A Study on B-Spline Wavelets and Wavelet Packets

In this paper, we discuss the B-spline wavelets introduced by Chui and Wang in [1]. The definition for B-spline wavelet packets is proposed along with the corresponding dual wavelet packets. The properties of B-spline wavelet packets are also investigated.


Introduction
Spline wavelet is one of the most important wavelets in the wavelet family.In both applications and wavelet theory, the spline wavelets are especially interesting because of their simple structure.All spline wavelets are linear combination of B-splines.Thus, they inherit most of the properties of these basis functions.The simplest example of an orthonormal spline wavelet basis is the Haar basis.The orthonormal cardinal spline wavelets in ( ) 2 L  were first constructed by Battle [2] and Lemarié [3].Chui and Wang [4] found the compactly supported spline wavelet bases of ( ) 2 L  and developed the duality principle for the construction of dual wavelet bases [1] [5].
Wavelets are a fairly simple mathematical tool with a variety of possible applications.If L  , then ψ is called a wavelet.Usually a wavelet is derived from a given multiresolution analysis of ( )

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L  .The construction of wavelets has been discussed in a great number of papers.Now, considerable attention has been given to wavelet packet analysis as an important generalization of wavelet analysis.Wavelet packet functions consist of a rich family of building block functions and are localized in time, but offer more flexibility than wavelets in representing different kinds of signals.The main feature of the wavelet transform is to decompose general functions into a set of approximation functions with different scales.Wavelet packet transform is an extension of the wavelet transform.In wavelet transformation signal de-composes into approximation coefficients and detailed coefficients, in which further decomposition takes place only at approximation coefficients whereas in wavelet packet transformation, detailed coefficients are decomposed as well which gives more wavelet coefficients for further analysis.
For a given multiresolution analysis and the corresponding orthonormal wavelet basis of ( )

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L  , wavelet packets were constructed by Coifman, Meyer and Wickerhauser [6] [7].This construction is an important generalization of wavelets in the sense that wavelet packets are used to further decompose the wavelet components.There are many orthonormal bases in the wavelet packets.Efficient algorithms for finding the best possible basis do exist.Chui and Li [8] generalized the concept of orthogonal wavelet packets to the case of nonorthogonal wavelet packets.Yang [9] constructed a scale orthogonal multiwavelet packets which were more flexible in applications.Xia and Suter [10] introduced the notion of vector valued wavelets and showed that multiwavelets can be generated from the component functions in vector valued wavelets.In [11], Chen and Cheng studied compactly supported orthogonal vector valued wavelets and wavelet packets.Other notable generalizations are biorthogonal wavelet packets [12], non-orthogonal wavelet packets with r-scaling functions [13].
The outline of the paper is as follows.In §2 , we introduce some notations and recall the concept of B-splines and wavelets.In §3 , we discuss the B-spline wavelet packets and the corresponding dual wavelet packets.

Preliminaries
In this Section, we introduce B-spline wavelets (or simply B-wavelets) and some notions used in this paper.
Every mth order cardinal spline wavelet is a linear combination of the functions Here the function m N is the mth order cardinal B-spline.Each wavelet is constructed by spline multiresolution analysis.Let m be any positive integer and let m N denotes the mth order B-spline with knots at the set  of integers such that The cardinal B-splines m N are defined recursively by the equations The Fourier transform of the scaling function m N is given by ( ) For each , j k ∈  , we set .Then m N is said to generate spline multiresolution analysis if the following conditions are satisfied. 1) is a Riesz basis of m j V .Following Mallat [14], we consider the orthogonal complementary subspaces These subspaces , m j W j∈  , are called the wavelet subspaces of ( ) where { } k p is some sequence in 2  .Taking the Fourier transform on both sides of (2), we obtain ( ) Substituting the value of ( ) ˆm N ω from (1) into (3), we have This gives So, (2) can be written as which is called the two scale relation for cardinal B-splines of order m .
Chui and Wang [1], introduced the following mth order compactly supported spline wavelet or B-wavelet W j∈  .To verify that m ψ is in 1 m V , we need the spline identity So, substituting (6) into (5), we have the two scale relation where, ( ) ( ) with the corresponding two scale sequence { } For the scaling function m N , we define its dual m N  by ( ) ( ) Now, we have Taking the Fourier transform of (13), we have where, ( ) A necessary and sufficient condition for the duality relationship (12) A proof of this statement is given in ([15], Theorem 5.22).Also from ( 7) and ( 9), we have where, ( ) ( ) We say that ( ) Again by Lemma 1 and (22), we have On the other hand, let (24) holds.Now,

B-Spline Wavelet Packets and Their Duals
Following Coifman and Meyer [6] [7], we introduce two sequences of 2

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the sequence of B-spline wavelet packets induced by the wavelet m ψ and its correspond- denotes the corresponding sequence of dual wavelet packets.By applying the Fourier transformation on both sides of (25), we have where, , .
So, (24) can be written as Similarly, taking the Fourier transformation on both sides of (26), we have where, , .
Using these conditions we can write We are now in a position to investigate the properties of B-spline wavelet packets.x N x =  satisfies (35) for 0 n = .We may proceed to prove (35) by induction.Suppose that (35) holds for all n , where 0 2 r n ≤ < , r a positive integer and , where [ ] x denote the largest integer not exceeding x .By induction hypothesis and Lemma 1, By using ( 27), ( 30) and (36), we obtain , n m x  be a B-spline wavelet packet with respect to the orthonormal scaling function Proof By (27), ( 30) and (36), for k ∈  we have For the family of B-spline wavelet packets , consider the family of subspaces , ,

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. We observe that holds for all l ∈  .
Proof In order to prove the two scale relation, we need the following identity, see ([15], Lemma 7.9) Taking the right-hand side of (38), and applying the identity (39), we have , Proof We will prove (41) by induction on n .The case 0 n = is the same as our assumption (12) on the dual scaling functions . Suppose that (41) holds for all n where 0 2 r n ≤ < , where r is a positive integer.Then for for some { } 0,1 λ ∈ , according to the proof of Theorem 7.24 in [15].From the Fourier transform formulations of equations (25) and (26) and using (34) we have , .
Thus, we have

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We use the following convention for the Fourier transform, i wavelet function associated with orthogonal scaling function ( ) by (13) is an orthonormal scaling function.Assume that 24) Proof Let us suppose that ( ) m x ψ is an orthonormal wavelet function associated with ( ) m N x .By Lemma 1 and (21), we have

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the above result for other values of n can be written as For the B-spline wavelet packets, the following two scale relation

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This completes the proof of the theorem.Next, we discuss the duality properties between the wavelet packets { } For all , k l ∈  and n + ∈  ,

Proof
By applying the Fourier transform formulations of Equations (25) and (26) and using (42) and (34), we have as in the proof of Lemma 2 that