The Construction Of The Biorthogonal Periodic Interpolatory Wavelets And The Implementation Of The Corresponding Algorithms

Wavelets analysis is a new branch of mathematics developed from 80's last century which has been widely used in numerical analysis, engineering technology and other fields. It has become a focus of many subjects.The classical wavelets are mostly defined on the whole space. But, many practical problems (such as the numerical solution for differential and integral equations, signal (image) processing and other fields) encountered in applications a.re always on finite region which may lead extra computations and boundary error when we deal with these problems with the wavelets defined on the whole space. Therefore, it is important to construct the wavelets on the finite region which have some good properties (such as smoothness, symmetry, cardinal interpolatory). Various efforts have been made to construct the wavelets on finite region ([1, 2]), among which constructing periodic wavelets is a important approach.Periodic wavelets were studied in Meyer and Daubcchies' first by periodizing the known wavelets. Subsequently, many researchers contributed to the establishment and development of periodic wavelet theory. Chui-Mhaskar constructed the trigonometric wavelets. The general theory of periodic wavelet was studied first by Plonka-Tashe, Koh-Lee-Tan and Narcowich-Ward. They defined the periodic multiresolution analysis first, and then, constructed different periodic scaling functions a.nd wavelets. The scaling functions given in [6] are semi-orthogonal while the scaling functions obtained in [7] arc orthogonal but not equivalent by translated. Narcowich-Ward obtained the scaling functions which are invariable by translated. The decomposition and reconstruction algorithms involve only 2terms for the last two methods. Plonka and Chen et al did a lot of important work in constructing periodic wavelets. (see[9, 10, 11-17]). In [12], Chen et. al. constructed a class of periodic orthogonal wavelets which are real-valued. The decomposition and reconstruction algorithms involve only 4 terms. Subsequently, Chen and Xiao et. al. constructed a class of biorthogonal periodic wavelets which are real-valued, symmetric, cardinal interpolatory and local. (see[15-17]). But, the construction of the dual periodic scaling functions and wavelets and therefore the decomposition and reconstruction algorithms further need to compute the inner product of the original scaling functions. The computational complexity is very hard.The main topics of this paper is the construction of periodic wavelets and its applications. This paper consists of two parts. Developing the construction method in [15] is the subject of the first part. In the section part, we gave the examples of the wavelets constructed in part one and discuss their applications in image processing.In chapter two, we review the definitions and their properties of B-splines and orthogonal periodic wavelets with respect to B-splines. Let Bm{x) De the central B-spline function with even degree. LetCm(u) = Bm(2uj)/Bm(Lj), nm(w) = ^|BTn(w + 2fc7r)j2. (1)then, Cm(u)) is innegative and flm(w) has a positive boundary.Let T be a positive integer. For any integer j > 0, let Tj = 2jT, Z(Tj) = {0,1, ■ ? ? , 7) -1} and TT = [0, T]. Denote the set of all the square integrable functions on TT with period 2tt by Ll{TT).Definition 1 A subspace sequence {V}}j>o in L2t{TT) is called a periodic multireso-lution analysis, if1) VrcVj+1,j>0.2) \Jj>0Vj is dense inLl{TT).3) For any j > 0, there exists a function j) € V, such that {fj{- - 2~jk);k € Z{Tj)} forms a set basis of Vj.Periodizing Bm(x) as follows:^;*), (2)then B?k{x) is periodic with period T. For convenient, we omit the superscript m. For instance, S?fc(a;) denoted by Bj^x).Denote g%(x) = ei27rnx/T.DefineV^spanfB^W;^^;)}, j > 0. (3)ThenProposition 1 The subspace sequence {Vj}j>o forms a periodic multiresolution analysis.Definition 2 For any j > 0, k 6 Z(Tj), define$>j,fc(x) = aAfc ^ 9J(fe/2>)BJiP(i), (4)w/tere(5)with nm(a') defined in (1).ThenThrorem 1 TTie function system {t+>j^(x);k 6 ￡(;/)} forms a set of orthogonal basis of Vj and satisfy the following equation:Where<*j,k.v =----^-----Cm((27r* + vTj)/Tj+i), v = 0,1. (6)"j+i.fc+i/j;,iwtA Oj.fc defined in (5).For any j > 0, let Wj be the orthogonal complement of Vj in V3+1, that is V^- ± Wj, and Vj- ? IV,- = Vj+i. For any j >0,je Z(j), definej (X), (7)where Qj,k,v, v = 0,1 is defined in (6). ThenThrorem 2 For any j > 0, Vo _L Wj;/or j,n > 0, j # n, W} _L Wn, and L2m(TT) = Vq ?j>o Wj. The function system {4'j,k(x);k ￡ Z(7))} forms a set of orthogonal basis of Wj. Furthermore, { 0} form a set of orthonormal basis ofLl(TT).In chapter three, we develop the construction method presented in [15]. We presented a new method which can avoid the integral operations.For any j > 0, k € Z(Tj), defineThenThrorem 3 For any j > 0, the function systems {Lj^x), k € Z{Tj)} and {Ljtk(x), k € Z(Tj)} form a pair of biorthogonal basis ofVj. The functions Lj^{i) satisfy the following cardinal interpolatory property:Furthermore, the following equations hold:Ljlfc(i)=with aj:qiV defined in (6).For any j > 0, k e Z{Tj), defineWhere rfKp = (T^>.p(2-?+1>))~1 flj(-2^fc), p G Z('J)). ThenThrorem 4 The function systems {H^k{x),k G Z(7j)} and {Hj,k{x),k G Z{Tj)}form a pair of biorihogonal basis of Wj andFurthermore, the following equations hold: Hjik(x) = Ylp€Z(T3+i)WTiereatj,q,u and Vp q are defined in (6) and proposition 3 respectively.At the end of this chapter, we prove that the scaling functions and the wavelets constructed in this chapter are real-valued and symmetrical.In chapter four, we establish the compositon and reconstruction algorithms first. Then, by using the FFT algorithm, we establish the fast implementation of the corresponding compositon and reconstruction algorithms. Finally, we applied the wavelets constructed with our method in image processing and give a example illustrated by the form of figure.