`
`
`
`_
`Box 6:
`Muliiresoluiion analysis
`The. concept ofmuittresoiution approzctmauon of
`
`_i‘un::tl-ms was introduced by Meyer and Msiist
`
`lMi=.l.89a. mtsac. ii'iE\__'90i end provides a power-
`
`ful framework to uriidleratend wavelet decomposi-
`
`tions. The basic idea is..§_i.he.t of successive
`
`approudmation. together with_:_ti1st o£'addecl detail‘
`
`as one goes from one apprwittnstitin to the 11631.
`
`. finer one. We here give thejntuitioin behind the
`ill‘
`
` _ A aiadderéofepaoejssuch that:
`in
`5
`.
`V1 cllfilfo t: V11 C
`c ...
`
`then
`‘perty thlit
`Ii‘ 41:): V;
`Vi.g.caJ1'£1v.r¢-use
`
`
`
`Pig. M. Tiz.nscaJ.a.1qfthelD4 tunnel2td:Itisl'iy‘l5.1"lLLI.se!tJ
`fururtniorts is orthogonal.
`
`APPLICATIONS OF WAVELETS IN
`SIGNAL PROCESSING
`
`From the derivation of the wavelet transform as an
`
`alternative to the S'I‘F'l‘. it is clear that one oi’ the main
`
`applications will be in non-stauonsry signal analysis.
`While conceptually. the CWT‘ is a classical I:onai.a.nt—Q
`
`analysis. its simple definition {based on a single rune,-
`
`tion rather than multiple flltersi allows powerful
`
`analytical derivations and has already lead both to new
`
`insights and new theoretical results [WAV89|.
`
`Applications ofwavelet decompositions in numerical
`
`analysis. e.g. for solving ps.t1.ia.| differential equations.
`
`seem very promising because of the 'zoornln,g" pmperiy
`
`which allows a very good representation of discon-
`
`tinuities. unlike the Fourier tranafurrn IEEYBBS.
`Perhaps the biggest potential of wavelets has been
`
`-v
`.
`'
`
`
`claimed for signal compression. Since discrete vvavelet
`i.ransi'orms are essentially subband coding systems.
`ie .c
`the suc-
`
`and since subband coders have been successful
`in
`
`apfilzogdmauon or -« ;i.
`- numlhfwavelets
`
`speech and Image compression. it is clear that wavelets
`scale.
`'
`
`
`will find immediate application In compression
`problems. The only difference with traditional subbarld
`generate wavelet bases. The converse is also true. That
`coders is the fact that filters are designed to be regular
`is. orthonormal sets of scaling functions and wavelets
`[that is. lhey have many zeroes at z - 0 or z = xi. Note
`can be used to generate perfect reconstruction iilier
`that although classical subhand filters are nqt regular
`banks iDAUB8. MAi.89a. MA.l.8Bc|.
`[see Box 5 anti Fig. 12]. they have been designed to have
`Extension oi’ the wavelet concept to multiple dimen-
`good stopbanda and thus are close to being "regular".
`at least for the first Few octaves ofsuhband decomposiv
`sions. which is useful. e.g. for image coding. is sl'iown
`Lion.
`In Box 7.
`
`
`
`
`
`
`
`
`Page 164 of 437
`
`Page 164 of 437
`
`

`

`
`
`Hg. 15. Orrhonomaai wavelet genemtedfierna length-18 regularjl|ler.iDAU88J.T71e Ilrnejuneiton is shown an an Lefiami tho
`speclntrnlsonlherlght
`
`It is therefore deer that dranie improvements of
`compression will not be achieved an easily simply be-
`eause wavelets are used. However. wavelets bring new _
`ideas and insights. In this respect. the use of wavelet
`deeotnposliions in eannectlan with other techniques
`(like vector quantization [ANI9D| or multiacale edges
`[MA.L89dl] are promising compression techniques
`which make use of the elegant theory ofwavclete.
`New develeprnenm. based on wavelet concepts. have
`
`already appeared. For example. statistical signal
`processing using wavelets is a promising field. Multi-
`scaie models of stochastic processes [M589]. [CH091|.
`and analysts and synthesis of U1‘ noise [GAC91l.
`[WOR9D| are examples where wavelet analysis hasbeen
`successful. ‘wavelet packets’ [WICB9]. which cer-
`feapnmi
`to arbitrary adaptive ute—stn.u:l:umd' filter
`banks. are another promising example.
`
`
`
`I6‘. Blarihoganal urauzfeig gnerlerutedfrurri i8-lap reg-uiapfliters JVSMODL (flJ Analysts uuzixiei. ['11) Synthesis Luquelei. The
`Fig.
`rimefllncilnn is shown on time left and the spectrum is on the right.
`
`DHDEEI IW
`
`IEEE Sf Halfillllli
`
`Page 165 of437
`
`Page 165 of 437
`
`

