Abstract: The notion of orthogonal nonseparable trivariate wavelet packets, which is the generalization of orthogonal univariate wavelet packets, is introduced. An approach for constructing them is presented. Their orthogonality properties are discussed. Three orthogonality formulas concerning these wavelet packets are obtained. The orthonormal bases of space L2 (R3) is presented.
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),
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,
| (2) |
| (3) |
Define a subspace Vj⊂L2 (R3) by:
| (4) |
The trivariate function Υ (t) in (Eq. 1) yields a multiresolution analysis {Vj} of L2 (R3), if the sequence {Vj}jεz, defined in Eq. 4 satisfies: (a) Vj⊂Vj+1, ∀j∈Z; (b) ∩j∈Z Vj = {0}; ∪j∈Z Vj is dense in L2 (R3); (c) Υ (t)εVj⇔Υ (2t)εVj+1, ∀j∈Z; (d) the family {2jΥ(2j.-k):kεZ3} is a Riesz basis for Vj (jεZ). Let Wj (jεZ) denote the orthogonal complementary subspace of Vj in Vj+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 Wj, 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 ψv (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 {A8n+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 Λ8n+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<8r0 (r0 is a positive integer) is a positive integer). Consider the case of 0 8r0≤n<8r0+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 ⟨Λm (.), Λn (.-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 mr, nrεΩ0 and λ3, μ3εΩ0. If αr = βr then λr≠μr Similar to Case 1, (10) follows. If αr≠βr, 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 Γn⊂L2 (R3). For any n∈Z+, It is defined
| (23) |
where the family {Λn (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, Γv = W(v) where,
v ε {0,1,2,…,7}.
Lemma 5: The space DΓn can be orthogonally decomposed into spaces Γ8n+v, vεΩ0. i.e.,
| (24) |
where, ⊕ denotes the orthogonal sum (Cheng, 2007). For arbitrary j∈Z+ define the set
Theorem 4: The family {Λn (.-k), nεjΔ, kεZ3} forms an orthogonal basis of DjW0. In particular, {Λn (.-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 Vj+1 = Vj⊕Wj thus it follows that DjΓ0 = Dj-1Γ0⊕Dj-1 W0. From this formula and Theorem 1, it leads to:
| (25) |
By Theorem 3, for n∈Z+,the family {Λn (.-k), nεjΔ, kεZ3} is an orthogonal basis of DjW0 Moreover, according to (25), {Λn (.-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 {Λn (2j t-k), nεSm, 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).