Orthogonal Wavelets via Filter Banks Theory and Applications. Joab R. Winkler

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1 Orthogoal Wavelets via Filter Bas Theory ad Applicatios Joab R. Wiler The Uiversity of Sheffield Departmet of Computer Sciece Reget Court 211 Portobello Street Sheffield S 4DP Uited Kigdom j.wiler~dcs.shef.ac.u Abstract Wavelets are used i may applicatios, icludig image processig, sigal aalysis ad seismology. The critical problem is the represetatio of a sigal usig a small umber of computable fuctios, such that it is represeted i a cocise ad computatioally efficiet form. t is show that wavelets are closely related to filter bas (sub bad filterig) ad that there is a direct aalogy betwee multiresolutio aalysis i cotiuous time ad a filter ba i discrete time. This provides a clear physical iterpretatio of the approximatio ad detail spaces of multiresolutio aalysis i terms of the frequecy bads of a sigal. Oly orthogoal wavelets, which are derived from orthogoal filter bas, are discussed. Several examples ad applicatios are cosidered. 1 troductio The fudametal aim of sigal processig is the costructio of a set of basis (or more geerally expasio) fuctios that allow a cocise, efficiet ad iformative represetatio of a sigal. Clearly, the choice of basis fuctios depeds o the sigal; the Fourier basis (complex expoetials) is usually satisfactory for smooth sigals but it is poor if the sigal has discotiuities or has portios of both low ad high frequecies. More specifically, sice the Fourier basis fuctios have ifiite support i the time domai, they have poor time resolutio properties ad therefore do ot reveal the istats ti at which chages i a sigal f(t) occur. The short time Fourier trasform (STFT) improves this situatio by placig a widow roud the expoetial basis fuctios, thereby allowig idividual portios of the sigal to be aalysed ad revealig more local iformatio about the sigal. The mai disadvatage of the STFT is that the width of the widow is costat ad must therefore capture iformatio i both the low ad high frequecy portios of the sigal. A wavelet basis is superior because it ca adapt to local chages i the sigal; arrow widows ca be used i the high frequecy regios of the sigal ad wider widows ca be used i the low frequecy regios. The wavelet trasform of a fuctio f (t) is the represetatio of f (t) i a wavelet basis, ad the elemets of the basis are scaled ad traslated forms of a mother wavelet 'fjj(t).. t is show that the calculatio of the coefficiets of the dilated ad traslated versios of 'fjj(t) i the represetatio of f(t) ca be be performed recursively ad implemeted i a 1

2 perfect recostructio filter ba, that is, a filter ba i which the output is exactly equal to the iput. Two-chael filter bas are cosidered i sectio 2 ad the coditios for perfect recostructio are derived. A orthogoal filter ba is a particular class of filter ba, ad the coditios for perfect recostructio i this type of filter ba are obtaied i sectio 3. The multiresolutio aalysis of f(t) ito ested subspaces is cosidered i sectio 4 ad it is show that the equatios that gover this wavelet decompositio of f (t) are idetical to the equatios that defie perfect recostructio i a orthogoal filter ba. This aalogy betwee filter bas, whose iput is a discrete sigal, ad the multiresolutio aalysis of the cotiuous fuctio f(t) allows the coefficiets of the represetatio of f(t) i a wavelet basis to be implemeted i a filter ba. Several examples of the applicatios of wavelets are cosidered i sectio 5. Attetio is restricted to orthogoal filter bas which lead to orthogoal wavelets, ad the Daubechies wavelets are the best examples of this class of wavelet. More geeral biorthogoal filter bas have greater desig freedom i the choice of filters that lead to perfect recostructio, ad two multiresolutios are required, oe for the aalysis ba ad oe for the sythesis ba. 2 Two-chael filter bas A two-chael filter ba is show i figure 1. t cosists of two lowpass filters Ho ad Fa, two highpass filters H; ad F l, two dowsamplers..!. 2, ad two upsamplers t 2. The dowsamplers remove all the odd-umbered samples, ad the upsamplers isert a zero betwee every pair of samples. The filters Ho ad H are called aalysis filters, ad the filters Fa ad Fl are called sythesis filters. X Ho! 2 l- v 0 - t 2 Fo - y H! t 2 F f.-- ANALYSS BANK V SYNTHESS BANK Figure 1: A two-chael filter ba. The iput sigal x = {x()} is separated ito two frequecy bads, a low frequecy bad correspodig to the upper chael, ad a high frequecy bad correspodig to the lower chael. The sigal i each bad betwee the aalysis ad sythesis stages may be coded for trasmissio or storage, ad compressio may occur i which case iformatio is lost. Perfect recostructio requires that the aalysis ba be coected directly to the sythesis ba, ad thus o compressio occurs. The dowsamplers improve the efficiecy of the filter ba because each chael carries oly half the data (iformatio); the exclusio of the dowsamplers implies that all the iput goes ito each chael, thus doublig the wor that is performed by the filter ba, but with o icrease i iformatio. The aalysis of the filter ba i figure 1 requires that the upsamplers ad dowsamplers be formally defied. Thus if r = {r()}, s = {s()}, ad t = {t()} the 2

