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Orthogonal filters and the implications of wrapping on discrete wavelet transforms

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Title:
Orthogonal filters and the implications of wrapping on discrete wavelet transforms
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Bleiler, Sarah K
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Filter
Matrix transform
Fourier series
Smoothness
Zero moments
Dissertations, Academic -- Mathematics -- Masters -- USF   ( lcsh )
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non-fiction   ( marcgt )

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Abstract:
ABSTRACT: Discrete wavelet transforms have many applications, including those in image compression and edge detection. Transforms constructed using orthogonal filters are extremely useful in that they can easily be inverted as well as coded. We review the major properties of three well-known orthogonal filters, namely, the Haar, Daubechies, and Coiflet filters. Subsequently, we analyze the Fourier series that corresponds to each of those filters and recall some important results about the smoothness of the modulus of those Fourier series. We consider a specialized case in which the length of the discrete wavelet transform is not much longer than the length of the filter used in its construction. For this case, we prove the existence of additional degrees of freedom in the system of equations used in the construction of the aforementioned orthogonal filters. We suggest a modified Coiflet filter which takes advantage of the extra degrees of freedom by imposing further conditions on the derivative of the Fourier series.
Thesis:
Thesis (M.A.)--University of South Florida, 2008.
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Includes bibliographical references.
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by Sarah K. Bleiler.
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Orthogonal filters and the implications of wrapping on discrete wavelet transforms
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ABSTRACT: Discrete wavelet transforms have many applications, including those in image compression and edge detection. Transforms constructed using orthogonal filters are extremely useful in that they can easily be inverted as well as coded. We review the major properties of three well-known orthogonal filters, namely, the Haar, Daubechies, and Coiflet filters. Subsequently, we analyze the Fourier series that corresponds to each of those filters and recall some important results about the smoothness of the modulus of those Fourier series. We consider a specialized case in which the length of the discrete wavelet transform is not much longer than the length of the filter used in its construction. For this case, we prove the existence of additional degrees of freedom in the system of equations used in the construction of the aforementioned orthogonal filters. We suggest a modified Coiflet filter which takes advantage of the extra degrees of freedom by imposing further conditions on the derivative of the Fourier series.
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System requirements: World Wide Web browser and PDF reader.
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Fourier series
Smoothness
Zero moments
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by SarahK.Bleiler Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofArts DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:CatherineBeneteau,Ph.D. ThomasBieske,Ph.D. DmitryKhavinson,Ph.D. DateofApproval: November18,2008 Keywords:lter,matrixtransform,Fourierseries,smoothness,zeromoments cCopyright2008,SarahK.Bleiler

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Iwouldliketothankmyadvisorandmentor,Dr.CatherineBeneteau,forallofherhelpandsupportthroughoutmytimeinthemathematicsdepartmentatUniversityofSouthFlorida.Herenthusiasm,guidance,andadvicewereinvaluableinthewritingofthisthesisandinmyacademicjourneythusfar. IwouldalsoliketothankbothDr.ThomasBieskeandDr.DmitryKhavinsonforalltheiradviceandsuggestionsduringtheformulationofthisthesis.Dr.Bieskewasparticularlywonderfulinassistingmewiththetechnicalformattingofthispaper,andDr.Khavinson'sreviewandsuggestionswereinvaluableinkeepingthesubjectmatterfocusedandinformative. Specialthankstomymotherandfather,withoutwhomIwouldneverbeinthegreatplaceIamtoday.Yourloveandsupporthavebeenmygreatestmotivation. Lastly,Iwanttothankmybestfriendinthisworld,Ray.Withoutyou,Isurelywouldneverhavemadeittothispoint.Iloveyou.

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ListofFiguresi Abstractiii 1Introduction1 2OrthogonalFilters14 2.1HaarFilter...................................14 2.2DaubechiesFilters...............................15 2.3CoietFilters..................................19 3Smoothness23 3.1SmoothnessofjH(w)j2.............................23 3.2SmoothnessofjH(w)j.............................25 4Wrapping33 4.1Non-wrappingRows..............................35 4.2WrappingRows.................................36 4.2.1Non-wrappingPortion.........................36 4.2.2WrappingPortion............................37 4.3EectsofWrappingonZeroOrthogonalityConditions...........39 4.4ANewFilterConstruction...........................43 5Conclusion48 5.1FutureResearchandStudy..........................48 5.2MathematicaProgramforConstructionofDaubechiesFilters.......49 i

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1.1jH(w)j(fromExample1.0.12)on[0;],hisalowpasslter.........8 1.2jG(w)j(fromExample1.0.12)on[0;],gisahighpasslter........9 1.3ImagebeforeandaftertheapplicationoftheHaarwavelettransform...11 3.1jH(w)jon[0;]forDaubechieslength4lter................30 3.2jH(w)j2on[0;]forDaubechieslength4lter................30 3.3jH(w)jandjH(w)j2on[0;4]forDaubechieslength4lter.........31 3.4jH(w)jforDaubechiesandCoietlters(length6).............31 3.5jH(w)jforDaubechiesandCoietlters(length12).............32 4.1Lowpassportionofa1414DWT(thetophalfofthematrix)......34 4.2Lowpassportionofa1010DWT(thetophalfofthematrix)withc>L.37 4.3Length12modiedlters...........................45 4.4Length18modiedlters...........................46 4.5ModuluscomparisonofDaubechieslength12,Coietlength12,andmodi-edlength12(K=2,c=3)lters......................47 4.6ModuluscomparisonofDaubechieslength18,Coietlength18,andmodi-edlength18(K=3,c=5)lters......................47ii

