A Study on B-Spline Wavelets and Wavelet Packets

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1 Applied Matheatics Published Online Noveber 4 in SciRes. A Study on B-Spline Wavelets and Wavelet Pacets Sana Khan Mohaad Kaliuddin Ahad Departent of Matheatics Aligarh Musli University Aligarh India Eail: sana7han53@gail.co ahad_aliuddin@yahoo.co.in Received 6 August 4; revised 4 Septeber 4; accepted 5 October 4 Copyright 4 by authors and Scientific Research Publishing Inc. This wor is licensed under the Creative Coons Attribution International License (CC BY). Abstract In this paper we discuss the B-spline wavelets introduced by Chui and Wang in []. The definition for B-spline wavelet pacets is proposed along with the corresponding dual wavelet pacets. The properties of B-spline wavelet pacets are also investigated. Keywords B-Splines Spline Wavelets Wavelet Pacets. Introduction Spline wavelet is one of the ost iportant wavelets in the wavelet faily. In both applications and wavelet theory the spline wavelets are especially interesting because of their siple structure. All spline wavelets are linear cobination of B-splines. Thus they inherit ost of the properties of these basis functions. The siplest exaple of an orthonoral spline wavelet basis is the Haar basis. The orthonoral cardinal spline wavelets in L ( ) were first constructed by Battle [] and Learié [3]. Chui and Wang [4] found the copactly supported spline wavelet bases of L ( ) and developed the duality principle for the construction of dual wavelet bases [] [5]. Wavelets are a fairly siple atheatical tool with a variety of possible applications. If ψ ( x ) L then ψ is called a wavelet. Usually a wavelet is derived fro a is an orthonoral basis of given ultiresolution analysis of L. The construction of wavelets has been discussed in a great nuber of papers. Now considerable attention has been given to wavelet pacet analysis as an iportant generalization of wavelet analysis. Wavelet pacet functions consist of a rich faily of building bloc functions and are localized in tie but offer ore flexibility than wavelets in representing different inds of signals. The ain feature of the wavelet transfor is to decopose general functions into a set of approxiation functions with different scales. Wavelet pacet transfor is an extension of the wavelet transfor. In wavelet transforation signal de- How to cite this paper: Khan S. and Ahad M.K. (4) A Study on B-Spline Wavelets and Wavelet Pacets. Applied Matheatics

2 S. Khan M. K. Ahad coposes into approxiation coefficients and detailed coefficients in which further decoposition taes place only at approxiation coefficients whereas in wavelet pacet transforation detailed coefficients are decoposed as well which gives ore wavelet coefficients for further analysis. For a given ultiresolution analysis and the corresponding orthonoral wavelet basis of L wavelet pacets were constructed by Coifan Meyer and Wicerhauser [6] [7]. This construction is an iportant generalization of wavelets in the sense that wavelet pacets are used to further decopose the wavelet coponents. There are any orthonoral bases in the wavelet pacets. Efficient algoriths for finding the best possible basis do exist. Chui and Li [8] generalized the concept of orthogonal wavelet pacets to the case of nonorthogonal wavelet pacets. Yang [9] constructed a scale orthogonal ultiwavelet pacets which were ore flexible in applications. Xia and Suter [] introduced the notion of vector valued wavelets and showed that ultiwavelets can be generated fro the coponent functions in vector valued wavelets. In [] Chen and Cheng studied copactly supported orthogonal vector valued wavelets and wavelet pacets. Other notable generalizations are biorthogonal wavelet pacets [] non-orthogonal wavelet pacets with r-scaling functions [3]. The outline of the paper is as follows. In we introduce soe notations and recall the concept of B-splines and wavelets. In 3 we discuss the B-spline wavelet pacets and the corresponding dual wavelet pacets.. Preliinaries In this Section we introduce B-spline wavelets (or siply B-wavelets) and soe notions used in this paper. ( Every th order cardinal spline wavelet is a linear cobination of the functions N ) ( x ). Here the function N is the th order cardinal B-spline. Each wavelet is constructed by spline ultiresolution analysis. Let be any positive integer and let N denotes the th order B-spline with nots at the set of integers such that supp N. The cardinal B-splines [ ] N are defined recursively by the equations N x χ x [ ] ( ) N x N N x N x t d t 3. We use the following convention for the Fourier transfor ( ) e it f f t d. t The Fourier transfor of the scaling function N is given by i e N ( ). () i For each we set ; ( N N x ) and for each let V denotes the L -closure of the algebraic span of { N ; }. Then N is said to generate spline ultiresolution analysis if the following conditions are satisfied. ) V V V ) clos V L L ; 3) V { } 4) for each { ; } N is a Riesz basis of V. Following Mallat [4] we consider the orthogonal copleentary subspaces W W W that is; 5) V V W. 6) W W. W 7) L. 3

