Series Expansion with Wavelets. Advanced Signal Processing Reinisch Bernhard Teichtmeister Georg

Transcription

1 Series Expasio wih Waveles Advaced Sigal Processig - 7 Reiisch Berhard Teicheiser Georg

2 Iroducio Series expasio Fourier Series: Eiher periodic or badliied sigals Tiedoai: No frequecy iforaio Fourierdoai: No ie iforaio Is here soehig bewee? ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio /6

3 Coes Basics of sigal represeaio Waveles Haar wavele Muliresoluio aalysis Cosrucio of he Sic - Wavele Waveles derived fro ieraed filer baks Haar case, Sic case, geeral cosrucio Wavele series ad is properies Pracical oulook iage processig ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 3/6

4 Recap of Series expasio Sigals are pois i a Vecorspace Tie-Doai: Basis fucios are ifiie shor ipulses Sigals ca be projeced oo oher basis fucios f ϕ u, k k f u ϕ u, k f u ϕ * ϕ u f u du k k ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 4/6

5 Possible Basis Fucios Fourierseries periodic Fourierrasfor badliied STFT f x[ ] Ifiie se of Fourier Trasfors Piecewise Fourier Series Waveles k π π π F[ k] e X e jω jπk/ T e jω dω ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 5/6

6 Shor Tie Fourier Trasfor Widow Sigal Copue he Fourier Trasfor Shif widow ad repea Specrogra, Periodogra ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 6/6

7 Tie ad Frequecy Resoluio Widow has Eergydisribuio i boh: Frequecy σ ω ad Tie σ. Uceraiy priciple: σ ω σ Opialiy is oly reached by Gaussia widow ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 7/6

8 STFT T/F-Resoluio Cosa over Tie ad Frequecy ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 8/6

9 Piecewise Fourierseries Fourier Series wih o-overlappig recagular widows i ie ad periodic expasio Why? Overlappig widows are reduda iforaio Good Tie Resoluio Represeaio of arbirary fucios Bad Frequecy Resoluio Errors a boudaries ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 9/6

10 Desired Feaures of Basis Fucios Siple characerizaio Localizaio Properies i Tie ad Frequecy Ivariace uder cerai operaios Soohess properies Moe properies ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio /6

11 Haar - Expasio Siples Wavele Expasio Scaled ad shifed Waveles: φ φ, Scale Tieshif / ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio /6

12 Dyadic Tilig Resoluio depeds o Frequecy ow φ / φ, ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio /6

13 Orhooral Basis for L? Two waveles o he sae Scale have o coo suppor Shorer wavele always averages o zero Shifig so ha jup aches, is o possible ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 3/6

14 Proof: Defiiios Cosider fucios which are cosa o [, ] ad have fiie suppor o [, ] Ca approxiae L arbirarily well We call i f ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 4/6

15 5/6 ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio Proof: Scalig fucio The scalig fucio Approxiaig he piecewise cosa fucio < oherwise, ϕ, N N f f N f f ϕ

16 Proof: Illusraio ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 6/6

17 Proof: Keysep Exaiaio of wo adjace Iervals Now [ [, ad, f ca be expressed as f ϕ, f, ϕ For,, his eas [,3 ad [3,4 f ϕ, f3 ϕ,3 ϕ, 3ϕ,3 ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 7/6

18 8/6 ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio Proof: Average ad Differece The fucio ca also be expressed as he average ad he differece over wo iervals f, f f ϕ, f f φ

19 9/6 ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio Proof: Coefficies Wih We ge f f d f f f,,,, d f f f φ ϕ ϕ ϕ

20 /6 ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio Proof: Fializaio Applyig all he higs we ca wrie Repeaig he Average/Differece schee for higher scales leads o / /, / /, N N N N d f d f f φ ϕ M M d d f f ε φ φ,,

21 Proof: Illusraio ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio /6

22 Muliresoluio Successive approxiaio Coarse approxiaio added deails Coarse ad deail subspace are orhogoal Leads o self-siilar Waveles i Scale Useful for applicaios ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio /6

