Wveles 65 PART V Wveles & Muliresoluion Anlysis ADDITIONAL REFERENCES: A. Cohen, Numericl Anlysis o Wvele Mehods, Norh-Hollnd, (003) S. Mll, A Wvele Tour o Signl Processing, Acdemic Press, (999) I. Dubechies, Ten Lecures on Wveles, SIAM, (99) www.wvele.org
Wveles 66 WAVELETS OVERVIEW (I) Wh is wrong wih FOURIER ANALYSIS??? All spil inormion is hidden in he PHASES o he epnsion coeiciens nd hereore no redily vilble Loclized uncions ( bumps ) end o hve very comple represenion in Fourier spce Locl modiicion o he uncion ecs is whole Fourier rnsorm I he dominn requency chnges in spce, only verge requencies re encoded in Fourier coeiciens Remedy need n nlysis ool h will encode boh SPACE (TIME) nd FREQUENCY inormion he sme ime Following he convenion, will work wih TIME () nd FREQUENCY (), rher hn wvenumber (k)
Wveles 67 WAVELETS OVERVIEW (II) From DISCRETE FOURIER TRANSFORM o INTEGRAL FOURIER TRANSFORM Consider he spce L o squre inegrble uncions deined on ; i L sisies suible decy condiions (which??), he DISCRETE FOURIER TRANSFORM cn be replced wih he INTEGRAL FOURIER TRANSFORM ˆ ˆ e i d e i d Ineresingly, he Fourier Trnsorms (boh discree nd inegrl) re consruced s superposiions o DILATIONS o he uncion w (w k k ) w Wn o consruc n inegrl rnsorm using bsis uncion ψ which is very loclized ( wvele ); we will hereore need: DILATIONS TRANSLATIONS e i
Wveles 68 WAVELETS GABOR TRANSFORM (I) The hisory begins wih WINDOWED FOURIER TRANSFORM known s he GABOR TRANSFORM (946) G α b e i where he WINDOW FUNCTION is given by g α g α b d πα e 4α wih α Noe h he Fourier rnsorm o Gussin uncion is noher Gussin uncion, i.e., i e 4 d e π e 0 Noe lso h he window uncion hs he ollowing normlizion b d gα db gα Thereore, or he Gbor rnsorm we obin G α b dbˆ Thus, he se o Gbor rnsorms o decomposes he Fourier rnsorms ˆ o ecly o give is LOCAL specrl inormion G α b : b
Wveles 69 WAVELETS GABOR TRANSFORM (II) The WIDTH o he window uncion cn be chrcerized by employing he noion o he STANDARD DEVIATION Noe h or α Proo: 0 gα gα g α α gα g α 8πα 4 cn be evlued seing 0 nd α epression or he Fourier rnsorm o Gussin uncion gα d in he d cn be evlued diereniing wice he Fourier rnsorm o Gussin uncion nd gin seing 0 nd α Insed o loclizing he Fourier rnsorm o, he Gbor rnsorm my equivlenly be regrded s windowing wih he WINDOW FUNCTION G α b G α b G α b Gb α G αb d e i πα e 4α
Wveles 70 WAVELETS GABOR TRANSFORM (III) Using he Prsevl ideniy nd noing h Ĝ α b η e ib η α e η we obin or he Gbor rnsorm G α b π e e G α b ib πα ˆ π ib πα G 4α η e ib ˆ e ibη ˆ ˆ η b Ĝα b α e η g η 4α dη η dη The hird line ( in red ) indices h up o muliplicive cor he WINDOWED FOURIER TRANSFORM o wih g α b, he WINDOWED INVERSE FOURIER TRANSFORM o ˆ wih g η ARE EQUAL! π α e 4α ib
Wveles 7 WAVELETS UNCERTAINTY PRINCIPLE (I) Consider more generl window uncions w requiremen w I cn be shown h w w L L L he Fourier rnsorm ŵ is coninuous ŵ L Noe, however, h in generl ŵ be FREQUENCY WINDOW FUNCTION L L which sisy he, hereore w my no in generl I w L is chosen so h boh w nd ŵ sisy he bove condiion, hen he window Fourier rnsorm G b e i w b d Wb where W b eiw b, is clled SHORT TIME FOURIER TRANSFORM
b Wveles 7 WAVELETS UNCERTAINTY PRINCIPLE (II) We cn deine he CENTER w w d nd RADIUS w o w s w w w d Then, G b gives locl inormion on in he TIME WINDOW b w w We cn deermine he CENTER nd he RADIUS ŵ o he (requency) window uncion ŵ using ormule similr o he bove Deining V b η η window uncion wih he cener he Prsevl ideniy) πŵb G b π eib e ibηŵ η, which is lso nd rdius ŵ, we cn wrie (using Wb ˆ Vb Thus, window G b lso gives locl specrl inormion bou in he requency ŵ ŵ
b Wveles 73 WAVELETS UNCERTAINTY PRINCIPLE (III) Thereore by choosing w L, such h w L ŵ L, o deine windowed Fourier rnsorm loclizion in TIME FREQUENCY WINDOW G b nd we obin b w w ŵ ŵ wih re equl o 4 w ŵ In c, here is relion beween possible ime nd requency windows which is mde precise in he ollowing heorem HEISENBERG UNCERTAINTY PRINCIPLE Le w h w nd ŵ. Then L L L be chosen so w ŵ Furhermore, equliy is ined i nd only i w ceiα gα b where c0, α 0, nd b.
