Pattern Recognition Techniques Applied to Biomedical Signal Processing

Transcription

1 Patter Recogitio Techiques Applied to Biomedical Sigal Processig PhD Course- Prague, October 3 Dr.Ig. David Cuesta Frau dcuesta@disca.upv.es Polytechic Uiversity o Valecia- Spai

2 Course Syllabus Patter Recogitio Systems. Amplitude samples Trace segmetatio Polygoal aprox. Wavelet Trasorm Max-Mi k-meas k-medias Sesors Preprocessig Feature Extractio Clusterig Wavelet Trasorm Dyamic Time Warpig Hidde Markov Models Dissimilarity Measures

3 Schedule 6//3 7//3 8//3 9//3 // :3 Wavelet trasorm Feature extractio Clusterig algorithms Hidde Markov Models 4.3 Dyamic Time Warpig 3//3 4//3 5//3 6//3 7//3 3

4 Course Structure Lectures. Laboratory exercises. Assigmets. Total Credits:- 4

5 Sigal Processig Itroductio to Wavelets Dr.Ig. David Cuesta Frau Polytechic Uiversity o Valecia(Spai) dcuesta@disca.upv.es

6 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 6

7 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 7

8 Sigal Processig..- Itroductio. Extract iormatio rom a sigal. Time domai is ot always the best choice. Frequecy domai: Fourier Trasorm: X ( ) x( t) jπt e dt Easily implemeted i computers: FFT. 8

9 x() t Fourier Aalysis..- Itroductio. Ca ot provide simultaeously time ad requecy iormatio:time iormatio is lost. Small chages i requecy domai cause chages everywhere i the time domai. For statioary sigals: the requecy cotet does ot chage i time, i.e.: X ( ) t 9

10 Fourier Aalysis..- Itroductio. Basis Fuctios are set: siusoids. May coeiciets eeded to describe a edge or discotiuity. I the sigal is a siusoid, it is better localized i the requecy domai. I the sigal is a square pulse, it is better localized i the time domai. x() t X ( ) t Short Time Fourier Trasorm.

11 .- Itroductio. Short Time Fourier Aalysis. Time-Frequecy Aalysis. Widowed Fourier Trasorm (STFT). FT Basis Fuctios STFT Basis Fuctios

12 .- Itroductio. Short Time Fourier Aalysis. ( t, ω ) ( t) g( t t ) ( ω t)dt STFT si Discrete versio very diicult to id. No ast trasorm. Fixed widow size (ixed resolutio): Large widows or low requecies. Small widows or high requecies. Wavelet Trasorm.

13 .- Itroductio. Wavelet Trasorm. Small wave. Eergy cocetrated i time: aalysis o ostatioary, trasiet or time-varyig pheomea. Allows simultaeous time ad requecy aalysis. 3

14 .- Itroductio. Wavelet Trasorm. Fourier: localized i requecy but ot i time. Wavelet: local i both requecy/scale (via dilatios) ad i time (via traslatios). May uctios are represeted i a more compact way. For example: uctios with discotiuities ad/or sharp spikes. Fourier trasorm: O(log ()). Wavelet trasorm: O(). 4

15 Wavelet Trasorm..- Itroductio. WT t b a ( a, b) ( t) dt STFT Basis Fuctios WT Basis Fuctios 5

16 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 6

17 .- Deiitios. Mathematical Backgroud. Topic o pure mathematics, but great applicability i may ields. Liear decompositio o a uctio: i () t c () i t i Iteger idex or the iite or iiite sum ci Expasio Coeiciets () t Expasio Set i Decompositio Uique: { () t } Basis or ( t) i 7 i

18 .- Deiitios. Mathematical Backgroud. Basis orthogoal: () t, () t () t () l k l t () t () t dt k dt k l k, The the coeiciets ca be calculated by : c Sice: k k l, l () t, () t () t () t k k dt () t () () () k t dt ci i t k t dt c i 8 i k k

