INTRODUCTION

Wavelet analysis has been studied extensively in both theory and applications during the last two decades. The main advantage of wavelets is their time-frequency localization property. Construction of wavelet bases is an important aspect of wavelet analysis and multiresolution analysis method is one of important ways of constructing various wavelet bases. Wavelet transform is a simple mathematical tool that cuts up data or functions into different frequency components and then analyzes each component with a resolution matched to its scale.

The main feature of the wavelet transform is to hierarchically decompose general functions, as a signal or a process, into a set of approximation functions with different scales. Engineers in fact have discovered that it can be applied in all environments where the signal analysis is used. In order to implement the wavelet transform, we need to construct various wavelet functions. Though orthogonal wavelets have many desired properties such as compact support, good frequency localization and vanishing moments, they lack symmetry as demonstrated by Daubechies (1992). Vector-valued wavelets are a class of generalized multiwavelets (Yang et al., 2002). Xia and Suter (1996) and Xia and Zhang (1993) introduced the notion of vector-valued wavelets which have led to exciting applications in signal analysis (Telesca et al., 2004), fractal theory (Iovane and Giordano, 2007), image processing (Zhang and Wu, 2006) and so on. It is showed that multiwavelets can be generated from the component functions in vector-valued wavelets. Vector-valued wavelets and multiwavelets are different in the following sense. For example, prefiltering is usually required for discrete multiwavelet transforms but not necessary for discrete vector-valued wavelet transforms (Xia et al., 1996). In real life, Video images are vector-valued signals. Vector-valued wavelet transforms have been recently studied for image coding by Li (1991). Hence, studying vector-valued wavelets is useful in multiwavelet theory and representations of signals. Chen and Cheng (2007) studied orthogonal finitely supported vector-valued wavelets with 2-scale. Similar to uni-wavelets, it is more complicated and meaningful to investigate vector-valued wavelets with five-scale. Inspired by Chen and Huo (2009), we are about to investigate existence and construction of a sort biorthogonal finitely supported vector-valued wavelets with five-scale and propose a constructive algorithm for designing biorthogonal finitely supported vector-valued wavelets. Nowadays, wavelet packets, due to their nice characteristics, have attracted considerable attention, which can be widely applied in science (Chen and Zhi, 2008) and engineering (Leng et al., 2006), as well as optimal weight problem (Li and Fang, 2009). Coifman et al. (1992) firstly introduced the notion of orthogonal wavelet Packets which were used to decompose wavelet components. Chen and Wei (2009) generalized the concept of orthogonal wavelet wraps to the case of non-orthogonal wavelet wraps so that wavelet wraps can be applied to the case of the spline wavelets and so on. The introduction for biorthogonal wavelet wraps was attributable to Cohen and Daubechies (Behera, 2007; Zhang, 2007), Zhang and Saito (2009) and Chen et al. (2009c) constructed 4-scale biorthogonal vector wavelet wraps, which were more flexible in applications. We will generalize the concept of univariate biorthogonal wavelet wraps to vector-valued wavelet wraps with multi-scale and investigate their biorthogonality property.

THE VECTOR-VALUED FUNCTION SPACE

Let be a constant and s≥2. The space L² (, C^SxS) is defined to be the set of all multiple vector -valued functions , i.e:

where, f_{l, v} (t) εL² (), l, v = 1, 2,... s. Examples of multiple vector-valued signals are video images in which f_{l, v} (t) is the pixel at the time the lth row and the vth column. For any F (t)εL² (, C^SxS):

and its integration is defined to be:

i.e., the matrix of the integration of every scalar function f_{j, l} (t), j, l = 1, 2, …, s. For any F (t)εL² (, C^SxS), its Fourier transform is defined by:

(1)

For any F (t) (t)εL² (, C^SxS), their symbol inner product is defined by:

(2)

where, ^* means the transpose and the conjugate.

Definition 1: We say that a family of multiple vector-valued function:

is an orthonormal basis in L² (, C^SxS), if it satisfies: and G (t)∈L² (, C^SxS), there exists a sequence of sxs constant matrice Q_k such that where, I_s denotes the sxs identity matrix and δ_{j, 1} = 1 when j = 1 and δ_{j, 1} = 0 when j ≠ 1.