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SarahK.Bleiler ABSTRACT Discretewavelettransformshavemanyapplications,includingthoseinimagecompres-sionandedgedetection.Transformsconstructedusingorthogonalltersareextremelyusefulinthattheycaneasilybeinvertedaswellascoded.Wereviewthemajorpropertiesofthreewell-knownorthogonallters,namely,theHaar,Daubechies,andCoietlters.Subsequently,weanalyzetheFourierseriesthatcorrespondstoeachofthoseltersandrecallsomeimportantresultsaboutthesmoothnessofthemodulusofthoseFourierseries.Weconsideraspecializedcaseinwhichthelengthofthediscretewavelettransformisnotmuchlongerthanthelengthofthelterusedinitsconstruction.Forthiscase,weprovetheexistenceofadditionaldegreesoffreedominthesystemofequationsusedintheconstructionoftheaforementionedorthogonallters.WesuggestamodiedCoietlterwhichtakesadvantageoftheextradegreesoffreedombyimposingfurtherconditionsonthederivativeoftheFourierseries. iii

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Discretewavelettransforms(DWTs)arematricesusedtotransformsetsofdiscretedataintodatathatiseasiertoworkwithandmanipulate.Thistransformation,orpro-cessing,ofdataisdonethroughmatrixmultiplication.OneexampleofsuchprocessingistheapplicationofaDWTtoanimage.Forinstance,wecanuseDWTstoaidintheprocessofimagecompression.Todothis,wewouldneedtochooseanappropriateDWT,onewhichisknowninpracticetobeusefulinimagecompression,andthenapplyittothematrixcontainingthepixelintensitiesoftheimagewewishtocompress.Theresulting(transformed)imageshouldthenbebettersuitedtotheprocessofimagecompression.AnothersuchimageprocessingdomainforwhichDWTshaveprovenusefulisedgedetec-tion.Wewillexploretheseimageprocessingapplicationsinmoredetailattheendofthissection(seeFigure1.3). Wetakesometimetodiscussthehistoryofwaveletsandwhytheycameabout.Fourieranalysishasbeenusedforcenturiesasatoolforanalyzingandtransformingsignalsandimages.Fourierseriesarebuiltusingsineandcosinefunctions,whichareperiodicwavesthatcontinueinbothdirectionsforever.Therefore,whenprocessingdatawhichisnotitselfperiodicand/ortime-independent,butratheristransientandcontainsabruptchanges,Fouriertransformsarenotveryecient.Waveletswereconstructedasthetoolneededtosuccessfullyworkwithsuchtransientdata.In1946,GaborexaminedFourierserieswhichweretakenonaniteinterval,suchthatbothtimeandfrequencycouldbeconsidered[11].1

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Wenowoutlinethestructureofthisthesis.FirstnotethatthebuildingblocksofDWTsarecalledlters(essentiallytherowsoftheDWT),andtheywillbeourmainmathematicalobjectsofdiscussion.Wewillbegin,inChapters1and2,byintroducingsometerminologyandthebasicsbehindtheconstructionofsomeofthemostwell-knownlters,namelytheHaar,Daubechies,andCoietlters.Thenwewillseehowtheycanbeusedinapplicationssuchasimagecompressionandedgedetection.WewillclassifytheconstructionoftheseltersintermsoftheircorrespondingFourierseriesanddiscusswhythisisuseful.InChapter3,wewillconsiderthe\smoothness"ofthegraphofthemodulusoftheFourierseriesforsomegenerallters.Thiswillhelpustobetterunderstandtheimplicationsofeachlterconstructionaswellastoapproximatehowclosealteristobeing\ideal."WenoticethatforanyDWTcomposedofltersoflengthgreaterthantwo,therewillalwaysbewhatiscalled\wrapping"throughoutthematrix.OurmainquestionforthisthesisistondoutwhatimplicationstheprocessofwrappinghasontheconstructionconditionsthatwehighlightinChapters1and2.Therefore,inChapter4ourfocuswillbeonexploringwrapping,identifyinginterestingoutcomesforDWTscontainingsignicantamountsofwrapping,andconsideringanewlterconstructionthatresultsfromthoseoutcomes.Thisnewlterproducesacorrespondinggraphwhichis\smoother",thusimplyingthatthelteriscloserto\ideal." Webeginwiththedenitionofalter.Denition1.0.1

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2;1 2;0;0;:::),wherehk=1 2fork=0;1,andhk=0otherwise.Ifxisanysequence,thenthecomponentsoftheconvolutionproductcanbewrittenasyn=1 2xn+1 2xn1.Therefore,the\processing"ofdatainthiscasecanbethoughtofasasimpleaveragingofconsecutiveentriesoftheinputdatasequencex.Someltersareusefulforidentifyinghomogeneous(i.e.,locallyconstant)portionsoftheinputdata,whileothersareusefulforidentifyinglargedierencesintheinputdata.Therefore,thelteronechoosesdependsgreatlyontheapplication,andthetypeofprocessingthatisdesired. Throughoutthisthesis,wewillonlybeconcernedwithnitelengthlters.Suchaltercanbewrittenash=(hl;hl+1;:::;hL1;hL),wherehl6=0,hL6=0.Thismeansthateachcomponentoftheconvolutionproductwillbeaconvergentseries,representedbyyn=LXk=lhkxnk: Inordertogainabetterunderstandingofhowdierentlterswillprocessdata,itisoftenusefultoconsidertheircorrespondingFourierseries.Denition1.0.4 Itisbenecialtoconsideralterinthe\Fourierdomain"forseveraldierentreasons,suchasthefollowing:

