ANALYTIC WAVELETS AND MULTIRESOLUTION ANALYSIS A NOTE ON CERTAIN ORTHOGONALITY CONDITIONS

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1 Proceedigs, Fculty of Mechicl Egieerig, Sope, ol 3, No, pp 4 47 (4) CODEN: ZTFSEH 35 ISSN eceived: September 9, 4 UDK: Accepted: November 9, 4 Short commuictio ANALYTIC WAELETS AND MULTIESOLUTION ANALYSIS A NOTE ON CETAIN OTHOGONALITY CONDITIONS oz Aces Fculty of Mechicl Egieerig, Ss Cyril d Methodius Uiversity, POBox 464, MK- Sope, epublic of Mcedoi roz@mfuimedum The mi disdvtges of the Fourier series d trsforms re left behid by ew tool: wvelets! The properties of wvelets re well preseted, prt from cotiuity, by the oldest exmple the Hr wvelets This wor dels with expdig wvelets o the complex ple usig their lytic represettios Here re reviewed lytic wvelets d their bsic properties The correspodig multiresolutio lysis, however, does ot preserve the ortogolity it hd o the rel lie Here is give cosequece regrdig the ortogolity of the bsis geerted by the sclig fuctio Uder some coditios, the orthogolity is preserved, s see with the Sho wvelets Key words: wvelets; multiresolutio lysis; sclig fuctio; lytic wvelets; orthogolity EIEW ON WAELETS I the lst twety yers the Fourier series d trsforms re filly replced by ew tool: wvelets! Wvelet expsios hve quite few properties ot vilble i Fourier expsios (or y other expsios) To see this i the simplest cotext, cosider rel-vlued fuctio ( x) f o the itervl [,] Uder some coditios, it c be expded i Fourier series f ( x) ( cos x b si πx) or i Hr fuctio series where (see Fig ) π () ( x) c c ( x ) f ψ, (),, x < 5, ψ ( x), 5 x < (3), otherwise - Fig Hr wvelet Both series re exmples of expsios i terms of orthogol fuctios i L ([,] ), the spce of squre itegrble fuctios o the itervl [,] There re simple formuls for clcultig the coefficiets But the Fourier series is ot well loclized i spce; if you re iterested i the behvior of f o subitervl [ b], you eed to ivolve ll the Fourier coefficiets O the other hd, the Hr series is very well loclized: to restrict the ttetio to the subitervl [, b] oly te the sum i () over those idices for which the support (closure of

2 4 Aces the ozero re for fuctio) of ( ) ψ itersects [, b] Further more, the prtil sum of the Hr series (summig N) clerly represets pproximtio to f tig ito ccout detils N o the order of mgitude or greter Aythig smller will ot be cosidered These two properties, locliztio d sclig, re the ttributes of wvelet expsios I dditio, the Hr fuctios re creted out of sigle fuctio ψ by dydic diltios d iteger trsltios The lst property hs to be icluded i the defiitio of y fmily of wvelets The wvelet expsios c be thought of s geerliztios of the Hr series, i which the fuctio ψ is replced by smoother fuctios Before turig to the exct properties these fuctios eed to hve d how to costruct them, it is useful to bctrc d see exctly how the Hr fuctios rise It will tur out to be esier if we cosider the whole lie s the domi of our fuctios THE SIMPLE HAA WAELETS Let ( ) [ ]( x) x, Surely this is oe of the simplest fuctios oe c imgie, but it is chose becuse it hs two importt properties: (i) the trsltes of by itegers, ( ) form orthoorml set of fuctios for L ( ) ; (ii) is self-similr, ie if you cut the grph i hlf you c use ech hlf to recover the whole grph Or, lgebriclly expressed: ( x) ( x) ( x ) () which is clled the sclig idetity Note: ecll tht the sclr product of two fuctios, f d g, is defied by f, g : f ( x) g( x) d these two fuctios re orthogol if f, g Specilly, if ll fuctios from the set { f } stisfy, f, f,, the the set of fuctios is clled orthoorml The eergy, ie the orm of fuctio is defied by / f : f, f Bc to the fuctio ( ) [ ]( x) x, ; it is clled the sclig fuctio, which is quite proper cosiderig the sclig property () The sigificce of tht property is the followig: Let deote the lier sp of ( ), Z, cosistig of ll piecewise costt fuctios with ump discotiuities t the itegers This is turl spce to cosider, i view of (i) Of course, is ot ll of Oe c get lrger spce by sclig Let L ( ) be the spce of ll piecewise costt fuctios with umps t, Z f iff It is cler tht ( ) ( ) ; the fuctios / ( x ) f form orthoorml bsis for The sclig idetity () gives Now, by itertig up d dow the dydic scle, icresig sequece of subspces, Z, is produced Of course: The sequece, Z, is clled multiresolutio lysis There re two iterestig properties, mely d Z () ( ) L (3) Z I view of (3) it is temptig to combie ll / Z ; the orthoorml bses { ( ) } but, lthough, the bsis of is ot cotied i the bsis of! This ïve ttempt for obtiig orthoorml bsis for L ( ) hs filed! This is obvious by ust looig t the grph of ( x) [, ]( x) d comprig it to the grph of ( x) [, 5]( x), elemet of the bsis for, obtied by simple diltio: ( ) ( x) [ ]( x) x, 5 The sclr product of these two is ot zero, so they re ot orthogol Proc Fc Mech Eg Sope, 3,, 4 47 (4)

