The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients

Transcription

1 OpeStax-CNX module: m The Scalig Fuctio ad Scalig Coefficiets, Wavelet ad Wavelet Coefficiets C. Sidey Burrus This work is produced by OpeStax-CNX ad licesed uder the Creative Commos Attributio Licese 3.0 We will ow look more closely at the basic scalig fuctio ad wavelet to see whe they exist ad what their properties are [9], [21], [14], [16], [15], [20], [6]. Usig the same approach that is used i the theory of dieretial equatios, we will examie the properties of φ (t) by cosiderig the equatio of which it is a solutio. The basic recursio here 1 that comes from the multiresolutio formulatio is φ (t) = h () 2 φ (2t ) (1) with h () beig the scalig coeciets ad φ (t) beig the scalig fuctio which satises this equatio which is sometimes called the reemet equatio, the dilatio equatio, or the multiresolutio aalysis equatio (MRA). I order to state the properties accurately, some care has to be take i specifyig just what classes of fuctios are beig cosidered or are allowed. We will attempt to walk a e lie to preset eough detail to be correct but ot so much as to obscure the mai ideas ad results. A few of these ideas were preseted i Sectio: Sigal Spaces 2 ad a few more will be give i the ext sectio. A more complete discussio ca be foud i [33], i the itroductios to [34], [36], [1], or i ay book o fuctio aalysis. 1 Tools ad Deitios 1.1 Sigal Classes There are three classes of sigals that we will be usig. The most basic is called L 2 (R) which cotais all fuctios which have a ite, well-deed itegral of the square: f L 2 f (t) 2 dt = E <. This class is importat because it is a geeralizatio of ormal Euclidea geometry ad because it gives a simple represetatio of the eergy i a sigal. The ext most basic class is L 1 (R), which requires a ite itegral of the absolute value of the fuctio: f L 1 f (t) dt = K <. This class is importat because oe may iterchage iite summatios ad itegratios with these fuctios although ot ecessarily with L 2 fuctios. These classes of fuctio spaces ca be geeralized to those with it f (t) p dt = K < ad desigated L p. Versio 1.3: Feb 11, :15 pm "A multiresolutio formulatio of Wavelet Systems", (13) < 2 "A multiresolutio formulatio of Wavelet Systems": Sectio Sigal Spaces <

2 OpeStax-CNX module: m A more geeral class of sigals tha ay L p space cotais what are called distributios. These are geeralized fuctios which are ot deed by their havig values" but by the value of a ier product" with a ormal fuctio. A example of a distributio would be the Dirac delta fuctio δ (t) where it is deed by the property: f (T ) = f (t) δ (t T ) dt. Aother detail to keep i mid is that the itegrals used i these deitios are Lebesque itegrals which are somewhat more geeral tha the basic Riema itegral. The value of a Lebesque itegral is ot aected by values of the fuctio over ay coutable set of values of its argumet (or, more geerally, a set of measure zero). A fuctio deed as oe o the ratioals ad zero o the irratioals would have a zero Lebesque itegral. As a result of this, properties derived usig measure theory ad Lebesque itegrals are sometime said to be true almost everywhere," meaig they may ot be true over a set of measure zero. May of these ideas of fuctio spaces, distributios, Lebesque measure, etc. came out of the early study of Fourier series ad trasforms. It is iterestig that they are also importat i the theory of wavelets. As with Fourier theory, oe ca ofte igore the sigal space classes ad ca use distributios as if they were fuctios, but there are some cases where these ideas are crucial. For a itroductory readig of this book or of the literature, oe ca usually skip over the sigal space desigatio or assume Riema itegrals. However, whe a cotradictio or paradox seems to arise, its resolutio will probably require these details. 1.2 Fourier Trasforms We will eed the Fourier trasform of φ (t) which, if it exists, is deed to be Φ Φ (ω) = ad the discrete-time Fourier trasform (DTFT) [24] of h () deed to be H (ω) = = φ (t) e iωt dt (2) h () e iω (3) where i = 1 ad is a iteger ( Z). If covolutio with h () is viewed as a digital lter, as deed i Sectio: Aalysis - From Fie Scale to Coarse Scale 3, the the DTFT of h () is the lter's frequecy respose, [24], [25] which is 2π periodic. If Φ (ω) exits, the deig recursive equatio (1) becomes which after iteratio becomes Φ (ω) = 1 2 H (ω/2) Φ (ω/2) (4) Φ (ω) = k=1 { 1 ( ω ) H 2 2 k }Φ (0). (5) if h () = 2 ad Φ (0) is well deed. This may be a distributio or it may be a smooth fuctio depedig o H (ω) ad, therefore, h ()[33], [6]. This makes sese oly if Φ (0) is well deed. Although (1) ad (5) are equivalet term-by-term, the requiremet of Φ (0) beig well deed ad the ature of the limits i the appropriate fuctio spaces may make oe preferable over the other. Notice how the zeros of H (ω) determie the zeros of Φ (ω). 3 "Filter Baks ad the Discrete Wavelet Trasform": Sectio Aalysis From Fie Scale to Coarse Scale <

3 OpeStax-CNX module: m Reemet ad Trasitio Matrices There are two matrices that are particularly importat to determiig the properties of wavelet systems. The rst is the reemet matrixm, which is obtaied from the basic recursio equatio (1) by evaluatig φ (t) at itegers [23], [7], [8], [31], [30]. This looks like a covolutio matrix with the eve (or odd) rows removed. Two particular submatrices that are used later i Sectio 10 (Calculatig the Basic Scalig Fuctio ad Wavelet) to evaluate φ (t) o the dyadic ratioals are illustrated for N = 6 by h h 2 h 1 h h 2 4 h 3 h 2 h 1 h h 5 h 4 h 3 h 2 h h 5 h 4 h 3 φ 0 φ 1 φ 2 φ 3 φ 4 = φ 0 φ 1 φ 2 φ 3 φ 4 (6) which we write i matrix form as h 5 φ 5 φ 5 M 0 φ = φ (7) with M 0 beig the 6 6 matrix of the h () ad φ beig 6 1 vectors of iteger samples of φ (t). I other words, the vector φ with etries φ (k) is the eigevector of M 0 for a eigevalue of uity. The secod submatrix is a shifted versio illustrated by h 1 h h 3 h 2 h 1 h h 2 5 h 4 h 3 h 2 h 1 h h 5 h 4 h 3 h h 5 h φ 0 φ 1 φ 2 φ 3 φ 4 φ 5 = with the matrix beig deoted M 1. The geeral reemet matrix M is the iite matrix of which M 0 ad M 1 are partitios. If the matrix H is the covolutio matrix for h (), we ca deote the M matrix by [ 2] H to idicate the dow-sampled covolutio matrix H. Clearly, for φ (t) to be deed o the dyadic ratioals, M 0 must have a uity eigevalue. A third, less obvious but perhaps more importat, matrix is called the trasitio matrixt ad it is built up from the autocorrelatio matrix of h (). The trasitio matrix is costructed by φ 1/2 φ 3/2 φ 5/2 φ 7/2 φ 9/2 φ 11/2 T = [ 2] HH T. (9) This matrix (sometimes called the Lawto matrix) was used by Lawto (who origially called it the Wavelet- Galerki matrix) [15] to derive ecessary ad suciet coditios for a orthogoal wavelet basis. As we will see later i this chapter, its eigevalues are also importat i determiig the properties of φ (t) ad the associated wavelet system. 2 Necessary Coditios Theorem 1 If φ (t) L 1 is a solutio to the basic recursio equatio (1) ad if φ (t) dt 0, the h () = 2. (10) (8)