`

`---Box Z: Mulllltilmenslonul filler banks and wavelets
`In order to apply wavelet decompositions to multi-
`pllngby 21:: each dimension. that is. overall subsum-
`din1ens_1qns.l signals [e.g.. Images]. multidimensional
`pllng by 4 (see Fig. 11"].
`More interesting [that is. non-trivial] mult[d|me'n-
`erctenstons of wayelets are required. An obvious way
`sional wavelet schemes are obtained when non-
`to do ti-nuts touse ‘separable wavelets‘ obtained from
`preductsxui one-cllmenslonsl wavelets and scaling
`separable subsampling is used [KOV92]. For
`example. a non-separable subsampltng by 2 of a
`fi.1ncI:lonsIM.r\1-399.. M.ALB9vc, MEYSDE. Let us consider
`double Indexed signal Jt(l'I.1.- ml is obtained by retelli-
`the ‘»'\lN'J~d_.1Infilsinna.l. da.s-e_I'o1"lta simplicity. Take a
`Lng only samples satisfying:
`- .scallng_fI_._rnci:ion ggbd ['15) and awavelet hub! [16]. one
`_ can
`two-d1m'ens1nn.sl firncuons :
`l$l=ll-‘mlli‘él-
`The resulting points are located on a so-called
`qulncunx sublsttlce of 22' NM». one can construct a
`perfect reconstruction filter barn]: Involving such sub-
`sampllng because Ii. resembles its one-dixnerralonal
`counterpart [KCN92]. The suhsnmpllng rate is 2
`[equal to the detennlnamoftlaemsmx in IE7. lll. and
`the filter bank has 2 channels. Iteratlnngf the filter
`bank on the lovwpass branch (see Fig. 18) leads to a"
`discrete wavelet transform. and if the inter is regular
`[which now depends on the matrbt representing the
`lattice IKOV92]]. one can construct-_norr'-separable
`wavelet bases for square integrable functxnns overjla
`with aresolution change by 2 [and not-‘lane in the
`separable cssel. An exarnple scaJJng'l_fu.ne'tlon 1: ple-
`zureclInn_g_ 19.
`.-
`
`(57.1)
`
`l1c{-1}?!-slit)
`
`orthogmlal to each otherwith respect to
`. (this fOIIOV@_- from the orthogonaflty of
`-
`- oomponentJ.The flmctlongglaeyl
`
`filter] u‘-huethe runeuuns hL°cx.yi are
`e set lrtltzw-. 2*;-a. l=1.2,3 andJ.k.l
`orthonm.-rnal basis for square intcgmble
`II‘ This solution corresponds to a
`Inna] filter bank with au.baa.m-
`
`I
`_
`
`such
`f§ _mwhmmnsmmmmaummmwmdMbuuw%mmmmcmmw4M
`.1!npw1!ibr:qf'Ure_frequ¢u'typlanelslnd.ic\ateclo41therghtrnandHnstand_farlourpassaI1dhu;h-
`9.
`
`Hg. 18. ltsruflnn qfn rmn_-aeparuflcfillarbunk based on
`nan-separable szrbsampung. This mnsmxction lead: to non-
`scpnnahlz |.r.u:|.I.velel.s.
`
`Fig‘. 19. 1109-cttmensmnal non-separable o'rtJwrmrma.I smi-_
`Lngjlmctian lKOV92l forttragonaluy is with respvet to integer
`If! each dII’l'lEl'l-
`s.i"I.y‘l5,l. The resolution change [5 tag 2 NE
`sioru. The matrix rrsedfar the subsamptmg is the onegtuen
`Ln {B11}.
`
`Itrx
`
`Page 166 of 437
`
`Page 166 of 437
`
`

`

`" -_CONCLUSION
`
`‘
`
`We have seen that the Short-}[‘irnc F'ourier‘[‘runsl'orm
`and the Waveiet'l‘1-ansforrn represent alternative ways
`to divide the time-frequency [or Li1n¢:—scalel plane. Two
`major advantages of Lhe Wavelet Transform are that it
`can zoom in to time discontinuities and that orthonor-
`rnal bases.
`localized in lime and frequency. can be
`constructed. In the discrete case. the Wavelet Trans-
`form is equivalent to a iogarilhrriic filter bank. with the
`added constraint of regularity on the lowpass filter.
`The theory of wavelets can be seen as a common
`framework [or techniques that had been developed
`independently in various fields. This conceptual
`unification furthers the understanding of the
`mechanisms Involved. quantifies trade-offs. and points
`Lo new potential applications. A number of questions
`remain open. however. and will require further invcs—
`tigations (e.g.. what is Lhc "optimal" wavelet for a par-
`ticular applicationfl
`While some see wavelets as a very promising brand
`‘new theory ICIPBGI. others express some doubt that it
`represents a major breakthrough. One reason for skep-
`ticism is that the concepts have been around for some
`time. under different names. For example. wavelet
`transforms can be seen as constant-Q analysis
`[\i'0U73l. Wide-band cross-ambiguity functions [SP'E6?.
`AUS90l. F‘razier«Jeweri.i1 transforms IFHA861. perfect
`reconstruction octave-band filter banks [MlN35.
`SMISBI. or a variation of Laplscian pyramid decomposi-
`tion lBURa3]. IBURSSII
`We thlrrk that the interest and merit ofwavelet theory
`is to unify all this into a oomrnon framework. thereby
`allowing new ideas and developments.
`
`ACKNOWLEDGMENTS
`
`The authors would like to thank C. Harley for many
`_ useful suggestions and for generating the continuous
`STFT and WV!‘ plots: and Profs. F. Boudreau.1(—Ba.rtels.
`M.J.T. Smith and Dr. P. Duhamel for useful suggestions
`on the manuscript. We t.ha.nlr B. Shakib HEM} for
`creating the three-dirnensional color rendering ofphasc
`and magnitude of wavelet transforms [so-called
`'phasemsgrams'l used in the cover picture and else-
`where: C.A. Plclrover [IBM] lathanked for his 3D display
`software and J.1.. _Mar1nior1 for his help on software
`tools. The second author would like to acknowledge
`support by NSF under grants ECO-88-1 ll l 1 and MIP-
`90- 14 189.
`
`
`
`Olivier lltoul was born in Strasbourg.
`‘_.: France on July 4. 1964. He received
`diplomas in Electrical Engineering
`from the [tonic Polytechniquo.
`Paiaiseau. France. and from Telecam
`University. Paris.
`in I93?’ and i989.
`respectively.
`Since 1989. he has been with the
`Centre National d'ELudes dcs Telecommunications
`[CN-E'l'l. issy—i..es-Moulineaux. France. where he is com-
`
`
`
`'.'-i-.-.-
`11"
`_
`picting work In ii1e'P'n.D. degree in Signal Processing at
`Télécolii Unlverslly. specializing in wavelet
`theory.
`iniaize coding. and fast signal algorithms.
`Martin Votterll was born in Switzer-
`land In I957. He received the Dip.
`El.-lng.
`degree
`from
`Lhe
`Eldgcnoesische
`Technische
`Hochschuie Ziirich. Switzerland.
`in
`1981:
`the Master of Science degree
`from Stanford University. Stanford CA.
`in 1982: and the Doctorat ea Science
`Degree from the -Ecoie Polytcchriique
`Federal: do Lausarinc. Switzerland. in 1936.
`in i932. he was a Research Assistant at Stanford
`University. and from 1983 to 1986 he was a researcher
`at the Ecole Polyiechniquc. He has worked for Siemens
`and AT-SIT Bell Lab-oratorlea. in 1986. he joined Colum-
`bis University in New York where he is currently as-
`sociate professor of Electrical Engineering. member of
`the Center for Telecommunications Research. and
`codirector of the image and Advanced Television
`laboratory.
`.
`He is a senior member ofthe'lEEE. a member ofSLAM
`and ACM. 21 member of the MD SP committee of the IEEE
`Signal Processing Society.-and ofthe editorial boards of
`Signal Processing and Image Communication. He
`received the Best Paper Award of EURASIP in 1984 for
`his paper on rnultidimeno tonal subbanricoding. and the
`Research Prize of the Brown B-overt Corporation [Swit-
`zerland] in 1986 for his thesis. His research interests
`include multirate signal processing. wavelets. computa-
`tional comp icxity. signal processing for teleeornrnunicav
`tions and digital video processing.
`'
`
`REFERENCES
`
`ttmaol mi. Alransu. RA. Haddsd. and H. Cagla-r.‘ 'l‘t1e
`Binomial QM?-Wavelet 'I‘ra.nsI'orm for Muluresoluiion Signal
`Deco1'nposiI.ion.' subrnltted IEEE 'i"l'un.-r. Signal Pmc..
`!99i'_'l.
`L'l].l.'.1"‘l'| J.B. Allen and L..R. Flabirier. ‘A Unified Approach In
`Short—‘i‘1rnc Fourier Analysis and Synthesis.‘ Prat. IEEEL Vol.
`65. No. 11. pp. 1553-1554. 1971.
`MNISDI M. Antoninl. M. Bariaud. P. Mathieu and [. Usu-
`bechles. ‘Image Coding Using Vector Quantization In the
`wavelet Transform Domain.‘ in Prcc. 1990 IEEE int. Corji
`Acoust. Speech. Sig-mi Pmc.. Albuquerque. NM.Apr.3-6. 1990.
`pp. 2297-2300.
`[nussoi i. ausiander and 1. oer-tner. ‘Wide-Barld Ambiguity
`Function and a.x+b group.‘ in Signal Processing. Part I: Signal
`Processing Theory. l. Auolarider. T. itailsth. 3. Mitter eds..
`Institute for Mathernaucs and Its A piicatinnn. Vol. 22.
`Springer Veriag. New York. pp.1-I2. l9
`.
`IBASBQ] M. Bssseville and A, Benveniste. ‘Muluscale Statisti-
`cal Signal Processing.‘ in Prue. I989 lEEE Int. Cont. Kl:0usL,
`Speech. Signal Pnoc.. Glasgow. Scotland. pp. 2055-2068. May
`2326. I939.
`IBERBBI J. Bertrand and P. Bertrand. 'I‘lrne-Frequency Rep-
`resentations of Broad-Band Signals.‘ in Proc. £958 IEEE‘ Int.
`Conji Acou.§i.. Speech. signal Free. New York. N‘(.Apr.l1-l4.
`I988. pp.2 196-2199.
`1
`IBEYBQL G. BL-y_Ik.in. I1, Colfrrmn and V, Rokhlln. 'Fasl'.Wr1velet
`'I‘rs1nsfurn1s and Numerical Algorithms.
`I.‘ spbmlited. Dec.
`I989.
`
`Ofillltl fill
`
`IEEESP lllfillllil
`
`35
`
`Page 167 of 437
`
`Page 167 of 437
`
`

