Sheng, Y. Wavelet Transform. The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000

Transcription

1 Sheg, Y. Wavelet Trasform. The Trasforms ad Applicatios Hadboo: Secod Editio. Ed. Alexader D. Poularias Boca Rato: CRC Press LLC,

2 Wavelet Trasform Yulog Sheg Laval Uiversity. Itroductio Cotiuous Wavelet Trasform Time-Frequecy Space Aalysis Short-Time Fourier Trasform Wiger Distributio ad Ambiguity Fuctios. Properties of the Wavelets Admissible Coditio Regularity Multiresolutio Wavelet Aalysis Liear Trasform Property Examples of the Wavelets.3 Discrete Wavelet Trasform Time-Scale Space Lattices Wavelet Frame.4 Multiresolutio Sigal Aalysis Laplacia Pyramid Subbad Codig Scale ad Resolutio.5 Orthoormal Wavelet Trasform Multiresolutio Aalysis Bases Orthoormal Bases Orthoormal Subspaces Wavelet Series Decompositio Recostructio Biorthogoal Wavelet Bases.6 Filter Ba FIR Filter Ba Perfect Recostructio Orthoormal Filter Ba Orthoormal Filters i Time Domai Biorthogoal Filter Ba.7 Wavelet Theory Orthoormality Two-Scale Relatios i Frequecy Domai Orthoormal Filters i Time Domai Wavelet ad Subbad Filters Regularity.8 Some Orthoormal Wavelet Bases B-Splie Bases Lemarie ad Battle Wavelet Basis Daubechies Basis.9 Fast Wavelet Trasform Wavelet Matrices Number of Operatios Time Badwidth Product. Applicatios of the Wavelet Trasform Multiresolutio Sigal Aalysis Sigal Detectio Image Edge Detectio Image Compressio ABSTRACT The wavelet trasform is a ew mathematical tool developed maily sice the middle of the 98 s. It is efficiet for local aalysis of ostatioary ad fast trasiet wide-bad sigals. The wavelet trasform is a mappig of a time sigal to the time-scale joit represetatio that is similar to the short-time Fourier trasform, the Wiger distributio ad the ambiguity fuctio. The temporal aspect of the sigals is preserved. The wavelet trasform provides multiresolutio aalysis with dilated widows. The high frequecy aalysis is doe usig arrow widows ad the low frequecy aalysis is doe usig wide widows. The wavelet trasform is a costat-q aalysis. by CRC Press LLC

3 The base of the wavelet trasform, the wavelets, are geerated from a basic wavelet fuctio by dilatios ad traslatios. They satisfy a admissible coditio so that the origial sigal ca be recostructed by the iverse wavelet trasform. The wavelets also satisfy the regularity coditio so that the wavelet coefficiets decrease fast with decreases of the scale. The wavelet trasform is ot oly local i time but also i frequecy. To reduce the time badwidth product of the wavelet trasform output, the discrete wavelet trasform with discrete dilatios ad traslatios of the cotiuous wavelets ca be used. The orthoormal wavelet trasform is implemeted i the multiresolutio sigal aalysis framewor, which is based o the scalig fuctios. The discrete traslates of the scalig fuctios form a orthoormal basis at each resolutio level. The wavelet basis is geerated from the scalig fuctio basis. The two bases are mutually orthogoal at each resolutio level. The scalig fuctio is a averagig fuctio. The orthogoal projectio of a fuctio oto the scalig fuctio basis is a averaged approximatio. The orthogoal projectio oto the wavelet basis is the differece betwee two approximatios at two adjacet resolutio levels. Both the scalig fuctios ad the wavelets satisfy the orthoormality coditios ad the regularity coditios. The discrete orthoormal wavelet series decompositio ad recostructio are computed i the multiresolutio aalysis framewor with recurrig two discrete low-pass ad high-pass filters, that are, i fact, the -bad parauitary perfect recostructio quadrature mirror filters, developed i the subbad codig theory, with the additioal regularity. The tree algorithm operatig the discrete wavelet trasform requires oly O(L) operatios where L is the legth of the data vector. The time badwidth product of the wavelet trasform output is oly slightly icreased with respect to that of the sigal. The wavelet trasform is powerful for multiresolutio local spectrum aalysis of ostatioary sigals, such as the soud, radar, soar, seismic, electrocardiographic sigals, ad for image compressio, image processig, ad patter recogitio. I this chapter all itegratios exted from to, if ot stated otherwise. The formulatio of the wavelet trasform i this chapter is oe-dimesioal. The wavelet trasform ca be easily geeralized to ay dimesios.. Itroductio.. Cotiuous Wavelet Trasform Defiitio Let L deote the vector space of measurable, square-itegrable fuctios. The cotiuous wavelet trasform of a fuctio f(t) Z is a decompositio of f(t) ito a set of erel fuctios h s,τ (t) called the wavelets: f(, )= () s, τ () W s τ * f t h t dt (..) where * deotes the complex cojugate. However, most wavelets are real valued. The wavelets are geerated from a sigle basic wavelet (mother wavelet) h(t) by scalig ad traslatio: h s,τ ()= t (..) where s is the scale factor ad τ is the traslatio factor. We usually cosider oly positive scale factor s >. The wavelets are dilated whe the scale s > ad are cotracted whe s <. The wavelets h s,τ (t) geerated from the same basic wavelet have differet scales s ad locatios τ, but all have the idetical shape. The costat s / i the expressio (..) of the wavelets is for eergy ormalizatio. The wavelets are ormalized as s h t τ s h t dt h t dt s,τ () = () = by CRC Press LLC

4 so that all the wavelets scaled by the factor s have the same eergy. The wavelets ca also be ormalized i terms of the amplitude: hs,τ () t dt = I this case, the ormalizatio costat is s istead of s /, ad the wavelets are geerated from the basic wavelet as h s,τ t s h t τ ()= s (..3) I this chapter, we cosider mostly the ormalizatio of the wavelet i terms of eergy. O substitutig (..) ito (..) we write the wavelet trasform of f(t) as a correlatio betwee the sigal ad the scaled wavelets h(t/s): = () Wf s,τ s * f t h t τ dt s (..4) Wavelet Trasform i Frequecy Domai The Fourier trasform of the wavelet is H ω s h t τ τ = exp jωt dt s s, = sh( sω) exp( jωτ) (..5) where H(ω) is the Fourier trasform of the basic wavelet h(t). I the frequecy domai the wavelet is scaled by /s, multiplied by a phase factor exp( jωτ) ad by the ormalizatio factor s /. The amplitude of the scaled wavelet is proportioal to s / i the time domai ad is proportioal to s / i the frequecy domai. Whe the wavelets are ormalized i terms of amplitude, the Fourier trasforms of the wavelets with differet scales will have the same amplitude, that is suitable for implemetatio of the cotiuous wavelet trasform usig the frequecy domai filterig. Equatio (..5) shows a well ow cocept that a dilatatio t/s (s > ) of a fuctio i the time domai produces a cotractio sω of its Fourier trasform. The term /s has a dimesio of frequecy ad is equivalet to the frequecy. However, we prefer the term scale to the term frequecy for the wavelet trasform. The term frequecy is reserved for the Fourier trasform. The correlatio betwee the sigal ad the wavelets, i the time domai ca be writte as the iverse Fourier trasform of the product of the cojugate Fourier trasforms of the wavelets ad the Fourier trasform of the sigal: = Wf s,τ s π * F ω H sω exp jωτ dω (..6) The Fourier trasforms of the wavelets are referred to as the wavelet trasform filters. The impulse respose of the wavelet trasform filter, sh( sω), is the scaled wavelet s / h(t/s). Therefore, the wavelet trasform is a ba of wavelet trasform filters with differet scales, s. by CRC Press LLC