4 OpeStax-CNX module: m The proof of this theorem requires oly a iterchage i the order of a summatio ad itegratio (allowed i L 1 ) but o assumptio of orthogoality of the basis fuctios or ay other properties of φ (t) other tha a ozero itegral. The proof of this theorem ad several of the others stated here are cotaied i Appedix A. This theorem shows that, ulike liear costat coeciet dieretial equatios, ot just ay set of coeciets will support a solutio. The coeciets must satisfy the liear equatio (10). This is the weakest coditio o the h (). Theorem 2If φ (t) is a L 1 solutio to the basic recursio equatio (1) with φ (t) dt = 1, ad φ (t l) = φ (l) = 1 (11) l l with Φ (π + 2πk) 0 for some k, the h (2) = h (2 + 1) (12) where (11) may have to be a distributioal sum. Coversely, if (12) is satised, the (11) is true. Equatio (12) is called the fudametal coditio, ad it is weaker tha requirig orthogoality but stroger tha (10). It is simply a result of requirig the equatios resultig from evaluatig (1) o the itegers be cosistet. Equatio (11) is called a partitioig of uity (or the Strag coditio or the Shoeberg coditio). A similar theorem by Cavaretta, Dahma ad Micchelli [3] ad by Jia [12] states that if φ L p ad the iteger traslates of φ (t) form a Riesz basis for the space they spa, the h (2) = h (2 + 1). Theorem 3 If φ (t) is a L 2 L 1 solutio to (1) ad if iteger traslates of φ (t) are orthogoal as deed by the φ (t) φ (t k) dt = E δ (k) = { E if k = 0 0 otherwise, h () h ( 2k) = δ (k) = { 1 if k = 0 0 otherwise, Notice that this does ot deped o a particular ormalizatio of φ (t). If φ (t) is ormalized by dividig by the square root of its eergy E, the iteger traslates of φ (t) are orthoormal deed by φ (t) φ (t k) dt = δ (k) = { 1 if k = 0 0 otherwise, This theorem shows that i order for the solutios of (1) to be orthogoal uder iteger traslatio, it is ecessary that the coeciets of the recursive equatio be orthogoal themselves after decimatig or dowsamplig by two. If φ (t) ad/or h () are complex fuctios, complex cojugatio must be used i (13), (14), ad (15). Coeciets h () that satisfy (14) are called a quadrature mirror lter (QMF) or cojugate mirror lter (CMF), ad the coditio (14) is called the quadratic coditio for obvious reasos. Corollary 1 Uder the assumptios of Theorem p. 4, the orm of h () is automatically uity. h () 2 = 1 (16) (13) (14) (15)

5 OpeStax-CNX module: m Not oly must the sum of h () equal 2, but for orthogoality of the solutio, the sum of the squares of h () must be oe, both idepedet of ay ormalizatio of φ (t). This uity ormalizatio of h () is the result of the 2 term i (1). Corollary 2 Uder the assumptios of Theorem p. 4, h (2) = h (2 + 1) = 1 2 (17) This result is derived i the Appedix by showig that ot oly must the sum of h () equal 2, but for orthogoality of the solutio, the idividual sums of the eve ad odd terms i h () must be 1/ 2, idepedet of ay ormalizatio of φ (t). Although stated here as ecessary for orthogoality, the results hold uder weaker o-orthogoal coditios as is stated i Theorem p. 4. Theorem 4If φ (t) has compact support o 0 t N 1ad if φ (t k) are liearly idepedet, the h ()also has compact support over 0 N 1: h () = 0 for < 0 ad > N 1 (18) Thus N is the legth of the h () sequece. If the traslates are ot idepedet (or some equivalet restrictio), oe ca have h () with iite support while φ (t) has ite support [27]. N These theorems state that if φ (t) has compact support ad is orthogoal over iteger traslates, 2 biliear or quadratic equatios (14) must be satised i additio to the oe liear equatio (10). The support or legth of h () is N, which must be a eve umber. The umber of degrees of freedom i choosig these N coeciets is the N 2 1. This freedom will be used i the desig of a wavelet system developed i Chapter: Regularity, Momets, ad Wavelet System Desig ad elsewhere. 3 Frequecy Domai Necessary Coditios We tur ext to frequecy domai versios of the ecessary coditios for the existece of φ (t). Some care must be take i specifyig the space of fuctios that the Fourier trasform operates o ad the space that the trasform resides i. We do ot go ito those details i this book but the reader ca cosult [33]. Theorem 5 If φ (t) is a L 1 solutio of the basic recursio equatio (1), the the followig equivalet coditios must be true: h () = H (0) = 2 (19) This follows directly from (3) ad states that the basic existece requiremet (10) is equivalet to requirig that the FIR lter's frequecy respose at DC (ω = 0) be 2. Theorem 6 For h () l 1, the h (2) = h (2 + 1) if ad oly if H (π) = 0 (20) which says the frequecy respose of the FIR lter with impulse respose h () is zero at the so-called Nyquist frequecy (ω = π). This follows from (4) ad here 4, ad supports the fact that h () is a lowpass digital lter. This is also equivalet to the M ad T matrices havig a uity eigevalue. Theorem 7 If φ (t) is a solutio to (1) i L 2 L 1 ad Φ (ω) is a solutio of (4) such that Φ (0) 0, the φ (t) φ (t k) dt = δ (k) if ad oly if Φ (ω + 2πl) 2 = 1 (21) 4 "Geeralizatios of the Basic Multiresolutio Wavelet System", (7) < l

