INTRODUCTION

Since 1986, wavelet analysis (Daubechies, 1992) has become a popular subject in scientific research. Its applications involve in many areas in natural science and engineering technology. The main advantage of wavelets is their time-frequency localization property. Many signals in areas like music, speech, images and video images can be efficiently represented by wavelets that are translations and dilations of a single function called mother wavelet with bandpass property. Wavelet packets, due to their nice properties, have attracted considerable attention. They can be widely applied many aspects in science (Qingjiang and Zhi, 2008) and engineering (Telesca et al., 2004), as well as optimal weight problem (Li and Fang, 2009). Researchers firstly introduced the notion of orthogonal wavelet packets which were used to decompose wavelet components. Qingjiang and Zhengxing (2007) generalized the concept of orthogonal wavelet packets to the case of non-orthogonal wavelet packets so that wavelet packets can be appllied to the case of the spline wavelets and so on. The tensor product multivariate wavelet packets has been constructed by Mallat (1998).

The introduction for the notion of nontensor productwavelet packets is attributable to Shen Zuowei. Since, the majority of information is multidimensional information, many researchers interest themselves in the investigation into multivariate wavelet theory. The classical method for constructing multivariate wavelets is that separable multivariate wavelets may be obtained by means of the tensor product of some univariate wavelets. But, there exist a lot of obvious defects in this method, such as, scarcity of designing freedom. Therefore, it is significant to investigate nonseparable multivariate wavelet theory. Nowadays, since there is little literature on orthogonal wavelet packets, it is necessary to investigate orthogonal wavelet packets. Inspired by Chen and Huo (2009), Chen and Qu (2009) and Chen et al. (2009a, b), we are about to generalize the concept of univariate orthogonal wavelet packets to orthogonal trivariate wavelet packets. The definition for nonseparable orthogonal trivariate wavelet packets is given and a procedure for constructing them is described. Next, the orthogonality property of nonseparable trivariate wavelet packets is investigated.

MULTIRESOLUTION ANALYSIS

We begin with some notations and definitions which will be used in this study. Z and Z₊denote all integers and nonnegative integers, respectively. Let R be the set of all real numbers. R³ stands for the 3-dimentional Euclidean space. L² (R³) denotes the square integrable function space on R³. Denote t = (t₁, t₂, t₃)εR³, k = (k₁, k₂, k₃), ω = (ω₁, ω₂, ω₃), . The inner product for two arbitrary f (t), and the Fourier transform of ħ (t) are defined by, respectively,

where, ω.t = ω₁t₁+ω₂t₂+ω₃t₃ and ħ (t) denotes the conjugate of ħ (t).

The multiresolution analysis (Behera, 2007) method is an important approach to obtaining wavelets and wavelet packets. The concept of multiresolution analysis of L² (R³) will be presented. Let Υ (t)ε L² (R³) satisfy the following refinement equation:

(1)

where, is a real number sequence and Υ (t) is called a scaling function. Taking the Fourier transform for both sides of Eq. 1 leads to

(2)

(3)

Define a subspace V_j⊂L² (R³) by:

(4)

The trivariate function Υ (t) in (Eq. 1) yields a multiresolution analysis {V_j} of L² (R³), if the sequence {V_j}_jεz, defined in Eq. 4 satisfies: (a) V_j⊂V_j+1, ∀_j∈Z; (b) ∩_j∈Z V_j = {0}; ∪_j∈Z V_j is dense in L² (R³); (c) Υ (t)εV_j⇔Υ (2t)εV_j+1, ∀j∈Z; (d) the family {2^jΥ(2^j.-k):kεZ³} is a Riesz basis for V_j (jεZ). Let W_j (jεZ) denote the orthogonal complementary subspace of V_j in V_j+1 and assume that there exist a vector-valued function Ψ (t) = (ψ₁ (t), ψ₂ (t),…,ψ₇ (t))^T (Ruilin, 1995) such that the translstes of its components form a Riesz basis for W_j, i.e.,