3 1. The th compoet of r = (~2) t is the (2)th compoet of t, r() = t(2). (1) t is show i [11J, chapter 3, that the a-domai form of (1) is (2) where R(z) ad T(z) are the z-trasforms of r ad t respectively. 2. The th compoet of s = (t 2) t is s(2) ad the z-domai form of (3) is [11], chapter 3, = t() s(2 + 1) = 0, (3) Although the filters Ho, Fo, H ad F are liear ad time-ivariat, it follows immediately from (1) ad (3) that dowsamplig ad upsamplig are liear but ot time-ivariat. The dowsamplers ad upsamplers create aliases ad images respectively, ad these are removed by the filters. particular, Ho ad H must be bad-limited to lower ad upper halfbads respectively, (4) Ho(w) = 0 H(w) = 0 if if 7r 2 ~ wl < 7r 7r o ~ wl < 2' to remove aliases. t is show i the ext sectio that the coditio for perfect recostructio implies that the sythesis filters are defied by the aalysis filters. 2.1 Perfect recostructio The coditios for perfect recostructio i a two-chael filter ba are derived, ad the costraits that they impose o the coefficiets of the filters are cosidered. The iput to the dowsampler i the upper chael i the filter ba i figure 1 is Ho(z)X(z) where X(z) is the z-trasform of the iput sequece {x()}, ad thus it follows from (2) that the z-trasform of the output of the dowsampler i the upper chael is Uo(z) = (~2) Ho(z)X(z) = ~ [Ho (A) X (A) +Ho (-A) X (-zt)], ad hece the output of the ups ampler i the upper chael is, from (4), Vo(z) = UO(z2) 1 = 2[Ho(z)X(z)+Ho(-z)X(-z)]. Similarly, the output of the upsampler i the lower: chael is 1 V (z) = - [Hdz) X (z) + Hd-z) X (-z)], 2 3

4 ad thus the z-trasform of the output y = {y()} of the filter ba is Y(z) = Fo(z)Vo(z) + F (z)vi (z) 1 1 = 2" (Fo(z)Ho(z) + F (z)h (z)] X(z) + 2" (Fo(z)Ho( -z) + F (z)h1 (-z)] X( -z). (5) Equatio (5) shows that Y(z) is a fuctio of X(z) ad X( -z), ad sice perfect recostructio requires that Y(z) = X(z), it follows that Fo(z)Ho(z) + F(z)H 1 (z) = 2, Fo(z)Ho(-z) + F(z)H(-Z) = O. (6) (7) Equatio (6) is the coditio for the absece of distortio ad (7) is the coditio for alias cacellatio, ad the two equatios defie the coditios that must be satisfied by the filters for perfect recostructio. They ca be stated i matrix form by maig the substitutio z -1- -z, i which case they become ad (6)-(9) ca be combied to yield or [ Fo( -z)ho( -z) + F (-Z)H( -z) = 2, Fo( -z)ho(z) + F (-z)h (z) - 0, Fo(z) F(Z)] [Ho(z) Ho( -z) ] _ [2 0] Fo (- z) F (- Z) H (z) H (- z) ' (8) (9) (10) where F m(z) is the sythesis modulatio matrix ad Hm(z) is the aalysis modulatio matrix. The frequecy domai forms of (6) ad (7) are obtaied by substitutig z = e iw, ad these equatios require that Fo(w = O)Ho(w = 0) = 2 ad Fl(w = 7r)Hl(W = 7r) = 2. t follows that the lowpass ad highpass filters must be ormalised so that Ho(w = 0) = Fo(w = 0) = viz ad (12) t is oted that their more usual values are (11) Ho(w = 0) = Fo(w = 0) = 1 ad t follows from (10) that dethm(z) = Ho (z)h(-z) -H(z)Ho(-z) = -dethm(-z), ad thus det Hm(z) is a odd fuctio of z, t also follows from (10) that ad thus [ ] 1 [ 0 ] [H1(-Z) -Ho(-z)] Fo(z) F1(z) = det Hm(z) 2 -H 1 (z) Ho(z), ( ) _ 2Hl(-Z) F ( ) _ - 2Ho(-z) Fo z - ad z -. det Hm(z) det Hm(z) f Ho(z) ad H(z) are fiite impulse respose (FR) filters, the Fo(z) ad Fl(Z) are also FR filters if det Hm(z) is of the form cz- l where l is a odd iteger ad c is a arbitrary costat. The choice c = 2 leads to ad (13) 4

5 ad these defiitios of the sythesis filters i terms of the aalysis filters guaratee that the alias cacellatio coditio (7) is satisfied. t is oted that the frequecy domai form of (13) is Fo(w) = e iwl H (w + 71') ad F(w) = _e iwl Ho(w + 71'), ad thus Fo(w) ad F(w) are lowpass ad highpass filters respectively if Ho(w) ad H(w) are lowpass ad highpass filters respectively, as required. The coditio for perfect recostructio ca be cast i a more coveiet form by substitutig (13) ito (8), which yields where Po(z) + Po( -z) = 2, (14) Po(z) = Fo(z)Ho(z), (15) is the lowpass product filter. t follows from (14) that all the eve coefficiets, apart from the costat coefficiet. of Po(z) are zero, The desig of a perfect recostructio three steps : two-chael filter ba has bee reduced to the followig 1. Desig a lowpass filter Po(z) that satisfies (14). The desig variables are the odd coefficiets of Po (z). 2. Factorise Po(z) ito Fo(z)Ho(z). 3. Calculate the highpass filters from the alias cacellatio coditio (13). There are several methods that ca be used to achieve the first step, ad the factorisatio i the secod step is ot uique because if Po(z) = (z - (Xi), the assigmet of the factors z - (Xi to Fo(z) ad Ho(z) must still be determied, ad thus there is cosiderable scope for the desig of filters that satisfy additioal properties. The ext sectio cosiders the alteratig flip, which simplifies the desig procedure because oly the lowpass filter Ho(z) is desiged ad the remaiig filters are the defied from it. Furthermore, the alteratig flip leads to orthogoal filter bas The alteratig flip The alteratig flip defies the coefficiets hl() of H(Z) i terms of the coefficiets ho() of Ho(z), =O... N, (16) where Ho(z) ad H (z) have N + 1 coefficiets. Furthermore, N must be odd sice it guaratees double shift orthogoality betwee ho = {ho ()} ~=o ad h = {h ()} ~=o, N L ho( - 2)hl () = 0, =O (17) for all itegers. t will be show that this double shtft orthogoality provides the li betwee perfect recostructio i a orthogoal filter ba ad multiresolutio aalysis. 5

