Pattern Recognition Techniques Applied to Biomedical

PhD Course- Prague, October 2003Dr.Ing. David Cuesta [email protected] University of Valencia- Spain

Pattern Recognition Techniques Applied to Biomedical Signal Processing

Pattern Recognition Techniques Applied to Biomedical Signal Processing

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Author: David Cuesta Frau 2

• Pattern Recognition Systems.

Sensors Preprocessing FeatureExtraction Clustering

DissimilarityMeasures

Course Syllabus

WaveletTransform

Amplitude samplesTrace segmentation

Polygonal aprox.Wavelet Transform

Max-Mink-Means

k-Medians

Dynamic Time Warping

Hidden MarkovModels

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Schedule

15:30•Hidden Markov Models

10.30•Feature extraction •Clustering algorithms14.30•Dynamic Time Warping

14.30•Wavelet transform

17/10/200316/10/200315/10/200314/10/200313/10/2003

10/10/200309/10/200308/10/200307/10/200306/10/2003


Course Structure

•Lectures.

•Laboratory exercises.

•Assignments.

•Total Credits:1-2

Signal ProcessingSignal Processing

Introduction to Wavelets

Dr.Ing. David Cuesta FrauPolytechnic University of Valencia(Spain)[email protected]


Index1.- Introduction.2.- Definitions.3.- Multiresolution Analysis.4.- Discrete Wavelet Transform.5.- Applications in Signal Processing.

5.1.- Denoising.5.2.- Abrupt Change Detection.5.3.- Long-Term Evolution.5.4.- Compression.

6.- The Matlab Wavelet Toolbox.7.- Other Topics.8.- Summary.9.- Bibliography.








1.- Introduction.• Signal Processing.

– Extract information from a signal.– Time domain is not always the best choice.– Frequency domain: Fourier Transform:

– Easily implemented in computers: FFT.


( ) ( )∫∞

∞−

−= dtetxfX ftj π2

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1.- Introduction.• Fourier Analysis.

– Can not provide simultaneously time and frequency information:Time information is lost.

– Small changes in frequency domain cause changes everywhere in the time domain.

– For stationary signals: the frequency content does not change intime, i.e.:


0f f

( )tx

t

( )fX

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1.- Introduction.• Fourier Analysis.

– Basis Functions are set: sinusoids. – Many coefficients needed to describe an “edge” or discontinuity.– If the signal is a sinusoid, it is better localized in the frequency

domain.– If the signal is a square pulse, it is better localized in the time

domain.


Short Time Fourier Transform.

t

( )tx

f

( )fX

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1.- Introduction.• Short Time Fourier Analysis.

– Time-Frequency Analysis.– Windowed Fourier Transform (STFT).


FT Basis Functions STFT Basis Functions

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1.- Introduction.• Short Time Fourier Analysis.

– Discrete version very difficult to find.– No fast transform.– Fixed window size (fixed resolution):

• Large windows for low frequencies.• Small windows for high frequencies.

– Wavelet Transform.


( ) ( ) ( ) ( )dttttgtftSTFT 0000 sin, ωω −= ∫

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1.- Introduction.• Wavelet Transform.

– Small wave.– Energy concentrated in time: analysis of non-

stationary, transient or time-varying phenomena.– Allows simultaneous time and frequency analysis.


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1.- Introduction.• Wavelet Transform.

– Fourier: localized in frequency but not in time.– Wavelet:

• local in both frequency/scale (via dilations) and in time (via translations).

• Many functions are represented in a more compact way. For example: functions with discontinuities and/or sharp spikes.

– Fourier transform: O(nlog2(n)).– Wavelet transform: O(n).


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1.- Introduction.Introduction to Wavelets

STFT Basis Functions WT Basis Functions

( ) ( ) dta

bttfbaWT

−= ∫ ψ,

• Wavelet Transform.

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2.- Definitions.• Mathematical Background.

– Topic of pure mathematics, but great applicability in many fields.

– Linear decomposition of a function:( ) ( )∑=

iii tctf ψ

( ) SetExpansion tsCoefficienExpansion

sum infiniteor finite for theindex Integer

≡≡≡

tci

i

i

ψ

( ) ( )tfti for Basis≡ψ


– Decomposition Unique:

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– Basis orthogonal:

( ) ( ) ( ) ( )∫ ≠== lkdttttt lklk ,0, ψψψψ


– Then the coefficients can be calculated by :

– Since:

( ) ( ) ( ) ( )∫== dtttfttfc kkk ψψ,

( ) ( ) ( ) ( )∫ ∑∫ =

==

kkkii

iik cdtttcdtttf ψψψ

( ) ( )∫ == lkdttt lk ,1ψψ

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– Linear decomposition of a vector:


( )123 ,,−=ar

( ) ( ) ( )100101020013 ,,,,,, +−=ar

( ) ( ) ( ) 100010001 ,,,,,,,,

Generator systemin R3

IndependentOrthogonal Basis in R3

( )123 ,,− Coordinatesin R3

Normal

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2.- Definitions.

