Lab 1 - Wavelet-based biomedical signal processing. Lab Task 1.1 (need 1 week) What & Why is wavelet?

Transcription

1 Lab - avelet-based bioedical sigal processig Lab Task. eed week hat & hy is wavelet? Learig goal: avelet tutorial; Cotet: Use Haar or other basic other wavelets to study the Tie-Frequecy doai eatures o a dowloaded EEG sigals. Lab Task. eed week How to use T wavelet trasor? Learig goal: avelet-based copressio. Cotet: check the copressio ratio by wavelets. Recostruct the copressed EEG strea. Lab Task.3 eed week hat do we eed i the sigal? Learig goal: Seek sigularity poits via LHE algorith. Cotet: Use TMM theory to id the LHE o the wavelet "doiat" eatures. hat & hy is wavelet?

2 . avelet Trasor Tutorial Matheatical trasors are applied to sigals to obtai ioratio that is ot readily available i the raw sigals. Most popular trasors iclude Fourier trasor ad wavelet trasor. Both two trasors are reversible liear trasors. a Fourier Trasor The Fourier Trasor is a atheatical operatio to traser the sigal ro tie to the requecy doai. e ca obtai requecy ioratio or sigal aalysis. X t e jt dt Iverse Fourier Trasor t X e jt d Discrete Fourier Trasor: F e j / Iverse Discrete Fourier Trasor:

3 3 / j e F The atri ipleetatio o Discrete Fourier Trasor: F F F F where j e /. The atri ipleetatio o Discrete iverse Fourier Trasor: F F F F V V V V V V V V where j V e /. Eaple:

4 ties FreqecyHz Figure Oe liitatio is that the Fourier trasor ca t localize the requecy eatures o tie doai ad deal ieectively with o-statioary sigals. b Short-tie Fourier trasor STFT STFT is a Fourier-related trasor used to deterie the siusoidal requecy ad phase cotet o local sectios o a sigal as it chages over tie. X t, w t e j d 4

5 where is a ask widow uctio. Figure Separate sigal i requecy-tie doai or ied resolutio. igure c avelet Trasor 5

6 avelet trasor is to decopose ad represet sigals by dieret wavelet uctios to etract useul ioratio we eed. It has leible resolutio i both tie ad requecy doais ad ca localize the ioratio easily. avelet trasors are broadly divided ito three classes: cotiuous, discrete ad ultiresolutio-based. Meyer Figure 3 Morlet t Mother wavelet. igure 3 t a t b a Scalig ad shitig. a, b t t a b a i. Cotiuous avelet Trasor CT For tie-requecy sigal aalysis we use CT. 6

7 t b wa, b a, b, t t a, b dt a a Iverse CT: t C where C w a, b w w a, b dadb t a dw ad w dw ii. Discrete avelet Trasor DT For ipleet wavelet trasor coputatio we use DT. DT: IDT: iii. Multiresolutio / w t, a t a t b dt,, ψ, t = a / ψa t b. w t t,, To reduce the uerical copleity, we use oe auiliary uctio arther uctio to represet the DT. Ad eawhile we get a ultiresolutio epressio o the sigal based o wavelet trasor. igure 4 7

8 a STFT Figure 4 b DT

9 9 avelet Trasor:,, V,, Iverse avelet Trasor:,,,, V here or DYADIC wavelet trasor:, is the scalig uctio ather uctio; ad, is the wavelet uctio other uctio. Ad is the orthogoal copleet o V to the V +. V + = V The atri ipleetatio o Discrete avelet Trasor Haar:,,,,,,,,,,,,,,,3,,,,,,, L L L K L L L where /, log K L, that is HF. The atri ipleetatio o iverse Discrete avelet Trasor:

10 F H T A alterative atri ipleetatio o Discrete avelet Trasor: where H H L L H H F H L K K, K K, O, M, M O I ad K, M log L, L [, log ] / The alterative atri ipleetatio o Iverse Discrete avelet Trasor: F T T H H H T L H T L For eaple, take the Haar wavelet as the base or a sigal o saples,, [ ], [, [ ], [ ], [ ], [ ], [ ],3 [ ] The atri ipleetatio o Discrete avelet Trasor is ]

11 The atri ipleetatio o Iverse Discrete avelet Trasor is

12 For the alterative atri ipleetatio o DT ad IDT, we have H H 3 H

13 It is easy to prove that H H 3H H iv. Filter Bak I practical ilter bak are ostly used or wavelet trasor ipleetatio. Figure 5 As show i igure 5, sigal Is iltered to low requecy approiatios ad the high requecy details. For aalysis with orthogoal wavelets the high pass ilter is calculated as the quadrature irror ilter QMF o the low pass, ad recostructio ilters are the tie reverse o the decopositio ilters. e ipleet the T atri by this.igure 6. Ad the recostructio is as the iverse directio ad use the upsaplig operatio istead o the dow-saplig i the decopositio. 3