5 I the defiitio of the wavelet trasform, the erel fuctio, wavelet, is ot specified. This is a differece betwee the wavelet trasform ad other trasforms such as the Fourier trasform. The theory of wavelet trasform deals with geeral properties of the wavelet ad the wavelet trasform, such as the admissibility, regularity, ad orthogoality. The wavelet basis is built to satisfy those basic coditios. The wavelets ca be give as aalytical or umerical fuctios. They ca be orthoormal or oorthoormal, cotiuous or discrete. Oe ca choose or eve build himself a proper wavelet basis for a specific applicatio. Therefore, whe talig about the wavelet trasform oe used to specify what wavelet is used i the trasform. The most importat properties of the wavelets are the admissibility ad regularity. As we shall see below, accordig to the admissible coditio, the wavelet must oscillate to have its mea value equal to zero. Accordig to the regularity coditio, the wavelet has expoetial decay so that its first low order momets are equal to zero. Therefore, i the time domai the wavelet is just lie a small wave that oscillates ad vaishes, as that described by the ame wavelet. The wavelet trasform is a local operator i the time domai. The orthoormality is a property that belogs to the discrete wavelet trasform. We discuss the discrete orthoormal ad bi-orthoormal wavelet trasforms i Sectios.3 to.9... Time-Frequecy Space Aalysis The wavelet trasform of a oe-dimesioal sigal is a two-dimesioal fuctio of the scale, s, ad the time shift, τ, that represets the sigal i the time-scale space ad is referred to as the time-scale joit represetatio. The time-scale wavelet represetatio is equivalet to the time-frequecy joit represetatio, which is familiar i the aalysis of ostatioary ad fast trasiet sigals. Nostatioary Sigals The wavelet trasform is of particular iterest for aalysis of ostatioary ad fast trasiet sigals. Sigals are statioary if their properties do ot chage durig the course of sigals. The cocept of the statioarity is well defied i the theory of stochastic processes. A stochastic process is called strict-sese statioary if its statistical properties are ivariat to a shift of the origi of the time axis. A stochastic process is called wide-sese (or wea) statioary if its secod order statistics is ivariat to shift i time ad depeds oly o time differece. Most sigals i ature are ostatioary. Examples of ostatioary sigals are speech, radar, soar, seismic, electrocardiographic sigals ad music. Two-dimesioal images are also ostatioary because the edges, textures, ad determiistic objects are distributed at differet locatios ad orietatios. The ostatioary sigals are i geeral characterized by their local features rather tha by their global features. Time-Frequecy Joit Represetatio A example of the ostatioary sigal is music. The frequecy spectrum of a music sigal chages with the time. At a specific time, for istace, a piao ey is oced, which the gives rise to a soud which has a specific frequecy spectrum. At aother time, aother ey will be oced geeratig aother spectrum. The otatio of music score is a example of the time-frequecy joit represetatio. A piece of music ca be described accurately by air pressure as a fuctio of time. It ca be equally accurately described by the Fourier trasform of the pressure fuctio. However, either of those two sigal represetatios would be useful for a musicia, who wats to perform a certai piece. Musicias prefer a two-dimesioal plot, with time ad logarithmic frequecy as axes. The music scores tell them whe ad what otes should be played. Fourier Aalysis of Nostatioary Sigals The Fourier trasform is widely used i sigal aalysis ad processig. Whe the sigal is periodic ad sufficietly regular, the Fourier coefficiets decay quicly with the icreasig of the frequecy. by CRC Press LLC

6 For operiodic sigals, the Fourier itegral gives a cotiuous spectrum. The Fast Fourier trasform (FFT) permits efficiet umerical Fourier aalysis. The Fourier trasform is ot satisfactory for aalyzig sigals whose spectra vary with time. The Fourier trasform is a decompositio of a sigal ito two series of orthogoal fuctios cosωt ad jsiωt with j = ( ) /. The Fourier bases are of ifiite duratio alog the time axis. They are perfectly local i frequecy, but are global i time. A sigal may be recostructed from its Fourier compoets, which are the Fourier base of ifiite duratio multiplied by the correspodig Fourier coefficiets of the sigal. Ay sigal that we are iterested i is, however, of fiite extet. Outside that fiite duratio, the Fourier compoets of the sigal, which are ozero, must be cacelled by their ow summatio. A short pulse that is local i time is ot local i frequecy. Its Fourier coefficiets decay slowly with frequecy. The recostructio of the pulse from its Fourier compoets depeds heavily o the cacellatio of high frequecy Fourier compoets ad, therefore, is sesitive to high frequecy oise. The Fourier spectrum aalysis is global i time ad is basically ot suitable to aalyze ostatioary ad fast varyig trasiet sigals. May temporal aspects of the sigal, such as the start ad ed of a fiite sigal ad the istat of appearace of a sigularity i a trasiet sigal, are ot preserved i the Fourier spectrum. The Fourier trasform does ot provide ay iformatio regardig the time evolutio of spectral characteristics of the sigal. The short-time Fourier trasform, or called the Gabor trasform, the Wiger distributio, ad the ambiguity fuctio are usually used to overcome the drawbac of the Fourier aalysis for ostatioary ad fast trasiet sigals. The Wiger distributio ad the ambiguity fuctio are ot liear, but are biliear trasforms...3 Short-Time Fourier Trasform Defiitio A ituitive way to aalyze a ostatioary sigal is to perform a time-depedet spectral aalysis. A ostatioary sigal is divided ito a sequece of time segmets i which the sigal may be cosidered as quasistatioary. The, the Fourier trasform is applied to each of the local segmets of the sigal. The short-time Fourier trasform is associated with a widow of fixed width. Gabor i 946 was the first to itroduce the short-time Fourier trasform which is ow as the slidig widow Fourier trasform. The trasform is defied as = () ( ) ( ) * S ω, τ f t g t τ exp jω t dt f where g(t) is a square itegrable short-time widow, which has a fixed width ad is shifted alog the time axis by a factor τ. Gabor Fuctios The Gabor trasform may also be regarded as a ier product betwee the sigal ad a set of erel fuctios, called the Gabor fuctios: g(t τ) exp(jω t). The Gabor basis is geerated from a basic widow fuctio g(t) by traslatios alog the time axis by τ. The phase modulatios exp(jω t) correspod to traslatios of the Gabor fuctio spectrum alog the frequecy axis by ω. The Fourier trasform of the basic Gabor fuctio g(t)exp(jω t) is expressed as gt () exp( jω t) exp( jωtdt ) = Gω ω The Fourier trasform G(ω) of the basic widow fuctio g(t) is shifted alog the frequecy axis by ω. The short-time Fourier trasform of a oe-dimesioal sigal is a complex valued fuctio of two real parameters: time τ ad frequecy ω i the two-dimesioal time-frequecy space. by CRC Press LLC