6 OpeStax-CNX module: m This is a frequecy domai equivalet to the time domai deitio of orthogoality of the scalig fuctio [21], [22], [6]. It allows applyig the orthoormal coditios to frequecy domai argumets. It also gives isight ito just what time domai orthogoality requires i the frequecy domai. Theorem 8 For ay h () l 1, h () h ( 2k) = δ (k) if ad oly if H (ω) 2 + H (ω + π) 2 = 2 (22) This theorem [14], [11], [6] gives equivalet time ad frequecy domai coditios o the scalig coef- ciets ad states that the orthogoality requiremet (14) is equivalet to the FIR lter with h () as coeciets beig what is called a Quadrature Mirror Filter (QMF) [29]. Note that (22), (19), ad (20) require H (π/2) = 1 ad that the lter is a half bad" lter. 4 Suciet Coditios The above are ecessary coditios for φ (t) to exist ad the followig are suciet. There are may forms these could ad do take but we preset the followig as examples ad give refereces for more detail [9], [21], [14], [16], [15], [20], [6], [19], [18], [17]. Theorem 9 If h () = 2 ad h () has ite support or decays fast eough so that h () (1 + ) ε < for some ε > 0, the a uique (withi a scalar multiple) φ (t) (perhaps a distributio) exists that satises (1) ad whose distributioal Fourier trasform satises (5). This [9], [14], [13] ca be obtaied i the frequecy domai by cosiderig the covergece of (5). It has recetly bee obtaied usig a much more powerful approach i the time domai by Lawto [17]. Because this theorem uses the weakest possible coditio, the results are weak. The scalig fuctio obtaied from oly requirig h () = 2 may be so poorly behaved as to be impossible to calculate or use. The worst cases will ot support a multiresolutio aalysis or provide a useful expasio system. Theorem 10 If h (2) = h (2 + 1) = 1/ 2 ad h () has ite support or decays fast eough so that h () (1 + ) ε < for some ε > 0, the a φ (t) (perhaps a distributio) that satises (1) exists, is uique, ad is well-deed o the dyadic ratioals. I additio, the distributioal sum φ (t k) = 1 (23) k holds. This coditio, called the fudametal coditio [31], [19], gives a slightly tighter result tha Theorem p. 6. While the scalig fuctio still may be a distributio ot i L 1 or L 2, it is better behaved tha required by Theorem p. 6 i beig deed o the dese set of dyadic ratioals. This theorem is equivalet to requirig H (π) = 0 which from the product formula (5) gives a better behaved Φ (ω). It also guaratees a uity eigevalue for M ad T but ot that other eigevalues do ot exist with magitudes larger tha oe. The ext several theorems use the trasitio matrix T deed i (9) which is a dow-sampled autocorrelatio matrix. Theorem 11 If the trasitio matrix T has eigevalues o or i the uit circle of the complex plae ad if ay o the uit circle are multiple, they have a complete set of eigevectors, the φ (t) L 2. If T has uity magitude eigevalues, the successive approximatio algorithm (cascade algorithm) (71) coverges weakly to φ (t) L 2 [13]. Theorem 12 If the trasitio matrix T has a simple uity eigevalue with all other eigevalues havig magitude less tha oe, the φ (t) L 2. Here the successive approximatio algorithm (cascade algorithm) coverges strogly to φ (t) L 2. This is developed i [31]. If i additio to requirig (10), we require the quadratic coeciet coditios (14), a tighter result occurs which gives φ (t) L 2 (R) ad a multiresolutio tight frame system.

7 OpeStax-CNX module: m Theorem 13 (Lawto) If h () has ite support or decays fast eough ad if h () = 2 ad if h () h ( 2k) = δ (k), the φ (t) L2 (R) exists, ad geerates a wavelet system that is a tight frame i L 2. This importat result from Lawto [14], [16] gives the suciet coditios for φ (t) to exist ad geerate wavelet tight frames. The proof uses a iteratio of the basic recursio equatio (1) as a successive approximatio similar to Picard's method for dieretial equatios. Ideed, this method is used to calculate φ (t) i Sectio 10 (Calculatig the Basic Scalig Fuctio ad Wavelet). It is iterestig to ote that the scalig fuctio may be very rough, eve fractal" i ature. This may be desirable if the sigal beig aalyzed is also rough. Although this theorem guaratees that φ (t) geerates a tight frame, i most practical situatios, the resultig system is a orthoormal basis [16]. The coditios i the followig theorems are geerally satised. Theorem 14 (Lawto) If h() has compact support, h () = 2, ad h () h ( 2k) = δ (k), the φ (t k) forms a orthogoal set if ad oly if the trasitio matrix T has a simple uity eigevalue. This powerful result allows a simple evaluatio of h () to see if it ca support a wavelet expasio system [14], [16], [15]. A equivalet result usig the frequecy respose of the FIR digital lter formed from h () was give by Cohe. Theorem 15 (Cohe) If H (ω) is the DTFT of h () with compact support ad h () = 2 with h () h ( 2k) = δ (k),ad if H (ω) 0 for π/3 ω π/3, the the φ (t k) satisfyig (1) geerate a orthoormal basis i L 2. A slightly weaker versio of this frequecy domai suciet coditio is easier to prove [21], [22] ad to exted to the M-bad case for the case of o zeros allowed i π/2 ω π/2[6]. There are other suciet coditios that, together with those i Theorem p. 6, will guaratee a orthoormal basis. Daubechies' vaishig momets will guaratee a orthogoal basis. Theorems,, ad show that h () has the characteristics of a lowpass FIR digital lter. We will later see that the FIR lter made up of the wavelet coeciets is a high pass lter ad the lter bak view developed i Chapter: Filter Baks ad the Discrete Wavelet Trasform 5 ad Sectio: Multiplicity-M (M-Bad) Scalig Fuctios ad Wavelets 6 further explais this view. Theorem 16 If h () has ite support ad if φ (t) L 1, the φ (t) has ite support [13]. If φ (t) is ot restricted to L 1, it may have iite support eve if h () has ite support. These theorems give a good picture of the relatioship betwee the recursive equatio coeciets h () ad the scalig fuctio φ (t) as a solutio of (1). More properties ad characteristics are preseted i Sectio 8 (Further Properties of the Scalig Fuctio ad Wavelet). 4.1 Wavelet System Desig Oe of the mai purposes for presetig the rather theoretical results of this chapter is to set up the coditios for desigig wavelet systems. Oe approach is to require the miimum suciet coditios as costraits i a optimizatio or approximatio, the use the remaiig degrees of freedom to choose h () that will give the best sigal represetatio, decompositio, or compressio. I some cases, the suciet coditios are overly restrictive ad it is worthwhile to use the ecessary coditios ad the check the desig to see if it is satisfactory. I may cases, wavelet systems are desiged by a frequecy domai desig of H (ω) usig digital lter desig techiques with wavelet based costraits. 5 The Wavelet Although this chapter is primarily about the scalig fuctio, some basic wavelet properties are icluded here. 5 "Filter Baks ad the Discrete Wavelet Trasform" < 6 "Geeralizatios of the Basic Multiresolutio Wavelet System": Sectio Multiplicity-M (M-Bad) Scalig Fuctios ad Wavelets <