3 These subspaces N ( x) V and V V we have where { p } is soe sequence in Substituting the value of N This gives So () can be written as W are called the wavelet subspaces of ( ) N x pn x S. Khan M. K. Ahad L relative to the B-spline N. Since (). Taing the Fourier transfor on both sides of () we obtain i e N p N. (3) fro () into (3) we have i i i i i pe e i i e e e p for otherwise. N x N x ( ) which is called the two scale relation for cardinal B-splines of order. Chui and Wang [] introduced the following th order copactly supported spline wavelet or B-wavelet with support [ ] that ψ x ( N N ) ( x ) (5) that generates W and consequently all the wavelet spaces W. To verify ψ is in V we need the spline identity N x N x. (6) ( ) ( ) ( ) So substituting (6) into (5) we have the two scale relation where Let ψ 3 ( x) qn ( x ) (7) with the corresponding two scale sequence { h } dual wavelet of ψ such that For the scaling function ( ). (8) q N ( x) h N ( x ) ψ (9). If ψ is a wavelet then there exists another ψ called the ψ. ψ. l δ l. () l N we define its dual N by ( ) N x g N x (). (4) 33

4 S. Khan M. K. Ahad such that Now we have Taing the Fourier transfor of (3) we have where δ N. N. l l. () l ( ) ( ) N x pn x N x g N x. N N N ( ) N. ( ) ( ) i p e g i e. A necessary and sufficient condition for the duality relationship () is that ( ) and ( ) scale sybols in the sense that (3) (4) (5) are dual two. (6) A proof of this stateent is given in ([5] Theore 5.). Also fro (7) and (9) we have where N We observe that ψ N ψ ( ) N. ( ) ( ) See ([5] Section 5.3). N x is an orthogonal scaling function then If We say that ( x ) ( x ) if 3 i q e h i e. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). δ (7) (8) (9) N. N. l l. () l ψ is orthogonal (o.n) B-wavelet function associated with orthogonal scaling function 34

5 S. Khan M. K. Ahad and ψ ( x l ) l is an orthonoral basis of W Lea Let N ( x) L ( ). Then N Proof See ([5] page no. 75]. N. ψ. l l () ( l) so we have ψ. ψ. δ l. () l x is an orthonoral faily if and only if N l N l. (3) l Theore Let N ( x ) defined by (3) is an orthonoral scaling function. Assue that ψ ( x ) L ( ) whereas ( ) and ( ) are defined by (5) and (8) respectively. Then ( x ) wavelet function associated with N ( x ) if and only if ψ is an orthonoral ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). Proof Let us suppose that ψ is an orthonoral wavelet function associated with and () we have ( ) ψ ( ) N l l l l ( l) N ( l) N ( l) ( l) ( ρ) N ( ) ρ ν N( ρ ν ) ( ρ) ρ ν ( ) ( ) ( ) ( ). Again by Lea and () we have ( l) ( l) ψ ψ l ( η) N ( ) η ν N( η ν ) ( η) η ν On the other hand let (4) holds. Now Also ( ) ( ) ( ) ( ). il N. ψ. l N e ψ( ) il N ψ( ) e d ( ν ) il ( ) ( ) e d N ψ ν ν il N ( ) ( ) e d ν ν ψ ν. ψ. ψ. ψ ν ψ ν e d δ by Lea. il ( l) ( ) ( ) [ ] l ν Thus N ( x ) and ψ ( x ) are orthogonal and ( x ) (4) N x. By Lea ψ is an orthonoral wavelet function associated with 35