7 Iverse Short-Time Fourier Trasform Whe τ ad ω are cotiuous variables, the sigal f(t) may be recostructed completely by itegratig the Gabor fuctios multiplied by the short-time Fourier trasform coefficiets: ad this holds for ay chose widow g(t). The iverse short-time Fourier trasform may be proved by the followig calculatio: provided that the widow fuctio is ormalized as Time ad Frequecy Resolutio f ()= t S f ( ω, τ) g( t τ) exp( jω t) dω dτ π ( ) ( ) S ω, τ g t τ exp jω t dω dτ f ( ) ( ) ( ) ( ) * = f t g t τ exp jω t g t τ exp jω t dω dτdt ( ) ( ) ( ) = () ( ) = () * = πδ t t f t g t τ g t τ dτdt πf t g t τ dτ πf t gt () dt= (..7) I the short-time Fourier trasform, the sigal is multiplied by a slidig widow that localizes the sigal i time domai, but results i a covolutio betwee the sigal spectrum ad the widow spectrum; that is, a blurrig of the sigal i the frequecy domai. The arrower the widow, the better we localize the sigal ad the poorer we localize its spectrum. The width t of the widow g(t) i time domai ad the badwidth ω of the widow G(ω) i frequecy domai are defied respectively as t = () () t g t dt gt dt ω = (..8) where the deomiator is the eergy of the widow i time ad frequecy domais. The two siusoidal sigals ca be discrimiated oly if they are more tha ω apart. Thus, ω is the resolutio i the frequecy domai of the short-time Fourier trasform. Similarly, two pulses i time domai ca be discrimiated oly if they are more tha t apart. Note that oce a widow has bee chose for the short-time Fourier trasform, the time ad frequecy resolutios give by (..8) are fixed over the etire time-frequecy plae. The short-time Fourier trasform is a fixed widow Fourier trasform. Ucertaity Priciple The time-frequecy joit represetatio has a itrisic limitatio, the product of the resolutios i time ad frequecy is limited by the ucertaity priciple: ω G ω dω G ω dω t ω (..9) This is also referred to as the Heiseberg iequality, familiar i quatum mechaics ad importat for time-frequecy joit represetatio. A sigal ca ot be represeted as a poit i the time frequecy space. Oe ca oly determie its positio i the time-frequecy space withi a rectagle of t ω. by CRC Press LLC

8 Gaussia Widow The time-badwidth product t ω must obey the ucertaity priciple. We ca oly trade time resolutio for frequecy resolutio or vice versa. Gabor proposed the Gaussia fuctio as the widow fuctio. The Gaussia fuctio has the miimum time-badwidth product determied by the ucertaity priciple (..9). The Fourier trasform of the Gaussia widow is still a Gaussia as gt which have a miimum spread. A simple calculatio shows that which satisfies the ucertaity priciple (..9) ad achieves the miimum time-badwidth product t ω = /. The short-time Fourier aalysis depeds critically o the choice of the widow. Its applicatio requires a priori iformatio cocerig the time evolutio of the sigal properties i order to mae a priori choice of the widow fuctio. Oce a widow is chose, the width of the widow alog both time ad frequecy axes are fixed i the etire time-frequecy plae. Discrete Short-Time Fourier Trasform Whe the traslatio factors of the Gabor fuctios alog the time ad the frequecy axes, τ ad ω, tae discrete values, τ = τ ad ω = mω with m ad Z, the discrete Gabor fuctios are writte as: ad their Fourier trasforms are The discrete Gabor trasform is t exp ad G( ω)= exp s ω π s s ()= t s = ad ω = s ()= ( ) g m, t g t τ exp jmωt [ ] Gm, ( ω)= G( ω mω) exp j( ω mω) τ = () ( ) ( ) * S m f t g t τ exp jmω t dt f, The sigal f(t) ca still be recovered from the coefficiets S f (m,), provided that τ ad ω are suitably chose. Gabor s origial choice was ω τ = π. Regular Lattice If the widow fuctio is ormalized as show i (..7) ad is cetered to the origi i the timefrequecy space, so that: () = = tg t dt ωgω dω the the locatios of the Gabor fuctios i the time-frequecy space are determied by: = tg m, () t dt= tg t τ dt τ by CRC Press LLC

9 ad, = ( ) = ωg ω dω ωg ω mω dω mω The discrete Gabor fuctio set will be represeted by a regular lattice with the equal itervals τ ad ω i the time-frequecy space, as will be show i Figure.a...4 Wiger Distributio ad Ambiguity Fuctios m The Wiger distributio fuctio ad the ambiguity fuctio are secod-order trasform or biliear trasforms that perform the mappig of sigals ito the time-frequecy space. Wiger Distributio Fuctio The Wiger distributio fuctio is a alterative to the short-time Fourier trasform for ostatioary ad trasiet sigal aalysis. The Wiger distributio of a fuctio f(t) is defied i the time domai as t * t Wf ( τω, )= f τ + f τ exp jωt dt (..) that is the Fourier trasform of the product, f(τ + t/)f * (τ t/), betwee the dilated fuctio f(t/) ad the dilated ad iverted fuctio f * ( t/). The product is shifted alog the time axis by τ. The Wiger distributio is a complex valued fuctio i the time-frequecy space ad is a time-frequecy joit represetatio of the sigal. I the frequecy domai the Wiger distributio fuctio is expressed as Wf τω F ω ξ * (, )= + F ω ξ exp( jτξ) dξ π (..) where F(ω) is the Fourier trasform of f(t). The iverse relatios of the Wiger distributio fuctio ca be obtaied from the iverse Fourier trasforms of (..) ad (..). With the chages of variables t = τ + t/ ad t = τ t/, the iverse Fourier trasform of the Wiger distributio of (..) gives [ ] f t f t W t t + * = f jt t d, ω exp ( ) ω ω π (..) Similarly, with the chages of variables ω = ω + (ξ/) ad ω = ω (ξ/) the iverse Fourier trasform of (..) gives [ ] F ω F ω Wf τ ω ω + * =, j ω ω τ dτ exp ( ) The sigal f(t) ca be recovered from the iverse Wiger distributio fuctio. Let t = t ad t =, (..) becomes () = f t f W t * f, ω exp jωt dω π where f * () is a costat. Hece, the fuctio f(t) is recostructed from the iverse Fourier trasform of the Wiger distributio fuctio, W f (t/,ω), dilated i the time domai. by CRC Press LLC