19 .- Deiitios. Mathematical Backgroud. Liear decompositio o a vector: a r a r ( 3,, ) (,, ) (,, ) + (, ) 3, {(,, )(,,, )(,,, )} ( 3, ), Coordiates i R 3 Geerator system i R 3 Idepedet Orthogoal Normal Basis i R 3 9

20 .- Deiitios. Mathematical Backgroud. () t, () t () t () t I dt,where m m m { () t } Orthogoal Basis or ( t) i c Otherwise : i () t, () t () t () t i { () t, () t } Biorthogoal Basis or ( t) i ~ i () t, () t () t c i () t ci ~ i i i i

21 () t.- Deiitios. Mathematical Backgroud. Example o liear decompositio o a uctio: 3 t () t i t () t ( t i)... i i + t () t () t m dt δ m m * () t () t () t dt m m

22 .- Deiitios. Mathematical Backgroud. Example o liear decompositio o a uctio: () t 3 t () t () t + ( t ) ( t ) c c c c i, otherwise

23 3.- Deiitios. Itroductio to Wavelets Example: Four dimesioal space (our o-ull coeiciets). Sythesis Formula: () () i i c i t t () () ( ) () () () ( ) () () () ( ) () () () ( ) () () () ( ) () c c c c

24 4.- Deiitios. Itroductio to Wavelets Example: () () ( ) () () () () () () () () () ( ) ( ) ( ) ( ) () () () () c c c c c

25 5.- Deiitios. Itroductio to Wavelets Example: () () () () () () () () () () () () () () () () () () ( ) () c c c c ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ c ~ T

26 .- Deiitios. Example: c ~ T ~ T ~ T c I colums are orthoormal: T T I Dual Basis T ~ T Sigle Basis 6

27 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 7

28 3.- Multiresolutio Aalysis. Problem deiitio: Time scale chage. ( t)? () t t t 8

29 3.- Multiresolutio Aalysis. Multiresolutio Formulatio. Ratioale: i a set o sigals ca be represeted by a weighted sum o ϕ(t-k), a larger set (icludig the origial), ca be represeted by a weighted sum o ϕ(t-k). Sigal aalysis at dieret requecies with dieret resolutios. Good time resolutio ad poor requecy resolutio at high requecies. Good requecy resolutio ad poor time resolutio at low requecies. Used to deie ad costruct orthoormal wavelet basis or L. Two uctios eeded: the Wavelet ad a scalig uctio. 9

30 3.- Multiresolutio Aalysis. Multiresolutio Formulatio. Eect o chagig the scale. Scalig uctio: To decompose a sigal ito ier ad ier details. Subspace : ν () t L { ()} t spa ϕ k L k () t c ϕ () () k k t t ν k Icrease the size o the subspace chagig the time scale o the scalig uctios: ϕ jk j () ( ) j t ϕ t k 3 ϕ () t

31 3.- Multiresolutio Aalysis. Multiresolutio Formulatio. Subspace : ν j { ( )} j ϕ t spa () t { } spa ϕ L 3 k () ( ) j t c () kϕ t + k t ν j k The spaed spaces are ested: ν ν ν L... ν ν... ϕ () t ν ( ) j t ν j+ () t h( ) ϕ ( t ), Z jk

32 3.- Multiresolutio Aalysis. Multiresolutio Formulatio. Example: Haar Scalig Fuctio. ϕ ϕ() t () t ϕ( t) + ϕ( t ) h () () h ϕ ( t) ϕ( t ) 3

33 3.- Multiresolutio Aalysis. Multiresolutio Formulatio. Icreasig j, the size o the subspace spaed by the scalig uctio, is also icreased. Wavelets spa the diereces betwee spaces w i. More suitable to describe sigals. Wavelets ad scalig uctios should be orthogoal: simple calculatio o coeiciets. ν ν w ν... L ν ν w w w w... 33

34 3.- Multiresolutio Aalysis. Multiresolutio Formulatio. w w w ν () t h [] ( t ), ℵ jk ϕ j () ( ) j t t k [] h[] l ϕ[ l] ϕ, l Z l Mother Wavelet Scale Fuctio 34