`

`
`
`IBOUBSJ G. F. Boudreaux-Bartels. ‘Time-Varying Signal
`Processing Using the WEgner\DIslrIbutIon'|‘l1'ne- Frequency Sig-
`nal Rep1'eser1lalian."ln min. in Geophysical Data Proc.. Vol. 2.
`pp. 33-79. Jal Press Inc.. 1965.
`IBUR83] P.J. Burt and EH. Adelaon. "I11: 'l.ap|acla.n Pyramid
`use Compactlmagecode.‘|EE!ET'r:.1u.onCo1-n.,Vol,3l.No.4.
`pp. 532-540. April 1933.
`
`P...I. Bun. ‘tiuliiresaiuuon Tecrmlques for image
`|BUR89i
`Representation. Analysis. and 'Srr|a.ri."I1-ansrniuson.‘ Prue.
`SPII‘-J Cool’. on Visuai Communioouon and Image Processing..
`pp. 2-15. Fhiisdelphil. PA. NW. I989.
`'
`IC.-A154] A. Calderon. ‘intermediate Spaces and Interpoiation.
`‘l
`.
`tltgeefomplex Method.‘ Studio In-i.ai‘.h.. Voi. 24. pp. _l t3-190.
`|Ct-109i] KC. Chou. 5. Golden. and 11.3. wiilslqr. ‘Modeling
`and Estimation oi’ Multiaesie Stochasilo Processes.‘ in
`P1-oc.l99I !E.E.'El'ni.'Con,F. Aeoust. Speech. Slgr1olFi"nc..'I‘oronto.
`Ontario. Canada. 1113- 1709-1?l2.Mny iii-I7. i991.
`ICIFQOI B.A. Cipm. ‘A New Wave in Applied Mathematics.‘
`Science. Research News. Vol. 249. Augustin’.-I. 1990.
`ICLASGJ
`'!'.A.C.M. Classen and W.l-‘.0. Mecltlenbraurr. The
`Wigner Distribution — A Tool for 't’1me-Frequen
`Signal
`1
`0.
`Agglysla. Part I. ll. ill.‘ Philip: J. Res. VDL35. 379-
`l7-8-89.
`[l:0H66l
`L. Cohen. ‘Generalized Phase-space Distribution
`l'-‘unctlona."J. Marh..Phy.r..VoI. 1. No. 5. pp. 781-786. 1966.
`[CDH89|
`i.. Cohen. ‘limo-Frequency Dlsirlboiion - ARevieur.'
`Proc.1£.'E.'.‘r'oi. 77. No. 7. pp. 941-931. 198-3.
`[CO1-190:1 A. Cohen. I. Dwbochies and -1.0. Feauveau. ‘Blar-
`thogonal Buses arc
`Supporined Wavelets.‘ to appear
`InGomnLn.uea.ri.d.nppiiedA€ndL
`ICOHSOIJI A. Cohen. ‘fionstnicizlocn do Bases d‘011rle|eltes
`Holderiennesfi Reuista Manemnflon i'bei-ooinerimno. Vol.6. No.
`5 y 4. 1990.
`ICRIE-'6] A. Croloier. D. Esteban. Ind C. Galarid. ‘Perfect
`Channel Splitting by Use of lnterpolaiion. Decimaiion. ‘Noe
`Decomposition Techniques.‘ Int. Cont’. on Iniomisuon Scien-
`cesfsystmns. Patras. pp. 443-446. Aug. 1976.
`ICFAOTBI R.E. crochiere. SA. Weber. ondJ.L Flanagan. “Digi-
`tai Coding oispeech in subbsnd.s.' Bciisysr. ‘Tech. .J.. voI.55.
`pp. 1069-1085. OcL197a.
`ICROBSI RE. Cruchiere. and LR. Rabinef. Multirole Digital
`Signal Processing. Frantic:-Hull. Ehgieorood C|J.iI's. NJ. I983.
`iDALI58|
`I. Daubonhies. ' I Bun oi Cnrmpactly
`Supp-o1'lI.'dWaveIet:i.' Cornm. lnHiruuJ'IdAppliedMctiiL. Vol.4].
`No.7. pp.909-996. 1958.
`IDAUBOa|
`I. Daubechlel. "I'll: Wavelet 'h'ni1aI'an11. Tirn:-—
`Frequency _i|'..nca1la:atlon and Signal Am.iys!I.“ i'E.':‘.E ‘Mme. on
`Info. Theory. Wt. 38. No.5. pp.961-I005. Sept. 1990.
`IDAU90b|
`l. Dnubeehles. “orthonormal Bones of Coinplctiy
`Supported Wavelets ii. ‘Variations on I 1'I1eme." submitted to
`SIAM J. Math. Anal.
`
`.
`
`l. Daubechlec and J.C. Lagariao. "'l\Vo£caJe DIlt’er-
`IDAUQOC!
`' once Equations ii. Local Reglrlndty.
`infinite Products of
`Matrices and Fractals.‘ submitted to 3.M.M..r. Mam. Anal. 1990,
`|DUF52] R.-J. Duifin and A.C. Sc-hnelTer.
`‘A Class ol'Nor1-
`han-nonic Fourier Series.‘ Trans. Am. Math. Soc.. Vol. 72. pp.
`341366. 1952.
`
`[55777] D. Esteban and C. Galand. 'A]:ipllcIi’.lon ofguadralure
`Mirror Filter: to Split-Band Voice Coding Schemes.‘ Int. Cont.
`Acoush. Speech. Signal Proc.. Harilord. Cannecllcul. pp. 191-
`.195. May 1977.
`
`|l-‘IASQI P. Flandrin. ‘Some Aspects of l'ln11-Stationary Signnl
`Processing with Emphasis on Time-Froquency and Time-Scale
`MeIJ1od5.'in lWAV89|. 1111.68-98. I989.
`[F1390]
`P. Fiondrin and O. Rloui. ‘Wavelets and Milne
`Smoothing orthe Wignei‘~Ville Distribution.‘ In Pme. IQBGIEBE
`
`ml Func-
`