Wveles 74 WAVELETS UNCERTAINTY PRINCIPLE (IV) Proo o he HEISENBERG UNCERTAINTY PRINCIPLE Le us ssume h he ceners cn modiy w s w i We observe h w ŵ e w w Using he Schwrz inequliy we ge nd re zero (i hey re no, hen we ) d d ŵ w 4 ŵ w w d d w ŵ 4 w w 4 4 w 4 w w w w w d w d w d
Wveles 75 WAVELETS UNCERTAINTY PRINCIPLE (V) Proo o he HEISENBERG UNCERTAINTY PRINCIPLE coninued Inegring by prs nd noing h lim w seen erlier) we obin L w ŵ 4 w 4 w 0 d 0 (since An equliy will be obined when he Schwrz inequliy becomes n equliy; his implies h here eiss b such h 4 w bw so h here eiss n such h w e b Thus he GABOR TRANSFORM hs he smlles possible ime requency window. The bove Heisenberg Unceriny Principle hs r reching consequences.
Wψ Wveles 76 INTEGRAL WAVELET TRANSFORM (I) The shor ime Fourier rnsorm hs RIGID ime requency window, in he sense h is widh ( w ) is unchnged or ll requencies nlyzed; his urns ou o be limiion when sudying uncions wih vrying requency conen The INTEGRAL WAVELET TRANSFORM provides window which: uomiclly nrrows when ocusing on high requencies, uomiclly widens when ocusing on low requencies I ψ L sisies he dmissibiliy condiion C ψ ˆψ hen ψ is clled BASIC WAVELET. Relive o every bsic wvele ψ. he INTEGRAL WAVELET TRANSFORM (IWT) in L is deined by d b ψ b d L 0 b
Wψ Wveles 77 INTEGRAL WAVELET TRANSFORM (II) Hereer we will ssume h ψ L nd ˆψ L, so h he bsic wvele ψ provides ime-requency window wih inie re From he bove ssumpion i lso ollows h ˆψ is coninuous uncion nd hereore inieness o C ψ implies ˆψ 0 0 ψ d 0 Seing he IWT cn be wrien s ψ b; b ψ b ψb; I he wvele ψ hs he cener nd rdius given by nd ψ, respecively, hen he uncion ψ b; hs is cener b nd rdius equl o ψ Thus, he IWT provides locl inormion bou he uncion in ime window b which nrrows down s 0. ψ b ψ
Wveles 78 INTEGRAL WAVELET TRANSFORM (III) Consider he Fourier rnsorm o bsic wvele π ˆψ b; π e i ψ b d π e ib ˆψ Suppose h ˆψ hs he cener nd rdius ˆψ. Deining η ˆψ we obin window uncion wih cener he origin nd unchnged rdius Applying he Prsevl ideniy o he deiniion o he IWT we obin b Wψ π ˆ e i η which, modulo muliplicion by consn cor nd liner requency shi, loclizes inormion bou he uncion o he FREQUENCY WINDOW ˆψ ˆψ d
Wveles 79 INTEGRAL WAVELET TRANSFORM (IV) Noe h he rio o he CENTER FREQUENCY ˆψ cener requency bndwidh ˆψ o he BANDWIDTH is independen o he scling ; hus, he bndwidh grows wih requency in n dpive shion ( consn Q ilering ) Reconsrucion o uncion rom is IWT Le ψ be bsic wvele, hen 0 Furhermore, or ny L C ψ 0 b Wψ g nd L Wψg b db d C ψ g which is coninuous Wψ b ψb; db d Proo using he Prsevl ideniy, inegring wih respec o d using he deiniion o C ψ Noe he role o he ADMISSIBILITY condiion or ψ nd