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LetusinvestigatetheimportanceofthemodulusjH(w)jthatisassociatedwithalterh.HowcanjH(w)jbeusedtopredictalter'sproperties?Proposition1.0.5 Proof.H(w)=LXk=lhkeik(w): H(w)=LPk=lhkeikw(sincehkisreal). SoH(w)= ByEuler'sformula,weknoweikw=coskw+isinkw.BecauseH(w)=LPk=lhkeikw=LPk=lhk(coskw+isinkw),itiseasytoseethatH(w)canbewrittenasapolynomialintermsofcoskwandsinkw,andhencejH(w)jis2-periodic.GiventhefactthatjH(w)jisbothevenand2-periodic,wecangatherallimportantinformationofthegraphbyconsideringonlytheportiondenedon[0;]. ByexaminingjH(w)jonthatinterval,wecangetagoodideaofthenatureoftheprocessingthatthecorrespondinglterhwillperform.Forinstance,ifthevalueofjH(w)jislargeatw=0,thenwhenweprocessthedatausinglterh,ifthereareportionsofthedatathatarelargelyhomogeneous,thelterwillpreservethoseportionsofthedata.Asanexample,considerthelterh=(h0;h1;h2)=(1 4;1 2;1 4).ThecorrespondingFourierseriesisH(w)=1 4+1 2eiw+1 4e2iw,andthusjH(0)j=1.Ifweusethisltertoprocessthesignalx=(:::;1;1;1;:::),thenhx=(:::;1;1;1;:::),thuspreservingthehomogeneityofthesignal.4

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4;1 2;1 4)asabove,notingthatjH()j=0.Ifweusethisltertoprocessthesignalx=(:::;1;1;1;1;:::),thenhx=(:::;0;0;0;:::),thusannihilatingtheoscillationofthesignal. Wecharacterizethistypeoflter(i.e.,onewhichpreserveshomogeneitywhilesimulta-neouslyannihilatinglargeamountsofoscillation)asalowpasslter.Example1.0.12willhelptoclarifythisnotionevenfurther.Wegivetheprecisedenitionsforlowpass(andhighpass)ltersbelow(see[20],p.143,147):Denition1.0.6 Forexample,ifhisaltersuchthatjH(w)j=p Thedenitionoftheidealhighpasslterisanalogoustothedenitiongivenabove. Ingeneral,weconsiderasimplicationofthedenitionsoflowpassandhighpassltersgivenabovebyrequiringonlythefollowingconditions:5

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IfhisalowpasslterthenwemusthavejH(0)j=p IfgisahighpasslterthenwemusthavejG(0)j=0andjG()j=p Thevaluep

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ADWTisusedtoprocessdatainmuchthesamewayaswasdescribedforasinglelter.Lookingcloser,ifxisavectorthatrepresents,say,asignal,theninmultiplyingxbyWN,weareessentiallycomputingeveryothercomponentoftheconvolutionproduct.FormoreinformationontherationalebehindtheconstructionofaDWTanditsrelationtoconvolution,see[20]. WeworkthroughanexamplebelowtoshowaspeciccaseofaDWT.Example1.0.12 2;p 2)andg=(p 2;p 2).Wecaneasilycheckthathisalowpasslter(usingthesimplieddenition):H(w)=p 2+p 2eiw=p 2(1+eiw)=p 2(1+cosw+isinw): 2(1+1)=p 2(1+(1))=0.Similarly,gisahighpasslter.ThegraphsofjH(w)jandjG(w)jaregiveninFigures1.1and1.2. Nowsupposewewanttoprocessthevectorv=(1;1;1;1;1;1;1;1)usingaDWTconstructedwithlowpasslterh=(p 2;p 2)andhighpasslterg=(p 2;p 2).Thevectorvisoflength8sotheDWTwewillusetoprocessthedatainvshouldbeamatrixof7

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Figure1.2:jG(w)j(fromExample1.0.12)on[0;],gisahighpasslter8

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2p 200000000p 2p 200000000p 2p 200000000p 2p 2p 2p 200000000p 2p 200000000p 2p 200000000p 2p 21CCCCCCCCCCCCCCCCCA 2p 200000000p 2p 200000000p 2p 200000000p 2p 2p 2p 200000000p 2p 200000000p 2p 200000000p 2p 21CCCCCCCCCCCCCCCCCA0BBBBBBBBBBBBBBBBB@111111111CCCCCCCCCCCCCCCCCA=0BBBBBBBBBBBBBBBBB@p

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Wecouldeasilycheckthatthehighpassportionofthematrixactsinexactlytheoppositemanner,preservingdierencesandannihilatingsimilarities.Remark1.0.13 2;p 2)andhighpassvectorg=(p 2;p 2).WewilllearninthenextchapterthatthenameofthisparticularDWTistheHaarwavelettransform.

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Weseeimmediatelyhowthisprocesscanaidinbothedgedetectionandimagecompres-sion.Asstatedabove,afterweperformthetransform,anapproximationoftheoriginalimageappearsinthetopleftcornerwhiletheotherthreeblocksrepresentthe\dierences"amongpixelintensitiesinthatimage.Ifthereisalargedierenceinvaluebetweentwoconsecutivepixelsintheoriginalimage,thentheresultingvalueinthe\dierence"blockswillalsobelarge.Thepixelintensitiesforatypical8-bitgrayscaleimagerangefrom0to255where0isblackand255iswhite.Thereforetheedgesofanimageareeasilydetectedbylookingforthelargestvalues(i.e.,thewhitestpixels)intheresultingthree\dierence"blocks. Also,notethattheresultingtransformedmatrixhassignicantlylessdetailthantheoriginalimage.Infact,manyofthepixelvalueshavebeenconvertedto0(black)orcloseto0.Thisislikelytooccurwithmostnaturalimagesbecausetheyusuallyhavelargehomogeneousregions,andthereforethedierencesinthoseareaswillbeminimal.Performingwhatiscalledlossycompression,wecanconvertto0allofthosepixelintensitieswhicharecloseto0.Thisway,wesignicantlyreducetheamountofstoragespaceneededinordertostoretheimageorsenditasaleovertheinternet.Wecallthislossycompressionbecausewhenwerestoretheimage(byperforminganinversetransform),someoftheoriginalinformationwillbelostduetotheslightconversionofthosepixelintensities.11