`int. Cory’. .-°.cOuSt.. Speech. Sig-mi Proc.. Albuquerque. NM.April
`3-6. 1990. pp. 2455-2458.
`WC)‘-J38] J. B. J. Fourier. Thoorie Analyiique de la Chaieurf
`in Oeuures do Fourier. tome premier. G.Darboi.iI<. Ed.. Paris:
`Gauthier:-Vilinrel, 1338.
`IFRAZBI P. Franklin, ‘A Set at Continuous Orthogo
`tlons." Math. Annai- Vol.100. pp.522-529. 1928.
`IFR.-W6] M. Frazier and B. Jawerth. "me 1i~'Ii-nnaform and
`Decomposition of Distributions.‘ Proc. Coqfi Function Spaces
`and Appt. Lund 1936. Lect. Notes Mom. Springer.
`[GAB-1.6] D. Gabor. ‘Theory oI'oommunicotlon.' J of the IEE.
`va1.93. pp.129-451. 19-16.
`'Frs1:1sI
`[GAC91] N. Cache. P. Flnndrin. and D. Garreau.
`Dimension Estimator: for Fractions: Brownie":-r Motions.‘ in
`Proc. I99 I 1'2!-2.E.‘Jnt. Coq,i'..Acou.rt., Speech. SignolProc.. Toronto.
`Ontario. Canada. pp. 3551-3530. May 14-11 1991.
`|GOU84] P. Goupiliaud. A. Gmmnann and J."i~‘!uu-let. ‘Cycle-
`Octave and Related ‘n-snsfonns In Seismic Signal Analysis.‘
`Gcoemioratiort Vol.23, pp.85-102. Elsevier Science Pub.,
`19B4.r'85.
`[Gfl08Iil A. Grossmsnn and J. Mm-let. ‘Decomposition of
`Hardy Functions into Square Integrable wavelets‘ o[Consla.r1l
`Shape.’ SIAM J.i\4nth.ArmJ.. Vol.15, No.4. p|'.I.723-1'35. July
`i984.
`-
`.
`IGROBBI A. Grooamnrin. R. Km-niend—Ma.rttnei. and J. Mariel.
`‘Reading and Understanding Continuous Wavelet Trans-
`forms.‘ in [WAVES]. pp.2-20. 1989.
`-
`[I-IAAIOI A. Hanr. 'Zu.r"i'hom-ie der Orthogonalen i-hnkiionem
`syaterr1e." lln German] Math. Annat. Vol. 99. pp. 331-371.
`mm.
`IHEIQOI C.E. Heil, ‘wavelets and 1-Yunnan.’ in signal Process-
`ing, Part. I: Signal Processinynieorgr. L. Auslonder. oi oi. edo..
`IMA. Vol. 22. Springer. New York. pp. 141- 160. I990.
`[HER71] O. Hernunrln. “On the Approiornntton Proimn in
`Nonfiecursiire Digital F'i.lterDcslg|1.'l'EEE1"I'ans.CircuitTheory.
`vo|.cr-1a.No.3. pp. 411-41a.1.iay 1971.
`[H921 IEEE Tmnsaciiorts on irgfiarrnomn Theory. Special issue
`on wsvcletTransi'orms and Muitiresolution Signs! Analysis. to
`appeardanuary. 1992.
`-
`IJOI-I60] JD. Johnston. ‘A Filter Pamiiy
`ed for Use In
`Guadnature Mirror Filter Banks,‘ in Proe. 1983 £EE.'Eln!. Carji
`rioou.st.. Speech. Signal Proc.. pp. 291-294. Apr. 1930.
`IKADEII S. Kadambe and GJ’. Boudreau:-Bu-tels. ‘A Compar-
`ison oithe Existence ol"C1-ass ‘hm-us‘ In the Wu-eletTrau1sforrn.
`Wigner Distribution and Sl'mri.~'l‘im: Fourier 'Ti'nnsI'o1'm.' Suh-
`mltted l‘E£E."l'lI'ur1.1. Signal Proc.. ‘Revisod Jan. 199].
`IKOVQZI J. Knvvlcettlc and M. Vettcrlt. ‘Non-separable Mulli-
`dlrnensional. Perfect Rciioiutrul:-i:Ion Filter Banks and Wavelet
`Bases for R".' JEEE Trans. on info. Theory. special Issue on
`wavelet u1I.nsi'or1ne and rriultiresolution aigna! analysis.
`to
`appeardon. 1992.
`iLlT37] J. Llltiewood and R. Paley. ‘theorems on Fourier Series
`and Power Series.‘ Pi-oc. London Math. Soc.. Vol.-I2. pp. 52-89.
`1937.
`‘A Thoory for Multiresoiution Signal
`S. Mailat.
`lMALB9nl
`Decomposition: the Wavelet Representation.‘ JEEE ‘mans. on
`Pattern Analysis o.ndM'ar:i1tv1e inlehi. Vol.11, No. 7. pp.6?'4-B93.
`July I989.
`IMAI-39b| S.Mn1lI1i. 'Mt1itlfrequcncy Channei Dccomposluons
`of images and Wavelet Models.’ IEEE 'l'l'ons. Amust. Speerli.
`Signal Pi'o<.‘.. Vol. 37. No.12. pp.2i]9l-2110. December 1989.
`[M.ALB9cI
`S. Millnt.
`‘in-iultiruolutlcin Approximations nnd
`Woreletorthonornial Bases orL*1R|.°. Tl'uns.Arner. MLIIJL Soc..
`\a'0I.315. No.1. pp.oo-57. September 1939.-
`[MALBEMI S. Mallet and S. Zhong. “Complete Signal Repre-
`sentation with Multiscttle Edges.’ submitted to IRES 'l'l'm1s.
`Pattern Analysis and Machine tnteit. 1939.
`
`Page 168 of 437
`
`Page 168 of 437
`
`