6 Example 2.1 f N = 5, the from (16), ad it is clear that ho = ho(o) ho(l) ho(2) ho(3) ho( 4) ho (5) h = ho(5) -ho(4) ho(3) -ho(2) ho(l) -ho(o), Lho()hl() = Lho(-2)hl() = Lho(-4)hl() =0, =O =2 =4 ad thus (17) is satisfied for all the double shifts of a filter with six coefficiets. However it is readily verified that (17) ot satisfied if N is eve. 0 t follows from (16) that (18) i the z-domai, ad i the frequecy domai, ad thus H(w) is highpass if Ho(w) is lowpass. The simplificatios i the desig of a perfect recostructio filter ba that follow from the applicatio of the alteratig flip are obtaied by substitutig (18) ito (13), Fo(z) = zl-n Ho (z-), where both l ad N are odd itegers. The choice l = N yields Fo(z) = Ho (z-) ad thus the lowpass product filter (15) is give by (19) Po(z) = Ho(z)Ho (z-), (20) which must still satisfy (14). The frequecy domai form of (20) is ad thus the frequecy domai form of (14) is -- 2 Po(w) = Ho (w)ho(w) = Ho(w)1, (21) Furthermore, it is easily verified from (19) that (22) H (w) Ho (w) + H (w + 71') Ho (w + 71') = 0, (23) Ho (w) H (w) + Ho (w + 71') H (w + 71') = O. (24) t is show i sectio 3 that (21) is the fudametal desig equatio of a orthogoal filter ba. Equatios (22)-(24) follow directly from the alteratig flip ad therefore do ot yield extra iformatio. However they are icluded because they will be refereced i sectio 3 where it will be show that the alteratig flip leads to a orthogoal filter ba. Compariso of (15) ad (20) shows that the applicatio of the alteratig flip implies that Po(z) is a fuctio of Ho(z) rather tha Ho(z) ad Fo(z), ad thus the desig of a orthogoal filter ba reduces to the desig of the lowpass aalysis filter Ho(z); the highpass aalysis filter H (z) is defied by the alteratig flip ad the sythesis filters Fo(z) ad F (z) are defied by (13). 6

7 3 Orthogoal filter bas Equatio (11) states that perfect recostructio i a arbitrary filter ba is achieved if the aalysis ba is iverted by the sythesis ba. Restrictios o the class of matrices limit the class of filters, ad i particular, a orthogoal filter ba is derived by restrictig Hm(z) to be uitary, up to a scalar multiplier, o the uit circle i the z-plae. This leads to parauitary matrices, which are extesios of uitary matrices [16]. Defiitio 3.1 The matrix H(z) is parauitary o the uit circle zl = 1 if where d is a arbitrary positive costat. This exteds to all z :f: 0 by o for all w, (25) HT (z-) H (z) = H (z) H (z) = dj. (26), The positive costat d is icluded so that the aalysis modulatio matrix, for which d = 2, ca be costraied to be parauitary. t follows from (25) that H(z) is parauitary o the uit circle i the z-plae, that is, i the frequecy domai, ad that its extesio (26) to the rest of the z-plae (apart from the origi) follows from z = z- = e- iw if zl = 1. t follows from (26) that the aalysis modulatio matrix Hm(z) is parauitary if [ Ho (z-) H (z-) ] [Ho(z) Ho(-z)] _ [2 0] Ho (_z-) H (_z-) H1(z) H(-z) ' ad sice the right had side is a multiple of the idetity matrix, the order of multiplicatio the left had side ca be reversed, [ Ho(z) Ho(-z)] [ Ho (Z-) H (z-) ] _ [2 0] H(z) H(-z) Ho (_z-) H (_Z-l) o (27) The frequecy domai form of (27) is [ Ho(w) Ho(w + rr) ] [ Ho (w) H (w) ] [2 0] H(w) H(w + rr) Ho (w + rr) H (w + rr) ' (28) ad thus (21)-(24) are reproduced. This establishes that if the sythesis filters are defied by (13), the the alteratig flip defiitio of the aalysis highpass filter (16) ecessarily leads to a orthogoal filter ba. t follows from (27) that (29) (30) (31) ad it is see that (32) follows from (31) by the substitutio z ~ z-. t is istructive to cosider the iverse z-trasform of each of these equatios because it will reveal the coditios (32) 7

8 that the coefficiets of Ho(z) ad H (z) must satisfy i a orthogoal filter ba. t is adequate to cosider the geeral fuctio P(Z)Q(Z-l) + P( -z)q( _z-l), because the left had sides of (29) ad (30) are obtaied by settig P(z) = Q(z). particular, if P(z) t-t p() ad Q(z) t-t q(), the ad thus the iverse z-trasforms P (z) Q (z-l) + P (-z) Q (_z-l) t-t 2 2:p()q( - 2), of (29)-(32) are 2: ho()ho ( - 2) = 6(), (33) 2: h1()ho( - 2) = O. These equatios defie the double shift orthogoality coditios, ad their satisfactio guaratees that the filter ba is orthogoal. Equatios (33) ad (34) are the ormalisatio coditio o the 2-orm of the coefficiets, ad (35) is exactly the same as (17), as expected. t is importat to emphasize that oly (33) eed be solved; the alteratig flip ecessarily implies that if this equatio is satisfied, the (34) ad (35) are also satisfied. Although the coefficiet forms (33)-(35) are equivalet to the z-domai forms (29)-(32), the former are preferred because it will be show that the equivalece betwee multiresolutio ad perfect recostructio i a orthogoal filter ba is clearer. Compariso of (10) ad (27) shows that the sythesis filters are give by or fo() = ho(-) ad (34) (35) ad (36) ad thus the sythesis filters are aticausal if the aalysis filters are causal. These results are summarised i the followig theorem. Theorem 3.1 The coefficiets of the filters i a orthogoal filter ba are give by h1() = (-ltho(n - ), fo() = ho( -), f() = h1(-). =O... N, The coefficiets ho() satisfy the double shift orthogoality coditio, 2: ho()ho( - 2) = 6(). o Example 3.1 The simplest example of a orthogoal filter ba is obtaied with Ho(z) 1 = y'2(1 + z-l), 8

