Rational Filter Wavelets*

Transcription

1 Ž Journal of Matheatical Analysis and Applications 39, Article ID jaa , available online at on Rational Filter Wavelets* Kuang Zheng and Cui Minggen Departent of Matheatics, Harbin Institute of Technology, Harbin , China Subitted by George Leitann Received March 4, 1999 In this paper, we introduce two types of rational filters for the construction of wavelets The rational filters are a natural developent of the polynoial filters We use rational filters to derive a large faily of the wavelets which can include Daubechies wavelets and BattleLearie wavelets Especially the II-type rational filter wavelets aong the faily have linear phases Furtherore, we analyze the regularity of the rational filter wavelets and estiate their regularity indices Soe exaples are also given 1999 Acadeic Press Key Words: wavelet, filter, rational filter, regularity, scaling function 1 ITRODUCTIO We have nown that a scaling function, which generates a ultiresolution analysis in L Ž R, satisfies the dilation equation Ý Ž x h Ž x, Ž 11 Z where Z is integer set When the filter coefficients h and the scaling function are available, the corresponding orthogonal wavelet is given by Ý Ž Ž 1 Ž Ž Z x 1 h x 1 Denote the Fourier transfor of by ˆ Then Ž 11 and Ž 1 give the relations ˆ Ž HŽ ˆ Ž Ž 13 * This wor is supported in part by HIT Science Fund and LASG Fellowship X99 $3000 Copyright 1999 by Acadeic Press All rights of reproduction in any for reserved

2 8 ZHEG AD MIGGE and ˆ Ž GŽ ˆ Ž, Ž 14 where Ý Ž Ž 1 i H h e 15 Z and Ý Ž 1 Ž 1 i i Ž Ž Z G 1 h e e H 16 The HŽ plays an iportant role in wavelet theory It is called the filter function or filter for short The orthonorality of the function Ž, Z can be expressed by the relation 1, p 13 If ˆ Ž 0 0, then Ž 13 leads to HŽ HŽ 1 Ž 17 HŽ 0 1 Ž 18 Further, Ž 17 and Ž 18 lead to HŽ 0 If HŽ has an -fold zero at, then HŽ can be factorized by where Ž Ž 13 and 19 will forally lead to 1 e i i HŽ FŽ e, Ž 19 FŽ e i Ý f e i Z j i Ł j1 ž / j i i e sinž Ž Ł FŽ e ˆ Ž 0 Ž 110 j1 1 e j i ˆ Ž FŽ e ˆ Ž 0

3 RATIOAL FILTER WAVELETS 9 The infinite product Ł FŽe i j aes sense under the conditions j1, p 951 and Ý Z Ž Ž sup i F e R f, for soe 0 Ž 11 The conditions Ž 111 and Ž 11 ensure that the scaling function corresponding to the filter Ž 19 exists The celebrated wor of Daubechies gives explicit construction of the copactly supported wavelets by using polynoial filters; ie, FŽ z in Ž 19 is a polynoial In this paper, we use rational rather than polynoial filters to generate wavelets Our principal purpose is to extend the construction of wavelets as well as yield good wavelets such as linear phase wavelets and ore regular wavelets In general, rational filters will lead to infinite support wavelets The strong lin between the construction of wavelets and rational filters has been investigated in 3, 4 The use of certain particular rational filters for the construction of wavelets was discussed in 5 Herley and Vetterli 4 gave a coplete constructive ethod which yields orthogonal two channel filter bans, where the filters have rational transfer functions In this paper, we introduce two types of rational filters which are designated as I-type and II-type, and derive linear phase wavelets and ore regular wavelets than Daubechies fro the In Section of this paper, we give the definition of the rational filters and discuss how to construct a rational filter In Section 3, the relations between rational filters and BattleLearie wavelets are investigated The fourth section relates to the estiation of regularity indices for rational filter wavelets In Section 5, we give soe exaples of rational filters and perfor the nuerical coputation RATIOAL FILTERS Ž ŽŽ i DEFIITIO For a filter H 1 e FŽe i, we call HŽ a rational filter if FŽ z is a real coefficient rational function or the odulus of a real coefficient rational function Assue FŽ z to be a real coefficient rational function Then FŽe i is a positive rational function of the cosines; therefore, it is also a rational function of y cos