Definition 2: A sequence of vector-valued functions is called a Riesz basis of U if (1) For any, G (t) ε L² (, ) there exists a unique sequence of sxs matrix {P_n}_k∈Z such that:

(3)

(2) there exist constants 0<C₁≤C₂<∞ such that, for any sxs constant matrix sequence {P_k}_k∈Z:

where, ||{P_k}||_* is the norm of the matrix seq {P_k}_k∈Z.

We begin with the following refinement equation and the multiple vector-valued multiresolution analysis, that is commonly used in the construction of wavelets. Assume that H (t)εL² (, C^SxS) is satisfied the following refinable equation:

(4)

where, {D_n}_n∈Z is an sxs sequence of matrice, which has only a finite nonzero terms. Define a closed subspace V_j ⊂ L² (,), j∈Z as follows:

where, j∈Z We say that H (t) in (3) generates a vector-valued multiresolution analysis {V_j}_j∈Z, of L² (,), if the sequence {V_j}_j∈Z is satisfied: (i) V_j ⊂ V_j+1, ∀ j∈Z; (ii) F(t) ∈ V₀<=>F (5t)∈V₁; (iii) ∩_j∈Z V_j = {O}; U_j∈Z V_j is dense in L² (,); (iv) The translations {H_n(t): = H (t-n), n∈Z} form a Riesz basis for V₀. Here H (t) is called a vector-valued scaling functions. Let W_j, j∈Z, stand for the complementary subspace of V_j in V_j+1 and there exists four vector-valued function Ψ_ι(t)∈L²(,), ∈ = {1, 2, 3, 4}, such that:

forms a Riesz basis of W_j. It is clear that Ψ_ι (t)∈W₀⊂V₁. Hence there exist four sequences of sxs matrices such that:

(5)

We say H (t), are a pair of biorthogonal multiple vector-valued scaling functions, if there is another multiple vector valued scaling functions such that:

(6)

In particular, H (t) is called an orthogonal one while the relation holds.

We call Ψ_ι (t), (, C^SxS) pairs of biorthogonal multiple vector-valued wavelets associated with a pair of biorthogonal multiple vector-valued scaling functions, if:

(7)

(8)

(9)

Similar to Eq. 4 and 5 also satisfy the following refinement equations:

(10)

(11)

Then, we can gain the following results by Eq. 5 and 8.

Theorem 1: Assume that , defined by Eq. 4 and 10, are a pair of biorthogonal vector-valued scaling functions. Then, for any ,we have:

(12)

Proof: Substituting Eq. 4 and 10 into the biorthogonality Eq. 6, we have

Theorem 2. Chen et al. (2006a): Assume Ψ_i (t) and , defined in Eq. 5 and 11, are vector-valued function in L² (, C^SxS). Then Ψi (t) and are pair of biorthogonal multiple vector-valued wavelet functions associated with a pair of biorthogonal vector-valued scaling functions H (t) and , then we have:

(13)

(14)

(15)

Thus, both Theorem 2 and (13-15) provide an approach for constructing compactly supported biorthogonal multiple vector-valued wavelets.

CONSTRUCTION OF THE BIORTHOGONAL MULTIPLE VECTOR-VALUED WAVELETS

Theorem 3: Let H (t) and be a pair of 6-coefficient biorthogonal multiple vector-valued finitely supported scaling functions satisfying the following equations:

(16)

(17)

Assume there is an integer l, 0≤l≤5 , such that the matrix P below is an invertible one:

(18)

Define:

(19)

where, εΛ. Then:

are pairs of biorthogonal multiple vector-valued wavelet functions associated with H(t) and .