9 whose output is a scaled form of the movig average of the iput, where the scale factor y'2 is icluded to satisfy (12). The filter H1(z) is a scaled from of the movig differece, ad the sythesis filters are, from (36), 1-1 H1(z) = y'2(1 - z ), 1 Fo(z) = y'2(1 + z) ad These filters, which satisfy (33)-(35) trivially, give rise to the Haar wavelet. t is easily verified that det Hm(z) = 2z- 1, as required sice N = 1. 0 Example 3.2 The coefficiets satisfy the double shift orthogoality coditio (33). They defie the Daubechies wavelet D4, which is a member of the Daubechies family of wavelets. This family of filters ad wavelets has maximum flatess at w = 7r ad therefore good approximatio properties. t is easily verified that det Hm(z) = 2z- 3, as required sice N = 3. 0 Theorem 3.1 shows that all the filters must be the same legth (N + 1), but this may be a disadvatage i some applicatios. This costrait is relaxed i biorthogoal wavelets, which leads to greater desig freedom. Furthermore, some applicatios require that the filter have liear phase, which implies that the filter coefficiets are symmetric or ati-symmetric, but this property ca oly be achieved by a very restricted class of orthogoal filter, as show i the followig example. Example 3.3 Cosider a ati-symmetric filter for which N = 5, ho = ha(0) ha(1) ha(2) -ha (2) -ha (1) -ha (0). Sice this vector must be orthogoal to its shift by two ad shift by four, it follows that ha(0)ha(2) -ha(1)ho(2) = 0 ad ha(o)ha(l) = o. Sice N = 5, it follows that ha(o) 1= 0, ad thus ha(l) hece oly the first ad last coefficiets of the filter are o-zero. 0 = 0, which implies that ha(2) = 0, ad The ext sectio cosiders multiresolutio aalysis, which is defied for a cotiuous rather tha discrete fuctio. t will be show that a multiresolutio aalysis yields the double shift orthogoality coditios (33)-(35), ad this eables the li betwee sub bad filterig ad multiresolutio aalysis to be cosidered i greater detail. 4 Multiresolutio aalysis Every measuremet or observatio is made o a particular scale, ad this defies the amout of iformatio that is cotaied i the measuremet. For example, whe a distat object is viewed, oly the low level (coarse) detail ca be observed ad hece a small scale is used. However as the distace betwee the observer ad object decreases, higher level (fier) details are observed ad the scale icreases. Aother example of scale occurs i maps, for which a small scale is used to cover a large area ad thus local detail is abset, but as the scale icreases, 9

10 a smaller physical area is covered ad more detail is icluded. These two simple examples show that measuremets ca be made o differet scales. Multiresolutio is a mathematical descriptio of a object (measuremet, image, etc.) i terms of scale, or level of detail. The scale i which a object is viewed is itrisically related to samplig a fuctio f(t). Specifically, if f (t) is sampled at 2 j samples per uit time, the scale of the discrete sigal f () is defied because details that occur at a frequecy that is higher tha that specified by the samplig theorem are ot represeted i f (). However if the samplig frequecy icreases to 2 j +l, the fier detail ca be captured i the discrete sigal. The chages i detail that occur as the samplig frequecy chages must be defied quatitatively, ad this leads to a multiresolutio decompositio of f (t). 4.1 Scale spaces The relatio betwee scale ad detail is formalised by defiig a subspace \tj c L2 (R) that cotais all fuctios fj(t) that are represeted at a scale 2 j, that is, fuctios that ca be recostructed by samplig at 2 j samples per uit time. Similarly, the subspace \tj+l cotais all fuctios f)+l(t) that ca be represeted at a scale 2)+1, ad sice the space \tj+l cotais fuctios tj.fj (t) that are ot i \tj, it follows that there exists a detail space Wj such that ad \tj Wj = {O}. (37) where tj.fj (t) E Wj. The zero itersectio coditio guaratees that oly the zero vector lies i both \tj ad Wj. By defiitio, \tj C \tj+l, ad this defies a family of ested subspaces,.., C V-2 C V- C Vo C Vi C V2... (38) Multiresolutio decomposes a fuctio f(t) ito a sum of fuctios fj(t) E \tj where each space \tj is associated with a scale or level of detail. particular, the recursive applicatio of (37) leads to or i terms of fuctios, fo(t) + tj.fo(t) + tj.ft(t) + tj.fz(t) tj.fj(t) = fj+l(t) (40) Equatios (39) ad (40) show that multiresolutio aalyses a sigal f(t) i terms of a low level approximatio that lies i a space VD ad successively fier details that lie i the spaces Wb ~ O. the frequecy domai, (40) states that the fuctio fo(t) E Vo is a low frequecy approximatio of f(t), ad higher frequecies are added as more detail spaces Wj are icluded, ad thus the approximatio of f(t) by fj(t) icreases as j icreases. This frequecy domai iterpretatio of scale spaces is show i figure 2 for j = 3, h(t) E V3 = V2$ W2 = Vi $ W $ W2 = Vo$ Wo $ W1 $ W2 t is see that the fuctio fo(t) E Vo has a lowpass frequecy spectrum but that each of the fuctios tj.fj(t) E Wj has a badpass frequecy spectrum. t follows from (39) that W is cotaied i \tj if < j, ad thus if it is assumed that Wj is orthogoal to \tj, it follows that Wj is orthogoal to W : (39) \tj 1.Wj ::::::;.Wj 1.W. (41) This coditio implies that if fj(t) is orthogoal to tj.fj(t), the tj.fj(t) is orthogoal to tj.f(t). Two properties of the spaces \tj are obtaied by cosiderig chages of scale ad traslatio of a fuctio fj(t) E Vi : 10