10 For the basic properties of the Wiger distributio fuctio we metio that the projectios of W f (τ,ω) alog the τ-axis i the time-frequecy space gives the square modulus of F(ω), because accordig to (..) the projectio alog the τ-axis is = + Wf τωdτ F ω ξ * F ω ζ, exp jτξ dτdξ F ω π = The projectio of W f (τ,ω) alog the ω-axis gives the square modulus of f(t), because accordig to (..) the projectio alog the ω-axis is = + = () t * t Wf τω, dω f τ f τ exp jωt dtdω π f t Also, there is the coservatio of eergy of the Wiger distributio i the time-frequecy joit represetatio: π W f ( τω, ) dτω d = F( ω) dω = f () t dt π Ambiguity Fuctio The ambiguity fuctio is also a mappig of a trasiet time fuctio f(t) ito the time-frequecy space. The ambiguity fuctio is defied i the time domai as: 3 t * t Af ( t,ω)= f τ + f τ exp jωτ dτ (..3) I the frequecy domai, the ambiguity fuctio is expressed as Af t,ω F ξ ω * = + F ξ ω exp( jτξ) dξ π The ambiguity fuctio ca be viewed as a time-frequecy auto-correlatio fuctio of the sigal with the time delay t ad the Doppler frequecy shift, ω. The ambiguity fuctio has foud wide applicatios for radar sigal processig. Accordig to the defiitios (..) ad (..3) the double Fourier trasform of the product f(τ + t/)f * (τ t/) with respect to both variables t ad τ gives the relatio betwee the Wiger distributio fuctio ad the ambiguity fuctio: A t, ω exp jωt dt W τ, ω exp jωτ dτ f = f The cross ambiguity fuctio is defied as the Fourier trasform of the product, f(τ)g * (τ) of two fuctios f(τ) ad g(τ): t * t At (,ω)= f τ + g τ jωτ dτ exp High values of A(t,ω) mea that the two fuctios are ambiguous. The fuctio g(τ) ca also be cosidered as a widow fuctio of fixed width that is shifted alog the time axis by t. Hece, the cross by CRC Press LLC

11 ambiguity fuctio is the fixed-widow, short-time Fourier trasform. The cross Wiger distributio fuctio is defied as that ca be see as the Fourier trasform of the sigal f(t) dilated by a factor of two ad multiplied with a iverted widow g( t) which is also dilated by a factor of two ad shifted by τ. Both the ambiguity fuctio ad the Wiger distributio fuctio are useful for active ad passive trasiet sigal aalysis. Both trasforms are biliear trasform. However, the mappig of a summatio of sigals f (t) + f (t) ito the time-frequecy space with the ambiguity fuctio or with the Wiger distributio fuctio produces cross-product iterferece terms that might be a uisace i the projectios i the time-frequecy space ad i the recostructio of the sigal.. Properties of the Wavelets I this sectio we discuss some basic properties of the wavelets. Oe of them is related to the fact that we must be able to recostruct the sigal from its wavelet trasform. This property ivolves the resolutio of idetity, the eergy coservatio i the time-scale space ad the wavelet admissible coditio. Ay square itegrable fuctio which has fiite eergy ad satisfies the wavelets admissible coditio ca be a wavelet. The secod basic property is related to the fact that the wavelet trasform should be a local operator i both time ad frequecy domais. Hece, the regularity coditio is usually imposed o the wavelets. The third basic property is related to the fact that the wavelet trasform is a multiresolutio sigal aalysis... Admissible Coditio t * t W( τω, )= f τ + g τ jωt dt exp Resolutio of Idetity The wavelet trasform of a oe-dimesioal sigal is a two-dimesioal time-scale joit represetatio. No iformatio should be lost durig the wavelet trasform. Hece, the resolutio of idetity must be satisfied, that is expressed as = ds dτ f h h f c f f (..), s, τ s, τ, h, s where <, > deotes the ier product ad c h is a costat. I the left-had side of (..) the extra factor /s i the itegral is the Haar ivariat measure, owig to the time-scale space differetial elemets, dτd(/s) = dτds/s. We have assumed positive dilatio s > ; usig (..6) for wavelet trasform i the Fourier domai, we have: ds s dτ f, h h, f s, τ s, τ 4π ds s * * jτω ( ω) dτ sf ω H sω F ω H sω e dω ω = * ds = F F H s s d ( ω ) ( ω ) ( ω ) ω π c h = F( ω) F * ( π ω ) d ω by CRC Press LLC

12 where we used the chage of variables ω = sω ad ds = dω/ ω, so that ds ad dω are of the same sig. Because s >, we have ds/s = dω/ ω, the we defied the costat: c = h H( ω ) dω ω Accordig to the Parseval s equality i the Fourier trasform, we have: π F ω F ω dω f t f t dt f, f * * Hece, the resolutio of idetity is satisfied o the coditio that = () () = H ( c = ω) h ω dω <+ (..) Admissible Coditio The coditio (..) is the admissible coditio of the wavelet, which implies that the Fourier trasform of the wavelet must be equal to zero at the zero frequecy: = H ω ω = (..3) Equivaletly, i the time domai the wavelet must be oscillatory, lie a wave, to have a zero-itegrated area, or a zero-mea value: htdt () = (..4) Eergy Coservatio Whe f = f, the resolutio of idetity, (..) becomes: W s d ds (..5) f(,τ) τ = c f t dt h () s This is the eergy coservatio relatio of the wavelet trasform, equivalet to the Parseval eergy relatio i the Fourier trasform. Iverse Wavelet Trasform By withdrawig from the both sides of the resolutio of idetity (..), we have directly: f f t W s c s h t τ d ds f,τ τ s s ()= h (..6) This is the iverse wavelet trasform. The fuctio f(t) is recovered from the iverse wavelet trasform by the itegratig i the time-scale space the wavelets h s,τ (t) weighted by the wavelet trasform coefficiets, W f (s,τ). The wavelet trasform is a decompositio of a fuctio ito a liear combiatio of the wavelets. The wavelet trasform coefficiets W f (s,τ) are the ier products betwee the fuctio ad the wavelets, which idicate how close the fuctio f(t) is to a particular wavelet h s,τ (t). by CRC Press LLC