4 30 ZHEG AD MIGGE Ž For the sae of considering the orthogonal condition 17, we let i FŽ e R cos, Ž 1 where RŽ y is a rational function For our purposes we need, however, FŽe i or FŽe i itself, not FŽe i So we ust deterine FŽe i by the given RŽ y in advance This needs the following lea LEMMA 1 Let M be a positie trigonoetric rational function of cosines; M is necessarily of the for L Ž Ý Ý K M a cos l b cos, with a, b R l0 l l 0 Then there exists a trigonoetric rational function L K il i Ž Ý pe l Ý qe, with p l, q R l0 0 Ž Ž such that M This lea can be easily verified by the Riesz lea Žsee 6 with respect to the positive polynoials of cosines Let i FŽ e P 1 cos Q 1 cos, Ž where P and Q are real coefficient polynoials and PŽ y QŽ y 0on 0, 1 According to Lea 1, we can deterine the rational function FŽ z by the chosen polynoial P and Q Putting Ž 19 and Ž together, we have Denote cos Ž 17 leads to P Ž 1 cos Ž HŽ cos Ž 3 QŽ 1 cos Ž by y Then substituting 3 into the orthonoral condition Ž Ž Ž Ž Ž Ž Ž Ž Ž y P 1 y Q y 1 y P y Q 1 y Q y Q 1 y 4

5 RATIOAL FILTER WAVELETS 31 Ž Ž Ž Ž Ž Ž Ž If we let S y P y Q 1 y and T y Q y Q 1 y, then 4 can be rewritten as y SŽ 1 y Ž 1 y SŽ y TŽ y Ž 5 Since PŽ 1 y QŽ 1 y SŽ 1 y TŽ 1 y and TŽ y TŽ 1 y,we obtain fro Ž and Ž 5 SŽ 1 y i FŽ e, y cos Ž 6 y SŽ 1 y Ž 1 y SŽ y Thus, for a given polynoial SŽ y, the equation Ž 6 deterines a rational function FŽ z Correspondingly we obtain the two different types of rational filters and 1 e i I i H Ž FŽ e Ž 7 1 e i II i H Ž FŽ e Ž 8 I Ž II We call H and H Ž a I-type rational filter and II-type rational filter, respectively Both of the satisfy Ž 17 and Ž 18 ext, we discuss the existence of the scaling functions corresponding to the rational filters For this, we consider conditions Ž 111 and Ž 11 According to Ž 111, the polynoial SŽ y in Ž 6 ust satisfy SŽ 1 y Ž 1 0, for y 0,1 Ž 9 y SŽ 1 y Ž 1 y SŽ y THEOREM 1 Let SŽ y be a real coefficient polynoial FŽ z is a rational function deterined by Ž 6 If SŽ y satisfies the inequality Ž 9, then the scaling functions corresponding to the I-type and II-type rational filter exist Proof In fact, we only need to verify the condition Ž 11 for FŽe i and FŽe i For this, we have to expand FŽe i and FŽe i Ž 9 and Ž 6 tell us that there is not any pole of FŽ z on the unit circle Let all the poles of FŽ z be z, z,, z Then FŽ z 1 r can be written as r Ý A i FŽ z i DŽ z, Ž 10 Ž z z i1 i

6 3 ZHEG AD MIGGE where integer 1, DŽ z i is a polynoial and Ai are real or coplex constants For a certain z, either z 1orz1 If z i i i i 1, then we have i A i Ž 1 A i i Ž z z z i Ž 1 zz i i i i / j i Ž Ž Ž 1 1 Ý ž Ai i i 1 i j 1 z z i i j1 j! zi Ž 11 If z 1, then we siilarly have i Ai Ai i Ž z z z i Ž 1 z z i i i A Ž 1 Ž j 1 z j i i i i i 1 Ý Ž 1 z i j! z j1 Cobining Ž 11 and Ž 1 with Ž 10, we obtain the expansion of FŽe i and this expansion naturally satisfies Ž 11 so that the scaling functions corresponding to the I-type rational filters exist ext, we consider the expansion of a II-type rational filter Since the FŽe i is an even function with period, FŽe i can be expanded to a Fourier series where i F e i Ý f e, 13 Z Ž Ž 1 i f f FŽ e cos d, 0,1, Ž 14 H 0 When SŽ y satisfies Ž 9, the rational function FŽ z deterined by Ž 6 has neither zeros nor poles on the unit circle This iplies that the FŽe i is two ties continuously differentiable According to the Fourier series theory, f O Ž so that the expansion Ž 13 also satisfies Ž 11