23 Axioaic Defiiio Sequece of ebedded closed subspaces K V V V V V K Upward Copleeess U V Z L R Dowward Copleeess I V Z {} ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 3/6

24 Axioaic Defiiio Scale Ivariace f V f V Shif Ivariace f V f V Z Exisace of a orhooral Basis No-orhogoal Basis ca be orhogoalized ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 4/6

25 Orhogoal Coplees V is a subspace of V - We defie W he orhogoal subse of V i V - V V W V is he space of he scalig fucios W he space of he waveles By repeaig we ge L R Z W M ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 5/6

26 Cosrucig he Sic Wavele Now he scalig fucios will be he space of badliied fucios V is badliied bewee [-π, π], V - bewee [-π, π] W he fucios badliied o [-π, -π] cobied wih [π, π] V V W ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 6/6

27 Scalig fucio The scalig fucio is give by siπ ϕ π ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 7/6

28 8/6 ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio Represeaio of φ V belogs o V - φ ca be represeed by basis fucios of V - Wihou proof, ] [ ; ] [ ] [ g g g ϕ ϕ ϕ ϕ Z g g g ] [ ] [ ] [ ϕ φ

29 Cosrucio Kerel g is give by g [ ] G e jω si π / π / e jω Ad fially he wavele π ω oherwise π si π / φ cos3π / π / ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 9/6

30 Sic Wavele: Illusraio ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 3/6

31 Ieraed filer baks Uil ow we cosruced waveles by scalig ad shifig of orhooral fucio failies Based o uliresoluio aalysis Differe approach by filer baks Ieraio leads o a wavele Key proberies regulariy degree of regulariy ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 3/6

32 Haar case Low- ad Highpass g, ; g, Ierae he filer bak o he lowpass chael Mulirae sigal processig resuls ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 3/6

33 Haar case size-8 discree Haar rasfor Nuber of coefficies growh expoeially Coiuous ie fucio Legh bouded, piecewise cosa ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 33/6

34 Sic case Ipulse resposes low ad highpass filer g si π / [ ] ; g[ ] π / g [ ] Fourier rasfor Now cosider he ieraed filer bak Upsaplig filer ipulse Eulae he Haar cosrucio wih g [], g [] Ad defie a scalig fucio ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 34/6

35 35/6 ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio Sic case For furher aalysis: ipora par is i he brackes This produc is i π periodic -> i he ed i s oly a perfec lowpass sic scalig fucio / i / / / i / / si : rewrie ca we M : shor... : where / / si of rasfor Fourier Φ Φ i i j i k k j j j j j i i i j j i i i i i i i e M e G e G e G e G e G e e G ω ω ω ω ω ω ω ω ϕ ω ω ω ω ω ω ω ω

36 Sic case 3 Cubersoe way Bu we have gaied a ore geeral cosrucio The key is he ifiie produc Does his produc coverge ad o wha Coverge o wha kid of scalig fucio ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 36/6

37 Ieraed filer baks co. Geeral cosrucio Two chael orhogoal filer bak g [], g [] are low- ad highpass filer Ierae o he brach of he lowpass filer ad process his o ifiiy Express he wo filers afer i-seps Mulirae coclusios Filerig wih G i z followed by upsaplig by is equivale o upsaplig by followed by filerig wih G i z ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 37/6

38 Ieraed filer baks co. Discree ie ieraed filers cobied wih he coiuous ie fucios Noralizaio ad rescalig Graphical fucio piecewise cosa halvig he iervall ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 38/6

39 Ieraed filer baks co. 3 Fourier doai ac as above I he ieraio schee we are ieresig i covergece This will lead us o regulariy discussio ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 39/6

40 Regulariy The exisece of he lii are criical codiios Liis exis if g [] are regular Regular filer leads hrough ieraio o a scalig fucio wih soe degree of soohess regulariy Bu o oly covergece is sufficie we eed also L covergece o build orhooral bases A lo of sufficie codiios, differe approaches ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 4/6

41 Wavele series ad properies Eueraio of soe geeral properies of basis fucios f, Z F[, ] Wavele F[, ] ψ ψ,,, f ψ,, f d Lieariy, Shif, Dyadic saplig ad ie frequecy ilig, Scalig, Localizaio, decay properies ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 4/6

