z1ð ÞeLðz1ÞSðz21ÞeSð�z2

1Þ þ1þ z1

2

� �r1þ �z1

2

� �reLð�z1ÞLð��z1ÞQ2 1� z1

2

� �2r1� �z1

2

� �2r

þ 1� z1

2

� �r1� �z1

2

� �r

Lð��z1ÞeLð�z1ÞSðz21ÞeSð�z2

1Þ þ1� z1

2

� �r1� �z1

2

� �reLðz1ÞLð�z1ÞQ2 1þ z1

2

� �2r1þ �z1

2

� �2r

¼ Sðz21ÞeSð�z2

1Þ þ Q2 1þ z1

2

� �r1þ �z1

2

� �r1� z1

2

� �r1� �z1

2

� �r1þ z1

2

� �r1þ �z1

2

� �r

Lðz1ÞeLð�zÞ�þ 1� z1

2

� �r1� �z1

2

� �r

Lð�z1ÞeLð��zÞ�¼ Sðz2

1ÞeSð�z21Þ þ Q2 1� z2

1

4

� �r1� z�2

1

4

� �r

¼ 1:

On the other hand, by Lemma 1,

z2k12 z2k2


3 Aðz1ÞeBðz1Þ þ z2k12 z2k2

3 Bð�z1ÞeAð�z1Þ þ z�2k12 z�2k2

3 Að�z1ÞeBð�z1Þ

¼ z2k12 z2k2

3 Bðz1ÞeAðz1Þ þ Bð�z1ÞeAð�z1Þn o

þ z2k12 z2k2

3eBðz1ÞAðz1Þ þ eBð�z1ÞAð�z1Þn o

¼ 0:

It follows



3 Aðz1ÞeBðz1Þ þ Að�z1ÞeAð�z1Þ þ Bð�z1ÞeBð�z1Þ

þ z2k12 z2k2


3 Að�z1ÞeBð�z1Þ ¼ 1:

Similarly, we can show that




þ z2k32 z2k4


3 Að�z2ÞeBð�z2Þ ¼ 1

and




þ z2k52 z2k6


3 Að�z3ÞeBð�z3Þ ¼ 1:

Therefore,
X7
j¼0

P ðxþ pmjÞeP ðxþ pmjÞ ¼ 1:


In order to show that uðxÞ; euðxÞ are biorthogonal, we need to verify that PðxÞ; eP ðxÞ satisfies Cohen’s condition.Now let P(x) = 0, then


3 Bðz1Þ ¼ 0;

or


3 Bðz2Þ ¼ 0;

or


3 Bðz3Þ ¼ 0;

and jA(z1)j = jB(z1)j or jA(z2)j = jB(z2)j or jA(z3)j = jB(z3)j. There are three cases to consider.If jA(z1)j = jB(z1)j, then jA(z1)j2 = jB(z1)j2. Thus,

1� 1� z21

4

� �r1� z�2

1

4

� �r

Lð�1ÞeLð�1Þ� �

Lðz1ÞLð�z1Þ ¼1� z1

2

� �2r1� �z1

2

� �2r

Lð�1ÞeLð�1ÞeLð�z1ÞeLð��z1Þ:

That is

Lðz1ÞLð�z1Þ ¼1þ z1

2

� �r1þ �z1

2

� �r1� z1

2

� �r1� �z1

2

� �r

Lð�1ÞeLð�1ÞLðz1ÞLð�z1Þ

þ 1� z1

2

� �r1� �z1

2

� �r1þ z1

2

� �r1þ �z1

2

� �r

Lð�1ÞeLð�1ÞLð�z1ÞLð��z1Þ:

By (23),

Lðz1ÞLð�z1Þ ¼1� z1

2

� �r 1� �z1

2

� �r

Lð�1ÞeLð�1Þ:

Since

Lðz1ÞLð�z1Þ ¼Xr�1

j¼0

r þ j� 1

j

� �1� cos x1

2

� �j

¼Xr�1

j¼0

r þ j� 1

j

� �sin2 x1

2

� �j;

we have

1� z1

2

� �r 1� �z1

2

� �r

Lð�1ÞeLð�1Þ ¼Xr�1

j¼0

r þ j� 1

j

� �sin2 x1

2

� �j:

From Lemma 2, it follows that x1 = p.Similarly, jA(z2)j = jB(z2)j implies x2 = p and jA(z3)j = jB(z3)j implies x3 = p. So we can deduce that if x 5 (p,p,p),

then P(x) 5 0. The result also holds for eP ðxÞ. Noting that 12T � ½�p; p�3, we have, for j P 1, x 2 T,

jP ð2�jxÞjjeP ð2�jxÞj > 0:

Consequently, there exists a constant C > 0 such that

fjP ð2�jxÞjjeP ð2�jxÞj; j P 1; x 2 Tg > C;

which implies that P ðxÞ; eP ðxÞ satisfies Cohen’s condition.