(5)

Eq. 5, it is clear that ψ₁ (t), ψ₂ (t),… , ψ₇ (t)εW₀⊂V₁. Therefore, there exist seven real sequences {d^(v)(k)} (v = 1,2,…7, k∈Z³) such that:

(6)

Refinement Eq. 6 can be written in frequency domain as follows:

(7)

where, the symbol of the real sequence {d^(v)(k)} (v = 1,2,…7, kεZ³) is

(8)

A scaling function Υ (t)εL² (R³) is orthogo-gonal, if it satisfies:

(9)

The above function Ψ (t) = (ψ₁ (t), ψ₂ (t),…, ψ₇ (t))^T is called an orthogonal trivariate vector-valued wavelets associated with the scaling function Υ (t), if they satisfy:

(10)

(11)

Orthogonal trivariate wavelet packets: To construct wavelet packets, we introduce the following notation:

(12)

where, v = 1,2,…7. A family of orthogonal nonseparable trivariate wavelet packets.is about to be introduced.

•	Definition 1: A family of functions

n = 0,1,2,…, v = 0,1,…,7} is called a nonseparable trivariate wavelet packets with respect to the orthogonal scaling function Υ (t), where

(13)

Implementing the Fourier transform for Eq. 13 leads to

(14)

where, v = 0,1,2,…7 and

(15)

Lemma 1: Let ħ (t)εL² (R³) Then, ħ (t) is an orthogonal function if and only if

(16)

Proof: It follows from the assumption that

This leads to Eq. 14. The converse is obvious.

Lemma 2: Assuming that Υ (t) is an orthogonal scaling function. B (z₁, z₂, z₃)is its symbol of the sequence {b (k)} defined in Eq. 3. Then,

(17)

Proof: If Υ (t) is an orthogonal trivariate function, then

Therefore, by Lemma 1 and formula Eq. 2, it follows that

This complete the proof of Lemma 2.

Similarly, Lemma 3 from formulas in Eq. 2, 7, 12 and 16 can be obtained.

Lemma 3: If ψ_v (t) (v = 0,1,…7) are orthogonal wavelet functions associated with Υ (t). Then

(18)

For an arbitrary nεZ₊, expand it by

(19)

Lemma 4: (Jin-song et al., 2006) Let nεZ₊ and n be expanded as Eq. 19. Then it follows that

The following findings can be obtained and proved.

Theorem 1: If the family {A_8n+v (t): n = 0,1,2,…, v = 0,1,…,7} is a nonseparable trivariate wavelet packets with respect to the orthogonal scaling function Υ (t).Then for nεZ₊, kεZ³, it follows that

(20)

Theorem 2: For every kεZ³ and, nεZ₊, v ε {0,1,2,…,7}, it holds that:

(21)

Theorem 3: If the family Λ_8n+v (t): n = 0,1,2…, v = 0,1,…,7}

is a nonseparable trivariate wavelet packets with respect to the orthogonal scaling function Υ (t).Then for every kεZ³ and m, nεZ₊, it follows that:

(22)

Proof of Theorem 1: Formula (Eq. 9) follows from Eq. 10 as n = 0. Assume formula (Eq. 20) holds for the case of 0≤n<8^r0 (r₀ is a positive integer) is a positive integer). Consider the case of 0 8^r0≤n<8^r0+1. For v ∈{0,…,7}, by the induction assumption and Lemma 1, Lemma 3 and Lemma 4, it follows that

Thus, the proof of theorem 1 is completed.