35 3.- Multiresolutio Aalysis. Multiresolutio Formulatio. g g + h h ( ) ( ) ( ) h( ) scalig uctio coeiciets accouts or orm () t g() t dt g() t + t t dt ( t) g( t) dt g( t) dt g( u) g() t u t t du 35

36 3.- Multiresolutio Aalysis. Multiresolutio Formulatio. Example: Haar Wavelet (the oldest ad simplest!). h h h h () () () () () t φ( t) φ( t ) 36

37 3.- Multiresolutio Aalysis. Multiresolutio Formulatio. Haar Wavelet ad Scalig Fuctio: () t, t, t [, [ [,[ ϕ jk () t, t [,[ j () ( ) j t t k {, j Z, k Z} jk { ϕ j j, k Z} jk, jk, Orthoormal basis o L 37

38 3.- Multiresolutio Aalysis. Multiresolutio Formulatio. L ν w w... () t c ϕ () t + d () t k k k j k jk jk Scale Fuctio Wavelet c k () t, ϕ () t Approximatio k d jk () t, () t jk Details 38

39 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 39

40 4.- Discrete Wavelet Trasorm. Wavelet Trasorm: Buildig blocks to represet a sigal. Two dimesioal expasio. Wavelet set: c jk jk Time-Frequecy localizatio o the sigal. The calculatio o the coeiciets ca be doe eicietly. 4 () t () t c () jk t Discrete Wavelet k j jk Trasorm o () t

41 4.- Discrete Wavelet Trasorm. Wavelet Trasorm: Wavelet Systems are geerated rom a mother Wavelet by scalig ad traslatio. jk j () ( ) j t t k, j, k Z Multiresolutio coditios satisied. Filter bak: the lower resolutio coeiciets ca be calculated rom the higher resolutio coeiciets by a tree-structured algorithm. 4

42 4.- Discrete Wavelet Trasorm. Wavelet Trasorm: The eect o the Wavelet Trasorm is a covolutio to measure the similarity betwee a traslated ad scaled versio o the Wavelet ad the sigal uder aalysis. 4

43 4.- Discrete Wavelet Trasorm. Wavelet Trasorm: Wavelet Trasorm represetatio: Time-Scale Plae: 43

44 4.- Discrete Wavelet Trasorm. Wavelet Trasorm: Scale Sigal Time-Scale Plae t 44

45 4.- Discrete Wavelet Trasorm. Some Mother Wavelets: Compact Support 45

46 4.- Discrete Wavelet Trasorm. Eect o choosig dieret Mother Wavelets: The Wavelet trasorm i essece perorms a correlatio aalysis, thereore the output is expected to be maximal whe the iput sigal most resembles the mother wavelet. Depedig o the applicatio, mother wavelet should be as similar as possible to the sigal or to the sigal portio uder aalysis. For example, the Mexica Hat Wavelet is the optimal detector to id a Gaussia. 46

47 4.- Discrete Wavelet Trasorm. Eect o choosig dieret Mother Wavelets: xˆ () t x() t t xˆ () t t Adaptive Approach. t 47

48 4.- Discrete Wavelet Trasorm. Wavelet Trasorm Calculatio:. Choose a mother wavelet.. Give two values o j ad k, calculate the coeiciet accordig to: WT ( () t ) c ( t) ( jk jk t)dt t k jk j j () t 3. Traslate the Wavelet ad repeat step util the whole sigal is aalyzed. 4. Scale the Wavelet ad repeat steps ad 3. 48

49 4.- Discrete Wavelet Trasorm. Discrete Wavelet Trasorm Calculatio: Follow steps -4 deied previously. Chage the expressios: DWT c jk [] [] jk jk j [] [ ] j k DWT [] c [] jk j N k N jk 49

50 4.- Discrete Wavelet Trasorm. Discrete Wavelet Trasorm Calculatio: Usig Multiresolutio Aalysis: DWT ( [] ) h c d k jk [], ϕ [] [] ϕ [] [], [] [] [] 5 k jk [] h [] l ϕ[ l] l, l [] l ( ) h[ l] [] h[] l ϕ[ l] l l Z ϕ, l Z k jk