7 RATIOAL FILTER WAVELETS 33 Ž ow, we give soe choices of S y Let 1 1 P y Ý y 15 0 Ž Ž Ž Ž P y is a polynoial solution of the equation 16 Ž 1 y PŽ y y PŽ 1 y 1 Ž 16 Ž Ž 1 and 0 P y for y 0, 1 Žsee 1, Lea 714 If SŽ y P Ž y, then Ž 6 becoes i FŽ e P sin Ž 17 It just corresponds a Daubechies wavelet Žsee 1 This eans that Daubechies wavelets are included in the faily of the wavelets derived fro I-type rational filters If SŽ y 1, then Ž 6 becoes i FŽ e cos sin, Ž 18 which corresponds to a good regularity wavelet for sufficiently large Žsee Section 4 Especially when 1, Ž 18 corresponds to Haar wavelet ext, we give a ethod by which SŽ y can be chosen ore easily Let SŽ y P Ž 1 y rž y r Ž 1 y, Ž 19 where P Ž y is the polynoial introduced in Ž 15 and rž y is a polynoial Substituting Ž 19 into Ž 6 leads to P Ž 1 y rž y r Ž 1 y i FŽ e, y cos 1 rž y r Ž 1 y y Ž 1 y Ž 0 THEOREM If and rž y 0 for y 0, 1, then the FŽe i deterined by the equation Ž 0 satisfies Ž 111 Proof Since rž y 0 for y 0, 1, we have 1 rž y r Ž 1 y rž y r Ž 1 y 1

8 34 ZHEG AD MIGGE Ž Ž 1 Besides for y 0, 1, y 1 y Considering the relation Ž Ž Ž 1 16 and inequality 0 P y, we have Hence 0 P 1 y r y r 1 y Ž Ž Ž Ž 1 ½ Ž Ž Ž 5 1 r y r 1 y y 1 y P Ž 1 y rž y r Ž 1 y Ž rž y r Ž 1 y y Ž 1 y Rear When rž y 0, Daubechies filter Ž 17 can also be obtained fro Ž 0, but filter Ž 18 cannot 3 II-TYPE RATIOAL FILTERS AD BATTLELEMARIE WAVELETS II For a II-type filter H Ž, it satisfies H II Ž e i H II Ž Ž 31 This iplies that the scaling function corresponding to a II-type rational filter has linear phase In fact, cobining Ž 13 and Ž 8 with Ž 31, we easily obtain ˆ Ž e i ˆ Ž Ž 3 Ž 3 shows that Ž x is syetric about x0 Fro the relations Ž 14, Ž 16, and Ž 31, we can siilarly derive ˆ 1 e ˆ 33 i Ž Ž Ž Ž This eans that the wavelet function Ž x is syetric about x0 1 when is even or antisyetric about x0 1 when is odd ext we discuss the relations between II-type rational filters and BattleLearie wavelets We first recall BattleLearie wavelets referring to Battle s paper 7 and Chui s boo 8 An order B-spline function Ž x is defined as x x x 34 Ž Ž Ž Ž l l

9 RATIOAL FILTER WAVELETS 35 where Ž x is the indicator function of the interval 0, 1 l, and the sign denotes a convolution operation The Fourier transfor of is Let / 1 e i ˆ Ž Ž 35 ž i U Ž ˆ Ž Ž 36 Ý Z Then, perforing a orthonoralization tric, we obtain the Fourier transfor of the BattleLearie scaling function ' Ž ˆ Ž UŽ Ž 37 ˆ Thus the filters of BattleLearie wavelets are ( i ˆ ˆ 1 e UŽ H Ž Ž Ž Ž 38 UŽ Let ' Ž Ž Ž Ž i F e U U 39 Ž Ž According to the forula 418 in 8, U can be written as iž 1 e i U Ž E 1Ž e, Ž 310 Ž 1! where E Ž z 1 is a EulerFrobenius polynoial of degree E Ž z 1 has real zeros 1,,, the distribution of which is 0 1 Ž see 8, Chap and they satisfy

