PART V. Wavelets & Multiresolution Analysis

Transcription

1 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)

2 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)

3 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

4 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

5 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

6 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

7 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

8 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

9 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.

10 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

11 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.

12 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

13 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

14 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

15 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

16 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

17 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

18 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

19 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

20 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 ( )

21 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

22 Wveles 86 THE END