8 OpeStax-CNX module: m Theorem 17 If the scalig coeciets h () satisfy the coditios for existece ad orthogoality of the scalig fuctio ad the wavelet is deed by here 7, the the iteger traslates of this wavelet spa W 0, the orthogoal complimet of V 0, both beig i V 1, i.e., the wavelet is orthogoal to the scalig fuctio at the same scale, φ (t ) ψ (t m) dt = 0, (24) if ad oly if the coeciets h 1 () are give by h 1 () = ±( 1) h (N ) (25) where N is a arbitrary odd iteger chose to coveietly positio h 1 (). A outlie proof is i Appedix A. Theorem 18 If the scalig coeciets h () satisfy the coditios for existece ad orthogoality of the scalig fuctio ad the wavelet is deed by here 8, the the iteger traslates of this wavelet spa W 0, the orthogoal complimet of V 0, both beig i V 1 ; i.e., the wavelet is orthogoal to the scalig fuctio at the same scale. If φ (t ) ψ (t m) dt = 0 (26) the h () h 1 ( 2k) = 0 (27) which is derived i Appedix A, here 9. The traslatio orthogoality ad scalig fuctio-wavelet orthogoality coditios i (14) ad (27) ca be combied to give h l () h m ( 2k) = δ (k) δ (l m) (28) if h 0 () is deed as h (). Theorem 19 If h () satises the liear ad quadratic admissibility coditios of (10) ad (14), the h 1 () = H 1 (0) = 0, (29) H 1 (ω) = H (ω + π), (30) H (ω) 2 + H 1 (ω) 2 = 2, (31) ad ψ (t) dt = 0. (32) The wavelet is usually scaled so that its orm is uity. The results i this sectio have ot icluded the eects of iteger shifts of the scalig fuctio or wavelet coeciets h () or h 1 (). I a particular situatio, these sequeces may be shifted to make the correspodig FIR lter causal. 7 "A multiresolutio formulatio of Wavelet Systems", (24) < 8 "A multiresolutio formulatio of Wavelet Systems", (24) < 9 "Appedix A", (40) <

9 OpeStax-CNX module: m Alterate Normalizatios A alterate ormalizatio of the scalig coeciets is used by some authors. I some ways, it is a cleaer form tha that used here, but it does ot state the basic recursio as a ormalized expasio, ad it does ot result i a uity orm for h (). The alterate ormalizatio uses the basic multiresolutio recursive equatio with o 2 φ (t) = h () φ (2t ). (33) Some of the relatioships ad results usig this ormalizatio are: h () = 2 h () 2 = 2 h () h (h 2k) = 2 δ (k) h (2) = h (2 + 1) = 1 H (0) = 2 H (ω) 2 + H (ω + π) 2 = 4 (34) A still dieret ormalizatio occasioally used has a factor of 2 i (33) rather tha 2 or uity, givig h () = 1. Other obvious modicatios of the results i other places i this book ca be worked out. Take care i usig scalig coeciets h () from the literature as some must be multiplied or divided by 2 to be cosistet with this book. 7 Example Scalig Fuctios ad Wavelets Several of the moder wavelets had ever bee see or described before the 1980's. This sectio looks at some of the most commo wavelet systems. 7.1 Haar Wavelets The oldest ad most basic of the wavelet systems that has most of our desired properties is costructed from the Haar basis fuctios. If oe chooses a legth N = 2 scalig coeciet set, after satisfyig the ecessary coditios i (10) ad (14), there are o remaiig degrees or freedom. The uique (withi ormalizatio) coeciets are h () = { 1 2, 1 2 } (35) ad the resultig ormalized scalig fuctio is φ (t) = { 1 for 0 < t < 1 0 otherwise. (36) The wavelet is, therefore, ψ (t) = { 1 for 0 < t < 1/2 1 for 1/2 < t < 1 0 otherwise. (37)

10 OpeStax-CNX module: m Their satisfyig the multiresolutio equatio here 10 is illustrated i Figure: Haar ad Triagle Scalig Fuctios. Haar showed that traslates ad scaligs of these fuctios form a orthoormal basis for L 2 (R). We ca easily see that the Haar fuctios are also a compact support orthoormal wavelet system that satisfy Daubechies' coditios. Although they are as regular as ca be achieved for N = 2, they are ot eve cotiuous. The orthogoality ad estig of spaed subspaces are easily see because the traslates have o overlap i the time domai. It is istructive to apply the various properties of Sectio 5 (The Wavelet) ad Sectio 8 (Further Properties of the Scalig Fuctio ad Wavelet) to these fuctios ad see how they are satised. They are illustrated i the example i Figure: Haar Scalig Fuctios ad Wavelets 11 that Spa V j through Figure: Haar Fuctio Approximatio i V j. 7.2 Sic Wavelets The ext best kow (perhaps the best kow) basis set is that formed by the sic fuctios. The sic fuctios are usually preseted i the cotext of the Shao samplig theorem, but we ca look at traslates of the sic fuctio as a orthoormal set of basis fuctios (or, i some cases, a tight frame). They, likewise, usually form a orthoormal wavelet system satisfyig the various required coditios of a multiresolutio system. The sic fuctio is deed as si (t) sic (t) = (38) t where sic (0) = 1. This is a very versatile ad useful fuctio because its Fourier trasform is a simple rectagle fuctio ad the Fourier trasform of a rectagle fuctio is a sic fuctio. I order to be a scalig fuctio, the sic must satisfy here 12 as sic (Kt) = h () sic (K2t K) (39) for the appropriate scalig coeciets h () ad some K. If we costruct the scalig fuctio from the geeralized samplig fuctio as preseted i here 13, the sic fuctio becomes sic (Kt) = ( π sic (KT ) sic RT t π ) R. (40) I order for these two equatios to be true, the samplig period must be T = 1/2 ad the parameter K = π R (41) which gives the scalig coeciets as ( π ) h () = sic 2R. (42) We see that φ (t) = sic (Kt) is a scalig fuctio with iite support ad its correspodig scalig coeciets are samples of a sic fuctio. It R = 1, the K = π ad the scalig fuctio geerates a orthogoal wavelet system. For R > 1, the wavelet system is a tight frame, the expasio set is ot orthogoal or a basis, ad R is the amout of redudacy i the system as discussed i this chapter. For the orthogoal sic scalig fuctio, the wavelet is simply expressed by ψ (t) = 2 φ (2t) φ (t). (43) 10 "A multiresolutio formulatio of Wavelet Systems", (13) < 11 "A multiresolutio formulatio of Wavelet Systems", Figure 15 < 12 "A multiresolutio formulatio of Wavelet Systems", (13) < 13 "Bases, Orthogoal Bases, Biorthogoal Bases, Frames, Tight Frames, ad ucoditioal Bases", (35) <