( ) ( ) ( ) ( )∫ ≠== nmdttttt nmnm where,, If 0ψψψψ

( ) ( )tfforBasisOrthogonalti ≡ψ


• Mathematical Background.

( ) ( ) ( ) ( )∫== ttfttfc iii ψψ,

:Otherwise( ) ( ) ( )tfforBasisalBiorthogontt ii ~, ≡ψψ

( ) ( )ttfc ii ψ~,= ( ) ( )∑=i

ii tctf ψ



– Example of linear decomposition of a function:


( )tf

t

2

1

1−1 2 3

( )tψ

t

1

1( ) ( )itti −=ψψ

t

1

i 1+i...

( ) ( )∫ = mnnm dttt δψψ

( ) ( ) ( ) 1== ∫ dtttt mmm*ψψψ

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– Example of linear decomposition of a function:


( )tf

t

2

1

1−1 2 3

( ) ( ) ( ) ( )212 −−−+= ttttf ψψψ

otherwise , 01

21

2

1

0

=−=

==

icccc

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2.- Definitions.Introduction to Wavelets

• Example:• Four dimensional space (four non-null coefficients).

Synthesis Formula:

( ) ( )∑=i

ii tctf ψ

( )( )( )( )

( )( )( )( )

( )( )( )( )

( )( )( )( )

( )( )( )( )

+

+

+

=

3210

3210

3210

3210

3210

3

3

3

3

3

2

2

2

2

2

1

1

1

1

1

0

0

0

0

0

ψψψψ

ψψψψ

ψψψψ

ψψψψ

cccc

ffff



• Example:

( )( )( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

=

3

2

1

0

3210

3210

3210

3210

3333222211110000

3210

cccc

ffff

ψψψψψψψψψψψψψψψψ

ψcf =



• Example:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( )( )( )( )