42 Lieariy suppose operaor T T[f] F[,] ψ he for ay a,b R T[ a f,, f b g ] at f bt g The wavele series is liear. The proof follows fro he lieariy of he ier produc ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 4/6

43 43/6 ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio Shif For Fourier rasfor pair: f, Fω f-au, e -jωau Fω Now for he wavele series [ ] [ ] [ ] k F k f Z k k Z d f F d f F < ',,, or,, ' ' / ', ' τ τ τ ψ τ ψ

44 Scalig For Fourier rasfor F F pair: f, Fω fa, /a*fω/a ' ' [, ] ψ f a [ ] /, / a ψ f a f k k, k Z k, F [ k, ] d a d ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 44/6

45 Dyadic saplig ad ie frequecy ilig I is ipora o locae he basis fucios i he ie-frequecy plae saplig i ie, a scale, wih period ψ ψ,, The frequecy is he iverse of scale, we fid if he wavele is ceered aroud ω he: Ψ, ω is ceered aroud ω / This leads o dyadic saplig of ie frequecy plae ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 45/6

46 Dyadic saplig The dos idicae he ceer of he waveles The scale axis is logarihic ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 46/6

47 Tie localizaio Suppose we are ieresed i he sigal aroud Which values of F[,] carry iforaio abou sigal f a > f Suppose wavele ψ is suppored o he ierval [-, ] Ψ, is suppored o [-, ] Ψ, is suppored o [-, ] ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 47/6

48 Tie localizaio A scale, wavele coefficies wih idex saisfy - ca be rewrie - ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 48/6

49 Frequecy localizaio Suppose ow i localizaio, bu ow i frequecy doai ψ, F[, ] / ψ he Fourier rasfor is / Ψ ω F[, ] e ψ, π j ω f / F d j ω ω Ψ ω e dω ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 49/6

50 Frequecy localizaio Suppose ha he wavele vaishes i he Fourier doai ouside he regio [ω i,ω ax ] A specific scale, he suppor of Ψ, ω will be [ω i /,ω ax / ] Therefore, a frequecy copoe ω iclueces a scale ωi ω ω rewrie log ω ω i ax ASP 7 Teicheiser G. Reiisch B. log ω ω ax Wavele cosrucio 5/6

51 Decay properies Fourier series ca be used o characerize he regulariy of a sigal decay of he rasfro coefficies Global regulariy The wavele rasfor ca be used i a siiliar way Local regulariy ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 5/6

52 Mulidiesioal waveles A way o build ulidiesioal waveles is o use esor producs of heir oe diesioal couerpars This will lead o differe oher waveles Scale chages are ow represeed i arices offers diagoal scalig bu is also ore resriced ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 5/6

53 Pracical aspecs Waveles i alab Iages Iage copressio Edge deecio De-oisig ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 53/6

54 Malab Malab wavele oolbox Coad lie Help wavele Gui ool waveeu D waveles aalysis D waveles aalysis De-oisig Iage Fusio Copressio ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 54/6

55 Exaple D wavele rasfor Discree/coiuous wavele rasfor coefs cws, SCALES, wae ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 55/6

56 Copressio Wavele calculaio Fid sall coefficies ad discard he Sore oly reaiig coefficies Lossy copressio Good copressio wih a fas covergece speed of he wavele ad good decay of he coefficies ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 56/6

57 Copressio ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 57/6

58 Edge deecio origial decoposiio approxiaio is se o zero recosrucio ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 58/6

59 Refereces [] Waveles ad Subbad Codig, Mari Veerli, Jelea Kovacevic, ISBN [] Waveles prakische Aspeke, Markus Graber, VO6 [3] AK Copuergrafik Bildverarbeiug ud Musererkeug WS 6/7 ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 59/6

60 Series Expasio wih Waveles Thak you for you aeio! Advaced Sigal Processig 7 Teicheiser Georg Reiisch Berhard ASP 7 Teicheiser G. Reiisch B. Wavele cosrucio 6/6