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ThesignicantbenetofdealingwithorthogonalDWTsisthatwecaneasilyinvertthetransformprocesssincetheinverseofWNissimplyitstranspose.Inaddition,insteadofcomputingtheentirematrixmultiplication,wecantakeadvantageofthesparsenessofWNandwriteanalgorithmwhichecientlycomputestheDWT.HavingorthogonalityinourDWTsmakesiteasytowriteanearlyidenticalalgorithmfortheinversetransformprocess(see[20]foranexample). Inthenextchapter,weshowhowtoconstructsomespecicnitelengthlterswhichcometogethertoformanorthogonalDWT. 12

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2;p 2)andg=(p 2;p 2)whichwediscussedinChapter1areactuallythosecoecients.ThesetwoltersformthebuildingblocksfortheHaarwavelettransform.Recallthathisalowpasslterandgisahighpasslter.Therefore,wecanplacetheselters(andtheir2-translates)intheirrespectivepositionsofthelowpassandhighpassportionsofaDWT.The(lengthN)HaarwavelettransformWNiswrittenasfollows:0BBBBBBBBBBBBBBBBB@p 2p 2000000N 2p 20000rows0000......00000000p 2p 2p 2p 2000000N 2p 20000rows0000......00000000p 2p 21CCCCCCCCCCCCCCCCCA13

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NextwecheckfororthogonalityoftheHaarwavelettransform.Remember,WNisorthogonalifandonlyifWNWTN=IN.Inotherwords,thedotproductofrowiwithrowjmustbe1ifi=jand0ifi6=j.Itisenoughtoconsiderthedotproductofonlyrow1withallotherrowsofWN.Giventhestructureofh,g,andWNitiseasytoseethatallotherdotproductswillgivethesamesetofresults. 2p 2+p 2p 2=1 2+1 2=1. 2p 2+p 2(p 2)=0. Therefore,wehaveshownthattheHaarwavelettransformisorthogonal.2.2DaubechiesFilters Supposewewanttoconstructltershandgofarbitraryevenlengthsuchthathislowpass,gishighpass,andtheresultingDWTisorthogonal.Inthissectionweassume14

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2(correspondingly,LPk=0g2k=1andLPk=2mgkgk2m=0form=1;2;:::;L1 2),thenthesetoflowpass(highpass)rowsformanorthonormalset.Remark2.2.1 2.Note2.2.3 jH(0)j=p jH()j=0(lowpass).15

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jH(w)j2+jH(w+)j2=2(orthonormality). IfwesolvethissystemforalterwithL=1(i.e.,h=(h0;h1)),thenoneofoursolutionswillbetheHaarlowpasslter.IfwesolvethesystemaboveforanygreateroddvalueofL,thenweobtainaninnitenumberofsolutions[8].Therefore,weneedtoaddmoreconditionstooursystemofequations.Theequationswewilladdcanbeconsidered\smoothness"conditions.TheycomefromtakingderivativesoftheFourierseriesatw=andhenceatten(orsmooth)thegraphofjH(w)jatthatpoint.IfL=3,weonlyneedtoaddoneofthesederivativeconditionstothesystem,thatis,werequireH0()=0.ForL=5,weneedtwoderivativeconditions,namelyH0()=0andH00()=0.Ingeneral,forlterh=(h0;h1;:::;hL)oflengthL+1,weneedthederivativeconditionsH(m)()=0form=1;2;:::;L1 2,[8].Infact,Daubechiesshowsthatthisisthemaximalnumberofderivativesthatcanbetaken.Remark2.2.4WewillseeinChapter3howthesederivativeconditionsatplayanimportantrolenotonlyinthesmoothnessofjH(w)jatw=,butalsoatw=0. 2derivativeconditionsdescribedabove.Therefore,thesystemofequationsweneedtosolveinordertogetanitenumberofsolutionsforpossibleDaubechiesltersisthefollowing: jH(0)j=p jH()j=0(lowpass). jH(w)j2+jH(w+)j2=2(orthonormality). 2(derivativeconditions). Onceweobtainanitesetofsolutionstothissystem,wehavetonarrowitdowntooneparticularsolutionthatwewillcalltheDaubechieslter.Daubechies[8]explainsherrationaleforpickingtheDaubechieslterfromthisnitesetofsolutions.Fortheinterestedreader,wehaveincludedaMathematica[22]programwhichoutlinestheprocessfornding16

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SofarinthissectionwehavedescribedhowtoconstructtheDaubechieslowpasslter.Onceweconstructthislter,theonlyremainingproblemistoconstructthecorrespondinghighpasslterg=(g0;g1;:::;gL)thatcompletestheorthogonalDWT.Itiseasytocheckthatifweletgk=(1)khLk,thengishighpass(i.e.,satisesG(0)=0;jG()j=p 2.TheseconditionsgiveorthonormalitybetweenrowsinthehighpassandthelowpassportionsofaDWT.Thefollowingpropositiongivesanequivalentconditionfororthonormality(betweenlowpassandhighpassportions)intermsoftheFourierseriesH(w)andG(w)(see[20],p.292):Proposition2.2.5 2.Note2.2.6

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Proof.ByProposition2.2.2,weknowthatjH(w)j2+jH(w+)j2=2impliesthatthelowpassrows(ofanyDWTconstructedusingh)formanorthonormalset.SupposethatlandLhavethesameparity.IflandLarebothodd,thenhhasLl+1(i.e.,anoddnumberof)entries.ThesameistrueiflandLarebotheven. FortheorthonormalityofthelowpassrowstoholdforallN(i.e.,foralltransformmatricesWN),everyrowtranslatedbytwoentriesanddottedwithrow1mustgivezero.However,sincethereareanoddnumberofentriesinh,whenwetranslateforthe(Ll WewillseelaterthatCoietlterscanbedenedbytheFourierseriesH(w)=p HerewepresentaformulafortheresultingstartingandstoppingindicesoftheltercorrespondingtosuchaFourierseries(see[9]or[20],p.307):Proposition2.3.2 2)19