`

`.
`
`;'t
`
`. _.r
`
`l_-
`
`_
`
`-
`
`lME';'l'B53l 1'. Maya. ‘O1-I.honoI1'nol wm1¢1.u.' mfuvnviosqfi '
`"21-37.19119.
`'
`-
`1ME‘:'9Dl
`1r. Meyer. Ondoletms at ‘opemucm. Tame 1. ome-
`letles. Henmann ed.. Paris. 1990.
`|M1N35] ll’. M1nt2.er.“Flll.era for flillflfiinn-Free'N'D-Band Mul-
`l.|reI.e filler Banks.’ lE.'.E.*E 1‘|'a.r1.a. on Aoourt. Speech. Signal
`Pros. VaL33. pp.62aB-B3O..Jur1e 1985.
`IPORBOI Mil. Porlnoff.
`'I'lme-Fnsquency Reprcscnutlon of
`Dlgllal Signals and Syutewu Based on Short-‘Dime Fourier
`Analysis.‘ IEEE 1":-w1s.on Aoauat. Speedi. Signal P1-oc.. Vol.2B.
`pp.55-69. Feb. 1930.
`lRAM8B| T.A. Ra:-natad and T. Sammalu. 'E'.l!ic1ent Mulch-ate
`Realization for Narrow '11-nnaluon-Band FIR Filters.‘ In.Proc.
`1953 |'EEElr1.L 5ymp.CIrI:u1ts anasgszams. Hezasnu. Finland.
`pp. 2019-2021. 1933.
`IRl(}90al 0. Euoul and P. Ftnndrlrl. ‘l‘I.1ne-Scale Energy Dea-
`Lr!bu1.Iono:AGene1'aJ Clue E:xtamdI.r1g\lfaveletT‘r1n1:l'orrns.' to
`appear In IEEE ‘hum. Slgnal Pmc.
`[R1090b1 O. Rloul. ‘A Discrete-1'l.:11e Multlreoolutlon Thoory
`Unifying Octave-Band ‘Flilnf Bulka. Pyran-dd and Wavelet
`Transforms.” aubrrlltlod to IEEETMM. Signal Hue.
`lRl09la| O. Rloul and P. Duharncl. "Fhot.Algo»rlLl'1.1'na for Ols-
`erete and Conunuwa W11.vel.et‘I‘nnsl'or1:n.s." submitter! ln IEEE
`‘Trans.
`lrgfon-mttor1 Theory. Special lamp on W'aveJeLTI'a11s-
`rum. nnd Multlresoluuon signal Nmlyul.
`lRlO91bl O. - Rlfllll.
`'D§'ld-11: Up-Soallns Scllernes: Simple
`C1-lterla for Regulantv.‘ aubmlrted to 3MMJ.' Math. Anal.
`[REE-84-I A. R.osen.fe1d ed.. MI.LlIirunh.11.l.on Techniques in Corn-
`puter Vlslon. Springer-Verlng. NewYurk IBM.
`[5M136| M.d.'I'. Smith and ‘I2‘.E-‘. BIHIIAIEIL “End Emmanue-
`uon for Tree-Structured Subband. Codm.‘ {SEE 'l'Iu.ns. on
`.-\co1.1.st.. Speech a.n.d Slgnal Prue. VoL ASSPA34. pp. 434-4“.
`June 1936.
`lSHE9°l MJ. Shanon. ‘The DlIcl'lEVIIVBl»e1TIIl.IlBfil'I'l'I: Wed-
`ding the aT'roua and Mal.1.I.l. Algor|fhn'u.' Iubrrlllud to JEEE
`‘nuns. on A-oou.sL. Speech. and Signal Proc. 1990.
`ISPE-67] ..I.M. Spelaet. ‘mac-Ba.ndMn
`tyruncuonu.‘ E38
`.
`1‘mns.pnIr1_|b.‘I1'neory.pp. 122-128. I
`lVAIB7l P.P. Vasdymausnn. '0uadnt:.u1: I-l.I.rror rune: Banlu.
`M—bo,ncl Euenalons and Rdm- n Tenluuquea."
`IEEE.‘ ASSP Magazine. Vol. 4. No. 3. pp.-I-20. July 1957.
`WAJ83'|
`19.13. V
`stl-Inn end P.-Q. Huang. "lame: Stuc-
`tureo for 0‘P!l.I.'l'l.I.|
`ealgn and Rnbult lmplemmuflnn o{'No-
`Bnnd Perfect Reconstruction EMF Blnln.‘ IEEE mm. on
`fleousc. Speechond Sbrlflf Proc. Vol. AS6113-B. No. i. p1I.31v
`9-1-.Jan. 1988.
`N31139: RP. Vnidylnalllln and 2.. Dbfilflllfl. ‘The Role of
`mules: System: ln Modern Dlgllal Slcnll Prucoooulg.‘ IE3-‘E-‘
`.1'Y¢n.s. Eduoamn. Special loan: on Circuits Ind Sync:-n-Ia. Vol.
`32. No.3. Aug. I989. pp.].S1-197'.
`IVETEGI M. Vetterli. ‘Filler Bnnlua Allowing Perfect Remn-
`s£.1'ucl.lon.' Sb11a.ll"I'ooe.IIln9. Vol.10, No.3. Mart] 1986. 1:19.219-
`244.
`IVETBBI M. Vetlerll and D. L: G:ll.‘1h-ketflunnsuucuon FIR
`Flller Banka:3orr1e Pmpertles and hctortntlom. LEEE ‘nuru.
`an ;'l_mIu-‘IL. Speech Sandi Fr'oc.. \v'oL37. No.7‘. 911.1057-1071.
`July 1999.
`I’VE.‘T9CIaJ M. Veilerli and C. He:-lay ‘llfnveletl and Ftllorflanlur
`Relationships and New mum.“ In mac. l990l'£EEl'r1.: Can}
`Aonu.sL. Speech Signal P1-oc.. Albuquerque, NM. pp 1’!'.1.&
`L726. Apr. 3-6. L990.
`IVE'l‘90‘b] M. Vetterll and C. Harley. ‘Wavelet: and Filter