11 (a) (b) (c) frequecy r , ,,, 1 W2 V 2 V 2 W2l,, frequecy , , , ,,,,, W 'W1 : 2 :, W1: W2:,,,, frequecy (d) r----f , t W2 1 W 11 Wo Vo Vo Wo 1 W 1 i W2 i frequecy Figure 2: The frequecy domai iterpretatio of multiresolutio: (a) Vg, (b) Vg = V2 $ W2, (c) V3 = V $ W 1 $ W2, ad (d) V3 = Vo $ Wo $ W 1 $ W2. 1. Cosider the effect of a chage i scale, or more geerally, sice scalig i time causes a iverse scalig i frequecy,. 1 (W) jet) f+ F(w) <==> j (2Jt) f+ V F V ' (42) where F(w) is the Fourier trasform of jet). 2. The traslatio of jj (t) to ij (t - ) does ot cause a chage i the space Vi to which it belogs, because the traslatio of a fuctio oly causes a chage i the phase of its Fourier trasform, jet) f+ F(w) <==> j (t - ) f+ e- iw F (w). The two extreme limits o the space Vi are obtaied by cosiderig j ~ ±oo. (43) 11

12 1. As j -t -00, the space Vj cotais the costat fuctio, ad sice Vj C L2 (R), the costat fuctio must be the zero fuctio,,lim )-+-00 ) V = {O} or 00 Vj = {O}. j=-oo (44) This is called the emptiess coditio ad implies that llfi(t) -t 0 as j -t As j -t 00, all possible scales or levels of detail are cotaied i Vj ad the completeess coditio, im Vj = L2 (~) )-+00 or Closure (900 Vi) ~ ' (R), (45) is satisfied. The closure guaratees that the limits of all vectors i the subspaces Vj are icluded [8], page 52. particular, the uio V-oo of all the ested subspaces Vj is ot the same as the space L2 (~) but is dese i L2 (~) sice for every fuctio f(t) E L2 (~), there exists a fuctio i V-oo that is arbitrarily close to f (t). t follows from recursive applicatio of (37) that Vj+l = v., EB w., EB W-j+l EB EB W j -1 EB W j, ad the emptiess coditio (44) ad completeess coditio (45) yield 00 L2 (~) = EB Wj. j=-oo (46) t follows that if the orthogoality coditio (41) of the subspaces Wj is satisfied, the these spaces form a orthogoal decompositio of the space of square itegrable fuctios. This sectio has cosidered some properties that the spaces Vj must satisfy but there exists oe more requiremet that must be cosidered for a complete descriptio of a multiresolutio aalysis. particular, it is assumed that there exists a fuctio cfjo(t) = cfj(t) such that the iteger traslates {cfj(t - ) : E Z} of cfj(t) form a orthoormal basis for VG. t will be show that this eables a orthoormal basis for the space Vj to be developed. These requiremets of a multiresolutio aalysis are collected together i the followig defiitio. Defiitio 4.1 A multiresolutio aalysis cosists of a sequece of ested subspaces (38) such that the followig coditios are satisfied : 1. Scale ivariace: Equatio (42). 2. Traslatio ivariace: Equatio (43). 3. Emptiess: Equatio (44). 4. Completeess: Equatio (45). 5. There exists a fuctio cfjo(t) = cfj(t) such that {cfj(t - ) : E Z} is a orthoormal basis of Vo. 0 The ext sectios cosider the dilatio ad wavelet equatios ad the decompositio relatio, which eable expressios for cp(t) ad a orthoormal basis for Wj to be developed. 12

13 4.2 The dilatio equatio The ested property of the subspaces Vj implies that if j(t) = 4>(t - ) E VD, where {4>(t - ) : E Z} is a orthoormal basis for VD, the j (2 j t) = 4> (2 j t - ) E Vj, ad thus is a orthoormal basis for Vj. The abbreviated otatio (47) deotes the th traslate of the jth basis fuctio. The scale factor 2t is icluded so that the basis fuctios are ormalised for all scales j ad traslatios, By assumptio, Vo "C V ad sice 4>(t) E VD, it follows that 4>(t) E V ad thus there exist costats c() such that 4>(t) = V2: c()4>(2t - ). This is the dilatio equatio, also called the two-scale or refiemet equatio because it relates the basis fuctios at level j = 0 to the basis fuctios at level j = 1. t will be show i sectio 4.5 that the coefficiets c() are exactly equal to the coefficiets ho() of the lowpass filter Ho(z) i a orthogoal filter ba, ad thus the dilatio equatio must be solved for the scalig fuctio 4>(t) for give costats c(). This equatio is usually solved by the cascade algorithm, which is a iterative method. The equivalece betwee the costats c( ) ad filter coefficiets ho () is suggested by cosiderig the coditios that these costats must satisfy to guaratee that the fuctios {4>o(t)} = {4>(t - )} are orthoormal. particular, it follows from (48) that ad hece 4>(t - m) = V2: c()4>(2t - 2m - ), ( 4>(t), 4>(t - m)) = i:4>(t)4>(t - m) dt = 2 f,1 c()c(l) i:4>(2t - )4>(2t - 2m -l) dt (48) = 2: c()c(l) 100 4>(t)4>(t + - 2m - l) dt,l -00 = 2: c()c(l)o(2m + l - ),l = 2: c(l)c(2m + l). (49) Sice the set of fuctios {.j;(t - }} are orthoormal, it follows that ( cjj(t), cjj(t - m) ) = L:c()c( - 2m) = o(m), (50) 13