=

3210

3210321032103210

3333

2222

1111

0000

3

2

1

0

ffff

cccc

ψψψψψψψψψψψψψψψψ

~~~~~~~~~~~~~~~~

fψc T~=



• Example:

fψc T~=

ψcf =fψψf T~= 1T ψψ −=~

Dual Basis

If columns are orthonormal:fIψψ =T fψψf T= TT ψψ =~

Single Basis

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3.- Multiresolution Analysis.• Problem definition:

Time scale change.


( )tf 2

t

2

1

1−1 2

( )tψ

t

1

1

?


• Multiresolution Formulation.– Rationale: if a set of signals can be represented by a

weighted sum of ϕ(t-k), a larger set (including the original), can be represented by a weighted sum of ϕ(2t-k).

– Signal analysis at different frequencies with different resolutions.

– Good time resolution and poor frequency resolution at high frequencies.

– Good frequency resolution and poor time resolution at low frequencies.

– Used to define and construct orthonormal wavelet basis for L2.

– Two functions needed: the Wavelet and a scaling function.


3.- Multiresolution Analysis.

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• Multiresolution Formulation.– Effect of changing the scale. Scaling function:– To decompose a signal into finer and finer details.


( )tϕ

( ) 2Ltf ∈( ) 2

0 Ltspank

k ⊂= ϕν :Subspace

( ) ( ) ( ) 0νϕ ∈∀= ∑ tftctfk

kk

– Increase the size of the subspace changing the time scale of the scaling functions:

( ) ( )ktt jjk

j

−= −−

22 2 ϕϕ


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• Multiresolution Formulation.


( ) ( ) 22 Ltspantspan jkj

kj ⊂== ϕϕν :Subspace

( ) ( ) ( ) jk

jk tfktctf νϕ ∈∀+= ∑ 2

– The spanned spaces are nested:2

21012 L⊂⊂⊂⊂⊂⊂⊂ −− ...... ννννν( ) ( ) 12 +∈⇔∈ jj tftf νν

( ) ( ) ( ) Znntnhtn

∈−= ∑ ,22ϕϕ





– Example: Haar Scaling Function.

( ) ( ) ( )122 −+= ttt ϕϕϕ

( )tϕ

( )t2ϕ ( )12 −tϕ

( )2

10 =h

( )2

11 =h


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– Increasing j, the size of the subspace spanned by the scaling function, is also increased.

– Wavelets span the differences between spaces wi. More suitable to describe signals.

– Wavelets and scaling functions should be orthogonal: simple calculation of coefficients.

...

...

⊕⊕⊕=

⊕⊕=⊕=

1002

1002

001

wwL

www

ν

νννν





0ν0w1w2w

( ) [ ] ( ) ℵ∈−= ∑ nntnhtn

, 221 ϕψ

( ) ( )ktt jjk

j

−= −−

22 2 ψψMother Wavelet


[ ] [ ] [ ]∑ ∈−=l

Zllnlhn ,22ϕϕ Scale Function





( )( ) ( ) ( )

1norm for accounts

tscoefficien function scaling

=≡

−−=

≡

2

111 nhnh

nhn

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )tgduugdttgdttgtg

dttgdttgtg

t

tu

t

t

21

2222

121

1

0

2

20

22

0

22

====

==

∫∫∫

∫∫

=

∞+

∞−

∞+

∞−




( ) ( ) ( )122 −−= ttt φφψ

– Example: Haar Wavelet (the oldest and simplest!).


( )

( )

( )

( )211

210

211

210

1

1

−=

=

=

=

h

h

h

h




ZkZjjk ∈∈ ,,ψ

– Haar Wavelet and Scaling Function:


( ) [ [[ [

∈−∈

=1,,1

,0,1

2121

tt

tψ ( ) ( )ktt jjk

j

−= −−

22 2 ψψ

( ) [ [1,0,1 ∈= ttϕ

2LOrthonormal basis of

Zkjjjkkj ∈≥ ,, 0,0ψϕ

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( ) ( ) ( )∑ ∑∑∞

=

∞

−∞=

∞

−∞=+=

0j kjkjk

kkk tdtctf ψϕ


...⊕⊕⊕= 1002 wwL ν

Scale Function Wavelet

( ) ( )

( ) ( )ttfd

ttfc

jkjk

kk

ψ

ϕ

,

,

=

= Approximation

Details








• Wavelet Transform:• Building blocks to represent a signal.• Two dimensional expansion. Wavelet set:

• Time-Frequency localization of the signal.• The calculation of the coefficients can be done

efficiently.

( )tjkψ

( ) ( )∑∑=k j

jkjk tctf ψ

( )tfc jk of Transform WaveletDiscrete≡

4.- Discrete Wavelet Transform.

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• Wavelet Transform:• Wavelet Systems are generated from a mother

Wavelet by scaling and translation.

• Multiresolution conditions satisfied.• Filter bank: the lower resolution coefficients can

be calculated from the higher resolution coefficients by a tree-structured algorithm.

( ) ( ) Zkjktt jjk

j

∈−= −−

, , 22 2 ψψ


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• Wavelet Transform:• The effect of the Wavelet Transform is a