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See[3],Section6.9,foradiscussionofsomeofthebenetsandapplicationsofusingalterthathasabalanceofderivativeconditionsbetweenw=0andw=.Remark2.3.3 So,inreview,thecharacteristicswhichdeneaCoietlterareasfollows:1. Aswehaveseen,theFourierseriescorrespondingtotheCoietlterisalongandcompli-catedone.WecansimplifytheprocessofsolvingforCoietltersbyusingthestarting20

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WeareinterestedinthesmoothnessofjH(w)jbecauseitgivesusagoodideaofhowcloseourlterisincomparisontotheideallter(seeChapter1).Thatis,thesmootherthegraphofjH(w)jnearw=andw=0,thecloseritsrepresentativelteristobeingideal.Thisisimportantforimageprocessingapplicationssuchasedgedetection.Forinstance,ahighpasslterwhichisclosetoidealwilldoagoodjobofpreservingoscillatory(i.e.,dierent)andannihilatinghomogeneous(i.e.,similar)portionsofthedata.Therefore,theedges(largestdierences)willbeclearlyvisibleinthetransformedimage. In[9],Daubechiesconsidersthesmoothnessresultsintroducedbelow.Wenotethattheresultsaswrittenhereareourownproofs.Note3.0.5

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BytheLeibnizproductrule,(d2m+1 Fork=0;:::;m,ourassumptionconditionsH(j)()=0forj=0;1;:::;mimplythatthecorrespondingtermsinthesummation(3.1.1)arezero.Wenowonlyneedtoconsiderthetermsofthesummationwhichcorrespondtok=m+1;:::;2m+1,namely,2m+1m+1H(m+1)(w) Therefore,(d2m+1 RecallProposition2.2.2whichstatesthatifwehaveareal-valuedlterh=(hl;hl+1;:::;hL1;hL)whoseFourierseriessatisesjH(w)j2+jH(w+)j2=2,thentheorthonormalityconditionsLXk=0h2k=1andLXk=2mhkhk2m=0form=1;2;:::;Ll1 2aresatised.Wewillneedtoassumethatorthonormalityforthenextfewresults.Lemma3.1.2

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Noticethat(d2m+1 WeknowfromLemma3.1.1thatthesecondtermin(3.1.2)mustbe0.Therefore,wecanconcludethat(d2m+1 Proof.(Inductiononm) dwjH(w)j)jw=0=0. WeknowfromLemma3.1.2that(d dwjH(w)j2)jw=0=0. Wecanwritethisas(2jH(w)jd dwjH(w)j)jw=0=0andsincejH(0)j=p dwjH(w)j)jw=0=0.

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Wewillonlyshow(d2m+1 Considerd2m+1 dwjH(w)j2=d2m dwjH(w)j=2d2m dwjH(w)j: dwjH(w)j(2mk)=22mXk=02mk(jH(w)j)(k)(jH(w)j)(2mk+1): Recall,byLemma3.1.2,weknowthatd2m+1 25

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dwjH(w)j)=0.(Itistrivialthatlimw!jH(w)j=0.) Considerd dwjH(w)j2.Wecanwritethistwodierentways,asfollows: dwjH(w)j2=2jH(w)jd dwjH(w)j. dwjH(w)j2=H(w) Therefore,limw!(2jH(w)jd dwjH(w)j)=limw!(H(w) Becausewearetakingalimitasw!,wecandividethroughby2jH(w)jtogetlimw!d dwjH(w)j=limw!H(w) 2jH(w)j 2jH(w)jH0(w): 2jH(w)jand 2jH(w)jareequalto1 2. Thereforewehavelimw!(d dwjH(w)j)=limw!(1 2 2H0(w))=0+0=0. Considerdm dwjH(w)j)=2dm1 dwjH(w)j). UsingtheLeibnizproductrule,wecanrewritetheseasfollows: dwjH(w)j)(m1k)=2m1Pk=0m1k(jH(w)j)(k)(jH(w)j)(mk).

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2jH(w)j( 2jH(w)j!: 4p 4p 4p 4p TheDaubechiesltersatisesalltheconditionsinLemma3.1.1,Lemma3.1.2,Theorem3.2.1,andTheorem3.2.3.Therefore,inthegures,weseethatthe\smoothness"ofjH(w)j2isthesameatw=0asitisatw=.Evenmoreinteresting,thegraphofjH(w)jisabouttwiceas\smooth"atw=0asitisatw=,eventhoughthederivativeconditionsaretakenatw=. Nowlet'sconsiderwhathappensifwehavethederivativeconditionsH(j)(0)=0forj=1;2;:::;m,alongwithorthonormality,asisthecaseforhalfthederivativeconditionsimposedonCoietlters.WhatimplicationsdotheseconditionshaveonthesmoothnessofjH(w)jandjH(w)j2?Itiseasytoshow(similartotheargumentsinLemma3.1.2andTheorem3.2.1)thatwiththeseconditions,weonlyget(dj

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Figure3.2:jH(w)j2on[0;]forDaubechieslength4lter29

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WhencomparingDaubechiesltersandCoietltersofthesamelength,wewouldexpectthegraphofthemodulusfortheDaubechiesltertobesmootheratbothw=0andw=thanthegraphofthemodulusoftheCoietlterwouldbenearthosepoints.Figure3.4andFigure3.5comparethemodulusoftheFourierseriescorrespondingtotheDaubechiesandCoietltersoflength6andlength12,respectively.WewillcomebacktothiscomparisonattheendofChapter4whenwesuggestanewlterconstructionthatblendsthecharacteristicsofboththeDaubechiesandCoietlterconstructions.