11 OpeStax-CNX module: m The sic scalig fuctio ad wavelet do ot have compact support, but they do illustrate a iitely dieretiable set of fuctios that result from a iitely log h (). The orthogoality ad multiresolutio characteristics of the orthogoal sic basis is best see i the frequecy domai where there is o overlap of the spectra. Ideed, the Haar ad sic systems are Fourier duals of each other. The sic geeratig scalig fuctio ad wavelet are show i Figure 1. Figure 1: Sic Scalig Fuctio ad Wavelet 7.3 Splie ad Battle-Lemarié Wavelet Systems The triagle scalig fuctio illustrated i Figure: Haar ad Triagle Scalig Fuctios is a special case of a more geeral family of splie scalig fuctios. The scalig coeciet system h () = { 1 2, 2 1 1, 2 2, 0} 2 gives rise to the piecewise liear, cotiuous triagle scalig fuctio. This fuctio is a rst-order splie, beig a cocateatio of two rst order polyomials to be cotiuous at the juctios or kots". A quadratic splie is geerated from h = {1/4, 3/4, 3/4, 1/4}/ 2 as three sectios of secod order polyomials coected to give cotiuous rst order derivatives at the juctios. The cubic splie is geerated from h () = {1/16.1/4, 3/8, 1/4, 1/16}/ 2. This is geeralized to a arbitrary Nth order splie with cotiuous (N 1)th order derivatives ad with compact support of N + 1. These fuctios have excellet mathematical properties, but they are ot orthogoal over iteger traslatio. If orthogoalized, their support becomes iite (but rapidly decayig) ad they geerate the Battle-Lemarié wavelet system" [6], [31], [4], [5]. Figure 2 illustrates the rst-order splie scalig fuctio which is the triagle fuctio alog with the secod-, third-, ad fourth-order splie scalig fuctios.

12 OpeStax-CNX module: m Figure 2: Splie Scalig Fuctios 8 Further Properties of the Scalig Fuctio ad Wavelet The scalig fuctio ad wavelet have some remarkable properties that should be examied i order to uderstad wavelet aalysis ad to gai some ituitio for these systems. Likewise, the scalig ad wavelet coeciets have importat properties that should be cosidered. We ow look further at the properties of the scalig fuctio ad the wavelet i terms of the basic deig equatios ad restrictios. We also cosider the relatioship of the scalig fuctio ad wavelet to the equatio coeciets. A multiplicity or rak of two is used here but the more geeral multiplicity-m case is easily derived from these (See Sectio: Multiplicity-M (M-Bad) Scalig Fuctios ad Wavelets 14 ad Appedix B 15 ). Derivatios or proofs for some of these properties are icluded i Appedix B 16. The basic recursive equatio for the scalig fuctio, deed i (1) as φ (t) = h () 2 φ (2t ), (44) 14 "Geeralizatios of the Basic Multiresolutio Wavelet System": Sectio Multiplicity-M (M-Bad) Scalig Fuctios ad Wavelets < 15 "Appedix B" < 16 "Appedix B" <

13 OpeStax-CNX module: m is homogeeous, so its solutio is uique oly withi a ormalizatio factor. I most cases, both the scalig fuctio ad wavelet are ormalized to uit eergy or uit orm. I the properties discussed here, we ormalize the eergy as E = φ (t) 2 dt = 1. Other ormalizatios ca easily be used if desired. 8.1 Geeral Properties ot Requirig Orthogoality There are several properties that are simply a result of the multiresolutio equatio (44) ad, therefore, hold for orthogoal ad biorthogoal systems. Property 1 The ormalizatio of φ (t) is arbitrary ad is give i (13) as E. Here we usually set E = 1 so that the basis fuctios are orthoormal ad coeciets ca easily be calculated with ier products. 2 φ (t) dt = E = 1 (45) Property 2 Not oly ca the scalig fuctio be writte as a weighted sum of fuctios i the ext higher scale space as stated i the basic recursio equatio (44), but it ca also be expressed i higher resolutio spaces: φ (t) = h (j) () 2 j/2 φ ( 2 j t ) (46) where h (1) () = h () ad for j 1 h (j+1) () = k h (j) (k) h (j) ( 2k). (47) Property 3 A formula for the sum of dyadic samples of φ (t) ( ) k φ 2 J = 2 J (48) k Property 4 A partitio of uity" follows from (48) for J = 0 φ (m) = 1 (49) Property 5 A geeralized partitio of uity exists if φ (t) is cotiuous m φ (t m) = 1 (50) Property 6 A frequecy domai statemet of the basic recursio equatio (44) m Φ (ω) = 1 2 H (ω/2) Φ (ω/2) (51) Property 7 Successive approximatios i the frequecy domai is ofte easier to aalyze tha the time domai versio i (44). The covergece properties of this iite product are very importat. This formula is derived i (74). Φ (ω) = k=1 { 1 ( ω ) H 2 2 k }Φ (0) (52)

14 OpeStax-CNX module: m Properties that Deped o Orthogoality The followig properties deped o the orthogoality of the scalig ad wavelet fuctios. Property 8 The square of the itegral of φ (t) is equal to the itegral of the square of φ (t), or A 2 0 = E. [ 2 φ (t) dt] = φ(t) 2 dt (53) Property 9 The itegral of the wavelet is ecessarily zero ψ (t) dt = 0 (54) The orm of the wavelet is usually ormalized to oe such that ψ (t) 2 dt = 1. Property 10 Not oly are iteger traslates of the wavelet orthogoal; dieret scales are also orthogoal. 2 j/2 ψ ( 2 j t k ) 2 i/2 ψ ( 2 i t l ) dt = δ (k l) δ (j i) (55) where the orm of ψ (t) is oe. Property 11 The scalig fuctio ad wavelet are orthogoal over both scale ad traslatio. 2 j/2 ψ ( 2 j t k ) 2 i/2 φ ( 2 i t l ) dt = 0 (56) for all iteger i, j, k, l where j i. Property 12 A frequecy domai statemet of the orthogoality requiremets i (13). It also is a statemet of equivalet eergy measures i the time ad frequecy domais as i Parseval's theorem, which is true with a orthogoal basis set. Φ (ω + 2πk) 2 = k 2 Φ (ω) dω = φ (t) 2 dt = 1 (57) Property 13 The scalig coeciets ca be calculated from the orthogoal or tight frame scalig fuctios by h () = 2 φ (t) φ (2t ) dt. (58) Property 14 The wavelet coeciets ca be calculated from the orthogoal or tight frame scalig fuctios by h 1 () = 2 ψ (t) φ (2t ) dt. (59) Derivatios of some of these properties ca be foud i Appedix B 17. Properties i equatios (1), (10), (14), (53), (51), (52), ad (57) are idepedet of ay ormalizatio of φ (t). Normalizatio aects the others. Those i equatios (1), (10), (48), (49), (51), (52), ad (57) do ot require orthogoality of iteger traslates of φ (t). Those i (14), (16), (17), (22), (20), (53), (58) require orthogoality. No properties require compact support. May of the derivatios iterchage order of summatios or of summatio ad itegratio. Coditios for those iterchages must be met. 17 "Appedix B" <