Proof: For convenience, let 1= 1. By Theorem 2 and formulas (13-15), it suffices to show that the set of matrices:

satisfies the following equations:

(20)

(21)

(22)

(23)

(24)

(25)

(26)

If are given by Eq. 19, then Eq. 20, 22, 23 hold from Eq. 12. For Eq. 21, we obtain from Eq. 12 and 19 that:

Similarly, Eq. 24 and 25 can be obtained. Now we will prove that Eq. 26 follows:

Corollary 1. Chen et al. (2009a): If H (t) defined in Eq. 4 is a 6-coefficient orthogonal vector-valued scaling function and there exists an integer 1, 0≤1≤5, such that the matrix P, defined in Eq. 27 is not only invertible but also Hermitian matrix:

(27)

(28)

Then Ψ (t) = 5 is an orthogonal multiple vector-valued wavelets with H (t):

Example: Let H(t), ε L² (R,C) and supp H (t) = [0, 5] be a pair of 5-coefficient biorthogonal vector-valued scaling functions satisfying the below equations (Wang et al., 2008):

where,

Let l = 1. By using Eq. 19 and 20, we get:

By Theorem 3, we have:

are biorthogonal multiple vector-valued wavelets associated with H (t) and .

THE PROPERTIES OF MULTIPLE VECTOR-VALUED WAVELET WRAPS

To introduce the notion of multiple vector-valued wavelet wraps, we set

For any α εΖ₊ and the given biorthogonal multiple vector-valued scaling functions Φ₀ (t) and, (t) iteratively define, respectively:

(29)

(30)

where, εΛ₀ = Λ∪ {0}, σεZ₊is the unique element such that α = 5σ+ εΛ₀ follows.

Definition 3: We say that two families of multiple vector-valued functions {Φ_5σ+1 (t): σεZ₊, εΛ₀} and {_5σ+1 (t): σεZ₊, εΛ₀} are multiple vector valued wavelet wraps with respect to a pair of biorthogonal multiple vector-valued scaling functions Φ₀ (t) and (t), respectively, where Φ_5σ+ι (t) and _{5σ+ι (t)} are given by Eq. 29 and 30, respectively.

Definition 4: A family of multiple vector-valued functions {Φ_5σ+ι (t):σεZ₊, εΛ₀} is called multiple vector-valued wavelet wraps with respect to an orthogonal multiple vector-valued scaling functions Φ₀ (t), where Φ_5σ+ι (t) are iteratively derived from Eq. 29.

Taking the Fourier transform for the both sides of Eq. 29 and 30, yields, respectively:

(31)

(32)

where:

(33)

(34)

We are now in a position to characterizing the biorthogonality property of the wavelet wraps.

Lemma 1. Cheng et al. (2007): Let F(t), εL² (, )

So they are biorthogonal ones if and only if :

(35)

Lemma 2. Chen et al. (2006b): Assume that εΛ, Φ_ι(t) _ι (t) ε L² (,) are pairs of biorthogonal multiple vector-valued wavelets associated with a pair of biorthogonal multiple scaling functions H (t) and . Then, for μ, v εΛ₀, we have:

Lemma 3. Chen et al. (2009b): Suppose that {Φ_α (t), αεZ₊} and {_α (t), αεZ₊} are multiple vector-valued wavelet wraps with respect to a pair of biorthogonal multiple vector-valued functions Φ₀ (t) and (t). Then, for α ε Z₊, we have:

(36)

Proof: The result (36) follows from (6) as α = 0. Assume that (36) holds when α<η, where η is a positive integer and α ε Z₊ For the case of α ε Z₊, α = η, we will prove that Eq. 36 holds. Order α = 5β+ρ where β ε Ζ₊, ρ ε Λ₀then β<α.

By induction assumption, we have:

Therefore, the result is established.

Theorem 4: Assume that {Φ_n (t), n ε Z₊} and {_n (t), n ε Z₊} are multiple vector-valued wavelet wraps associated with a pair of biorthogonal scaling functions Φ₀ (t) and ₀. Then, for any n ε Ζ₊ , , v ε Λ₀, we get that:

(37)

Proof: Since the set has the following partition:

where, u₁≠u₂, u₁, u₂ ε then by Lemma 1, we have:

This completes the proof of Theorem 4.