10 36 ZHEG AD MIGGE Thus we have Ž ) i F e iž 1 i i i e e 1 e e i i i e 1 e e Ž Ž Ž Ž Ž Ž ) i 1 1 i e e Ł i Ł i Ž 311 e e 1 1 Ž 311 shows that the filters of BattleLearie wavelets are the II-type rational filters Furtherore, we can also prove that F Že i satisfies Ž 111 In fact, for, we have fro forula Ž 410 in 8 1 U Ž sin Ý Z Ž sin sin Ý Ž Ž Z 0,1 1 1 sin Ž 1 1 in sin 0, Ž Ž 1 Since U Ž 1 Žcf 8, Chap 4, we obtain F e i 1 31 Ž Ž That is, the condition Ž 111 for F Že i holds Rear Actually, a better estiate than Ž 31 has been given in 1, Chap 5 Our purpose here is only to state that BattleLearie wavelets belong to the II-type rational filter faily restricted by Ž 111 Fro the above discussion, we have nown that the faily of the II-type rational filter wavelets include BattleLearie wavelets and have linear

11 RATIOAL FILTER WAVELETS 37 phases It is the reason we propose and investigate the II-type rational filters 4 REGULARITY AALYSIS Ž ˆŽ Ž Define C f x H f 1 d 4 R and call the regularity index of fž x Since a filter HŽ can be factorized as the for of Ž 19, HŽ and FŽe i depend on Let us add subscript to the in this section That is, they are denoted by H and F, respectively, and the corresponding scaling function and wavelet function are denoted by and, respectively The regularity index of Daubechies wavelet has been estiated by Daubechies 1, and Voler 9 By investigating the polynoial filters, Voler gives the asyptotic result and log ž / ž / log 3 log log 1 O 0075 O Ž 41 log 3 li 1 Ž 4 log Regularity eans soothness One is interested in finding wavelets as sooth as possible which leads to the proble in estiating the upper bounds of F Že i s ow, we approach regularity fro the rational filters Denote B sup F Že i We apply the general estiate Ž 43 R log Blog C 1 ˆ Ž Ž Ž 43 given by Daubechies 1, Lea 71 to analyze the regularity of rational filter wavelets According to Ž 43, we only need to estiate B in the faily of rational filters LEMMA Denote SŽ ysž y is a polynoial with real coefficients, and SŽ y 0 for y 0, 1 4 Then for a positie integer SŽ 1 y 1 inf sup Ž 44 s y SŽ 1 y Ž 1 y SŽ y y 0,1

12 38 Proof ZHEG AD MIGGE For any SŽ y, we have sup SŽ 1 y y0,1 y SŽ 1 y Ž 1 y SŽ y SŽ 1 1 ; Ž 1 SŽ 1 Ž 1 SŽ 1 hence SŽ 1 y 1 inf sup Ž 45 S y SŽ 1 y Ž 1 y SŽ y y 0,1 Ž Besides, for S y 1 SŽ 1 y 1 1 sup sup y SŽ 1 y Ž 1 y SŽ y y Ž 1 y y 0,1 y 0,1 This iplies SŽ 1 y 1 inf sup Ž 46 S y SŽ 1 y Ž 1 y SŽ y y 0,1 Cobining Ž 45 with Ž 46, we iediately prove the lea ote that Ž 44 also holds for the faily of the real functions larger than zero on the interval 0, 1 For a fixed, Lea tells us that Ž 1 is the least value of B in the faily of the F Že i s deterined by Ž 6 This least value is reached when SŽ y 1 in Ž 6 It corresponds the rational filter or 1 e i I i HŽ FŽ e Ž 47 1 e i II i H Ž FŽ e, Ž 48 Ž i Ž Ž where F e is deterined by 18 In such a case, 43 is written as Ž 1 C 1 ˆ Ž Ž Ž 49