15 OpeStax-CNX module: m Parameterizatio of the Scalig Coeciets The case where φ (t) ad h () have compact support is very importat. It aids i the time localizatio properties of the DWT ad ofte reduces the computatioal requiremets of calculatig the DWT. If h () has compact support, the the lters described i Chapter: Filter Baks ad the Discrete Wavelet Trasform are simple FIR lters. We have stated that N, the legth of the sequece h (), must be eve ad h () must satisfy the liear costrait of (10) ad the N 2 biliear costraits of (14). This leaves N 2 1 degrees of freedom i choosig h () that will still guaratee the existece of φ (t) ad a set of essetially orthogoal basis fuctios geerated from φ (t). 9.1 Legth-2 Scalig Coeciet Vector For a legth-2 h (), there are o degrees of freedom left after satisfyig the required coditios i (10) ad (14). These requiremets are ad which are uiquely satised by h (0) + h (1) = 2 (60) h 2 (0) + h 2 (1) = 1 (61) h D2 = {h (0), h (1)} = { 1 2, 1 2 }. (62) These are the Haar scalig fuctios coeciets which are also the legth-2 Daubechies coeciets [6] used as a example i Chapter: A multiresolutio formulatio of Wavelet Systems ad discussed later i this book. 9.2 Legth-4 Scalig Coeciet Vector For the legth-4 coeciet sequece, there is oe degree of freedom or oe parameter that gives all the coeciets that satisfy the required coditios: h (0) + h (1) + h (2) + h (3) = 2, (63) ad h 2 (0) + h 2 (1) + h 2 (2) + h 2 (3) = 1 (64) h (0) h (2) + h (1) h (3) = 0 (65) Lettig the parameter be the agle α, the coeciets become h (0) = (1 cos (α) + si (α)) / ( 2 2 ) h (1) = h (2) = (1 + cos (α) + si (α)) / ( 2 2 ) (1 + cos (α) si (α)) / ( 2 2 ) (66) h (3) = (1 cos (α) si (α)) / ( 2 2 ).

16 OpeStax-CNX module: m These equatios also give the legth-2 Haar coeciets (62) for α = 0, π/2, 3π/2 ad a degeerate coditio for α = π. We get the Daubechies coeciets (discussed later i this book) for α = π/3. These Daubechies-4 coeciets have a particularly clea form, h D4 = { , , , } (67) 9.3 Legth-6 Scalig Coeciet Vector For a legth-6 coeciet sequece h (), the two parameters are deed as α ad β ad the resultig coeciets are h (0) = [(1 + cos (α) + si (α)) (1 cos (β) si (β)) + 2si (β) cos (α)] / ( 4 2 ) h (1) = [(1 cos (α) + si (α)) (1 + cos (β) si (β)) 2si (β) cos (α)] / ( 4 2 ) h (2) = [1 + cos (α β) + si (α β)] / ( 2 2 ) h (3) = [1 + cos (α β) si (α β)] / ( 2 2 ) h (4) = 1/ 2 h (0) h (2) h (5) = 1/ 2 h (1) h (3) Here the Haar coeciets are geerated for ay α = β ad the legth-4 coeciets (66) result if β = 0 with α beig the free parameter. The legth-4 Daubechies coeciets are calculated for α = π/3 ad β = 0. The legth-6 Daubechies coeciets result from α = ad β = The iverse of these formulas which will give α ad β from the allowed h () are ( α = arcta 2 h(0) 2 + h(1) 2) 1 + (h (2) + h (3)) / 2 2 (h (1) h (2) h (0) h (3)) + (69) 2 (h (0) h (1)) ( ) h (2) h (3) β = α arcta h (2) + h (3) 1/ 2 As α ad β rage over π to π all possible h () are geerated. This allows iformative experimetatio to better see what these compactly supported wavelets look like. This parameterizatio is implemeted i the Matlab programs i Appedix C 18 ad i the Aware, Ic. software, UltraWave [2]. Sice the scalig fuctios ad wavelets are used with iteger traslatios, the locatio of their support is ot importat, oly the size of the support. Some authors shift h (), h 1 (), φ (t), ad ψ (t) to be approximately cetered aroud the origi. This is achieved by havig the iitial ozero scalig coeciet start at = N rather tha zero. We prefer to have the origi at = t = 0. Matlab programs that calculate h () for N = 2, 4, 6 are furished i Appedix C 19. They calculate h () from α ad β accordig to (62), (66), ad (68). They also work backwards to calculate α ad β from allowable h () usig (70). A program is also icluded that calculates the Daubechies coeciets for ay legth usig the spectral factorizatio techiques i [6] ad Chapter: Regularity, Momets, ad Wavelet System Desig of this book. Loger h () sequeces are more dicult to parameterize but ca be doe with the techiques of Polle [26] ad Wells [35] or the lattice factorizatio by Viadyaatha [32] developed i Chapter: Filter Baks ad Trasmultiplexers. Selesick derived explicit formulas for N = 8 usig the symbolic software system, Maple, ad set up the formulatio for loger legths [28]. It is over the space of these idepedet parameters that oe ca d optimal wavelets for a particular problem or class of sigals [6], [10]. 18 "Appedix C" < 19 "Appedix C" < (68) (70)

17 OpeStax-CNX module: m Calculatig the Basic Scalig Fuctio ad Wavelet Although oe ever explicitly uses the scalig fuctio or wavelet (oe uses the scalig ad wavelet coef- ciets) i most practical applicatios, it is elighteig to cosider methods to calculate φ (t) ad ψ (t). There are two approaches that we will discuss. The rst is a form of successive approximatios that is used theoretically to prove existece ad uiqueess of φ (t) ad ca also be used to actually calculate them. This ca be doe i the time domai to d φ (t) or i the frequecy domai to d the Fourier trasform of φ (t) which is deoted Φ (ω). The secod method solves for the exact values of φ (t) o the itegers by solvig a set of simultaeous equatios. From these values, it is possible to the exactly calculate values at the half itegers, the at the quarter itegers ad so o, givig values of φ (t) o what are called the dyadic ratioals Successive Approximatios or the Cascade Algorithm I order to solve the basic recursio equatio (1), we propose a iterative algorithm that will geerate successive approximatios to φ (t). If the algorithm coverges to a xed poit, the that xed poit is a solutio to (1). The iteratios are deed by φ (k+1) (t) = N 1 =0 h () 2 φ (k) (2t ) (71) for the k th iteratio where a iitial φ (0) (t) must be give. Because this ca be viewed as applyig the same operatio over ad over to the output of the previous applicatio, it is sometimes called the cascade algorithm. Usig deitios (2) ad (3), the frequecy domai form becomes Φ (k+1) (ω) = 1 ( ω ) H 2 2 ( Φ (k) ω ) 2 ad the limit ca be writte as a iite product i the form [ [ 1 ( ω ) ]] Φ ( ) (ω) = 2 H 2 k Φ ( ) (0). (73) k=1 If this limit exists, the Fourier trasform of the scalig fuctio is [ [ 1 ( ω ) ]] Φ (ω) = 2 H 2 k Φ (0). (74) k=1 The limit does ot deped o the shape of the iitial φ (0) (t), but oly o Φ (k) (0) = φ (k) (t) dt = A 0, which is ivariat over the iteratios. This oly makes sese if the limit of Φ (ω) is well-deed as whe it is cotiuous at ω = 0. The Matlab program i Appedix C 20 implemets the algorithm i (71) which coverges reliably to φ (t), eve whe it is very discotiuous. From this scalig fuctio, the wavelet ca be geerated from here 21. It is iterestig to try this algorithm, plottig the fuctio at each iteratio, o both admissible h () that satisfy (10) ad (14) ad o iadmissible h (). The calculatio of a scalig fuctio for N = 4 is show at each iteratio i Figure 3. Because of the iterative form of this algorithm, applyig the same process over ad over, it is sometimes called the cascade algorithm [31], [30]. 20 "Appedix C" < 21 "A multiresolutio formulatio of Wavelet Systems", (24) < (72)

