空间l2（Z×Z）上正交小波基的频域特征刻画与算法.pdf

小波基的频域特征刻画有助于小波基的构造。首先给出了函数空间l2(ZZ)上正交小波基的频域特征刻画，再根据正交小波基的频域特征刻画，可以容易验证：一维空间的两个正交小波基的乘积是二维空间的正交小波基。最后还给出了完美重构条件以及快速分解重构算法。关键词：正交小波基；卷积；滤波器组；完美重构中图分类号：O244文献标志码：ADOI：10.3969/j.issn.1674-8085.2023.04.001Received date：2022-09-08；Modified date：2023-04-12.Foundation item:The Scientific Research Foundation of the Education Bureau of Jiangxi Province(GJJ201009,GJJ211027)；Doctoral Research Startup Projectof Jinggangshan University(JZB2014)；Jian Science and Technology Support Project(JSKJZ2014:36-13).Biographies:*Yi Hua(1973-),male,born in Songzi,Hubei Province,doctor,associate professor,major in wavelet analysis and its application research(E-mail:)0 IntroductionThe wavelet transforms1-6have played animportant role in the applications such as signalprocessing,image processing.The construction oforthonormal bases of 2 is given by Daubechies7-8.The construction of orthonormal wavelet systems onl2(Z)and l2(ZZ)have been systematically investigatedby Frazier9.Algorithm implementation and examplesof Framlet packets on l2(Z)are studied in2.In thispaper,we generalize some results to thatof l2(ZZ).This paper is organized as follows.Section 1gives some notations,definitions,and lemmas weshall use.In section 2,we firstly give the definition oforthonormal wavelet basis of l2(ZZ).Secondly,thefrequency domain description of orthonormal waveletbasis is proved.Thirdly,we give decomposition andreconstruction algorithms based on filter banks.Lastly,an example of orthonormal wavelet basis is given.1 PreliminariesWe begin by introducing some notations anddefinitions we shall use.Definition 1.1 l2(ZZ)is a function space defined第44卷第4期Vol.44 No.4井冈山大学学报(自然科学版)2023年7月Jul.2023Journal of Jinggangshan University(Natural Science)1井冈山大学学报（自然科学版）2as follows9,1222121212()(,),|(,)|nZ nZlZZzz n nn nZz n n (1)For2,()z wlZZ,define complex inner productand norm as follows,121212,(,)(,),.nZ nZz wz n n w n nzz z(2)We say thatzandware orthogonal if,0z w.Definition 1.2 The Fourier transform on l2(ZZ)is the map:l2(ZZ)2()L,definedforzl2(ZZ)by91 12 2121212,(,)(,)ininnZ nZzz n n ee,(3)wheretheseriesisinterpretedasitslimitin2()L,.Definition1.3 Forasignal2()z l Z Zand12,k kZdefine912,121122,(,)k kRz n nz nk nk,(4)for12,n nZ.We call12,k kRzthe translation ofzby12,.k kDefinition1.4Definethedown-samplingoperator22:()()D l ZZl ZZbysetting9,for2()z l Z Z,1212()(,)(2,2)D z n nznn(5)for12,n nZ.Definition 1.5 Define the up-sampling operator22:()()U l ZZlZZby setting9,for2()z l Z Z,12()(,)U z n n121212(/2,/2)if and are even0 if or is odd.z nnnnnn(6)for12,n nZ.Definition 1.6 For any2()zlZZ,define9the conjugate reflection ofz:1212(,)(,)for all z n nznnn.(7)Also define1*121212(,)(1)(,),(,)nz n nz n nzn n 212*121212(1)(,),(,)1)(,()nnnz n nzn nz n n(8)We can easily verify that*1()4U D zzzzz121212,if and are even,0,if or is odd.z n nnnnn(9)Definition 1.7 For2,()z wlZZ,define912(,)zw m m12112212(,)(,)nZ nZz mn mn w n n(10)for all12,m mZ.A simple calculation can lead to the followingresults.Lemma 1.1 Suppose2,()z wlZZ.Then(i)2,()z z zzl ZZand122,k kRzlZ Z12for all,k kZ.