13 RATIOAL FILTER WAVELETS 39 and for the regularity index the filter Ž 47 or Ž 48, we have and of the scaling function corresponding to 1 05 O Ž 410 li 05 Ž 411 Ž Ž The estiate 410 is a great iproveent as copared with 41 The above discussion has proven the following result THEOREM 3 For the waelets deried fro the rational filter Ž 47 or Ž 48, their regularity indices satisfy the asyptotic estiate Ž 410 and Ž EXAMPLES AD COMPUTATIO In this section, we give the details of the expansion of a rational filter In exaple 1, we expand the I-type rational filter Ž 47 for 3 and 4 In exaple, we expand II-type rational filter Ž 48 for 3 and 4 EXAMPLE 1 For 3, we first solve the F Ž z in the filter Ž 47 3 by the relation i FŽ e cos sin Ž cos Let F Ž z Ža aza z 1 and substitute it into Ž Then we copare the corresponding coefficients of two sides of Ž Ž 5 i i a ae a e 5 3 cos 0 1 Consequently, we obtain a 34, a 0, a 14 Hence i F3Ž e i i 3 e 1 e i Ž 1 e Ý Ž 53

14 40 ZHEG AD MIGGE Substituting the expansion Ž 53 into Ž 47, we obtain the filter coefficients h of this I-type rational filter For 4, we can obtain the filter coefficients in the sae way Table I lists the values of the filter coefficient sequences TABLE I The Filter Coefficients of I-type Rational Filters Ž 47 for 3 and 4 K h

15 RATIOAL FILTER WAVELETS 41 We use the algorith given in 10 to carry out nuerical coputations of the corresponding scaling and wavelet functions They are plotted in Fig 1 These plots clearly show that they are soother than Daubechies ones for a fixed EXAMPLE integrals Let us expand II-type filter Ž 48 We need to calculate the 1 cos d H f, 0,1,, Ž 54 0 cos ( ž sin / They ay be calculated by the Roberg integration We can easily prove that f 0 for odd The filter coefficients h for 3 and 4 are listed Ž FIG 1 The functions, corresponding to I-type filter 47 for 3, 4

16 4 ZHEG AD MIGGE TABLE II The Filter Coefficients of II-type Rational Filter Ž 48 for 3 and 4 The h Satisfy h h K h h 6 10 in Table II ote that the coefficients h satisfy h h Their scaling functions and wavelet functions are plotted in Fig 6 DISCUSSIO The variety of applications deands the variety of the wavelets available This directly leads to the necessity of extending the construction of wavelets Indeed, the rational filters extend the construction of wavelets

17 RATIOAL FILTER WAVELETS 43 Ž FIG The functions, corresponding to II-type filter 48 for 3, 4 Especially rational filter wavelets include two iportant failies: Daubechies wavelets and BattleLearie wavelets Linear phase plays an iportant role in the reconstruction of a copressed iage Because II-type filter wavelets have linear phases, they deserve to be recoended for the reconstruction of signals REFERECES 1 I Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 199 I Daubechies, Orthonoral bases of copactly supported wavelets, Co Pure Appl Math XLI Ž 1988, G Malgouyres, Ondelettes a support copact et analyse ulti-resolution sur l intervalle, in Progress in Wavelet Analysis and Applications Ž Y Meyer and S Roques, Eds, pp 71570, Frontieres, Gif-sur-Yvette, C Herley and M Vetterli, IEEE Trans Signal Process 41 Ž 1993,

18 44 ZHEG AD MIGGE 5 G Evangelista, Wavelet transfors and wave digital filters, in Wavelets and Applications Ž Y Meyer, Ed, pp 39641, Masson, Paris, G Polya and G Szego, Aufgaben und Lehrsatze aus der Analysis, Springer-Verlag, Berlin, 1971, Vol II 7 G Battle, A bloc spin construction of ondelettes, Co Math Phys 110 Ž 1987, C K Chui, An Introduction to Wavelets, Acadeic Press, ew Yor, H Voler, On the regularity of wavelets, IEEE Trans Infor Theory 38, o Ž 199, Z Kuang and L Yunhui, A construction ethod of wavelets with linear phases, J Harbin Inst Techn E-3, o 4 Ž 1996