13 Reproducig Kerel The iverse wavelet trasform shows that the origial sigal may be sythesized by summig up all the projectios of the sigal oto the wavelet basis. I this sese, the cotiuous wavelet trasform behaves lie a orthogoal trasform. We refer to this property of the cotiuous wavelet trasform as the quasiorthogoality. Obviously, the set of the wavelet erel fuctios h s,τ (t) with cotiuously varyig scalig ad shift is ot orthogoal, but is heavily redudat. Applyig the wavelet trasforms i the two sides of (..6) yields: where the reproducig erel: W s W s K s s d ds f(, τ)= f(, τ) (, ; τ, τ) τ s Ks (,; s τ, τ)= c h ss t τ h h t τ dt * s s is ot zero with cotiuously varyig factors s, s, ω, ad ω that describes the itrisic redudacy betwee the values of the wavelets at (s,τ) ad at (s,τ ). Ay square itegrable fuctio satisfyig the admissible coditio may be a wavelet. Whe the wavelets satisfy the admissible coditio, the sigal ca be recovered by the iverse wavelet trasform. No sigal iformatio is lost... Regularity The wavelets should be local i both time ad frequecy domais. This is achieved by applyig the regularity coditio to the wavelet. The regularity is ot a obligated coditio, but is usually required as a importat property of the wavelet. Regularity of Wavelet For the sae of simplicity, let the traslatio of the wavelet τ = ad cosider the covergece to zero of the wavelet trasform coefficiets with icreasig of /s ad decreasig of s. The sigal f(t) is expaded ito the Taylor series at t = util order. The wavelet trasform coefficiets become 4 W ( s, )= f() t h s where the remaider i the Taylor series is f t s dt p f s p = * p t p h t s dt R t h t! + () s dt (..7) Rt ()= t ( t t )! ( + ) f t dt ad f (p) () deotes the pth derivative. Deotig the momets of the wavelets by M p : M P = () p t h t dt by CRC Press LLC

14 it is easy to show that the last term i the right-had side of (..7) which is the wavelet trasform of the remaider, decreases as s +. We the have a fiite developmet as: f f f W f ( s, )= f ( ) M s+ Ms + Ms + K+ M s s!!! + O( s ) (..8) Accordig to the admissible coditio of the wavelet, M =, the first term i the right-had side of (..8) must be zero. The speed of covergece to zero of the wavelet trasform coefficiets W f (s,τ) with decreasig of the scale s or icreasig of /s is the determied by the first ozero momet of the basic wavelet h(t). It is i geeral required that the wavelets have the first + momets util order, equal to zero: p M p = t h() t dt = for p =,,, K, (..9) The, accordig to (..8) the wavelet trasform coefficiet W f (s,τ) decays as fast as s +(/) for a smooth sigal f(t). The regularity leads to localizatio of the wavelet trasform i the frequecy domai. The wavelet satisfyig the coditio (..9) is called the wavelet of order. I frequecy domai, this coditio is equivalet to the derivatives of the Fourier trasform of the wavelet h(t) up to order to be zero at the zero frequecy ω = : H ( p ) ( )= for p =,,, K. (..) The Fourier trasform of the wavelet has a zero of order +. The order (+) is a measure of the flatess of the wavelet i the frequecy domai about ω =. Time Badwidth Product While the wavelet trasform of a oe-dimesioal fuctio is two-dimesioal, the wavelet trasform of a two-dimesioal fuctio is four-dimesioal. As a cosequece, we would have a explosio of the time badwidth product with the wavelet trasform, which is i cotradictio with the restrictios of may applicatios, such as data compressio ad patter classificatio, where the sigals eed to be characterized efficietly by fewer trasform coefficiets. We usually impose the regularity property to the wavelets such that the wavelet trasform coefficiets decrease fast with decreasig of the scale, s ad icreasig of /s. For this purpose, the Fourier trasform, H(ω), of the basic wavelet should have some smoothess ad cocetratio i frequecy domais, accordig to the wavelet trasform i the frequecy domai (..6). The wavelet trasform should be a local operator i frequecy domai...3 Multiresolutio Wavelet Aalysis The wavelet trasform performs the multiresolutio sigal aalysis with the varyig scale factor, s. The purpose of the multiresolutio sigal aalysis is decomposig the sigal i multiple frequecy bads, i order to process the sigal i multiple frequecy bads differetly ad idepedetly. Hece, we eed the wavelet to be local i both time ad frequecy domais. Historically, looig for a erel fuctio which is local i both time ad frequecy domais has bee a hard research topic ad was coducted to ivet the wavelet trasform. Example Figure. shows a typical wavelet multiresolutio aalysis for a electrical power system trasiet sigal. The sigal is decomposed with differet resolutios correspodig to differet scale factors of the by CRC Press LLC

15 FIGURE. Multiresolutio wavelet aalysis of a trasiet sigal i the electrical power system. (From Robertso, D. C. et al., Proc. SPIE, 4, 474, 994. With permissio.) wavelets. The sigal compoets i multiple frequecy bads ad the times of occurrece of those compoets are well preseted i the figure. This figure is a time-scale joit represetatio, with the vertical axis i each discrete scale represetig the amplitude of wavelet compoets. More detailed discussio will be give i Sectio... Localizatio i Time Domai Accordig to the admissible coditio, the wavelet must oscillate to have a zero mea. Accordig to the regularity coditio the wavelet of order has first + vaishig momets ad decays as fast as t. Therefore, i the time domai, the wavelet must be a small wave that oscillates ad vaishes, as that described by the ame wavelet. The wavelet is localized i the time domai. Localizatio i Frequecy Domai Accordig to the regularity coditio the wavelet trasform with a wavelet of order,, decays with s, as s +(/), for a smooth sigal. Accordig to the frequecy domai wavelet trasform, (..6) whe the scale s decreases the wavelet, H(sω) i the frequecy domai is dilated to cover a large frequecy bad of the sigal Fourier spectrum. Therefore, the decay with s as s +(/) of the wavelet trasform coefficiet by CRC Press LLC