6 S. Khan M. K. Ahad N ( x ). 3. B-Spline Wavelet Pacets and Their Duals de- Following Coifan and Meyer [6] [7] we introduce two sequences of fined by where n When λ and n we have and for λ and n we have We call { n } ( λ ) n n L functions { n } and { n } λ x ζ x λ (5) ( λ ) n n λ x γ x λ (6) ζ p N γ g N ζ γ q ψ. h ψ the sequence of B-spline wavelet pacets induced by the wavelet ψ and its corresponding scaling function N whereas { n } denotes the corresponding sequence of dual wavelet pacets. By applying the Fourier transforation on both sides of (5) we have where So (4) can be written as ( λ ) n λ n (7) ( ) (8) λ ( λ) i ζ e. (9). Siilarly taing the Fourier transforation on both sides of (6) we have where Using these conditions we can write (3) ( λ ) n λ ( ) n (3) ( ) (3) λ ( λ) i γ e. (33) λ µ λ µ. δ λ µ (34) We are now in a position to investigate the properties of B-spline wavelet pacets. x its corresponding faily of Theore Let ( x ) be any orthonoral scaling function and { } n λµ 36

7 S. Khan M. K. Ahad B-spline wavelet pacets. Then for each n we have Proof Since ( x) N ( x).. δ. (35) n n satisfies (35) for n. We ay proceed to prove (35) by induction. Suppose that (35) holds for all n where n < r r r r a positive integer and n <. We have r n r < we have where [ x ] denote the largest integer not exceeding x. By induction hypothesis and Lea [ ] [ ] ( ) δ [ ] n n ν n [ n ] ( ν )... (36) By using (7) (3) and (36) we obtain ν ( ν ) ( ν ) n n ν ( λ ) [ ] [ ] ( λ ) ν ν ν ν ν ( λ ) [ ] ( ) n [ n ] ( ) ρ n n ( λ ) ρ ρ ρ ρ λ λ λ λ. Hence by Lea (35) follows. Theore 3 Let { ( x n )} be a B-spline wavelet pacet with respect to the orthonoral scaling function N( x) ( x). Then for every n we have Proof By (7) (3) and (36) for we have n. n.. (37) i n. n. n n ( ) e d i e d n n ( ) ( ) ( n n ) i ( ) e d ( ν ) ( ) ( { } ) i e d ν n n ν For the faily of B-spline wavelet pacets ( x ) N generated by { n } where { } consider the faily of subspaces ( ) ( ) i e d n ν n ν ν ( ) ( ) ( ) ( ) i ( ) e d. { n } corresponding to soe orthonoral scaling function n : clos W x : n L. We observe that V is the MRA of n V W L generated by N and { } W is the sequence of orthogonal 37

8 S. Khan M. K. Ahad copleentary (wavelet) subspaces generated by the wavelet ψ. Then the orthogonal decoposition ay be written as V V W. A generalization of the above result for other values of n can be written as n n n. Theore 4 For the B-spline wavelet pacets the following two scale relation ( x l) { p ( x ) q ( x ) } (38) n l n l n holds for all l. Proof In order to prove the two scale relation we need the following identity see ([5] Lea 7.9) { pr pl qr ql } δr l. (39) Taing the right-hand side of (38) and applying the identity (39) we have { pl n ( x ) ql n ( x ) } { plpl qlql } n ( x l) l t { pt pl qt ql } n ( x t) t [ pt pl qt ql ] n ( x t) ( x t) δtl n. ( n ) This copletes the proof of the theore. Next we discuss the duality properties between the wavelet pacets { W n } and { n } t l x l Lea For all l and n n n l t W. (4).. l δ. (4) Proof We will prove (4) by induction on n. The case n is the sae as our assuption () on the dual scaling functions N and N. Suppose that (4) holds for all n where n < r r r n where r is a positive integer. Then for n < we can write n λ for soe λ { } according to the proof of Theore 7.4 in [5]. Fro the Fourier transfor forulations of equations (5) and (6) and using (34) we have il ( ) n. n. l n ( ) n ( ) e d n n Since < r e d l and this is equivalent to ( λ) ( λ) il ( ) n n e d. 4 il ( ) ( λ) ( λ) n n for all it follows fro the induction hypothesis that (. ) (. l) δl n n 38