51 4.- Discrete Wavelet Trasorm. Discrete Wavelet Trasorm Calculatio: Usig Multiresolutio Aalysis: DWT IDWT [] c ϕ [] + d [] k k k j k jk jk DWT IDWT Dyadic grid: [] c ϕ [] + d [] k j k,, 4,... j k, 4, 8, j k j k j j k jk jk

52 4.- Discrete Wavelet Trasorm. Discrete Wavelet Trasorm Calculatio: Example: [] {,,,, 3,,,, } Haar Scalig Fuctio: ϕ() t ϕ [] δ [] + δ [ ] t 5

53 4.- Discrete Wavelet Trasorm. Discrete Wavelet Trasorm Calculatio: Haar Wavelet: h h h h () () () () ϕ[ ] [] ϕ[ ] ϕ[ ] [] 53

54 4.- Discrete Wavelet Trasorm. Discrete Wavelet Trasorm Calculatio: d d d d 4 6 DWT ( [] ) For j, k,, 4, 6 c d k jk [], ϕ [] [] ϕ [] [], [] [] [] [] ( δ[] δ[ ] ) [] [] [] ( δ[ ] δ[ 3] ) [] [] 3 [] 4 [] 5 3 [] 6 [] 7 54 k jk [] {,,,, 3,,, } [] δ [] δ [ ] k jk

55 55 Itroductio to Wavelets Discrete Wavelet Trasorm Calculatio: [] [] [ ] ( ) [] [] [] [ ] [ ] ( ) [] [] [] [ ] [ ] ( ) [] [] [] [ ] [ ] ( ) [] [] c c c c δ δ δ δ δ δ δ δ [] { } 3,,,,,,, [] [] [ ] + δ δ ϕ 4.- Discrete Wavelet Trasorm. 4 6,,,, For k j

56 4.- Discrete Wavelet Trasorm. Discrete Wavelet Trasorm Calculatio: [] c k d k 56

57 57 Itroductio to Wavelets Discrete Wavelet Trasorm Calculatio: 4.- Discrete Wavelet Trasorm. 4,, For k j [] [] [ ] [ ] [ ] ( ) [] [] [] [] [ ] [ ] [ ] [ ] [ ] ( ) [] [] [] [] d d δ δ δ δ δ δ δ δ [] [ ] [] [ ] k j jk j

58 58 Itroductio to Wavelets Discrete Wavelet Trasorm Calculatio: 4.- Discrete Wavelet Trasorm. [] ϕ [ ] ϕ 3 [] [ ] k j jk j ϕ ϕ 4,, For k j [] [] [ ] [ ] [ ] ( ) [ ] [ ] [ ] [ ] [ ] ( ) c c δ δ δ δ δ δ δ δ

59 4.- Discrete Wavelet Trasorm. Discrete Wavelet Trasorm Calculatio: j Complete Decompositio (without actor ): [] 3 c k d k c k d k c k d k 59

60 6 Itroductio to Wavelets Discrete Wavelet Trasorm Calculatio: Recostructio: 4.- Discrete Wavelet Trasorm. From level j: [] [] [] ( ) [] d c k k k k k k ,,,,,,, ),,,,,,, ( ),,,,,,, (,,,,,, ϕ [] [] [] + j j k jk jk k k j k j d c ϕ

61 4.- Discrete Wavelet Trasorm. Discrete Wavelet Trasorm Calculatio: Wavelet Haar.. Daubechies-4 ( 3) + ( 3 3 ) + ( 3 3 ) ( ) Daubechies Order o the Wavelet: umber o ozero coeiciets. Value o wavelet coeiciets is determied by costraits o orthogoality ad ormalizatio. 6

62 4.- Discrete Wavelet Trasorm. Discrete Wavelet Trasorm Calculatio: Calculatio to be implemeted i a computer. Is there ay method, such as FFT, or DWT eiciet calculatio? Pyramid Algorithm Basic Idea: DWT (direct ad iverse) ca be thought o as a ilterig process. 6