`Banks:T'heo:y1nd Design.‘ to appear I.rI l€«E-‘Efrain. anstgnal
`Proc.. 1992.
`N-'AVB9l Wavelets. WM-Fleqlaenry Methods and Phase Space.
`Proc. Int. Cont‘. Mareellle. France. Dec. L4-lB. I937. J.M.
`Corrror.-3 at al. eds.. Inverse Problems and Theoretical Imaglng.
`
`I
`'§'pF1ngcr. a 15 pp; -19691;
`|'W|CB9] M.V. Wlckedmuaer. ‘Acoustic S|g1a_'| Compruelon
`with wave Packets.’ prcprlnt Yale Unlvemry. 1989.
`EWO-053| PM. Woodward. Pmboblkly and Injgrrnaltron Theory
`with Appllcrrflan ta Radar. Pergamon Press. London. 1953.
`lWC|R.90] G,W.Worne1l. ‘A Karhunen-Loewrlike Eatparulon [or
`1.ffPmces11ea ma wavelets.‘ IEEE Trans. Iqfo. Theory. Vol. 36.
`No. I'l.‘pp.859-83L -July 1990.
`IYOUTBI J.E.‘{r.1unbe1'g. S.F.BoI1. 'cu‘:ns1.m1-gs1gna1.Iu1a1ys1s
`and Synthesis.‘ In Prue. 1973 IEEE: Int. com. on Aoouan.
`Speech. and S1gn.a‘1 P:oc..‘h1|sa. OK. pp. 375-378. 1978.
`
`EXTENDED REFERENCES
`
`3.1-my of Wu-olntl:
`YOU‘!B,GOUB4].
`
`sec [HJ\l\10. FRASB, LIT37, CAL64,
`-
`
`Bean on wnveleuz tact also IwAv1_39. MEYQOIII
`L Dauhcchles. Ten Immms an Wauelen. CBMS. SLAM publ,
`to appear.
`Wavelets and lhelr npplwatwns. 11.11. Cowman. 1. Daubenauca.
`S. Mallet. 1''. Meyer scientific 315.. LA. Raphael. ll-LB. Ruakal
`1'11a.nag[.ngedn.. June! and Bertlel pub.. to appear. 1991.
`ntoflun on Wtvolou: [soc also M439. GROB9. I-lM4fi9b.
`MEYBBII
`-
`RR. Coifman. ‘Wavelet
`s and Slgnnl Processing.‘ in
`Slgnal Pmcemlng. Part I: Signal Pmceuslng'I‘l'1-eary. L Aualandv
`er et al. eds. IMA. Vol. 22. Springer. New York. I990.
`C.E. Hell and DJ‘. Walnut. ‘Contlmwus and Discrete Wiaivelet
`'I‘nn.sfor1111!.' SL-1M Reulelu. Vol. 31. No. 4. pp 628-663. Dec.
`1939.
`Y. Meyer. 3. Jnifanl. 0. Rlorul. ‘L‘Analyae pa: Ondeletleaf {Ln
`Fre1‘n:]'l]PtlI-ll’ In Stfflrlct. ND.1 I9. pp.28-37. Sept 1937.
`G. 511-nng. ‘Wavelet: and Dilation Bqu1l.Iona:A Brlel'1ntroduc-
`tlon.‘ 3lAMR¢1:leur. Vol. 3!. No. 4. pp. 614-621. Dec. 1939.
`Ilatllemntlu. Halhamatlnnl Phnlel and Qnnntum
`Mechanic:
`[see also IGROBI. DAUEB. ME'f‘9I'-H]
`'l‘l1e
`6. Bottle. ‘A Block 5pl.n cumtrucunn of O1'ldeletteII. 11.
`Qum1.urnF!eld'l'!looryl0P'l1 Conne1:uan.'Corn1'r1. Math. Pl1ys..
`vol. 114. pp. 93-102. 1983.
`.
`‘
`WM. l-IMML ‘fleoealary and Sufficlent Condlunna for Con-
`ac:-ueung Onhonormal Wavelu. Basel.‘ Aware Tech. Report I
`ADB00402.
`'Dnde]cl.tea ol: Bases H1]-
`F-‘.6. Lemlrle and Y. Meyer.
`bertlea-1.r.ea.' [Ln French]. Reutsta Mnternalim lberoontericana.
`Vol.2. No.1a1.2. pp. 1- 18. 1958.
`T. Paul. ‘Milne Coherent States and H1: Radial Schrodinger
`Equauon l. Radial Hanhorllc Ooclllninr and the Hydrogen
`Atom.‘ to appear in Anrt. l'n.aL l-Ll‘olm'.u.1'¢!.
`I'LL Resnllmlf. "E'uLmdal.luna or Ar[l.l'1rl'|eflc1.1.I11 Analysts: Cum-
`pacuy Supported Wavelet! and the Wave‘1el'Group.‘ Aware
`Tech. Report I ADBQDEOV.
`lhogulu Wnnlotl: we IUMJSB. DAUQDIJ. DAUQOC.-COH90b.
`9-TU9.'lb]
`"computer-slated Geometric Design using Rqular Interpola-
`lnrn‘
`
`S.Dubue. ‘ln1e:poh1tlonTl1roI.Lgh smIter11tlveschemc.‘.J'. Math
`Ano.lysI'.sAppl., Vol.1 14. pp. 135204. 1986.
`N. Dyn and D. Levln. “Uniform subdmaian schemes {or the
`Generallon of curves and Surfaces.‘ Conslrucllw: Approsdmm
`tion to appear.
`Nmaorloll Annlyull: {see also ll3E'l’39|}
`R.R. Coil'n1or1. ‘multlruolutlon Analysis in Nonl-mmngencoua
`Media.‘ In [WAVES]. pp. 259-262. I989.
`-
`
`Dflfllfl m1
`
`IRIS? Jllrliullllf
`
`3?
`
`
`
`Page 169 of 437
`
`Page 169 of 437
`
`