18 OpeStax-CNX module: m Iteratig the Filter Bak A iterestig method for calculatig the scalig fuctio also uses a iterative procedure which cosists of the stages of the lter structure of Chapter: Filter Baks ad the Discrete Wavelet Trasform which calculates wavelet expasios coeciets (DWT values) at oe scale from those at aother. A scalig fuctio, wavelet expasio of a scalig fuctio itself would be a sigle ozero coeciet at the scale of j = 1. Passig this sigle coeciet through the sythesis lter structure of Figure: Two-Stage Two-Bad Sythesis Tree 22 ad here 23 would result i a e scale output that for large j would essetially be samples of the scalig fuctio Successive approximatio i the frequecy domai The Fourier trasform of the scalig fuctio deed i (2) is a importat tool for studyig ad developig wavelet theory. It could be approximately calculated by takig the DFT of the samples of φ (t) but a more direct approach is available usig the iite product i (74). From this formulatio we ca see how the zeros of H (ω) determie the zeros of Φ (ω). The existece coditios i Theorem 5 (p. 5) require H (π) = 0 or, more geerally, H (ω) = 0 for ω = (2k + 1) π. Equatio (74) gives the relatio of these zeros of H (ω) to the zeros of Φ (ω). For the idex k = 1, H (ω/2) = 0 at ω = 2 (2k + 1) π. For k = 2, H (ω/4) = 0 at ω = 4 (2k + 1) π, H (ω/8) = 0 22 "Filter Baks ad the Discrete Wavelet Trasform", Figure 7 < 23 "Filter Baks ad the Discrete Wavelet Trasform", (17) <

19 OpeStax-CNX module: m Figure 3: Iteratios of the Successive Approximatios for φd4

20 OpeStax-CNX module: m at ω = 8 (2k + 1) π, etc. Because (74) is a product of stretched versios of H (ω), these zeros of H ( ω/2 j) are the zeros of the Fourier trasform of φ (t). Recall from Theorem 15 (p. 7) that H (ω) has o zeros i π/3 < ω < π/3. All of this gives a picture of the shape of Φ (ω) ad the locatio of its zeros. From a asymptotic aalysis of Φ (ω) as ω, oe ca study the smoothess of φ (t). A Matlab program that calculates Φ (ω) usig this frequecy domai successive approximatios approach suggested by (74) is give i Appedix C 24. Studyig this program gives further isight ito the structure of Φ (ω). Rather tha startig the calculatios give i (74) for the idex j = 1, they are started for the largest j = J ad worked backwards. If we calculate a legth-n DFT cosistet with j = J usig the FFT, the the samples of H ( ω/2 j) for j = J 1 are simply every other sample of the case for j = J. The ext stage for j = J 2 is doe likewise ad if the origial N is chose a power of two, the process i cotiued dow to j = 1 without calculatig ay more FFTs. This results i a very eciet algorithm. The details are i the program itself. This algorithm is so eciet, usig it plus a iverse FFT might be a good way to calculate φ (t) itself. Examples of the algorithm are illustrated i Figure 4 where the trasform is plotted for each step of the iteratio. Figure 4: Iteratios of the Successive Approximatios for Φ (ω) 24 "Appedix C" <

21 OpeStax-CNX module: m The Dyadic Expasio of the Scalig Fuctio The ext method for evaluatig the scalig fuctio uses a completely dieret approach. It starts by calculatig the values of the scalig fuctio at iteger values of t, which ca be doe exactly (withi our ability to solve simultaeous liear equatios). Cosider the basic recursio equatio (1) for iteger values of t = k φ (k) = h () 2 φ (2k ), (75) ad assume h () 0 for 0 N 1. This is the reemet matrix illustrated i (6) for N = 6 which we write i matrix form as M 0 φ = φ. (76) I other words, the vector of φ (k) is the eigevector of M 0 for a eigevalue of uity. The simple sum of h () = 2 i (10) does ot guaratee that M 0 always has such a eigevalue, but h (2) = h (2 + 1) i (12) does guaratee a uity eigevalue. This meas that if (12) is ot satised, φ (t) is ot deed o the dyadic ratioals ad is, therefore, probably ot a very ice sigal. Our problem is to ow d that eigevector. Note from (6) that φ (0) = φ (N 1) = 0 or h (0) = h (N 1) = 1/ 2. For the Haar wavelet system, the secod is true but for loger systems, this would mea all the other h () would have to be zero because of (10) ad that is ot oly ot iterestig, it produces a very poorly behaved φ (t). Therefore, the scalig fuctio with N > 2 ad compact support will always be zero o the extremes of the support. This meas that we ca look for the eigevector of the smaller 4 by 4 matrix obtaied by elimiatig the rst ad last rows ad colums of M 0. From (76) we form [M 0 I] φ = 0 which shows that [M 0 I] is sigular, meaig its rows are ot idepedet. We remove the last row ad assume the remaiig rows are ow idepedet. If that is ot true, we remove aother row. We ext replace that row with a row of oes i order to implemet the ormalizig equatio φ (k) = 1 (77) k This augmeted matrix, [M 0 I] with a row replaced by a row of oes, whe multiplied by φ gives a vector of all zeros except for a oe i the positio of the replaced row. This equatio should ot be sigular ad is solved for φ which gives φ (k), the scalig fuctio evaluated at the itegers. From these values of φ (t) o the itegers, we ca d the values at the half itegers usig the recursive equatio (1) or a modied form φ (k/2) = h () 2 φ (k ) (78) This is illustrated with the matrix equatio (8) as M 1 φ = φ 2 (79) Here, the rst ad last colums ad last row are ot eeded (because φ 0 = φ 5 = φ 11/2 = 0) ad ca be elimiated to save some arithmetic. The procedure described here ca be repeated to d a matrix that whe multiplied by a vector of the scalig fuctio evaluated at the odd itegers divided by k will give the values at the odd itegers divided by 2k. This modied matrix correspods to covolvig the samples of φ (t) by a up-sampled h (). Agai, covolutio combied with up- ad dow-samplig is the basis of wavelet calculatios. It is also the basis of digital lter bak theory. Figure 5 illustrates the dyadic expasio calculatio of a Daubechies scalig fuctio for N = 4 at each iteratio of this method.