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Wenowhaveathoroughunderstandingofthelterconstructionsforsomeofthemostwell-knownlters,namelytheDaubechiesandCoietlters.Hence,wethoughtitwouldbeinterestingtoconsiderwhywrapping,whichoccursinthemajorityofDWTs,doesnotseemtoaecttheconstructionconditionsoftheselters.Morespecically,sincetheselterswereconstructedinawaywhichassuredorthogonalityoftheircorrespondingDWTs,wequestionifandhowwrappingaectstheorthogonalityconditionsastheywereoriginallystatedinChapter2. SupposethatweconsiderthelowpassportionofNNtransformmatriceswhichareformedusingltersoflengthL+1.Inthissection,weassumethatNandL+1arebotheven.Forsimplicity,wewilluseltersoftheformh=(h0;h1;:::;hL)(sameasDaubechieslters),butwenotethateverythingthatfollowsinthischaptercaneasilybeextendedtoevenlengthlterswithanyindices(aswithCoietlterswhoseindicesarebothnegativeandpositive).Figure4.1givesanexampleofthelowpassportionofaDWToflengthN=14withlterlengthL+1=9+1=10.Takenotethatinrow1thereare4zerosfollowingtheltercoecientsofh.Ingeneral(i.e.,forlterlengthL+1andmatrixlengthN),wesaythattherearec1zerosplacedintheremainingentriesofrow1sothatN=L+c=(L+1)+(c1).Denition4.0.5 2orfewertimes(i.e.,c1orfewerspacestotheright)fromtheiroriginalpositioninrow1,wherec=NL,andL+1isthelengthofthelter.32

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2ormoretimes(i.e.,c+1ormorespacestotheright)fromtheiroriginalpositioninrow1,wherec=NL,andL+1isthelengthofthelter. Inotherwords,awrappingrowhasatleasttwononzeroentrieswrappedaroundtotherstcolumnsofthetransformmatrix.Note4.0.7 Itisclearfromthedenitionofnon-wrappingrowsthattherearec1 2non-wrappingrowsinthelowpassportionofanytransformmatrix.InthefollowingLemma,weshowthatthenumberofwrappingrowsinthelowpassportionofaDWTdependssolelyonthelengthofthelter.Lemma4.0.8 2. Proof.ThelowpassportionofaDWThasN 2)andanother1(sincewedonotincludeR1).Therefore,there33

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21=L+c 22 2=L1 2wrappingrowsinthelowpassportionofthetransformmatrix. WerecallthezeroorthogonalityconditionswhichcharacterizethesolutionofDaubechiesltersoflengthL+1:LXk=2mhkhk2m=0form=1;2;:::;L1 2:(4.0.1) Wecanthinkoftheseconditionsasdotproductsoftheoriginallterwithallpossible2-translatesofthelter.Note4.0.9 2non-wrappingrows(ofatransformmatrix)withR1arethesameastherstc1 2standardzeroorthogonalityconditions.Inotherwords,wecangeneralizethedotproductofRNWnwithR1asfollows:RNWnR1=LXk=2nhkhk2nforn=1;2;:::;c1 2:(4.1.2)Example4.1.1 2=4 2=2non-wrappingrowsinthis34

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NextwewillgeneralizeaformulaforthedotproductofthewrappingrowswithR1.4.2WrappingRows 2withR1,wemustconsiderboththewrappingportionandthenon-wrappingportionofeachdotproduct.Denition4.2.1 2times(eachtimeshiftingtwospaces)sothateachlterelementhasbeenshiftedtotherightc+1spacesfromitsoriginalposition(asinR1).Therefore,thenon-wrappingportionofthedotproductR1RW1isreallyjustthe(c+1 2)thstandardzeroorthogonalitycondition.Itisclearthatthisextendsforthenon-wrappingportionofeachofthedotproductsR1RW2;R1RW3;:::;R1RWL1 2.Thatis,thenon-wrappingportionofR1RWnforn=1;2;:::;L1 2isgivenbyLXk=2mhkhk2mform=n+c1 2(4.2.3) (i.e.,form=c+1 2;c+3 2;:::;c+L2 2):35

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2;:::;c+L2 2,therewillbenoresultingorthogonalityconditions.Inotherwords,thenon-wrappingportionsofR1RWLc+2 2;:::;R1RWL1 2areallzero. 2.Todothis,wesubstitutem=n+c1 2togetLXk=2mhkhk2m=LXk=2(n+c1 2)hkhk2(n+c1 2)forn=1;2;:::;L1 2:(4.2.4)4.2.2WrappingPortion 2.Next,weconsiderthewrappingportionofthedotproductofR1witheachofthewrappingrowsRW1;RW2;:::;RWL1 2.WeobservethatRW1haspreciselytwolterelements(h1andh0)whicharewrappedaroundtotherstcolumnsoftheDWT,RW2hasfourlterelements(h3;h2;h1,andh0)whicharewrappedaround,andsooninincrementsoftwountilRWL1 2hasL1lterelements(hL2;:::;h0)wrappedaround.Therefore,theorthogonalityconditionthatweobtainfromthewrappingportionofR1RW1ish1hL+h0hL1,fromthewrappingportionofR1RW2ish3hL+h2hL1+h1hL2+h0hL3,andsoonuntilthewrappingportionofR1RWL1 2whichishL2hL+hL3hL1++h1h3+h0h2.36