Proof of Theorem 2: By Lemma 1 and lemma 3 and

formulas Eq. 14 and relation Eq. 21 follows, since

Proof of Theorem 3: For the case of m = n, Eq. 22 follows from Theorem 1. As m≠n and m, nεΩ₀ the result Eq. 22 can be established from Theorem 2, where Ω₀ = {0,1,…,7}. In what follows, assuming that m is not equal to n and at least one of {m, n}doesn’t belong to Ω₀ rewrite m, n as m = 8m₁+λ₁, n = 8+μ₁ where m₁, n₁εZ₊ and λ₁, μ₁εΩ₀. Case 1 If m₁ = n₁ then λ₁≠μ₁. By Eq. 14, 16, 18 and 22) follows, since,

Case 2: If m₁≠n₁ we order m. = 8 m₂+λ₂, n₁ = 8 n₂+μ₂where m₂, n₂εZ₊ and λ₂, μ₂εΩ₀. If m₂ = n₂ then λ₂≠μ² Similar to Case 1, it holds that ⟨Λ_m (.), Λ_n (.-k)⟩.

That is to say, the proposition follows in such case. Since, m₂≠n₂ then order m₂ = 2 m₃+λ₃, n₂ = 2n₃+μ₃, n₂ = 2n₃+μ₃ once more, where m₃, n₃εZ₊and λ₃, μ₃εΩ₀ Thus, after taking finite steps (denoted by r), we obtain m_r, n_rεΩ₀ and λ₃, μ₃εΩ₀. If α_r = β_r then λ_r≠μ_r Similar to Case 1, (10) follows. If α_r≠β_r, Similar to Lemma 1, we conclude that

for ∀ωεR³. Therefore,

THE ORTHONORMAL BASES OF L² (R³)

First of all, a dilation operator is introduced, (Dħ) (t) = ħ (2t), where ħ (t) and set DΓ = {Dħ (t): ħ (t)εL² (R³) where Γ_n⊂L² (R³). For any n∈Z₊, It is defined

(23)

where the family {Λ_n (t), nεZ₊} is a nonseparable trivariate wavelet packets with respect to the orthogon- -al scaling function Υ (t) and l² (Z³) = {P: Z³→C^∞, ||P||₂ = {ΣkεZ³|p (k)|²)^1/2. Therefore, it follows that Γ₀ = V₀, Γ_v = W^(v) where,

v ε {0,1,2,…,7}.

Lemma 5: The space DΓ_n can be orthogonally decomposed into spaces Γ_8n+v, vεΩ₀. i.e.,

(24)

where, ⊕ denotes the orthogonal sum (Cheng, 2007). For arbitrary j∈Z₊ define the set

Theorem 4: The family {Λ_n (.-k), nεjΔ, kεZ³} forms an orthogonal basis of D^jW₀. In particular, {Λ_n (.-k), nεZ₊, kεZ³} constit- -utes an orthogonal basis of L² (R³) .

Proof: According to formula Eq. 24, it follows that

where, Ω = {1,2,…,7}, therefore DΓ₀ = V₀⊕W₀. It can inductively be proved by using Eq. 24 that

Due to V_j+1 = V_j⊕W_j thus it follows that D^jΓ₀ = D^j-1Γ₀⊕D^j-1 W₀. From this formula and Theorem 1, it leads to:

(25)

By Theorem 3, for n∈Z₊,the family {Λ_n (.-k), nεjΔ, kεZ³} is an orthogonal basis of D^jW₀ Moreover, according to (25), {Λ_n (.-k), nεZ₊, kεZ³} constituteses an orthogonal forms an orthogonal basis of L² (R³).

For an nonngative integer m, denoting

Corollary 1: The family of wavelet packet functions {Λ_n (2^j t-k), nεS_m, jεZ, kεZ³} constitutes an orthonormal basis of space L² (R³) (Chen et al., 2006).

CONCLUSION

The orthogonality property of nonseparable wave-let packets in L² (R³) is discussed. Three orthogon-ality formulas concerning the wavelet packets are obtained. The orthonormal bases of space L² (R³) is presented. Relation to optimal weight problem is also discussed (Chen and Qu, 2009).

ACKNOWLEDGMENTS

This work was supported by the Science Research Foundation of Education Department of Shaanxi Province (No. 08JK340), and also supported by the Natural Science Foundation of Shaanxi Province (2009JM1002).