Theorem 5: If {Φ_α (t), αεZ₊} and are multiple vector-valued wavelet wraps with respect to a pair of biorthogonal multiple vector-valued functions Φ₀ (t) and , Then, (t) for α, σεZ₊, we have:

(38)

Proof: When α = σ, Equation 38 follows by Lemma 3. As α≠σ and α, σεΛ₀, it follows from Theorem 1 that Eq. 38 holds, too. Assuming that α is not equal to σ, as well as at least one of {α, σ} doesn’t belong to Λ₀, we rewrite α, σ as α = 5α₁+ι₁, σ = 5σ₁+μ₁, where ρ₁, μ₁εΛ. Case 1. If α₁ = σ₁, then ₁ ≠ μ₁. Equation 38 follows by virtue of Eq. 31, 38 as well as Lemma 1 and Lemma 2, i.e.,

Case 2: If α₁≠σ₁, order α₁ = 5α₂+₂, σ₁ = 5σ₂ +μ₂, where and ι₂, μ₂εΛ₀. Provided that, then Similar to Case 1, (36) can be established. When, α₂≠σ₂ we order α₂ = 5α₃+ι₃, σ₂ = 5σ₃+μ₃, where, ι₃, μ₃εΛ₀ Thus, after taking finite steps (denoted by κ), we obtain α_κεΛ and ι_κ, μ_κεΛ₀. If α_κ = σ_κ, then ι_κ≠μ_κ. Similar to the Case 1, (33) follows. If α_κ≠α_κ, then it gets from Eq. 12 and 15:

Furthermore, we obtain:

Therefore, for any , result Eq. 38 holds.

Corollary 2: Let is a multiple vector-valued wavelet wraps with respect to the orthogonal multiple vector-valued function, Φ₀ (t) Then, for , it follows that:

(39)

In the following, we will decompose subspaces and by constructing a series of subspaces of multiple vector-valued wavelet wraps. Furthermore, we present the direct decomposition for space L² (, C^sxs). Let us define a dilation operator Δ, i.e., (ΔF) (t) = F (5t) where F(t)εL² (, C^sxs) and set ΔΩ = {ΔF(t): F(t) εΩ} where Ω⊂L²(, C^sxs). For any , denoted by:

Then . Assume that is a unitary matrix.

Lemma 4. Mallat (1999): For , the space Δ Ω n can be decomposed into the direct sum of i.e:

(40)

Similar to Eq. 40, we can establish the following result: For any σεN, define some sets:

Theorem 6: The family of multiple vector-valued functions forms a Riesz basis of Δ^σV₀. In particular, forms a Riesz basis of space L² (, C^sxs).

Proof: By virtue of Eq. 40, we have i.e.,. Since Ω₀ = V₀ and , then .

It can be inductively inferred by using (40) that:

(41)

Since , therefore, we have:

By Eq. 41 and Theorem 5, we have

(42)

In the light of Theorem 3, The family is a Riesz basis of Δ^σV₀. Moreover, according to (42), forms a Riesz basis of space .

Corollary 3: For every nεN, the family of multiple vector-valued functions constitutes a Riesz basis of space .

Proof: Now that the family {Φ_α(t-u), u, jεZ, αεΓ_σ} forms a Riesz basis of ΔⁿV₀, then for every jεZ, the sequence {Φ_α (5^j t-u), uεZ} constitutes a Riesz basis of subspace Δ^jΔ^σv₀ = Δ^σ+J v₀. Consequently, for every σεN, we have . Therefore, {Φ_α(5^j t-u), u, jεZ, αεΓ_σ} constitutes a Riesz basis of space L² (, C^sxs).

CONCLUSION

A necessary and sufficient condition on the existence of biorthogonal multiple vector-valued wavelets is presented by means of paraunitary vector filter bank theory time-frequency analysis method. An algorithm for constructing a sort of biorthogonal multiple vector-valued finitely supported wavelets is provided. We characterize the biorthogenality traits of these wavelet wraps. We also establish three biorthogonality formulas concerning the wavelet wraps. In the final part, we obtain two new Riesz bases of space L² (, C^sxs) from these wavelet wraps.