63 4.- Discrete Wavelet Trasorm. Pyramid Algorithm: DWT Filterig: ( [] ) c d k jk [] h[] [][ k h k] k [], ϕ [] [] ϕ [] k [], [] [] [] jk k jk Sigal is passed through a halbad lowpass ilter Sigal is passed through a halbad highpass ilter Samplig Frequecy: π Badwidth:π 63

64 4.- Discrete Wavelet Trasorm. Pyramid Algorithm: Ater ilterig, hal o the samples ca be elimiated: subsample the sigal by two. Subsamplig: Scale is doubled. Filterig: Resolutio is halved. Decompositio is obtaied by successive highpass ad lowpass ilterig. y y high low [] [][ k g k] k [] [][ k h k] k 64

65 4.- Discrete Wavelet Trasorm. Pyramid Algorithm: Filters h ad g are ot idepedet. Relatio betwee ilters: g [ ] L ( ) h[] ( L ilter legth) Quadrature Mirror Filters. Sigal legth must be a power o (due to dowsamplig by ). Maximum decompositio level depeds o sigal legth. 65

66 4.- Discrete Wavelet Trasorm. Pyramid Algorithm: [] Details Level [] g h[] [] g h[] Approximatio Level Details Level Approximatio Level

67 4.- Discrete Wavelet Trasorm. Pyramid Algorithm: Operators otatio. G H k k [][ k g k] [][ k h k] 67

68 4.- Discrete Wavelet Trasorm. Pyramid Algorithm: X [ Ω] A A A 4 D 4 D 3 D D A 3 π 4 π π Ω 68

69 4.- Discrete Wavelet Trasorm. Pyramid Algorithm: Recostructio. [] Details Level Details Level g [] h [] g [] h []... Approximatio Level 69

70 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 7

71 5..- Deoisig. Fiite Legth Sigal with Additive Noise: y [] x[] + σ r[], N Observatios o x + oise Noise : r [] ~ N(, ) Sigal Stadard Deviatio 7

72 5..- Deoisig. I the Trasormatio Domai: Y X + σ R where: Y Wy (W trasorm matrix). Xˆ estimate o X rom Y Diagoal liear projectio: Estimate: ( δ, δ,..., δ ), δ [ ] diag N i, xˆ W ˆ X W Y W Wy 7

73 5..- Deoisig. Geeral Method: ) Compute Y Wy ) Perorm Thresholdig i the Wavelet Domai. 3) Compute Questios: xˆ W Xˆ Which Thresholdig Method? Which Threshold? 73

74 5..- Deoisig. Thresholdig Method (Diagoal Liear Projectio): 74

75 5..- Deoisig. Threshold: Stei s Ubiased Risk Estimate (SURE). Dooho: :Sample size λ σ log( ) σ:oise scale Miimax Adaptive: based o sigular or smooth regios (i practice, heuristic idep. o sample size) 75

76 5..- Deoisig. Practical cosideratios: Sigal ormalizatio to avoid amplitude iluece over the process. Noise ca vary alog the sigal: widowed threshold. 76

77 5..- Deoisig. 77

78 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 78

79 5..- Abrupt Chage Detectio. Low level details cotai iormatio o high requecy compoets. Small support wavelets are better suited to detect such chages. Noise makes it diicult. Applicatio: QRS complex detectio. 79

80 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 8

81 5.3.- Log-Term Evolutio. Foudatios: Slow chages which take place i the sigal, i.e., low requecy variatios. I the time domai: Wavelet trasorm has high resolutio i time at low requecies. I the requecy domai: the higher the approximatio level, the arrower the low pass ilter is. Problem: depedig o the baselie(tred) requecy, the approximatio level may chage. The baselie is cosidered as additive oise. 8

82 5.3.- Log-Term Evolutio. Fiite Legth Sigal with baselie variatio: y [] x[] + b[], N Observatios o x + baselie Baselie Sigal 8