3 Alytic wvelets d multiresolutio lysis A ote o certi orthogolity coditios L ( ) W Z (6) Now the problems with the orthoormlity of two distict bses for W d W re over So we c gther ll the elemets ( ) / ψ ito oe grd orthoorml bsis for L ( ) To me it more cler, ust observe the grphs of the dilted wvelet cotied i tht cover the itervl [,] d how they combie with the Hr fuctio ψ ψ Its lier combitios give diltios d trsltios of the sclig fuctio ( x) [, ]( x) Fig Sclig fuctios from d respectively Sice bsis for (which fuctio would ψ be?), so tht the uio is orthoorml bsis for! So, the orthogol complemet of i, let it be deoted by W, would hve orthoorml bsis ψ ( ), Z Now we c write W, where deotes orthogol sum of the spces d ( ) is orthoorml, let s dd up some elemets ψ ( ) The swer is the Hr fuctio ψ Note tht for the Hr fuctio ( x) ( x) ( x ) ψ (4) Now we c rescle the spce W, so d ( x ) W (5) / ψ is orthoorml bsis for W Combiig (), (3) d (5) we get ( x) ( x) ψ ψ ψ ( x) ψ ( x ) Fig 3 Hr wvelets 3 MULTIESOLUTION ANALYSIS Defiitio A multiresolutio lysis (MA) ssocited with sclig fuctio, is icresig sequece of subspces of L ( ) stisfyig the followig coditios

4 44 Aces (i) L ( ) Z (ii) Z (iii) f ( ) iff f ( ) (iv) ( ), Z (3) forms orthoorml bsis for The oly disdvtge of the Hr wvelet is its discotiuity Cotiuous wvelets re sometimes more suitble for use It is show tht rpidly decresig fuctios re excellet choice for cotiuous sclig fuctios d wvelets sice it is esy wy to obti locliztio ecll tht fuctio, defied o is rpidly decresig of order r, if it is cotiuously differetible up to order r d for ech,,, r there exists costt c p, such tht for y c p, (3) ( ) ( x) x, p N ( x ) p Just lie i (4), y wvelet risig from MA is derived from the sclig fuctio: ( x) ( ) α ( x ( ) ) ψ (33) Z Furthermore, oe c provide orthoorml wvelet with compct support with y order of cotiuous differetibility tht rises from MA, 98 s result, give by Igrid Dubechies 4 ANALYTIC WAELETS Let the sclig fuctio be rpidly decresig fuctio; the hs Cuchy lytic represettio defied by ( z) πi ( x), Im z (4) x z It is lytic fuctio (ie hs cotiuous derivtives of y order) o the upper/lower complex hlf ple C, correspodig to the idex / We cll the fuctio lytic sclig fuc- tio o C ; it preserves most of the properties of Beig rpidly decresig, the fuctio is squre itegrble Its lytic cotiutio is Hrdy fuctio Note: A fuctio f ( x iy) is Hrdy fuctio, if the itegrl exists for y y d f ( x iy) / sup f ( x iy) < (4) y> The lst iequlity defies the orm, ie, the eergy of f Deote the spce of these fuctios by ( C ) H The sclr product of two Hrdy fuctios F, G is defied by F, G : lim F y ( x iy) G( x iy) The sclig fuctio forms multiresolutio lysis { } Z o L ( ), defied i the previous sectio It is iterestig to form multiresolutio lysis o the complex domi To eep it simple, the discussio is restricted to the upper hlf ple The results o the lower hlf ple re obtied by logy Let deote the set of lytic represettios of the fuctios i For the lytic represettio f ( z) of f ( t) (where z x iy ), pplyig the Cuchy iequlity, oe c derive So, if ( z ) y f f L z f, the f ( ) (*), for y Im z y > Let f d observe the sclig represettio f ( x) ( x ) ; lso, let C The Z f be the lytic represettio of f o ( z) ( z ) f y ( ) f (43) Proc Fc Mech Eg Sope, 3,, 4 47 (4)