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2)tothosethatwegetbyshiftingofrowsinthestandardzeroorthogonalityconditions,exceptinreverseorder.Thatis,thewrappingportionofR1RWnisgivenbythefollowing:LXk=2(L+1 2n)hkhk2(L+1 2n)forn=1;2;:::;L1 2:(4.2.5) Now,bycombiningtheresultsin(4.2.4)and(4.2.5),thecompletegeneralizationofR1RWncanbewrittenasfollows:LXk=2(L+1 2n)hkhk2(L+1 2n)+LXk=2(n+c1 2)hkhk2(n+c1 2)forn=1;2;:::;L1 2:(4.2.6) InordertohaveorthogonalityamongalltherowsofthelowpassportionofaDWT,alloftheexpressionsin(4.2.6)mustbesettozero.(Remember,wealsoneedtheconditionR1R1=1tohaveanorthonormalset,butwewillconcernourselvesonlywiththezeroorthogonalityconditionsatthispoint.)Example4.2.4 2=8 2=4wrappingrowsinthismatrix(seeLemma4.0.8),namelyrow4,row5,row6,androw7.ThedotproductsofthesefourrowswithR1areh9h1+h8h0+h3h9+h2h8+h1h7+h0h6;h9h3+h8h2+h7h1+h6h0+h1h9+h0h8;h9h5+h8h4+h7h3+h6h2+h5h1+h4h0;andh9h7+h8h6+h7h5+h6h4+h5h3+h4h2+h3h1+h2h0;respectively.Thesearethesameconditionsasin(4.2.6). Thecompletegeneralizationformulagivenin(4.2.6)showsthatthewrappingportionandthenon-wrappingportionareeachequivalenttooneofthestandardzeroorthogonality37

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2ofthestandardorthogonalityconditionsinordertoobtainorthogonalityinthelowpassportionofaDWT?Thatis,weknowthestandardzeroorthogonalityconditionsaresucientforanorthogonalDWTcontainingwrapping,butaretheynecessary?4.3EectsofWrappingonZeroOrthogonalityConditions 2(m1)holdsform=1;2;:::;c1 2. Proof.RecallfromNote4.2.3thatthenon-wrappingportionsofR1RWLc+2 2;:::;R1RWL1 2areallzero.SoforthelastL1 2Lc+2 2+1=c1 2wrappingrows,weobtainnoorthogonalityconditionsstemmingfromthenon-wrappingportionofthedotproduct.Therefore,weonlyneedtocomparetheorthogonalityconditionsfromthec1 2non-wrappingrowstotheorthogonalityconditionscomingfromthewrappingportionofthelastc1 2wrappingrows.38

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2non-wrappingrowsaregivenbyLXk=2mhkhk2mform=1;2;:::;c1 2:(4.3.7) Theconditionsfromthelastc1 2wrappingrowsaregivenbyLXk=2(L+1 2n)hkhk2(L+1 2n)forn=L1 2(m1)wherem=1;2;:::;c1 2:(4.3.8) Wesubstitutethevalueforninto(4.3.8)togetLXk=2(L+1 2n)hkhk2(L+1 2n)=LXk=2(L+1 2(L1 2(m1)))hkhk2(L+1 2(L1 2(m1)))=LXk=2mhkhk2mform=1;2;:::;c1 2: 2c1 2=Lc Proof.1. 2m)hkhk2(L+1 2m)+LXk=2(m+c1 2)hkhk2(m+c1 2)=LXk=L+12mhkhk(L+12m)+LXk=2m+c1hkhk(2m+c1):39

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2(Lc 2(Lc 2)hkhk2((Lc 2)=LXk=2m+c1hkhk(2m+c1)+LXk=L+12mhkhk(L+12m): 4ezeroorthogonalityconditionswhicharenecessaryinordertosatisfyorthogonalityofthelowpassportionofWN. Proof.Outofthec1rowsconsideredinLemma4.3.1(i.e.,c1 2non-wrappingrowsandc1 2wrappingrows),onlyc1 2uniqueorthogonalityconditionsresultfromtheirdotproductswithR1. Now,outoftheLc Weknowthatfortherstcase,whenLc 2+dLc 2+Lc 4e=dN2 4enecessaryzeroorthogonalityconditions. 40

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2ofthestandardzeroorthogonalityconditionstosatisfyorthogonalityinthelowpassportionofthematrix.Corollary4.3.5 2zeroorthogonalityconditions).Note4.3.6 Proof.RecallthatthereareL1 2standardzeroorthogonalityconditionsandonlydN2 4enecessaryzeroorthogonalityconditions.Toobtainthedegreesoffreedominoursystem,wesubtractthenumberofnecessaryconditionsfromthenumberofstandardconditions,asfollows:L1 2dN2 4e=2L2 4dL+c2 4e:(4.3.9) NotethatL+c2isalwayseven.Therefore,whenwedivideby4wegeteitheranintegeroranintegerandahalf.Thus,(4:3:9)=8<:2L2 4(L+c2 4);ifL+c2 42Z2L2 4(L+c2 4+1 2);ifL+c2 462Z 42ZLc 2;ifL+c2 462Z: 2isodd,thenLc 2fromtheoddnumberL+c2 2togeta(necessarilyodd)resultofLc 4isanintegeroranintegerandahalf). Thus,(4:3:9)=bLc 41

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2.42

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LetC=c1 2andL=Ll+1 2thenLPk=l+2(Ln)hkhk2(Ln)+LPk=l+2(n+C)hkhk2(n+C)=0forn=c+1 2;:::;Ll1 2. ForthestandardCoietlter,thevaluesofandareboth2K1.Therefore,forourmodiedlter,werstincreasethevalueofto2K,then,ifpossible,weincreasethevalueofto2K.Afterthatwewouldcontinueonebyoneincreasingthevalueofandthenincreasingthevalueof.Fromourempiricalobservations,weformthefollowingconjecture:Conjecture4.4.2 InFigure4.5,weprovideagraphicalcomparisonbetweenthemodulusgraphsofthestandardDaubechieslter,thestandardCoietlter,andthemodiedlter(K=2,c=3);allofthesearelength12lters.InFigure4.6,wecomparethemodulusoftheDaubechies,Coiet,andmodied(K=3,c=5)ltersoflength18.Inthesegraphicalrepresentations,itisinterestingtoseethatthegraphofjH(w)jcorrespondingtothemodiedlterconstruction\liesbetween"thegraphsofjH(w)jfortheDaubechiesandCoietltersofthesamelength. 43