83 5.3.- Log-Term Evolutio. With the Wavelet Approximatio o certai level, we estimate the baselie. It ca be used as extra iormatio about the sigal or to reduce the baselie oscillatio. WT xˆ ( y[] ) bˆ [] [] y[] bˆ [] 83

84 5.3.- Log-Term Evolutio. Applicatio: Electrocardiogram baselie waderig reductio x 4 84

85 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 85

86 Methods: Compressio. Quatizatio i the Wavelet domai. The set o possible coeiciets values is reduced: ĉ j c j 86

87 Methods: Compressio. Discard those coeiciets cosidered isigiicat : Threshold. Basis choice is very importat. Coeiciets Histogram: 87

88 5.4.- Compressio. Applicatio: ECG compressio. 88

89 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 89

90 6.- The Matlab Wavelet Toolbox. Matlab: GUI (type wavemeu). 9

91 6.- The Matlab Wavelet Toolbox. Matlab: Commad Lie. 9

92 6.- The Matlab Wavelet Toolbox. Fuctios: DWT: Sigle-level -D wavelet decompositio. [CA,CD] DWT(X,'wame'); Approximatio Coeiciets Detail Coeiciets Sigal Mother Wavelet Example: X[ ]; [CA,CD] DWT(X, haar'); CA CD

93 6.- The Matlab Wavelet Toolbox. Fuctios: IDWT: Sigle-level -D wavelet recostructio. X IDWT(CA,CD,'wame'); WAVEINFO: Provides iormatio about the wavelets i the toolbox. WAVEINFO( wame'); DWTMODE: Sets extesio mode to hadle border distortio. DWTMODE( status'); sym,zpd,sp,sp,ppd 93

94 6.- The Matlab Wavelet Toolbox. Fuctios: WAVEDEC: Multiple-level -D wavelet decompositio. [C,L] WAVEDEC(X,N,'wame') APPCOEF: Extracts -D approximatio coeiciets. A APPCOEF(C,L,'wame',N) DETCOEF: Extracts -D detail coeiciets. D DETCOEF(C,L,N) 94

95 6.- The Matlab Wavelet Toolbox. Fuctios: WRCOEF: Recostructs sigle brach rom -D coeiciets. X WRCOEF('type',C,L,'wame',N) WDEN: Automatic -D deoisig usig wavelets. [XD,CXD,LXD] WDEN(X,TPTR,SORH,SCAL,N,'wame') Deoised sigal Noisy sigal Threshold selectio rule: rigrsure heursure sqtwolog Threshold rescalig oe sl ml Mother Wavelet Decompositio Level Sot s or hard h thresholdig 95

96 6.- The Matlab Wavelet Toolbox. Example: Deoisig a ECG sigal usig Wavelets (WaveletExample). 96

97 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 97

98 Wavelet Packets. 7.- Other Topics. 98

99 7.- Other Topics. D Wavelet Trasorm. Applicatio to bidimesioal sigals: images. Noise reductio, compressio, etc. Image is cosidered a matrix: Wavelet trasorm is applied o rows ad colums. 99

100 Computer Graphics. 7.- Other Topics. Image illumiatio computatio. Figure aimatio. Surace modellig. Image compressio. Image query. Etc.

101 7.- Other Topics. Sel-Similarity Detectio(Fractals).

102 Feature Extractio. 7.- Other Topics. DWT ( [] ) c d k jk [], ϕ [] [] ϕ [] k [], [] [] [] jk k jk Approximatio o the iput sigal [] [], [],..., [ N ] { } c ( c N ) k k <<

103 7.- Other Topics. Applicatios i electrocardiology: ECG Compressio. ECG Patter Recogitio: detectio ad classiicatio o ECG waves, discrimiatig ormal ad abormal cardiac patters. ST segmet aalysis. Chage detectio. Heart rate variability: ECG is assumed to be ostatioary. High resolutio electrocardiography: detectio o late potetials. 3