(ii)1212(,)(,).zz (iii)121212(,)(,),(,)zzz 121212(,),(,)(,).zzz (iv)1 12 212,1212,ikikk kRzeez .(v)12121122,.k kjjjkjkRz Rwz Rw(vi)12,12,(,).k kz Rwzw k k(vii)1212,1,for all,),where 12121,=0,=0 (,)=0,othersnnn n.(viii)21212,www .(ix)Suppose2()zlZZand1()wl ZZ.Then2()zwlZZ,and1zwwz,(11)whereand1represent2l-norm and1l-norm井冈山大学学报（自然科学版）3respectively.Lemma 1.2 Suppose1,()w zl ZZ(i)The set122,212,kkRw k kZis an rthonormalset if and only if222121212,www 212,4w(12)for all12,0,).(ii)We have12122,22,21212,0 for all,kkjjRw Rzk kjjZIf and only if 1212121212121212,0 wzwzwzwz for all12,0,).Proof:For part(),we first observe that theelements122,2kkRwmust be distinct for12,k kZ,thatis,12122,22,2kkjjRwRwimplies1122,.kj kjNext,note that the quantity222121212,www 212,w is periodic with period,so the identity in equationholds for all12,if and only if it holds for all12,0,).It is easily verified that122,212,kkRwk kZis orthonormal if and only if12122,2(2,2),kkwwkkw Rw12121 if ,00 if 0 or 0.k kkk(13)By usingwith,zwwwe see that equation isequivalent to12(,)wwwwwwwwn n124(,).n nBy Fourier inversion and Lemma 1.1(vii),this isequivalent towwwwww12,4 ww for all12,.By Lemma 1.1(iii)and(viii),we get222121212,www 212,4.w The proof idea of part()is similar to that of part(i).2 Orthonormalwaveletbasisofl2(ZZ)Definition 2.1 An orthonormal basis for2lZZof the form1212122,21120122,22122,2,kkkkkkBRwk kZRwk kZRwk kZ123122,2,kkRwk kZ(14)for some10123,w w w wlZZ,is called anorthonormal wavelet basis of2lZZ.2.1Frequencydomaincharacterizationoforthonormal wavelet basis and decomposition,reconstruction algorithmsTheorem 2.1 Let10123,.w w w wl ZZDefine12,A,the system matrix of0123,w w w w,by01211201211212012112012112,1?,2,wwwwAwwww 212312212312212312212312,wwwwwwww (15)Then120122,2,kkBRwk kZ122,2112,kkRwk kZ井冈山大学学报（自然科学版）4122122,2 ,kkRwk kZ122,2312,kkRwk kZ(16)is an orthonormal wavelet basis for2lZZif andonly if12,A is unitary for12,0,).Proof：12,A is unitary for12,0,)ifand only if222121212212 ,4 =0,1,2,3iiiiwwwwi and1212121212121212,0,when.ijijijijwwwwwwwwij By Lemma 1.212120121122,22,2,kkkkRwk kZRwk kZ12122123122,22,2,kkkkRwk kZRwk kZis an orthonormal set for2lZZif and only if12,A is unitary.To complete the proof,we mustshow that if12,A is unitary for all12,0,),then the orthonormal setBis complete in2lZZ.Observethattheunitaryof12,A forall12,0,)implies the unitary for all12,Z because12(,),0,1,2,3iwi are2periodic for thefirst and second variable.We claim that30(),iiizwU D zw(17)for all2()zlZZ.To see this,by using we obtain3120()(,)iiiwU D zw 312121201(,)(,)(,)4iiiwzw 12121212(,)(,)(,)(,)iizwzw 1212(,)(,)izw32121203121212031212120312121201(,)|(,)|41(,)(,)(,)41(,)(,)(,)41(,)(,)(,).4 iiiiiiiiiiizwzwwzwwzww However,the unitarity of12,A implies the rowsof12,A are orthonormal,so the last expressionreduces to0,1,2,31212121212(,)1(,)0(,)0(,)0(,)zzzzz By Fourier inversion,this implies.Note that1212122,2()(,)(2,2),0,1,2,3.iikkiD zwk