9 S. Khan M. K. Ahad Thus we have ae... (4) n n 4 il ( ) ( λ) ( λ) n (. ) n (. l) e d il ( ) ( λ) ( λ) ( λ) ( λ) e d il ( ) e d δ l. r r This shows that (4) also holds for n <. Lea 3 For all l and n and λ µ { } with λ µ.. l. (43) n λ n λ Proof By applying the Fourier transfor forulations of Equations (5) and (6) and using (4) and (34) we have as in the proof of Lea that il ( ) n λ. n λ. l n λ ( ) n λ ( ) e d References ( λ) ( λ) il ( ) n n e d 4 il ( ) ( λ) ( λ) e n n d 4 il ( ) ( λ) ( λ) e d il ( ) ( λ) ( λ) ( λ) ( λ) e d il ( ) e δ λµ d δλµ δl λ µ l. [] Chui C.K. and Wang J.Z. (99) On Copactly Supported Spline Wavelets and a Duality Principle. Transactions of the Aerican Matheatical Society [] Battle G.A. (987) A Bloc Spin Construction of Ondelettes Part-I: Learié Functions. Counications in Matheatical Physics [3] Learié P.G. (988) Ondelettes á localisation exponentielles Journal de Mathéatiques Pures et Appliquées [4] Chui C.K. and Wang J.Z. (99) A Cardinal Spline Approach to Wavelets. Proceedings of the Aerican Matheatical Society [5] Chui C.K. and Wang J.Z. (99) A General Fraewor of Copactly Supported Splines and Wavelets. Journal of Approxiation Theory [6] Coifan R.R. Meyer Y. and Wicerhauser M.V. (99) Wavelet Analysis and Signal Processing. In: Rusai M.B. et al. Eds. Wavelets and Their Applications Jones and Barlett Boston [7] Coifan R.R. Meyer Y. and Wicerhauser M.V. (99) Size Properties of Wavelet Pacets. In: Rusai M.B. et al. Eds. Wavelets and Their Applications Jones and Barlett Boston [8] Chui C.K. and Li C. (993) Nonorthogonal Wavelet Pacets. SIAM Journal on Matheatical Analysis [9] Yang S. and Cheng Z. () A Scale Multiple Orthogonal Wavelet Pacets. Matheatica Applicata in Chinese 3 39

10 S. Khan M. K. Ahad [] Xia X.G. and Suter B.W. (996) Vector Valued Wavelets and Vector Filter Bans. IEEE Transactions on Signal Processing [] Chen Q. and Cheng Z. (7) A Study on Copactly Supported Orthogonal Vector Valued Wavelets and Wavelet Pacets. Chaos Solitons and Fractals [] Cohen A. and Daubechies I. (993) On the Instability of Arbitrary Biorthogonal Wavelet Pacets. SIAM Journal on Matheatical Analysis [3] Feng Z. and Chen G. () Nonorthogonal Wavelet Pacets with r-scaling Functions. Journal of Coputational Analysis and Applications [4] Mallat S.G. (989) Multiresolution Approxiations and Wavelet Orthonoral Bases of Aerican Matheatical Society [5] Chui C.K. (99) An Introduction to Wavelets. Acadeic Press Inc. Boston. L ( R ). Transactions of the 3

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