`

`V, Perrier. ‘Toward a M212-not‘! Losolve Partial Dl|‘|'crentlslB¢1ua-
`tions Using Wavelet Bases.‘ ln'|'l-VAVBQI. PD. 269-283. 1959.
`Mllltllesls Ststlstleil sigutl Frocullng; sec lB.ASE9.
`CHO91|.
`Fractals. T\.lrbuleru:-er [see also '.GAC9 1. WORSOH
`A. Arntodn. G. Gmsseau. and M. Hnlschtmdcr. ‘Wavelet
`’I‘rnns[or|11 dr Mult$frscI.a1s.' Phys. Review Letters. Vetal.
`No.20. P933281-22B4._ 1.933.
`1'-‘. Ju-gout. A. A:-needs. G. Orsssesu. Y. Gagne. E.J. Hopflngcr.
`and U. F1-Isch.
`'wn\rslst Analysis oFTurbulencs Reveals the
`Mulufrsctnl Nature of the Richardson Cascade.‘ Nnlum.
`Vol.33B. pp.5l-53..Marcl'l E989.
`OIII-D'llIl.lIIIi|'llI.I] 5lp|Il'Ln.IlyIl.I: {see also IOROSSI. IW'lC-
`39“
`C. D‘n.lu.snndro and -1.3. Llenard. “nemmpoalnun of the
`Speech Signal Into Short-Time Wavefomas Using Spectra] Seg-
`rne11t.aLIon.' in Free. l9B8I£EEln£. Cory’. Acosut. Speech. Signal
`I-‘l"oe._ New York. Apr.1l-'14. 193-B. 1111.35]. -354.
`S. Kndambe and G1-‘. Bmdrenux-Bands. ‘A comparison of
`Wavelet F‘una|:llom far Pltch Delectlan of Speech Signals.‘ In
`Prue. 1991 lE.E.El'n£. Can,|ZAooI.IsL.5peecl‘I. SlgrIa.lProc..‘I’orI:nto.
`Ontario. Canada. pp. 419-452. May. 144?. 1991.
`R. lironlsnd-lulsnzlnet. -J. Marla. and A. Grossmann. 'AJ1nJysI.s
`nfsound Patterns ‘Thrnugh Wavr.I:l.'I!'nnIlhm1.s.‘ Int J. Pnflem
`Remgnlllon a.nd.Artfl":.r.1n.lf . Vol.1. No.2, pp. 213-302.
`pp.97-126. 1987.
`.J.L,Larxun.11cul' andd. Marlct. ‘Wavelets and 5c1srn.tclnLcrp'rc-
`tatlun.' in waves). pp. 12513:. 1989.
`F. B. ‘muur. ‘wavelet Tnnsromuuens I.n Slmal Detecuonf
`1nPl-ac. I988 IEESIM. Con}: Amust. 5]:|¢£‘¢‘I'I. SlgnalPwc.. New
`York. NY. Apr. 11-11. l988. p'p.14:‘-5-1436. Also In IWAVEQI.
`pp. 132-153. 1989.
`fisdu,-‘Sena. Amlnlcwltr functions: see e.g. IALISQG. SFE67.
`WOO58]
`‘rlmo-Scale Reyreuntltlonl:
`RIOQOB}.
`rlltet flank1'heor_r: [see also ICRJM. CROT6. E5‘l'7T. CROB3.
`
`see IBBRBB. FEABQ. FLABO.
`
`'
`
`ID
`
`MIN-'35. SMIBB. VMB7. VMB8. \I".FLl3Q. VETSS. VETBQII
`J D. Johnston. ‘A Flltsr Family Designed for Use In Dusdralu re
`M11-rnr Filter Banks.‘ Pros ICASSP-80. pp.29l-254. Apnl 1980.
`T.Q. Nguyen and RP. Valdyahalhan. ‘Two-Channel FcrI=cl--
`. Reconsti-ucuon FIR QMF Siructureuvhich YIe|d'1.In:ar-Ph use
`Analysis and Synthesis Filters.‘ JEEE ‘mans. Acou.st.. Speech.
`Stgnalfiuwessmg. Vol. ASSP-37. No. 5. pp.S76-690. May 1939.
`M.J.'l'.'SI11l.!h and 11?. Bamwell. ‘A New Fllter Bnnk'l'he0rY 90'
`1‘lmc-Frequency Reprcsentstlunf JEEE ‘lhfims. an Acaust.
`Speech andslgnal Froc.. Vol. ASS!’-35, No.3. Mnnh 1937. pp.
`311-327.
`P.P. Valdysnathsn. Mulllrnle filler Banks. Pmntlc: Hall.
`appear.
`Prnruld ‘l‘rI.n|!m'm.s: one IEURB3. BUR89. ROSSH.
`uultldlmanuonn rum sun: {see also niov-321:‘
`G. Km-lsscn and M. Velterll. "'T‘hcory of Two-Dlmcnsiunall Mul-
`unite Filter Banks.‘ IEEE Trans. on Anoust. Speech. Signal
`Proc. Vul.38. No.6. pp.925-937. June 1990.
`M. Vetterll. ‘Malt:-Dimensional Subbmd Codlng‘; 5cmeT‘heary
`and Algo:1i'.hrns.' Sunni Processlrlg, Vol. 6. Nu.2. pp. 97-112.
`Feb. 1984.
`M.‘«’etI£r1l...I. Iccnraeevlc and D. lxcsll. ‘Perfect Reconstruction
`Filter Banks for HITTV Representation and Coding.‘ lrrwge
`Cwnmunleaflon Vol.2. Ne.3. Oet..1.990. pp.349-364.
`'
`E. Vtsclto and J. Allebanh. The Analysis and Denlgn nl’ Muld-
`dlrncnalnnal FIR Perfect Rzcmlatrununn Filter Balllm for Al'-
`bitrary Sampling l..nttlces.' IEEE 'l"ncm.s. Cinults and Systems.
`‘€01.35. p*p.2B--1.2. Jan. 1991.
`[see also MNIGU.
`lnltldlmsnslunsl Wsvalst-:
`MA1.3Ba. MAI£9b. Msuaflc. MA1.B9d|].
`J.C. Feau.ves.u_ 'A.I1,alyse Muluresoluliun pour 1:: Images ave:
`un Fhclcur cl: Résuhitlou 2.‘ [in FrenchI.Tl'lI.ll.e'I'ueJ1I. du Signal,
`Vol. 7. Na. 2. pp. 117-128. July 1990.
`K. Eh-dchenlg and w.R. Msdych.
`'Mulu.reso1uuan analysis.
`Has: bases and self-similar tlllngs of R".' submitted to IEEE
`‘l'l'uns. onlrgfa. Theory. Speeisl Issue on wavelet transforms and
`mttluresulutlnn signal smalysls. Jm.1992.
`
`l(0V92.
`
`Working with a competitive edge
`makes the difference.
`
`IEEE--The Institun ofEIeclrlcal and Electronics Engiiteers, Inc.-
`yourpersanal edge on technology.
`
`" FOR A F'REEMEMBE.RSH!P INFORMATION KIT. USE THIS COUPON.
`
`He rne
`Tille
`Firm
`Add res:
`
`?hunc
`
`Slnl:.‘CnunI.ry
`MAILJTO: IEEE. MEMEERSI-IIP DEVELOPMENT
`The lnslrlulc of Electrical Ind Electmnlu Engineers. Inc.
`443 Hoes Lane. F10. 301 I331
`!’is'.cnLaw:y.NJ DEI55-lnl. USA (903; 551.5524
`
`Page 170 of 437
`
`Page 170 of 437
`
`