14 ad thus the coefficiets c() have double shift orthogoality, which characterises the coditios o the coefficiets of a orthogoal ba, (33)-(35). The full equivalece betwee orthogoal filter bas ad multiresolutio will be established by cosiderig the wavelet equatio ad decompositio relatio. Equatio (48) is obtaied by cosiderig Vo C Vi., but it is valid for all subspaces, that is, Vj C Vj+' This follows by maig the substitutio t -+ 2 j t - m ad multiplyig both sides of the equatio by 2~, 2~<jJ(2jt - m) = 2i L c() <jj(2 j +1t - 2m - ). (51) The fuctio fj(t), defied by fj(t) = L aj<jjj(t), (52) is the orthogoal projectio of a fuctio f(t) E Vm, j < r, oto the space Vj. This projectio is a low level approximatio of f(t), ad it will be show i sectio 4.6 that a fast recursio eables the coefficiets aj i the space Vj to be calculated. 4.3 The wavelet equatio The derivatio of the wavelet equatio is similar to that of the dilatio equatio, except that the space Vo is replaced by the space Wo. particular, sice Wo C V, it follows that if'lj;(t) E Wo, the 'lj;(t) E V ad thus there exist costats d() such that 'lj;(t) = V2L d() <jj(2t - ). (53) t will be show that the costats d() are equal to the coefficiets of the highpass filter H(z) i a orthogoal filter ba, i the same way as the coefficiets c() of the dilatio equatio are equal to the coefficiets of the lowpass filter i the filter ba. Thus the wavelet equatio (53) yields the mother wavelet 'lj;(t) directly from the scalig fuctio. The coditio (50) o the coefficiets c() arises from the orthogoality of the basis fuctios {<jj(t - )}, ad similarly, a coditio o the coefficiets d() ca be deduced by imposig the orthogoality coditio (41). f the ier prod uct of both sides of (53) with <jj(t - m) is tae, the followig (49) idetically, the result L c()d( - 2m) = 0, is obtaied, ad it is oted that this equatio has the same form as (35). Although 'lj;(t) E Wo, a orthogoal basis for Wo has ot yet bee developed, but the decompositio relatio of <jj(t) ad 'lj;(t), which is cosidered i the ext sectio, esures that {'lj;(t - ) : E Z} geerates the whole space Wo. 4.4 The decompositio relatio t follows from (37) that if {'lj;(t - ) : E Z} geerates the whole space Wo, there must exist costats a(2) ad b(2) such that V2<jJ(2t) = L a(2)<jj(t - ) + L b(2)'lj;(t - ), (54) 14

15 is satisfied. The eve idexed coefficiets of the uow coefficiets are used because this simplifies the followig aalysis. Let 2 -t 2 - l, i which case (54) becomes ad the substitutio t -t t - ~ yields ad thus V2CP(2t -l) = 2: a(2 -l)cp(t - ) + 2: b(2 -l)1jj(t - ), 1 ( cp(t), cp(2t -l) ) = J2a( -l). (55) From the dilatio equatio, 1 (cp(t),cp(2t -l) ) = J2c(l), ad hece a(l) = c(-l), (56) which is the solutio for the coefficiets a(2) i (54). The expressio for the coefficiets b(2) follows by taig the ier product of (55) with 1jJ(t), ad from the wavelet equatio, ad thus V2( cp(2t -l), 1jJ(t) ) = 2: b(2 -l)( 1jJ(t- ), 1jJ(t) ), V2( cp(2t -l), 1jJ{t) ) = d(l), d(l) =L b(2 -l)( 1jJ(t- ), 1jJ(t) ). f 1jJ(t) is orthogoal to all its iteger traslates ad ormalised to uit magitude, the it follows from this equatio that b(l) = d( -l). (57) Equatios (56) ad (57) are the solutios of the decompositio relatio (54) if 1jJ( t) is orthogoal to all its iteger traslates. The substitutio of these solutios ito (55) yields V2cp(2t -l) = 2: c(l - 2)cp(t - ) + 2: d(l - 2)1jJ(t - ), ad the trasformatio t -t 2 j t followed by multiplicatio by 2 j yields 2i. CP(2 j +lt -) = 2t 2: c(l- 2)cp(2 j t - ) + 2tL d(l- 2)1jJ(2 j t - ), which is the decompositio relatio betwee levels j ad j + 1, that is, Vj+l = Vj $ W j. Thus (54) has a s~tl~io betwee levels j = 0 ad j = 1 ad hece there exists a solutio betwee ay two levels. t follows that (58) 15

16 is a orthoarmal basis for the space Wj. The substitutio yields t ~ 2 j t - m i the wavelet equatio ad multiplyig both sides by 2t 2t'jJ(2jt - m) = 2i:p L d()q;(2 j +lt - 2m - ), (59) which is a geeralisatio of the wavelet equatio from Wo C V l to Wj C Vj+ 1. This is equivalet to (51), which is the dilatio equatio betwee Vj ad Vj+l. The abbreviated otatio deotes the th traslate of the j th basis fuctio. Sice Wj.L W, it follows that (.1.. (t).1. (t)) = {1 if j = m ad = 'l'j, 'l'm 0 otherwise. The implicatios o the wavelet equatio of the orthoormality of the fuctios (58) must be cosidered. Followig the procedure i (49), it is easily show that ( 'jj(t), 'jj(t - m) ) = L d()d( - 2m) = 6(m), ad hece the orthoormality of the basis fuctios (58) implies a double shift orthogoality of the coefficiets de). Equatio (46) shows that the spaces Wj form a orthogoal decompositio of L2 (R), ad thus there exist costats bj such that J(t) = L L bj'jjj(t). j (60) The fuctios (58) are the wavelet basis fuctios at level i, ad (53) shows that the mother wavelet 'ljjoo(t) = 'jj(t) is derived directly from the wavelet equatio. The physical iterpretatio of (60) follows from figure 2, i which it is show that each basis fuctio i the set (58) has a badpass frequecy spectrum, ad thus the level of detail or approximatio icreases as j icreases. Fially, it will be show i sectio 4.6 the coefficiets bj ca be computed recursively ad quicly from the coefficiets aj i (52). The ext sectio combies the results o orthogoal filter bas ad multiresolutio aalysis, ad this eables their equivalece to be cosidered. 4.5 Multiresolutio ad orthogoal filter bas Three equatios, the dilatio equatio, the wavelet equatio ad the decompositio relatio have geerated three coditios o the coefficiets c() ad d( ). These coditios are summarised ad the idetificatio of the coefficiets c() ad de) with the filter coefficiets b«() ad hl () is established. 1. The dilatio equatio ad the orthoormality of the fuctios {q;(t - ) : E Z} : L c()c( - 2m) = 8(m). 2. The wavelet equatio ad the orthogoality of the spaces Vo ad Wo : L c()d( - 2m) = O. (61 ) (62) 16