16 implies that the Fourier trasform of the wavelet must decay fast with the frequecy, ω. The wavelet must be local i frequecy domai. Bad-Pass Filters I the frequecy domai, the wavelet is localized accordig to the regularity coditio, ad is equal to zero at the zero frequecy accordig to the admissible coditio. Therefore, the wavelet is itrisically a bad-pass filter. Ba of Multiresolutio Filters The wavelet trasform is the correlatio betwee the sigal ad the dilated wavelets. The Fourier trasform of the wavelet is a filter i the frequecy domai. For a give scale, the wavelet trasform is performed with a wavelet trasform filter sh( sω) i the frequecy domai, whose impulse respose is the scaled wavelet, h(t/s). Whe the scale s varies, the wavelet trasform performs a multiscale sigal aalysis. I the time-scale joit represetatio, the horizotal stripes of the wavelet trasform coefficiets are the correlatios betwee the sigal ad the wavelets h(t/s) at give scales. Whe the scale is small, the wavelet is cocetrated i time ad the wavelet aalysis gives a detailed view of the sigal. Whe the scale icreases, the wavelet becomes spread out i time ad the wavelet aalysis gives a global view ad taes ito accout the log-time behavior of the sigal. Hece, the wavelet trasform is a ba of multiresolutio filters. The wavelet trasform is a ba of multiresolutio bad-pass filters. Costat Fidelity Aalysis Scale chage of the wavelets permits the wavelet aalysis to zoom i o discotiuities, sigularities, ad edges ad to zoom out for a global view. This is a uique property of the wavelet trasform, importat for ostatioary ad fast trasiet sigal aalysis. The fixed widow short-time Fourier trasform does ot have this ability. With the ba of multiresolutio wavelet trasform filters, the sigal is divided ito differet frequecy subbads. I each subbad the sigal is aalyzed with a resolutio matched to the scales of the wavelets. Whe the scale chages, the badwidth, ( ω) s, of the wavelet trasform filter becomes, accordig to the defiitio of the badwidth (..8): ( ω) = s ω Hsω dω = Hsω dω = ( ω) sω H sω d sω s H( sω) d( sω) s The fidelity factor, Q, refers to, i geeral, the cetral frequecy divided by the badwidth of a filter. By this defiitio, the fidelity factor, Q, is the iverse of the relative badwidth. The relative badwidths of the wavelet trasform filters are costat because: Q ω s = = ( ω) s (..) which is idepedet of the scale, s. Hece, the wavelet trasform is a costat-q aalysis. At low frequecy, correspodig to a large scale factor, s, the wavelet trasform filter has a small badwidth, which implies a broad time widow with a low time resolutio. At high frequecy, correspodig to a small scale factor, s, the wavelet trasform filter has a wide badwidth, which implies a arrow time widow with high-time resolutio. The time resolutio of the wavelet aalysis icreases with decreases of the widow size. This adaptive widow property is desirable for time-frequecy aalysis. by CRC Press LLC

17 FIGURE. Coverage of the time-frequecy space with (a) the short-time Fourier trasform, where ω ad t are fixed i the whole plae; (b) the wavelet trasform, where the frequecy badwidth ω icreases ad the time resolutio t improves with icrease of (/s). Whe the costat-q relatio (..) is satisfied, the frequecy badwidth ω chages with the ceter frequecy, /s, of the wavelet trasform filter. The product ω t still satisfies the ucertaity priciple (..9). I the wavelet trasform, the time widow size t ca be arbitrarily small at small scale ad the frequecy widow size, ω, ca be arbitrarily small at large scale. Figure. shows the coverage of the time-scale space for the wavelet trasform ad, as a compariso, that for the short-time Fourier trasform. Scale ad Resolutio The scale is related to the widow size of the wavelet. A large scale meas a global view ad a small scale meas a detailed view. The resolutio is related to the frequecy of the wavelet oscillatio. For some wavelets, such as the Gabor wavelets, the scale ad frequecy may be chose separately. For a give wavelet fuctio, reducig the scale will reduce the widow size ad icrease the resolutio i the same time. Example Figure.3a shows the cos-gaussia wavelets i compariso with the real part of the Gabor trasform basis. Both fuctios cosist of a cosie erel with a Gaussia widow. The cos-gaussia wavelet is t ht ()= cos( ω t) exp π where ω = 5. The wavelets h m, (t) are geerated from h(t) by dilatio ad traslatio: h s,τ = s h t τ s with the discrete scale factor, s = m, ad the discrete traslatio factor, (τ/s) =. The discrete Gabor fuctio g m, (t) is defied as: ()= ( ) g m, t g t τ exp jmωt where g(t) is the Gaussia widow with a fixed width ad ω = π. by CRC Press LLC

18 I Figure.3a we see that the wavelets are with the dilated widow. All the dilated wavelets cotai the same umber of oscillatios. The wavelet trasform performs multiresolutio aalysis with high frequecy aalysis for arrow widowed sigals ad low frequecy aalysis for wide widowed sigals. This costat-q aalysis property maes the wavelet trasform surpass the fixed-widow short-time Fourier trasform for aalysis of the local property of sigals. FIGURE.3 (a) The cos-gaussia wavelets h m, (t) ad the real part of the Gabor fuctios g m, (t), with the scale factor s = m ad the traslatio factor τ = s for differet values of m. The wavelets have a dilated widow. The Gabor fuctios have a widow with fixed width. (b) Time-scale joit represetatio of the wavelet trasform ad time-frequecy joit represetatio of the Gabor trasform for a step fuctio. (From Szu, H. et al. Appl. Optics, 3(7), 99. Freema, M.O. Photoics New, August 995, 8-4. With permissio.) by CRC Press LLC

19 Figure.3b shows a compariso betwee the wavelet trasform ad the Gabor trasform for a step fuctio iput. The wavelets are with the dilated widows. The Gabor fuctios are with widows of fixed width. The time-scale joit represetatio, log s t, of the wavelet trasform ad the time-frequecy joit represetatio, log ω t, of the Gabor trasform are also show. The wavelet trasform with a very small scale, s, ad a very arrow widow is able to zoom i o the discotiuity ad to idicate the arrival time of the step sigal...4 Liear Trasform Property By defiitio the wavelet trasform is a liear operatio. Give a fuctio f(t), its wavelet trasform W f (s,τ), satisfies the followig relatios: Liear superpositio without the cross terms: Traslatio: Rescale: W s, τ W s, τ W s, τ f f f f + = + W s W s t f( t t )(, τ)= f( t), τ = W s, τ W αs ατ α f( αt) f( t), Differet from the stadard Fourier trasform ad other trasforms, the wavelet trasform is ot ready for closed form solutio apart from some very simple fuctios such as:. For f(t) =, from the defiitio (..4) ad the admissible coditio of the wavelets, (..4) we have W f ( s,τ )= The wavelet trasform of a costat is equal to zero.. For a siusoidal fuctio f(t) = exp(jω t), we have directly from the Fourier trasform of the wavelets (..6) that = ( ) ( ) * Wf s,τ sh sω exp jω τ The wavelet trasform of a siusoidal fuctio is a siusoidal fuctio of the time shift, τ. Its modulus W f (s,τ) depeds oly o the scale, s. 3. For a liear fuctio f(t) = t, we have Wf ( s,τ )= s th * t τ dt s = 3 * = s th t τ dt Hece, if the wavelet h(t) is regular ad of order so that its derivatives of first order is equal to zero at ω =, the wavelet trasform of f(t) = t is equal to zero. s j 3 dh * dω ( ω) ω = by CRC Press LLC