kzwkkz RwiHence,if122,2,0,0,1,2,3kkiz Rwifor all12,k kZ,then()0,iD zwi0,1,2,3.Hence,z=0,by equation,which proves the completeness ofB.Corollary 2.2 IfBdefined in is an orthonormalwavelet basis for2lZZ12121232,22,20,kkikkiikZ kZz Rw Rw 30()iiiwU D zw(18)Proof：The expansion ofzunder the orthonormalbasis B is12121232,22,20,.kkikkiikZ kZzz Rw Rw(19)Substituting into leads to.The calculation method of wavelet coefficients ofz represented in Fig.1,is known as filter bankapproach.To regainzfrom the output of the left filterbank in Fig.1,we follow up with a right filter bank asin the right half of Fig.2.Here 0,1,2,3iisare unknown.井冈山大学学报（自然科学版）5SupposeBdefined in is an orthonormal wavelet basisfor2lZZ.Then,0,1,2,3,iiswi(20)by.Next,we do not assume the conditions of Theorem2.1.Thus we have a more general result about filterbanksthatdonotnecessarilycorrespondtoorthonormal bases.Fig.1Fig.2Suppose22(),(),0,1,2,3jjwlZZ slZZj.In the jthbranch of the filter bank,with input2()zlZZ,we computejzw.Then we applydown-sampling.This is the decomposition stage of theprocess.In the reconstruction stage,we take theoutput of the jthbranch so far,namely()jD zw,applyU,andconvolvewithjs,giving()jjsU D zw.We have perfect reconstruction if the sum of thesealways equalsz,that is,if30()jjjzsU D zw(21)for all2zlZZ.Theorem 2.3 Let12,A be the matrix definedas.Wehaveperfectreconstructionifandonlyif01211212212312,2,0,0,0,ssAss for all120,.Proof：By using,we have()1()()().4jjjjjU D zwzwzwzwzwHence1212121212121212121(),4,.jjjjjU D zwzwzwzwzw Therefore,31203121212012121212121212121212 (),1,4,1,(,2,jjjjjjjjjsU D zwszwzwzwzwzz zzA ，01211212212312,.,ssss By Fourier inversion,we have perfect reconstructionif and only if,for all12,and2zlZZ,312120(),jjjsU D zwz ,then0121121212212312,2,0,for all ,0,).0,0,ssAss 井冈山大学学报（自然科学版）6It should be noted that some results for2()lZZarethe generalizations of those for2()NNlZZin9.However,completeness of orthonormal wavelet basismust be considered for2()lZZin this sectionbecause2()lZZis an infinite dimension space.2.2 Example of orthonormal wavelet basisThe easiest way to generate a basis of the type inis to form product wavelets9.Corollary 2.4 Suppose21212222 and kkkkk Zk Zk Zk ZR vR uR vR uare two orthonormal bases for 2lZ.Define0121122112112221211223121122,.wn nv n vnw n nun vnwn nv n unwn nun unProve that12120121122,22,2,kkkkRwk kZRwk kZ12122123122,22,2,kkkkRwk kZRwk kZis an orthonormal wavelet basis for2lZZ.Proof：Itsufficestoshowthat12,A defined inis unitary.Firstly,we show that1 12 212012012,(,)=ininnZ nZww n n ee 1 12 21211221122()()=ininnZ nZv n v n eevv(22)Similarly,we have1121122212,wuvw 11223121122,vuwuu(23)Substituting and into leads to121122112211221122112211221122112211221122112211221122112,=)()()()(1?)()()()()()()()(2)(Avvuvvuuuvvuvvuuuvvuvvuuuvvuv 211221122=)()(vuuu2222111122112222111111()()()()=,()()()()22 vuvuAAvuvu(24)where“”represents tensor product10.Since21kk ZR v21kk ZR uAnd22 kk ZR v22kk ZR 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