17 3. The decompositio relatio ad the orthoormality of the fuctios {'if;(t - ) : E Z} : L d()d( - 2m) = o(m). (63) The li betwee orthogoal filter bas ad multiresolutio aalysis follows directly by comparig (33)-(35) ad (61)-(63); both sets of equatios ivolve a double shift orthogoality, ad thus the coefficiets c() ad d() i a multiresolutio aalysis are also the filter coefficiets i a orthogoal filter ba. Moreover, the alteratig flip guaratees that if (33) is satisfied, the it ecessarily follows that (34) ad (35) are also satisfied, ad hece the alteratig flip ca also be used to defie the coefficiets d() i terms of the coefficiets c(), where N is odd. d() = (_l)c(n - ), =O... N, (64) The oly remaiig issue is the idetificatio of c() with the lowpass filter coefficiets ha () rather tha the highpass filter coefficiets h (). Sice a ecessary (but ot sufficiet) coditio for the covergece of the cascade algorithm for the scalig fuctio f/j(t) is that the filter C(~) = Ec()e- iw satisfies C(w = 7r) = 0, [11] page 234, it follows that the coefficiets c() must be idetified with the lowpass filter, ad thus the coefficiets d( ) are idetified with the highpass filter. 4.6 The fast wavelet trasform t has bee show that a multiresolutio aalysis of a fuctio requires two sets of spaces, the approximatio spaces Vi ad the detail spaces W j A fuctio i V;+ ca be writte as the sum of a fuctio i Vj ad a fuctio i W j, ad this process ca be repeated i a recursive maer. The fuctio ij-l(t) E Vi- is a low level approximatio to fj(t) E Vj, ad it is ecessary to calculate the coefficiets aj-, ad bj-, from aj. This calculatio ca be doe i a fast recursive maer called the fast wavelet trasform (FWT), which is derived i theorem 4.l. This is a aalysis trasform because it eables the coefficiets of a low level approximatio of fj(t) to be calculated. By cotrast, the iverse fast wavelet trasform (FWT), which is derived i theorem 4.2, is used for the sythesis of fj(t) from its low level approximatios. t will be show that these recursios ca be implemeted i a filter ba i which the coefficiets aj are geerated by the lowpass filter with coefficiets c(-) ad the wavelet coefficiets bj are geerated by the highpass filter with coefficiets d( -), that is, aiticausal filters are required. Two importat issues that must be cosidered for the implemetatio of the fast wavelet trasform are the iitialisatio of the recursio ad the effect of sigals of fiite legth. Both these poits will be discussed after the FWT ad FWT have bee derived. Theorem 4.1 A fuctio fj+l(t) = L aj+l,lf/jh,(t) E Vi+l = Vi $ Wj, (65) has coefficiets aj ad bj' aj = L c(l- 2)aj+l,1 i the spaces Vj ad Wj respectively. ad bj = L d(l - 2)aj+l,, (66) Proof Cosider the dilatio equatio (51), f/jjm(t) = L c(l - 2m)f/Jj+l,(t). 17 (67)

18 The equivalet equatio that is derived from the wavelet equatio is (59), 'l/jjm(t) = L del - 2m)<pj+l,(t). The recursio for the coefficiets aj i (66) is derived by taig the ier product of both sides of (67) with!j+l(t), which is defied i (65), (!J+l(t), <Pjm(t)) = L e(l - 2m) (!j+l(t),<pj+l,l(t) ) = L e(l - 2m) ( LaHl, <Pj+1,(t), <PHl,l(t) ) l = L e(l - 2m) LaHl, ( <Pj+1,(t), <Pj+1,l(t) ) l = L e(l - 2m) LaHl,a( - l) l (68) = L e(l - 2m)aHl,l. (69) The left had side of (69) is simplified by usig the decompositio of!j+1(t) portio i V; ad a portio i W j, E V;+1 ito a!j+1(t) = L aj+l,<phl,l(t) = L ajl<pjl(t) + L bjl'/jjl(t), (70) ad sice V; is orthogoal to Wj, (!Hl(t), <Pjm(t)) = (L ajl<pjl(t) + L bjl'/jjl(t), <Pjm(t) ) l = (L ajl<pjl(t), <Pjm(t) ) = L ajla(l - m) = ajm The combiatio of this result ad (69) establishes the recursio (66) for the coefficiets aj. The recursio for the wavelet coefficiets follows idetically, except that (68) is used istead of (67). 0 The recursios (66) represet the covolutio of {ah r} with the aticausal filters c(-) ad d( -), followed by dowsamplig, as show i figure 3. Theorem 4.2 The coefficiets aj+l,l i (65) ca be recostructed from their values at lower levels by aj+l,l = L e(l - 2)aj + L d(l- 2)bj (71) Proof By defiitio, <Pj(t) = 2t<p(2jt - ) ill = 2 2 L e() <p(2 j +1t ) = L:c()<pj+l,2+(t), (72) 18