20 For most fuctios the wavelet trasforms have o closed aalytical solutios ad ca be calculated oly by umerical computer or by optical aalog computer. The optical cotiuous wavelet trasform is based o the explicit defiitio of the wavelet trasform ad implemeted usig a ba of optical wavelet trasform filters i the Fourier plae i a optical correlator. FIGURE.4 Haar basic wavelet h(t) ad its Fourier trasform H(ω). (From Sheg, Y. et al. Opt. Eg., 3, 84, 99. With permissio.) Wavelet Trasform of Regular Sigals Accordig to what was discussed above, the wavelet trasform of a costat is zero, ad the wavelet trasform of a liear sigal is zero, if the wavelet has the first order vaishig momet: M =. The wavelet trasform of a quadratic sigal could be zero, if the wavelet has the first ad secod order vaishig momets: M = M =. The wavelet trasform of a polyomial sigal of degree m could be equal to zero, if the wavelet has the vaishig momets up to the order m. The wavelet trasform is efficiet for detectig sigularities ad aalyzig ostatioary, trasiet sigals...5 Examples of the Wavelets I this sectio we give some examples of the wavelets, useful maily for the cotiuous wavelet trasform. Examples of the wavelets for the discrete orthoormal wavelet trasform will be give i Sectio.8. Haar Wavelet The Haar wavelet was historically itroduced by Haar 5 i 9. It is a bipolar step fuctio: The Haar wavelet ca be writte as a correlatio betwee a dilated rectagle fuctio rect(t) ad two delta fuctios: The rectagular fuctio is defied as: ht ()= whe < t < whe < t < otherwise ht ()= t t 3 rect rect = rect( t) δ t δ t 4 4 rect t ()= whe < t < otherwise by CRC Press LLC

21 The Haar wavelet is real-valued ad atisymmetric with respect to t = /, as show i Figure.4. The wavelet admissible coditio (..4) is satisfied. The Fourier trasform of the Haar wavelet is complex valued ad is equal to the product of a sie fuctio ad a sic fuctio. ω ω ω H( ω)= j exp j sic si 4 4 ω ω = j cos 4 exp j ω (..) whose amplitude is eve ad symmetric to ω =. That is a bad-pass filter, as show i Figure.4. The phase factor exp( jω/) is related oly to the shift of h(t) to t = /, which is ecessary for the causal filterig of time sigals. The Haar wavelet trasform ivolves a ba of multiresolutio filters that yield the correlatios betwee the sigal ad the Haar wavelets scaled by factor, s. The Haar wavelet trasform is a local operatio i the time domai. The time resolutio depeds o the scale, s. Whe the sigal is costat, the Haar wavelet trasform is equal to zero. The amplitude of the Haar wavelet trasform has high pea values whe there are discotiuities of the sigal. The Haar wavelet is also irregular. It is discotiuous ad its first order momet is ot zero. Accordig to (..), the amplitude of the Fourier spectrum of the Haar wavelet coverges to zero very slowly as /ω. Accordig to (..8), the Haar wavelet trasform decays with icreasig of /s as (/s) 3/. The set of discrete dilatios ad traslatios of the Haar wavelets costitute the simplest discrete orthoormal wavelet basis. We shall use the Haar wavelets as a example of the orthoormal wavelet basis i Sectios.5 ad.7. However, the Haar wavelet trasform has ot foud may practical applicatios because of its poor localizatio property i the frequecy domai. Gaussia Wavelet The Gaussia fuctio is perfectly local i both time ad frequecy domais ad is ifiitely derivable. I fact, a derivative of ay order,, of the Gaussia fuctio may be a wavelet. The Fourier trasform of the th order derivative of the Gaussia fuctio is ω H( ω)= ( jω) exp that is the Gaussia fuctio multiplied by (jω), so that H() =. The wavelet admissible coditio is satisfied. Its derivatives up to th order H ( ) () =. The Gaussia wavelet is a regular wavelet of order. Both h(t) ad H(ω) are ifiitely derivable. The wavelet trasform coefficiets decay with icreasig of /s as fast as (/s) (/). Mexica Hat Wavelet The Mexica hat-lie wavelet was first itroduced by Gabor. It is the secod order derivative of the Gaussia fuctio: 7 ht ()= t ( t ) exp by CRC Press LLC

22 The Mexica hat wavelet is eve ad real valued. The wavelet admissible coditio is satisfied. The Fourier trasform of the Mexica hat wavelet is ω H ( ω)= ω exp that is also eve ad real valued, as show i Figure.5. The two-dimesioal Mexica hat wavelet is well ow as the Laplacia operator, widely used for zero-crossig image edge detectio. FIGURE.5 Mexica hat wavelet h(t) ad its Fourier trasform H(ω). (From Sheg, Y. et al. Opt. Eg., 3, 84, 99. With permissio.) Gabor Wavelet The Gabor fuctio i the short time Fourier trasform with Gaussia widow is t ht ()= ( j t) τ exp ω exp Its real part is a cosie-gaussia ad the imagiary part is a sie-gaussia fuctio. The Gaussia widow has a fixed width ad is shifted alog the time axis by τ. The Fourier trasform of the Gabor fuctio is a Gaussia widow which is shifted alog the frequecy axis by ω, as discussed i Subsectio..3. The Gabor fuctio ca also have a dilated widow as: ht exp j t ()= exp t s ω where s is the scale factor ad is the width of the Gaussia widow, exp(jω t) itroduces a traslatio of the spectrum of the Gabor fuctio. The Gabor fuctio with the dilatio of both widow ad Fourier erel is the Gabor wavelet. This wavelet was used by Martiet, Morlet, ad Grossma for aalysis of soud patters. The Morlet s basic wavelet fuctio is a multiplicatio of the Fourier basis with a Gaussia widow. Its real part is the Cosie-Gaussia wavelet, whose Fourier trasform cosists of two Gaussia fuctios shifted to ω ad ω, respectively: by CRC Press LLC