`

`
`
`Blorthogonal Wavelets
`
`Albert Cohen
`
`In this clnpter. we study the construction of blorthogonsl
`Abstract.
`bases of wavelets which generalize the orthonormal bases and have inter-
`esting properties in signal processing. We describe the class of aubbeod
`coding schemes mocisted with these wavelets and we give neoesssr§' sod
`suflicient conditions for frame bounds which ensure the stability of the
`decomposition-reconstruction algorithm. We finally present the example of
`compactly supported spline wavelets which can be generated by this tip-
`proach. The results presented in this chapter are mainly joint work with
`I. Daubechies and J. C. Resnvesu.
`
`§1. Introduction
`
`In recent years, orthonormal wavelet bases have revealed to be it powe.ri"u.l
`tool in applied mathematics and digital signal prooemiog. The possibility of
`dots cornprmion offered by a multiscaie decomposition leads to some very‘
`good results in speech [8] or image [1] coding or test numerical analysis of
`operators [2].
`One of the main reasons for this success is the existence of a Fast Wavelet
`‘Iinnsforrn algorithm {FWTJ which only requir a number of operations pro-
`portional to the size of the initial discrete data. This algorithm relates the
`orthonormal wavelet bases with more classic tools of digital signal processing
`such as subband coding schemes and discrete filters.
`We can describe in four steps the oonnections between these different
`domains:
`a) Wavelets bases are usually defined from the data. of s multiresointion
`analysis;
`r'.e., a ladder of approximation subspaces of L"(iR)
`
`‘l0l""'V1CV|:CV-—1"‘-vL:[R)
`
`which satisfy the following properties
`
`rm (5 mg ¢=> f(2s) e v,_, e. ;(2;.v‘=) e Va.
`Wnnlors-A Tutorial In ‘Theory and Application:
`c. Ir. cm: (-4.). pp. ms-its.
`Copyright H91 luv And-ml: Pull. Inc.
`All ruin: of reproduction In my Iorm nursed.
`ISBN G-I2-1?-Chm-I
`
`(1)
`
`(2)
`123
`
`'
`
`..
`
`Page 171 of437
`
`Page 171 of 437
`
`

`

`
`
`124
`
`A. Cohen
`
`There exists a scaling function 95(3) in Va such that
`
`{¢'l.}i-e‘«z = {3'm¢(3'5¢ * 5=)lke2z
`
`'
`
`(3)-
`
`The function in has to satisfy a two-scale dif-
`is an orthonormal basis for
`ference equation which expresses the embedded structure of the V,--spaces
`
`o>(m)=-:22 was-n1.
`BEE
`
`to
`
`The wavelet 1,’: is then defined by
`
`w<=).= 2 E (—1)"m-..¢{2= - n) = 2 Z 9..¢('-’= — on
`HER
`M53
`
`_
`
`(51
`
`and its integer translates {¢r(:t —