convolution to measure the similarity between a translated and scaled version of the Wavelet and the signal under analysis.


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• Wavelet Transform:• Wavelet Transform representation: Time-Scale

Plane:


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• Wavelet Transform:

Time-Scale Plane tSignal

Scale


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• Some Mother Wavelets:


Compact Support

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• Effect of choosing different Mother Wavelets:

The Wavelet transform in essence performs a correlation analysis, therefore the output is expected to be maximal when the input signal most resembles the mother wavelet.

Depending on the application, mother wavelet should be as similar as possible to the signal or to the signal portion under analysis. For example, the Mexican Hat Wavelet is the optimal detector to find a Gaussian.


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• Effect of choosing different Mother Wavelets:


Adaptive Approach.

t

t

( )tx

( )tx

( )tx

t



• Wavelet Transform Calculation: 1. Choose a mother wavelet.2. Given two values of j and k, calculate the coefficient

according to:

3. Translate the Wavelet and repeat step 2 until the whole signal is analyzed.

4. Scale the Wavelet and repeat steps 2 and 3.

( )( ) ( ) ( )dtttfctfWT jkjk ψ∫∞

∞−==

( )

−=jkt

jtjk ψψ 1


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• Discrete Wavelet Transform Calculation: • Follow steps 1-4 defined previously.• Change the expressions:

[ ] [ ]∑==n

jkjk nnfcDWT ψ

[ ] [ ]knn jjk

j

−= −−

22 2 ψψ

[ ] [ ]∑ ∑∈ ∈

− ==Nj Nk

jkjk ncnfDWT ψ1


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• Discrete Wavelet Transform Calculation: • Using Multiresolution Analysis:

[ ]( )[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

==

===

∑

∑

njkjkjk

nkkk

nnfnnfd

nnfnnfcnfDWT

ψψ

ϕϕ

,

,

[ ] [ ] [ ]

[ ] ( ) [ ]lhlh

Zllnlhn

ll

−−=

∈−= ∑

11

22

1

1 ,ϕψ

[ ] [ ] [ ]∑ ∈−=l

Zllnlhn ,22ϕϕ


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• Discrete Wavelet Transform Calculation: • Using Multiresolution Analysis:

[ ] [ ] [ ]∑ ∑∑∞

=

∞

−∞=

∞

−∞=

− +===0

1

j kjkjk

kkk ndncnfIDWTDWT ψϕ

[ ] [ ] [ ]∑ ∑∑∞

=

∞

−∞=

∞

−∞=

− +===0

00

1

jj kjkjk

kkjkj ndncnfIDWTDWT ψϕ

Dyadic grid:

...,...,, ,...,,

84014200

=⇒==⇒=

kjkj




• Discrete Wavelet Transform Calculation: • Example:

• Haar Scaling Function:

[ ] 111232111 ,,,,,,,,=nf

( )tϕ

t

1

1

[ ] [ ] [ ]1−+= nnn δδϕ

n

1

1




• Discrete Wavelet Transform Calculation: • Haar Wavelet:

[ ]n2ϕ

n

1

[ ] [ ] [ ]122 −−= nnn ϕϕψ

[ ]nψ

n

1

( )

( )

( )

( )211

210

211

210

1

1

−=

=

=

=

h

h

h

h

1−




• Discrete Wavelet Transform Calculation:

[ ]( )[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

==

===

∑

∑

njkjkjk

nkkk

nnfnnfd

nnfnnfcnfDWT

ψψ

ϕϕ

,

,

[ ] [ ] [ ]( ) [ ] [ ][ ] [ ] [ ]( ) [ ] [ ][ ] [ ][ ] [ ] 01176

1235413232

0101

06

04

02

00

=−=−==−=−=

−=−=−−−==−=−−=

ffdffd

ffnnnfdffnnnfd

δδδδ

[ ] 11232111 ,,,,,,,=nf[ ] [ ] [ ]1−−= nnn δδψ


64200 ,,,, For == kj




[ ] [ ] [ ]( ) [ ] [ ][ ] [ ] [ ]( ) [ ] [ ][ ] [ ] [ ]( ) [ ] [ ][ ] [ ] [ ]( ) [ ] [ ] 27676

5545433232

2101

6

4

2

0

=+=−+−==+=−+−==+=−+−=

=+=−+=

ffnnnfcffnnnfcffnnnfc

ffnnnfc

δδδδδδ

δδ

[ ] 11232111 ,,,,,,,=nf

[ ] [ ] [ ]1−+= nnn δδϕ


64200 ,,,, For == kj





n

nn

kc0

kd0

[ ]nf





401 ,, For == kj[ ] [ ] [ ] [ ] [ ]( )

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]( )

[ ] [ ] [ ] [ ] 3112376547654

121113210321

14

10

=−−+=−−+=−−−−−+−=

−=−−+=−−+=−−−−−+=

ffffnnnnnfd

ffffnnnnnfd

δδδδ

δδδδ

[ ]nψ

n

1

1−

[ ]2nψ

n

1

1−

[ ] [ ]knn jjk

j

−= −−

22 2 ψψ





[ ]nϕ

n

1

1

[ ]2nϕ

n

1

1 2 3

[ ] [ ]knn jjk

j

−= −−

22 2 ϕϕ

401 ,, For == kj

[ ] [ ] [ ] [ ] [ ]( )[ ] [ ] [ ] [ ] [ ]( ) 711237654

52111321

14

10

=+++=−+−+−+−==+++=−+−+−+=

nnnnnfcnnnnnfcδδδδ

δδδδ




Complete Decomposition (without factor ):


01-103-1-212

01-103-175

01-102532

11232111[ ]nf