23 ω ω ω ω H ( ω)= ππ + + exp exp that are real positive-valued, eve, ad symmetric to the origi ω =. The Gaussia widow is perfectly local i both time ad frequecy domais ad achieves the miimum time-badwidth product determied by the ucertaity priciple, as show by (..9). The cosie-gaussia wavelets are bad-pass filters i frequecy domai. They coverge to zero lie the Gaussia fuctio as the frequecy icreases. Figure.6 shows the cosie-gaussia wavelet ad its Fourier spectrum. FIGURE.6 Cos-Gaussia wavelet h(t) ad its Fourier trasform H(ω). (From Sheg, Y. et al. Opt. Eg., 3, 84, 99. With permissio.) The Gabor wavelets do ot satisfy the wavelet admissible coditio, because: H that leads to c h = +. But the value of H() is very close to zero provided that the ω is sufficietly large. Whe ω = 5, for example, 3 = exp H π that is of 5 order of magitude ad ca be practically cosidered as zero i umerical computatios..3 Discrete Wavelet Trasform The cotiuous wavelet trasform maps a oe-dimesioal time sigal to a two dimesioal time-scale joit represetatio. The time badwidth product of the cotiuous wavelet trasform output is the square of that of the sigal. For most applicatios, however, the goal of sigal processig is to represet the sigal efficietly with fewer parameters. The use of the discrete wavelet trasform ca reduce the time badwidth product of the wavelet trasform output. By the term discrete wavelet trasform, we mea, i fact, the cotiuous wavelets with the discrete scale ad traslatio factors. The wavelet trasform is the evaluated at discrete scales ad traslatios. The discrete scale is expressed as s = s i, where i is iteger ad s > is a fixed dilatio step. The discrete traslatio factor is expressed as τ = τ s i, where is a iteger. The traslatio depeds o the dilatio step, s i. The correspodig discrete wavelets are writte as: 5 by CRC Press LLC

24 h t s h s t τ s i i i i, ()= ( ) ( ) i i = s h s t τ (.3.) The discrete wavelet trasform with the dyadic scalig factor of s =, is effective i the computer implemetatio..3. Time-Scale Space Lattices The discrete wavelet trasform is evaluated at discrete times ad scales that correspod to a samplig i the time-scale space. The time-scale joit represetatio of a discrete wavelet trasform is a grid alog the scale ad time axes. To show this, we cosider localizatio poits of the discrete wavelets i the time-scale space. FIGURE.7 Localizatio of the discrete wavelets i the time-scale space. The samplig alog the time axis has the iterval, τ s i, that is proportioal to the scale, s i. The time samplig step is small for small-scale wavelet aalysis ad is large for large-scale wavelet aalysis. With the varyig scale, the wavelet aalysis will be able to zoom i o sigularities of the sigal usig more cocetrated wavelets of very small scale. For this detailed aalysis the time samplig step is very small. Because oly the sigal detail is of iterest, oly a few small time traslatio steps would be eeded. Therefore, the wavelet aalysis provides a more efficiet way to represet trasiet sigals. There is a aalogy betwee the wavelet aalysis ad the microscope. The scale factor s i correspods to the magificatio or the resolutio of a microscope. The traslatio factor, τ, correspods to the locatio where oe maes observatio with the microscope. If oe loos at very small details, the magificatio ad the resolutio must be large ad that correspods to a large ad egative i. The wavelet is very cocetrated. The step of traslatio is small, that justifies the choice, τ = τ s i. For large ad positive i, the wavelet is spread out, ad the large traslatio steps, τ s i are adapted to this wide width of the wavelet aalysis fuctio. This is aother iterpretatio of the costat-q aalysis property of the wavelet trasform, discussed i Sectio..3. The behavior of the discrete wavelets depeds o the steps, s ad τ. Whe s is close to ad τ is small, the discrete wavelets are close to the cotiuous wavelets. For a fixed scale step s, the localizatio poits of the discrete wavelets alog the scale axis are logarithmic as log s = i log s, as show i Figure.7. The uit of the frequecy samplig iterval is called a octave i music. Oe octave is the iterval betwee two frequecies havig a ratio of two. Oe octave frequecy bad has the badwidth equal to oe octave. The discrete time step is τ s i. We choose usually τ =. Hece, the time samplig step is a fuctio of the scale ad is equal to i for the dyadic wavelets. Alog the τ-axis, the localizatio poits of the discrete wavelets depeds o the scale. The itervals betwee the localizatio poits at the same scale by CRC Press LLC

25 are equal ad are proportioal to the scale s i. The traslatio steps are small for small positive values of i with the small scale wavelets, ad are large for large positive values of i with large scale wavelets. The localizatio of the discrete wavelets i the time-scale space is show i Figure.7, where the scale axis is logarithmic, log s = i log, ad the localizatio is uiform alog the time axis τ with the time steps proportioal to the scale factor s = i..3. Wavelet Frame With the discrete wavelet basis a cotiuous fuctio f(t) is decomposed ito a sequece of wavelet coefficiets: = () () = * W i, fth tdt fh, f i, i, (.3.) A questio for the discrete wavelet trasform is how well the fuctio f(t) ca be recostructed from the discrete wavelet coefficiets: f t A W f i, hi, t ()= i (.3.3) where A is a costat that does ot deped o f(t). Obviously, if s is close eough to ad τ is small eough, the wavelets approach as a cotiuum. The recostructio (.3.3) is the close to the iverse cotiuous wavelet trasform. The sigal recostructio taes place without orestrictive coditios other tha the admissible coditio o the wavelet h(t). O the other had, if the samplig is sparse, s = ad τ =, the recostructio (.3.3) ca be achieved oly for very special choices of the wavelet h(t). The theory of wavelet frames provides a geeral framewor that covers the above metioed two extreme situatios. It permits oe to balace betwee the redudacy, i.e., the samplig desity i the scale-time space, ad the restrictio o the wavelet h(t) for the recostructio scheme (.3.3) to wor. If the redudacy is large with high over-samplig, the oly mild restrictios are put o the wavelet basis. If the redudacy is small with critical samplig, the the wavelet base are very costraied. Daubechies 8 has prove that the ecessary ad sufficiet coditio for the stable recostructio of a fuctio f(t) from its wavelet coefficiets, W f (i,), is that the eergy, which is the sum of square moduli of W f (i,), must lie betwee two positive bouds: Af f, hi, Bf j, (.3.4) where f is the eergy of f(t), A >, B < ad A, B are idepedet of f(t). Whe A = B, the eergy of the wavelet trasform is proportioal to the eergy of the sigal. This is similar to the eergy coservatio relatio (..5) of the cotiuous wavelet trasform. Whe A B, there is still some proportioal relatio betwee the two eergies. Whe (.3.4) is satisfied, the family of erel fuctios {h i, (t)} with i, Z is referred to as a frame ad A, B are termed frame bouds. Hece, whe proportioality betwee the eergy of the fuctio ad the eergy of its discrete trasform fuctio is bouded betwee somethig greater tha zero ad less tha ifiity for all possible square itegrable fuctios, the the trasform is complete. No iformatio is lost ad the sigal ca be recostructed from its decompositio. Daubechies has show that the accuracy of the recostructio is govered by the frame bouds A ad B. The frame bouds A ad B ca be computed from s, τ ad the wavelet basis, h(t). The closer A ad B, the more accurate the recostructio. Whe A = B the frame is tight ad the discrete wavelets behave exactly lie a orthoormal basis. Whe A = B =, (.3.4) is simply the eergy coservatio equivalet by CRC Press LLC