5 Alytic wvelets d multiresolutio lysis A ote o certi orthogolity coditios 45 The lst iequlity follows from (*) So, if ( x) ( x ) f, Z the ( z ) f ( z) Z Property : Therefore, the opertor A mps oto, ie { ( ) } is lier sp of ( ) ) (see [], [3]) π { } is geertor set for ecll tht the Fourier trsform iξx ( ξ ): ( x) e trsfers the fuctio o the frequecy domi d exists for y squre itegrble fuctio For y rpidly decresig the lytic represet- tio is ( z) ( ) The set ( ) w e dw is ot lwys orthogol For exmple, for, the vlue of the sclr product is ( x i ) ( x i) iw ( w) e dw, π which could hve ozero vlue Still, ( ) is bsis o my occsios The followig property gives the coditios for bsis: Property : ( ) is bsis for ( w) lmost everywhere o [,), if By diltio d trsltio, oe c obti the elemets of the bsis of ll : ( ) ( t ) dt z, Z πi As coclusio follows: t z Theorem: Let be rpidly decresig sclig fuctio, let ( w) o [,) d let lmost everywhere be multiresolutio lysis for L ( ) ; the m m A m, m Z, form multiresolutio lysis for H ( C ), ie, the followig holds: ( C ) H, m d { ( ) } Z m, m Z ( ) H m C m Z is bsis for m (i geerl, the orthogolity is ot gurteed) Ad, m H C { ( )} m, Z is geertor set for ( ) The expsio o the upper hlf ple c be doe for wvelets s well, sice ( w) everywhere, if ( w) ψ lmost lmost everywhere Defiitio: The lytic represettio of the wvelet ψ ψ ( z) ( x) ψ πi x z is clled lytic wvelet o the upper/lower complex hlf ple Cosequece : Let S r be s i the theorem d let ψ be the wvelet The, ech subspce Wm A Wm hs bsis ψ m, d ech fuc- tio f H ( C ) f hs o uique represettio ( z) b m ( z) m,, ψ m, (44) 5 NOTE ON SOME EXTA CONDITIONS FO OTHOGONALITY Exmple: Let us loo t the Sho iξ / wvelets, defied by ψ ( ξ ) e I ( ξ ) I [ π, π ) ( π, π ], where The Fourier trsform of the sclig fuctio is ( ξ ) [ ]( ξ ) π, π Looig from the spce domi, the Sho wvelet d its sclig fuctio

6 46 Aces si re ( ) ( πx) cos( πx) ψ x, ( x) π ( x ) figures 4 d 5) ( πx) si (see πx Fig 4 Grph of the Sho sclig fuctio Fig 5 Grph of the Sho wvelet The sclr product of the lytic represettios o the -level is ( x i ) ( x i) π π π iw ( w) e dw iw π e πi π πi for iπ iπ ( e e ) So, the lytic MA, geerted by the Sho sclig fuctio, preserves the orthogolity o -th level This exmple implies modified defiitio of the lytic sclig fuctio *, the provided ( w) hs compct support i [, ), π d ( ) : * ( z) : ( w) e dw, Im z > π For the Sho sclig fuctio we hve, *( z) : ( w) e dw π π πiz πiz e π dw iπz iπz ( e e ) Cosequece : If ( w) compct support i [ ] ( ), the is bouded o its,, where π d * ( z) ( w) e dw (5) π provides orthogol bsis for { * ( ) } Z By logy, is orthogol bsis for Proof: The orthogolity of * ( z ) esured, becuse ( ), * ( ) * ( x i ) *( x i ) is * w π M πi M πi ( ) e iw dw iw e i i ( e e ) It is sufficiet to prove tht the zero fuctio hs hve compct support s i the coditios of the cocequece d let ( z ) The Z uique represettio Let ( w) Z π iw ( ) ( z w e ) dw iw ( w) e e dw Z Proc Fc Mech Eg Sope, 3,, 4 47 (4)

7 Alytic wvelets d multiresolutio lysis A ote o certi orthogolity coditios 47 The e ( w) χ[ )( w) iw, lmost everywhere By defiitio, ( w) [ )( w) χ, out of the support of ( w) So, iw e lmost everywhere Therefore,, Z CONCLUSION This wor dels with expdig wvelets o the complex ple usig their lytic Cuchy represettios Such expsio of wvelets o the complex ple exists The compct support of the Fourier trsformed sclig fuctio will provide orthogolity o the complex ple o the -th level, though It seems there is o reso gist orthogolity o vrious levels However, coclusio o this mtter hs ot bee chieved EFEENCES [] Weiss Herdez, A first course o wvelets, CC Press, 996 []!" # "$% &!' [3] Bremerm, Distributios, Complex ribles d Fourier Trsforms, Addiso Wesley Publ Co, 965 [4] Wlter, Alytic epresettios with Wvelet Expsios, Complex ribles, 6, (994) [5] Dubechies, Te Lectures o Wvelets, SIAM, Tilor Frcis, 99 ( )**,-*-///3/4//3! "#" $%%$ & roz@mfuimedum!"# $' $5 6'$!$ 6!' ('! $' (6 $' & ' $5 6$'$ 4'5$ $" $ 7! 5 $% 7! 5 "5 $" $ $5 '8 ' $5 55' wvelets9 ' $5 $:$ " %!5 $ 55$' $5 5$$% $/5!"($ 5!5; $' $5 $'$$ $! $ 5$ $' & ($ < $ $ $' & ' $5 '"$ $ 5$ % $5$ 55"5$!' ('! $' % ( $ ("!5 $'$ 55 5!"""$'" 55$'$% $ ( $ $ " $' & '$!$ # " $!'5 $'$ (&!% 5$ $' & ' $5 $,$$