k m d Wveles 80 DISCRETE WAVELET TRANSFORM (I) Consider he IWT discree se o smples some j where k Wψ k j j k j ψ j mus be chosen so h ψ j o ψ j is dense in L I ψ j k wih j k wih j k k j ψ ) ψ j j k orm Riesz bsis in L is RIESZ BASIS, he he relion k k j nd bk ψ j j or (i.e, he liner spn ψl m ψ j k δ j lδ k uniquely deines ANOTHER RIESZ BASIS ψ l Thus, every uncion L j k l m m known s he DUAL BASIS hs unique represenion k j ψ j k ψ j k
k Wveles 8 DISCRETE WAVELET TRANSFORM (II) For he bove represenion o quliy s WAVELET SERIES, he dul bsis ψ j mus be obined rom some bsic wvele ψ by ψ j k k ψ j, where k j ψ j ψ j k k In generl, ψ does no necessrily eis I ψ is chosen so h ψ does eis, he pir inerchngebly ψ ψ cn be used k j ψ j ψ j k k j ψ j k ψ j k ψ nd ψ re clled WAVELET nd DUAL WAVELET, respecively I he bsis ψ j k ORTHOGONAL WAVELET TRANSFORM k j k is orhogonl, i.e., ψ j ψ j ψ j k k or j ψ j k k, we obin n
Wj nd gl Wveles 8 DISCRETE WAVELET TRANSFORM (III) Consider wvele ψ nd he Riesz bsis ψ j W j denoe THE CLOSURE OF THE L INEAR SPAN o k i generes; or ech j k : k ψ j, le, i.e., W j clos L ψ j k : k Evidenly, L cn be decomposed s DIRECT SUM o he spces W j (dos over pluses indice direc sums ) L j W j W W0 W nd hereore every uncion L hs unique decomposiion g g0 g where g j Wj, j i ψ is n ORTHOGONAL WAVELET, hen he subspces W j MUTUALLY ORTHOGONAL W jw l, l j which mens h L re g j gl 0 l j where g j W l
Wveles 83 DISCRETE WAVELET TRANSFORM (IV) Thereore, in such cse, he direc sum becomes n ORTHOGONAL SUM L W j W W0 W j Thus, n orhogonl wvele ψ generes n ORTHOGONAL DECOMPOSITION o he spce L UNIQUE MUTUALLY ORTHOGONAL, s he uncions g j re
Wveles 84 MULTIRESOLUTION ANALYSIS (I) For every wvele ψ (no necessrily orhogonl) we cn consider he ollowing spce V j, j L V j Wj Wj The subspces V j hve he ollowing very ineresing properies:.. clos L 3. j 4. V j 5. V Vj Noe h j V j 0 Wj, j V j V0 V Vj L Vj, j In conrs o he subspces W j which sisy W Wl j sequence o subspces V j is NESTED ( ) 0, l, j he Every L cn be pproimed wih ARBITRARY ACCURACY by is projecions P j on V j ( )
Wveles 85 MULTIRESOLUTION ANALYSIS (II) I he reerence subspce V 0 is genered by single SCALING FUNCTION φ in he sense h L where V 0clos L k j φ j φ0 k : k hen ll he subspces V j re lso genered by he sme φ s φ j k V j clos L φ j k : k in he sme wy s he subspces W j re genered by he wvele ψ In he MULTIRESOLUTION ANALYSIS given scle ( j) he subspce V j represens he LARGE SCALE eures o he uncion he subspces W j represens he SMALL SCALE eures (deils) o he uncion
Wveles 86 THE END