19 a j + 1 c (-) ~ 2 a. J d (-) ~ 2 b. J Figure 3: The implemetatio of the fast wavelet trasform as the aalysis stage of a twochael filter ba. ad similarly, V;j(t) = L d() <Pj+1,2+ (t). (73) The ier product of (70) with <Pj+,m(t) ad the applicatio of (72) ad (73) yields L aj+l,l( <Pj+1,(t), <Pj+l,m(t) ) = L ajl L c()( <Pj+1,21+(t), <Pj+,m(t) ) + L bjl L d() ( <Pj+1,2l+(t), <Pj+1,m(t) ) = L ajl L c()6(m - 2l- ) 1 + L bjl Ld()6(m - 2l- ) = L c(m - 2l)ajl + L d(m - 2l)bjl. The left had side of this equatio is aj+l,m ad thus the result (71) is established. 0 The iverse fast wavelet trasform (71) is implemeted by ups amp lig {aj} ad {bj} ad the filterig with, respectively, filters with coefficiets c() ad d(), as show i figure 4. The combiatio of figures 3 ad 4 shows that the approximatio of fj+l(t) by fj(t), ad the its sythesis from fj(t) ad 6.fj(t) is implemeted by a orthogoal filter ba. More importatly, the fast wavelet trasform is implemeted by a tree of filter bas, as show i figure 5. t is see that the coefficiets {aj} are defied by the outputs of the lowpass filters ad the coefficiets {bj} are defied by the outputs of the highpass filters. The filters c(-) ad d(-) are the aticausal forms of the filters ho() ad h () i figure 1, ad thus a multiresolutio aalysis is implemeted as a tree of aalysis filter bas, where the filters satisfy the coditios of a orthogoal filter ba. There exist two further issues, the iitialisatio of the fast wavelet trasform ad the effect of sigals of fiite legth, that must be cosidered before the trasform ca be implemeted. The iitialisatio of the fast wavelet trasform eeds special attetio because the assumptio of multiresolutio aalysis of a fuctio f (t) requires that it lie i a space Vj. The simplest assumptio is that the sampled values f() of f(t) are used as iput to the filter ba. Assumig that the trasform is iitiated at level j = 0, it follows that there must exist 19

20 a J 1 2 c () a. J +1 b. J 1 2 d () Figure 4: The implemetatio two-chael filter ba. of the iverse fast wavelet trasform as the sythesis stage of a a' lja. J a j +1 b' J b. 1 J- Figure 5: The fast wavelet trasform implemeted as a tree of filter bas. a scalig fuctio <j;(t) such that fs(t) = 2: f()<j;(t - ), where fs(t) is the uderlyig cotiuous fuctio that is defied by the samples f(). This equatio is satisfied by the delta fuctio, <j;(t) = 6(t), but most scalig fuctios do ot satisfy it. A better approach requires that the coefficiets aj iterpolate exactly the fuctio values [2], chapter 6, [14], [15]. Cosider ow the implemetatio of the fast wavelet trasform for sigals of fiite legth. A basic assumptio of wavelet theory is that the sigals are of ifiite duratio, but all sigals are of fiite duratio. f the sigal is sufficietly log, the assumptio of a sigal of ifiite legth is adequate i the middle of the sigal where boudary effects eed ot be cosidered. However problems occur at the boudary of the sigal, where the sample values x() for < 0 ad > L are ot defied for a sigal of legth L+ 1. This fiite legth must be cosidered, ad the problem is filterig this class of sigal because the covolutio E h()x( - ) is ot always defied. t is ecessary to exted the sigal x() such that the covolutio is defied, ad there exist two possibilities, periodicity (wraparoud) ad extesio by reflectio (symmetric extesio) [11], [13]. t is oted that extesio by zeros (extrapolatio by zeros) yields poor results ad is ot cosidered satisfactory. 20

21 5 Examples Wavelets are used extesively may areas of applied mathematics, ad a few of these applicatios are listed: 1. mage ad video compressio [8]' [11]. 2. Speech aalysis, audio ad ECG compressio [4], [11]. 3. Medicie ad biology [1]. 4. Statistics [6],[9]. 5. Computatioal liear algebra ad the solutio of Poisso's equatio [12]. 6. The aalysis of ecoomic ad fiacial data [7]. 7. mage processig [3]. 8. Computer graphics [10]. 9. Vibratios ad acoustics [5]. Refereces [1] Aldroubi A., ad User M., Wavelets i Medicie ad Biology, Boca Rato, FL : CRC Press, (1996). [2] Chui C., Wavelets: A Mathematical Tool for Sigal Aalysis, Philadelphia, PA: SAM, (1997). [3] Kigsbury N., "mage processig with complex wavelets", Phil. Tras. R. Soc. Lad. A., 357, pp , (1999). [4] Kobayashi M., Saamoto M., Saito T., Hashimoto Y., Nishimura M., ad Suzui K., "Wavelet aalysis for a text-to-speech (TTS) system", i: Wavelets ad Their Applicatios: Case Studies (Kobayashi M., Ed.), Philadelphia, PA: SAM, pp , (1998). [5] Newlad D., "Harmoic wavelets i vibratios ad acoustics", Phil. Tras. R. Soc. Lad. A., 357, pp , (1999). [6] Ogde R., Essetial Wavelets for Statistical Applicatios ad Data Aalysis, Bosto, MA: Birhauser, (1997). [7] Ramsey J., "The cotributio of wavelets to ecoomic ad fiacial data", Phil. Tras. R. Soc. Lod. A., 357, pp , (1999). [8] Rao R., ad Bopardiar A., Wavelet Trasforms: troductio to Theory ad Applicatios, Readig, MA: Addiso-Wesley, (1998). [9] Silverma B., "Wavelets i statistics: beyod the stadard assumptios", Phil. Tras. R. Soc. Lod. A., 357, pp , (1999). [10] Stollitz E., DeRose T., ad Salesi D., Wavelets for Computer Graphics: Theory ad Applicatios, Sa Fracisco, CA: Morga Kaufma, (1996). 21

22 [11] Strag G., ad Nguye T., Wavelets ad Filter Bas, Wellesley, MA: Wellesley- Cambridge Press, (1997). [12] Taata N., "A wavelet based cojugate gradiet method for solvig Poisso equatios", i: Wavelets ad Their Applicatios: Case Studies (Kobayashi M., Ed.), Philadelphia, PA: SAM, pp , (1998). [13] User M., "A practical guide to the implemetatio of the wavelet trasform", i: Wavelets i Medicie ad Biology (A. Aldroubi ad M. User, Eds.), CRC Press, pp , (1996). [14] User M., Aldroubi A., ad Ede M., "A family of polyomial splie wavelet trasforms", Sigal Process., 30, pp , (1993). [15] User M., Aldroubi A., ad Ede M., "The l..qpolyomial splie pyramid", EEE Tras. Patter Aal. Mach. tell., 15, pp , (1993). [16] Vaidyaathe P., Multiraie Systems ad Filter Bas, Eglewood Cliffs, NJ: Pretice Hall, (1993). ' 22..