104 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 4

105 8.- Summary. Fourier trasorm is ot suitable to obtai time ad requecy iormatio: Wavelet Trasorm. Multiresolutio aalysis. It is possible to chage time ad scale. Discrete Wavelet Trasorm. Pyramidal Algorithm: Eiciet algorithm to calculate the Wavelet Trasorm i a computer. Applicatios i sigal processig: oise reductio, log-term evolutio, compressio, abrupt chages. Very importat to the aalysis o biological sigals sice most o the statistical characteristics o such sigals are ostatioary- Matlab Wavelet Toolbox: deiitios ad uctios to work with Wavelets. May other topics: Wavelet packets, Computer graphics, etc. 5

106 Idex.- Itroductio..- Deiitios. 3.- Multiresolutio Aalysis. 4.- Discrete Wavelet Trasorm. 5.- Applicatios i Sigal Processig Deoisig Abrupt Chage Detectio Log-Term Evolutio Compressio. 6.- The Matlab Wavelet Toolbox. 7.- Other Topics. 8.- Summary. 9.- Bibliography. 6

107 9.- Bibliography.. C.S.Burrus et al. Itroductio to Wavelets ad Wavelet Trasorms. Pretice Hall (998).. The Wavelet Tutorial. Robi Polikar T. Edwards. Discrete Wavelet Trasorms: Theory ad Implemetatio. (99) 4. Michael L. Hilto, "Wavelet ad Wavelet Packet Compressio o Electrocardiograms", IEEE Tras. o Biomedical Egieerig, Vol. 44, pp.394 4, (997). 5. Matlab Wavelet Toolbox User s Maual. 6. I.Provazik et al., Wavelet Trasorm i ECG sigal Processig. Proceedigs o the 5th Bieial Eurasip Coerece BIOSIGNAL. pp. -5, (). 7. M. User, Wavelets, statistics, ad biomedical applicatios. Proceedigs o 8th IEEE Sigal Processig Workshop o Statistical Sigal ad Array Processig. pp , (996) 8. Cuiwei Liyet al., Detectio o ECG Characteristic Poits Usig Wavelet Trasorms, IEEE Trasactios o Biomedical Egieerig, vol. 4, o., pp. -8. (995) 9. A. Bezarieas et al. Selective Noise ilterig o High Resolutio ECG Through Wavelet Trasorm, Computers i Cardiology pp , (996). H. Ioue; A. Miyazaki, A Noise Reductio Method or ECG Sigals Usig the Dyadic Wavelet Trasorm, IEICE Trasactios o Fudametals o Electroics, Commuicatios ad Computer Scieces, pp. -7, (998). Dooho D., De-oisig by sot-thresholdig, IEEE Tras. Iormatio Theory, vol. 4, o. 3, pp.6-67, (995) 7

108 9.- Bibliography.. B.Vidakovic ad P. Müller, Wavelets or Kids. A Tutorial Itroductio. Duke Uiversity. 3. J.P. Couderc ad W. Zareba, Cotributio o the Wavelet Aalysis to the No-Ivasive Electrocardiology. Uiversity o Rochester. 4. M.V. Wickerhauser, Lectures o Wavelet Packet Algorithms. Departmet o Mathematics, Washigto Uiversity (99). 5. K.Berker ad R.O.Wells, Wavelet Trasorms ad Deoisig Algorithms. Departmet o Mathematics, Rice Uiversity. IEEE (998). 6. L.V.Novikov, Adaptive Wavelet Aalysis o Sigals. 7. M. Kaapathipillai et al., Adaptive Wavelet Represetatio ad Classiicatio o ECG Sigals. IEEE (994). 8. B.Jawerth ad W.Sweldes, A Overview o Wavelet Based Multiresolutio Aalysis. Departmet o Mathematics, Uiversity o South Carolia. 9. R.R.Coima ad M.V. Wickerhauser, Etropy-Based Algorithms or Best Basis Selectio. Departmet o Mathematics, Yale Uiversity. 8

109 Sigal Processig Itroductio to Wavelets Laboratory Exercises Assigmets 9