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Figure4.6:ModuluscomparisonofDaubechieslength18,Coietlength18,andmodiedlength18(K=3,c=5)lters46

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Inthisthesis,wehavediscussedonlyorthogonallters.Sincetheconstructionoftheseltersrequiresalargenumberoforthogonalityconditions,therearenotmanydegreesoffreedomremaininginoursystemtobettersuittheapplicationweareworkingtowards.Therearehowevermanyothertypesofltersthathavebeenconstructed,includingthosecalledbiorthogonal.BiorthogonalltersareconstructedsothattheircorrespondingDWTisinvertible,butnotorthogonal[6].Theselterscanbeextremelyusefulinimageprocess-ingapplications.Actually,oneofthemostpowerfulandfrequentlyusedimagecompressiontoolsiscalledJPEG2000,anditusesbiorthogonalltersinitswavelettransforms.Forfurtherreadingontheconstructionofbiorthogonallters,see[6],andforfurtherreadingonJPEG2000anditsapplications,see[5].47

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Daub[L_]:=Module[{H,w,k,orthogsquare,orthog,m,i,remainingorthog,lowpasszero,lowpasspi,derivspi,derivpi,Solutions,R,n,P,z,zroot,mytable,mytable2},(*Definesamoduleandallofitslocalvariables*)H[w_]:=Sum[Subscript[h,k]*E^(I*k*w),{k,0,L}];(*TheFourierseriesforaDaubechiesfilteroflengthL+1*)orthogsquare=Sum[Subscript[h,k]^2,{k,0,L}]==1;(*ThefirstorthonormalityconditionfromProposition2.2.1*)orthog[m_]=Sum[Subscript[h,k]Subscript[h,k-2m],{k,2m,L}]==0/.{m->i};remainingorthog=Table[orthog[m],{i,1,(L-1)/2}];(*Thenext(L-1)/2orthonormalityconditionsfromProposition2.2.1*)lowpasszero=H[0]==Sqrt[2];(*ThelowpassconditionH(0)=Sqrt[2]*)lowpasspi=H[Pi]==0;48

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Endofcode

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G.Beylkin,R.Coifman,andV.Rokhlin,Fastwavelettransformsandnumericalalgorithms,CommunicationsonPureandAppliedMathematics44(1991),141-183.[2] A.BoggessandF.J.Narcowich,AFirstCourseinWaveletswithFourierAnalysis,PrenticeHall,UpperSaddleRiver,N.J.,2001.[3] C.S.BurrusandR.A.Gopinath,IntroductiontoWaveletsandWaveletTransforms:APrimer,PrenticeHall,UpperSaddleRiver,N.J.,1998.[4] C.S.BurrusandJ.E.Odegard,Coietsystemsandzeromoments,IEEETransactionsonSignalProcessing46(1998),no.3,761-766.[5] C.Christopoulos,A.Skodras,andT.Ebrahimi,TheJPEG2000stillimagecodingsystem:Anoverview,IEEETransactionsonConsumerElectronics46(2000),no.4,1103-1127.[6] A.Cohen,I.Daubechies,andJ.-C.Feauveau,Biorthogonalbasesofcompactlysup-portedwavelets,CommunicationsonPureandAppliedMathematics45(1992),no.5,485-560.[7] J.M.Combes,A.Grossman,andP.Tchamitchian,Eds.,Wavelets,Time-FrequencyMethodsandPhaseSpace,Springer-Verlag,Berlin,1989.[8] I.Daubechies,Orthonormalbasesofcompactlysupportedwavelets,CommunicationsonPureandAppliedMathematics41(1988),909-996.[9] I.Daubechies,Orthonormalbasesofcompactlysupportedwavelets,SIAMJournalonMathematicalAnalysis24(1993),no.2,499-519.[10] M.W.Frazier,AnIntroductiontoWaveletsThroughLinearAlgebra,Springer-Verlag,NewYork,N.Y.,1999.51

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D.Gabor,Theoryofcommunication,JournaloftheInstitutionofElectricalEngineers,PartIII93(1946),no.26,429-457.[12] A.GrossmanandJ.Morlet,DecompositionofHardyfunctionsintosquareintegrablewaveletsofconstantshape,SIAMJournalonMathematicalAnalysis15(1984),723-736.[13] A.Haar,Zurtheoriederorthogonalenfunktionensysteme,MathematischeAnnalen69(1910),331-371.[14] P.C.J.Hill,DennisGabor-Contributionstocommunicationtheory&signalprocess-ing,EUROCON,2007.TheInternationalConferenceon\ComputerasaTool"(2007),2632-2637.[15] Y.Meyer,Ed.,WaveletsandApplications,Springer-Verlag,Berlin,1992.[16] J.Morlet,Samplingtheoryandwavepropagation,IssuesonAcousticSignal/ImageProcessingandRecognition,C.H.Chen,Ed.,NATOASI,Springer-Verlag,NewYork,N.Y.,1983.[17] O.RioulandM.Vetterli,Waveletandsignalprocessing,IEEESignalProcessingMagazine8(1991),no.4,14-38.[18] M.B.Ruskai,G.Beylkin,R.Coifman,I.Daubechies,S.Mallat,Y.Meyer,andL.Raphael,Eds.,WaveletsandtheirApplications,JonesandBartlett,Boston,M.A.,1992.[19] J.TianandR.O.Wells,Vanishingmomentsandwaveletapproximation,TechnicalReportComputationalMathematicsLab,RiceUniversity,January1995.[20] P.J.VanFleet,DiscreteWaveletTransformations:AnElementaryApproachwithApplications,JohnWiley&SonsInc.,Hoboken,N.J.,2008.[21] D.F.Walnut,AnIntroductiontoWaveletAnalysis,Birkhauser,Cambridge,M.A.,2002.[22] WolframResearch,Inc.,MathematicaEdition:Version6.0,Champaign,I.L.,2007.52