(2) If ki = 0 for 1 6 i 6 6, then

P ðxÞ ¼ ðAðz1Þ þ Bðz1ÞÞðAðz2Þ þ Bðz2ÞÞðAðz3Þ þ Bðz3ÞÞ

and

eP ðxÞ ¼ eAðz1Þ þ eBðz1Þ� � eAðz2Þ þ eBðz2Þ

� � eAðz3Þ þ eBðz3Þ� �

;


which implies that a pair of uðxÞ; euðxÞ determined, respectively, by PðxÞ; eP ðxÞ is biorthogonal tensor product scalingfunctions.

(3) If there exists at least ki 5 0 for some i 2 {1,2,3,4,5,6}. Suppose that k2 5 0, then Aðz1Þ þ z2k23 Bðz1Þ is nonsep-

arable and u(x) is a nonseparable scaling function. Similarly, eP ðxÞ is also a nonseparable scaling function. Thiscompletes the proof. h

Having a pair of trivariate biorthogonal scaling functions, we can construct a pair of trviariate biorthogonal waveletfunctions. So we have the following.

Theorem 2. Let uðxÞ; euðxÞ be as defined in Theorem 1. Construct the corresponding functions wiðxÞ;fwiðxÞ ð1 6 i 6 7Þ as

follows: X
wiðxÞ ¼
b2Z3

qibuð2x� bÞ; ð24Þ

fwiðxÞ ¼Xb2Z3

eqibuð2x� bÞ; ð25Þ

where fqib; 1 6 i 6 7g; feqi

b; 1 6 i 6 7g are determined, respectively, by the following functions:

Q1ðxÞ ¼ Aðz1Þ þ z2k12 z2k2

3 Bðz1Þ�


3 Bðz2Þ�

z3 Að�z3Þ þ z2k51 z2k6

2 Bð�z3Þ�

;


3 Bðz1Þ�


3 Bð�z2Þ�


2 Bðz3Þ�

;


3 Bðz1Þ�

z2z3 Að�z2Þ þ z2k31 z2k4

3 Bð�z2Þ�

Að�z3Þ þ z2k51 z2k6

2 Bð�z3Þ�

;

Q4ðxÞ ¼ z1 Að�z1Þ þ z2k12 z2k2

3 Bð�z1Þ�


3 Bðz2Þ�


2 Bðz3Þ�

;

Q5ðxÞ ¼ z1 Að�z1Þ þ z2k12 z2k2

3 Bð�z1Þ�


3 Bðz2Þ�


2 Bð�z3Þ�

;

Q6ðxÞ ¼ z1z2 Að�z1Þ þ z2k12 z2k2

3 Bð�z1Þ�


3 Bð�z2Þ�


2 Bðz3Þ�

;

Q7ðxÞ ¼ z1z2z3 Að�z1Þ þ z2k12 z2k2

3 Bð�z1Þ�


3 Bð�z2Þ�


2 Bð�z3Þ�

and
eQ1ðxÞ ¼ eAðz1Þ þ z2k12 z2k2
3eBðz1Þ


3eBðz2Þ

� �z3eAð�z3Þ þ z2k5

1 z2k62eBð�z3Þ

� �;


3eBðz1Þ


1 z2k43eBð�z2Þ


2eBðz3Þ

� �;


3eBðz1Þ

� �z2z3

eAð�z2Þ þ z2k31 z2k4

3eBð�z2Þ

� � eAð�z3Þ þ z2k51 z2k6

2eBð�z3Þ

� �;

eQ4ðxÞ ¼ z1eAð�z1Þ þ z2k1

2 z2k23eBð�z1Þ


3eBðz2Þ


2eBðz3Þ

� �;

eQ5ðxÞ ¼ z1eAð�z1Þ þ z2k1

2 z2k23eBð�z1Þ


3eBðz2Þ


1 z2k62eBð�z3Þ

� �;

eQ6ðxÞ ¼ z1z2eAð�z1Þ þ z2k1

2 z2k23eBð�z1Þ


3eBð�z2Þ


2eBðz3Þ

� �;

eQ7ðxÞ ¼ z1z2z3eAð�z1Þ þ z2k1

2 z2k23eBð�z1Þ


3eBð�z2Þ


2eBð�z3Þ

� �:

Proof. Denote PðxÞ ¼ Q0ðxÞ; eP ðxÞ ¼ eQ0ðxÞ. By a simple calculation, we know that

MðxÞfM�ðxÞ ¼ I ;

where MðxÞ ¼ ðQiðxþ mjpÞÞ;fMðxÞ ¼ ðeQiðxþ mjpÞÞ ð0 6 i; j 6 7Þ. So wiðxÞ;fwiðxÞ ð1 6 i 6 7Þ are the correspondingtrivariate biorthogonal compactly supported wavelets. h

4. Conclusion

We obtain a sufficient condition on the existence of a pair of trivariate biorthogonal scaling functions and present analgorithm for constructing the corresponding pair of trivariate biorthogonal wavelets. And nonseparable or separablewavelets can be achieved from the algorithm.


Acknowledgement

The authors would like to express their gratitude to the referee for his (or her) valuable comments and suggestions.

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construction of trivariate biorthogonal compactly supported wavelets

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