14 Decopositio Recostructio Figure 6 Ipleetatio o wavelet decopositio ad recostructio with ilter bak. How to use T? 4

15 . EEG Sigal de-oisig ad copressio applicatios with T. I additio to sigal aalysis, based o this eatures the wavelet trasor T could also be used or sigal copressio ad oise reoval. a De-oisig 5

16 Base o the wavelet trasoratio, we ca cacel the sall coeiciets o sigal to reduce the oise iterereces. 5 4 Origial sigal uber o Saples 5 4 Recostructio sigal uber o Saples Figure 7 Origial ad recostructed EEG sigals usig wavelets 6

17 4 origial sigals sigals with oise Figure EEG sigals: a origial oe. b oisy oe 4 Recostructio sigal 7

18 Figure 9 EEG sigals ater T de-oisig b Copressio he sigals traserred ito wavelets doai they would be sparse or copressible. Thereore we could keep part o the large coeiciets to recostruct the origial sigals eawhile without uch quality loss. Figure

19 5 w wavelet coeiciects w wavelet coeiciects w reduced wavelet coeiciects Figure avelet coeiciets a b c 9

20 Figure avelet recostructio based o dieret copressio ratio hat do we eed i the sigal? 3. EEG Sigal structure eature etractio a Soothig

21 Dieret ro other wavelet trasor, i this lab we use uior saplig other tha dow saplig to see the derivatio detail i the whole ECG sigal. The we get the a value or TMM Figure soothed sigal by dieret scale T b avelet Trasor Modulus Maia. he a other wavelet is a gradiet o a soothig uctio, ulti-scale gradiets ca be coputed as wavelet trasor. Two wavelet uctios are the -order ad -order gradiets o a sooth, iite support uctio, a d b d, d d A uctio with dilatio s is deoted by

22 s s s. The wavelet trasor o a uctio is give by s s a a s *, s s b b s *, where * represets covolutio. Thereore, by detectig the odulus aia o the wavelet trasor o a sigal,, the structure ioratio o the sigal ca be captured. Use TMM or the sparse ad siple sigals, we ca detect it ad get the ioratio we eed eawhile deduce the size ad save the resources.

23 Figure 3 Sigal structure captured by TMM Figure 3 showed the structure eature etracted by TMM ater ulti-scale soothig. Matlab Eperiets: 3

24 Lab.: rite a piece o code that ca develop the atri ipleetatio i size o 5656 o DT ad IDT usig the Haar base. Lab.: Dowload a piece o EEG sigal ad peror DT ad IDT. Copare the wavelet coeiciets o dieret level wavelet trasor. Replace soe coeiciets o sall agitudes with zeros; plot the recostructio errors with respect the copressio ratio. Lab.3: Peror the wavelet trasor o a piece o EEG sigal ad detect the odulus aia to capture the structure o the EEG sigal. Matlab Code Saples:. Build ilter uctio or dieret wavelets. Build wavelet trasor atri 3. Load EEG sigals 4. Peror wavelet decopositio ad recostructio 5. TMM.. Build ilter uctio or dieret wavelets 4

25 uctio = MakeOFilterType,Par % Outputs % q quadrature irror ilter i strcptype,'haar', ed = [ ]./ sqrt; i strcptype,'beylki', = [ ]; ed i strcptype,'coilet', i Par==, = [ ]; ed Dieret ilter has dieret coeiciets to ake the QMF 5

26 . Build wavelet trasor atri uctio = avmath,, k, shit %--ake QM ilter G h=h:'; g = liplrh.* -.^:legthh; or k= k:-: clear gat; clear hat; ubjk = ^J-k; ubjk = ^J-k+; or jj= :ubjk or ii=:ubjk odulus = od+ii-*jj+shit,ubjk; odulus = odulus + odulus == *ubjk; hatii,jj = hodulus; gatii,jj = godulus; ed ed = [oldat * hat'; gat' ]; oldat = ; ed 6

27 3. Load EEG sigals 4. Peror wavelet decopositio ad recostructio 7

28 5.,TMM.

29 Reerece [] Stephae Mallat ad Sie Zhog, Characterizatio o Sigals ro Multiscale Edges, IEEE TRASACTIOS O PATIER AALYSIS AD MACHIE ITELLIGECE [] Stephae Mallat, A avelet Tour o Sigal Processig [3] Igrid Daubechies, couses o wavelet [4] Mallat, S. G.,A Theory or Multiresolutio Sigal Decopositio: The avelet Represetatio, IEEE Tras. PAMI, vol., o. 7, July 99, pp [5] [6] [7] Qi Hao, Fei Hu, A Copressive Eletroecephalography EEG Sesor Desig, IEEE Sesors 9