22 OpeStax-CNX module: m Figure 5: Iteratios of the Dyadic Expasio for Φ D4

23 OpeStax-CNX module: m Not oly does this dyadic expasio give a explicit method for dig the exact values of φ (t) of the dyadic ratioals (t = k/2 j ), but it shows how the eigevalues of M say somethig about the φ (t). Clearly, if φ (t) is cotiuous, it says everythig. Matlab programs are icluded i Appedix C 25 to implemet the successive approximatio ad dyadic expasio approaches to evaluatig the scalig fuctio from the scalig coeciets. They were used to geerate the gures i this sectio. It is very illumiatig to experimet with dieret h () ad observe the eects o φ (t) ad ψ (t). Refereces [1] Wavelets: A Elemetary Treatmet of Theory ad Applicatios. World Scietic, Sigapore, [2] Aware, Ic., Cambridge, MA. The UltraWave Explorer User's Maual, July [3] S. Cavaretta, W. Dahme, ad C. A. Micchelli. Statioary Subdivisio, volume 93. America Mathematical Society, [4] Charles K. Chui. A Itroductio to Wavelets. Academic Press, Sa Diego, CA, Volume 1 i the series: Wavelet Aalysis ad its Applicatios. [5] Charles K. Chui. Wavelets: A Tutorial i Theory ad Applicatios. Academic Press, Sa Diego, CA, Volume 2 i the series: Wavelet Aalysis ad its Applicatios. [6] Igrid Daubechies. Te Lectures o Wavelets. SIAM, Philadelphia, PA, Notes from the 1990 CBMS-NSF Coferece o Wavelets ad Applicatios at Lowell, MA. [7] Igrid Daubechies ad Jerey C. Lagarias. Two-scale dierece equatios, part i. existece ad global regularity of solutios. SIAM Joural of Mathematical Aalysis, 22: ;1410, From a iteral report, AT&T Bell Labs, Sept [8] Igrid Daubechies ad Jerey C. Lagarias. Two-scale dierece equatios, part ii. local regularity, iite products of matrices ad fractals. SIAM Joural of Mathematical Aalysis, 23: ;1079, July From a iteral report, AT&T Bell Labs, Sept [9] G. Deslauriers ad S. Dubuc. Iterpolatio dyadique. I Fractals, Dimesios No Eti[U+FFFD] et Applicatios, page ;45. Masso, Paris, [10] R. A. Gopiath, J. E. Odegard, ad C. S. Burrus. Optimal wavelet represetatio of sigals ad the wavelet samplig theorem. IEEE Trasactios o Circuits ad Systems II, 41(4): ;277, April Also a Tech. Report No. CML TR-92-05, April 1992, revised Aug [11] Hek J. A. M. Heijmas. Descrete wavelets ad multiresolutio aalysis. I Wavelets: A Elemetary Treatmet of Theory ad Applicatios, page ;80. World Scietic, Sigapore, [12] R. Q. Jia. Subdivisio schemes i spaces. Advaces i Computatioal Mathematics, 3: ;341, [13] W. Lawto. Private commuicatio. [14] Waye M. Lawto. Tight frames of compactly supported ae wavelets. Joural of Mathematical Physics, 31(8): , August Also Aware, Ic. Tech Report AD [15] Waye M. Lawto. Multiresolutio properties of the wavelet galerki operator. Joural of Mathematical Physics, 32(6): ;1443, Jue "Appedix C" <

24 OpeStax-CNX module: m [16] Waye M. Lawto. Necessary ad suciet coditios for costructig orthoormal wavelet bases. Joural of Mathematical Physics, 32(1):578211;61, Jauary Also Aware, Ic. Tech. Report AD900402, April [17] Waye M. Lawto. Iite covolutio products & reable distributios o lie groups. Trasactios of the America Mathematical Soceity, submitted [18] Waye M. Lawto, S. L. Lee, ad Z. She. Covergece of multidimesioal cascade algorithm. Numerische Mathematik, to appear [19] Waye M. Lawto, S. L. Lee, ad Z. She. Stability ad orthoormality of multivariate reable fuctios. SIAM Joural of Mathematical Aalysis, to appear [20] Waye M. Lawto ad Howard L. Resiko. Multidimesioal wavelet bases. Aware Report AD910130, Aware, Ic., February [21] S. G. Mallat. Multiresolutio approximatio ad wavelet orthoormal bases of. Trasactios of the America Mathematical Society, 315:6987, [22] S. G. Mallat. A theory for multiresolutio sigal decompositio: The wavelet represetatio. IEEE Trasactios o Patter Recogitio ad Machie Itelligece, 11(7): ;693, July [23] C. A. Micchelli ad Prautzsch. Uiform reemet of curves. Liear Algebra, Applicatios, 114/115: ;870, [24] A. V. Oppeheim ad R. W. Schafer. Discrete-Time Sigal Processig. Pretice-Hall, Eglewood Clis, NJ, [25] T. W. Parks ad C. S. Burrus. Digital Filter Desig. Joh Wiley & Sos, New York, [26] D. Polle. Daubechies' scalig fuctio o [0,3]. J. America Math. Soc., to appear. Also Aware, Ic. tech. report AD891020, [27] Amos Ro. Characterizatio of liear idepedece ad stability of the sfts of a uivariate reable fuctio i terms of its reemet mask. Techical report CMS TR 93-3, Computer Sciece Dept., Uiversity of Wiscosi, Madiso, September [28] Iva W. Selesick. Parameterizatio of orthogoal wavelet systems. Techical report, ECE Dept. ad Computatioal Mathematics Laboratory, Rice Uiversity, Housto, Tx., May [29] M. J. Smith ad T. P. Barwell. Exact recostructio techiques for tree-structured subbad coders. IEEE Trasactios o Acoustics, Speech, ad Sigal Processig, 34: ;441, Jue [30] G. Strag. Eigevalues of ad covergece of the cascade algorithm. IEEE Trasactios o Sigal Processig, 44, [31] Gilbert Strag ad T. Nguye. Wavelets ad Filter Baks. Wellesley8211;Cambridge Press, Wellesley, MA, [32] P. P. Vaidyaatha. Multirate Systems ad Filter Baks. Pretice-Hall, Eglewood Clis, NJ, [33] P. P. Vaidyaatha ad Igor Djokovic. Wavelet trasforms. I The Circuits ad Filters Hadbook, chapter 6, page ;219. CRC Press ad IEEE Press, Roca Rato, [34] Marti Vetterli ad Jelea Kova269;evi263;. Wavelets ad Subbad Codig. Pretice8211;Hall, Upper Saddle River, NJ, 1995.