**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 L^{2} (,
C^{SxS}) is defined to be the set of all multiple vector -valued functions
, i.e:

where, f_{l, v} (t) εL^{2} (), 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)εL^{2} (, 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)εL^{2} (, C^{SxS}),
its Fourier transform is defined by:

For any F (t)
(t)εL^{2} (,
C^{SxS}), their symbol inner product is defined by:

where, ^{*} means the transpose and the conjugate.

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

is an orthonormal basis in L^{2} (,
C^{SxS}), if it satisfies:
and G (t)∈L^{2} (,
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 L^{2} (,
)
there exists a unique sequence of sxs matrix {P_{n}}_{k∈Z}
such that:

(2) there exist constants 0<C_{1}≤C_{2}<∞ 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)εL^{2} (, C^{SxS}) is satisfied the following refinable equation:

where, {D_{n}}_{n∈Z} is an sxs sequence of matrice, which has only a finite nonzero terms. Define a closed subspace V_{j} ⊂ L^{2} (,), 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 L^{2} (,),
if the sequence {V_{j}}_{j∈Z} is satisfied: (i) V_{j}
⊂ V_{j+1}, ∀ j∈Z; (ii) F(t) ∈ V_{0}<=>F
(5t)∈V_{1}; (iii) ∩_{j∈Z} V_{j} = {O};
U_{j∈Z} V_{j} is dense in L^{2} (,);
(iv) The translations {H_{n}(t): = H (t-n), n∈Z} form a Riesz basis
for V_{0}. 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)∈L^{2}(,),
∈
= {1, 2, 3, 4}, such that:

forms a Riesz basis of W_{j}. It is clear that Ψ_{ι}
(t)∈W_{0}⊂V_{1}. Hence there exist four sequences of
sxs matrices
such that:

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:

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:

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

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:

**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 L^{2} (,
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:

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:

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

Define:

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:

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:

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

**Example:** Let H(t),
ε L^{2} (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 Φ_{0} (t) and,
(t) iteratively define, respectively:

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

**Definition 3:** We say that two families of multiple vector-valued functions
{Φ_{5σ+1} (t): σεZ_{+}, εΛ_{0}}
and {_{5σ+1}
(t): σεZ_{+}, εΛ_{0}}
are multiple vector valued wavelet wraps with respect to a pair of biorthogonal
multiple vector-valued scaling functions Φ_{0} (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_{+}, εΛ_{0}}
is called multiple vector-valued wavelet wraps with respect to an orthogonal
multiple vector-valued scaling functions Φ_{0} (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:

where:

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

**Lemma 1. Cheng ***et al*. (2007): Let F(t),
εL^{2} (,
)

So they are biorthogonal ones if and only if :

**Lemma 2. Chen *** et al*. (2006b): Assume
that εΛ,
Φ_{ι}(t) _{ι}
(t) ε L^{2} (,)
are pairs of biorthogonal multiple vector-valued wavelets associated with a
pair of biorthogonal multiple scaling functions H (t) and .
Then, for μ, v εΛ_{0}, 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 Φ_{0}
(t) and
(t). Then, for α ε Z_{+}, we have:

**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 Λ_{0}then
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 Φ_{0} (t) and
_{0}. Then, for any n ε Ζ_{+} , ,
v ε Λ_{0}, we get that:

**Proof:** Since the set has the following partition:

where, u_{1}≠u_{2}, u_{1}, u_{2} 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 Φ_{0} (t) and , Then,
(t) for α, σεZ_{+}, we have:

**Proof:** When α = σ, Equation 38 follows
by Lemma 3. As α≠σ and α, σεΛ_{0},
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 Λ_{0}, we rewrite α, σ as α
= 5α_{1}+ι_{1}, σ = 5σ_{1}+μ_{1},
where ρ_{1}, μ_{1}εΛ. Case 1. If α_{1}
= σ_{1}, then _{1}
≠ μ_{1}. Equation 38 follows by virtue of
Eq. 31, 38 as well as Lemma 1 and Lemma
2, i.e.,

**Case 2:** If α_{1}≠σ_{1}, order α_{1}
= 5α_{2}+_{2},
σ_{1} = 5σ_{2} +μ_{2}, where
and ι_{2}, μ_{2}εΛ_{0}. Provided
that, then Similar to Case 1, (36) can be established. When, α_{2}≠σ_{2}
we order α_{2} = 5α_{3}+ι_{3}, σ_{2}
= 5σ_{3}+μ_{3}, where, ι_{3}, μ_{3}εΛ_{0}
Thus, after taking finite steps (denoted by κ), we obtain α_{κ}εΛ
and ι_{κ}, μ_{κ}εΛ_{0}.
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, Φ_{0} (t) Then, for ,
it follows that:

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 L^{2} (,
C^{sxs}). Let us define a dilation operator Δ, i.e., (ΔF)
(t) = F (5t) where F(t)εL^{2} (,
C^{sxs}) and set ΔΩ = {ΔF(t): F(t) εΩ} where
Ω⊂L^{2}(,
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:

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^{σ}V_{0}. In particular,
forms a Riesz basis of space L^{2} (,
C^{sxs}).

**Proof:** By virtue of Eq. 40, we have
i.e.,. Since Ω_{0} = V_{0} and ,
then .

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

Since ,
therefore, we have:

By Eq. 41 and Theorem 5, we have

In the light of Theorem 3, The family
is a Riesz basis of Δ^{σ}V_{0}. 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 Δ^{n}V_{0},
then for every jεZ, the sequence {Φ_{α} (5^{j}
t-u), uεZ} constitutes a Riesz basis of subspace Δ^{j}Δ^{σ}v_{0}
= Δ^{σ+J} v_{0}. Consequently, for every σεN,
we have .
Therefore, {Φ_{α}(5^{j} t-u), u, jεZ, αεΓ_{σ}}
constitutes a Riesz basis of space L^{2} (,
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 L^{2} (, C^{sxs}) from these wavelet wraps.

**ACKNOWLEDGMENT**

The research is supported by National Natural Science Foundation of China (Grant No:10971160) and also supported by the Natural Science Foundation of Shaanxi Province, P. R. China (No. 2009JM1002).