kc0

kc1

kc2 kd2

kd1

kd0

22j−



• Discrete Wavelet Transform Calculation:Reconstruction:


From level j=0:

[ ] [ ] [ ]

( ) [ ]nf

ndncnfk

kkk

kk

2224642220011110022553322

642000

642000

⇒=−−+

=+= ∑∑==

,,,,,,,),,,,,,,(),,,,,,,(

,,,,,,ψϕ

[ ] [ ] [ ]∑ ∑∑∞

=

∞

−∞=

∞

−∞=+=

000

jj kjkjk

kkjkj ndncnf ψϕ

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0.035226-0.085441-0.1350110.4598770.8068910.332671Daubechies-6

Daubechies-4

1.01.0Haar

543210Wavelet

( )3141 + ( )334

1 + ( )3341 − ( )314

1 −

Order of the Wavelet: number of nonzero coefficients.Value of wavelet coefficients is determined by constraints of orthogonality and normalization.



• Discrete Wavelet Transform Calculation: • Calculation to be implemented in a

computer. • Is there any method, such as FFT, for DWT

efficient calculation?

Pyramid Algorithm

• Basic Idea: DWT (direct and inverse) can be thought of as a filtering process.


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• Pyramid Algorithm:

[ ] [ ] [ ] [ ]∑∞

−∞=−=∗

kknhkfnhnf

[ ]( )[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

==

===

∑

∑

njkjkjk

nkkk

nnfnnfd

nnfnnfcnfDWT

ψψ

ϕϕ

,

,

Filtering:

Sampling Frequency: π2

Bandwidth:π

Signal is passed through a half-band lowpass filterSignal is passed through a half-band highpass filter


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• Pyramid Algorithm:• After filtering, half of the samples can be

eliminated: subsample the signal by two.• Subsampling: Scale is doubled.• Filtering: Resolution is halved.

• Decomposition is obtained by successive highpass and lowpass filtering.

[ ] [ ] [ ]

[ ] [ ] [ ]∑

∑

∞

−∞=

∞

−∞=

−=

−=

klow

khigh

knhkfny

kngkfny

2

2


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• Pyramid Algorithm:• Filters h and g are not independent.• Relation between filters:

Quadrature Mirror Filters.

• Signal length must be a power of 2 (due to downsampling by 2).

• Maximum decomposition level depends on signal length.

[ ] ( ) [ ] ( )length filter ≡−=−− LnhnLg n11


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• Pyramid Algorithm:[ ]nf

[ ]ng [ ]nh

↓2 ↓2

[ ]ng [ ]nh

↓2 ↓2

... ...

DetailsLevel 1

DetailsLevel 2

ApproximationLevel 1



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• Pyramid Algorithm:Operators notation.

[ ] [ ]

[ ] [ ]∑

∑

∞

−∞=

∞

−∞=

−=

−=

k

k

knhkfHf

kngkfGf

2

2


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• Pyramid Algorithm:

Ω

1D

1A

2D

2A

3D4D

3A

4A

[ ]ΩX


π2π

4π



• Pyramid Algorithm: Reconstruction.


...


[ ]nf

↑2

[ ]ng [ ]nh

DetailsLevel 1

↑2

[ ]ng [ ]nhDetailsLevel 2

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5.1.- Denoising.

[ ] [ ] [ ] 10 −≤≤+= Nnnrnxny , σ

Standard Deviation

Signal

Observations of x + noise


• Finite Length Signal with Additive Noise:

[ ] ( )10,~ :Noise Nnr


5.1.- Denoising.

RXY σ+=


In the Transformation Domain:

where: (W transform matrix).WyY =

estimate of fromX X YDiagonal linear projection:

( ) [ ]10110 , , ,...,, ∈=∆ − iNdiag δδδδ

Estimate: WyWYWXWx ∆=∆== −−− 111 ˆˆ


5.1.- Denoising.Introduction to Wavelets

General Method:1) Compute2) Perform Thresholding in the Wavelet

Domain.3) Compute

Questions:Which Thresholding Method?Which Threshold?

WyY =

XWx ˆˆ 1−=



Thresholding Method (Diagonal Linear Projection):



Threshold:

Stein’s Unbiased Risk Estimate (SURE).

Donoho: n:Sample sizeσ:noise scale

MinimaxAdaptive: based on singular or smooth regions(in practice, heuristic indep. of sample size)

( )nlog2σλ =



5.1.- Denoising.Practical considerations:

• Signal normalization to avoid amplitude influence over the process.

• Noise can vary along the signal: windowed threshold.


http://www-dsp.rice.edu/edu/wavelet/denoise.html


5.1.- Denoising.







5.2.- Abrupt Change Detection.Introduction to Wavelets

• Low level details contain information of high frequency components.

• Small support wavelets are better suited to detect such changes.

• Noise makes it difficult.• Application: QRS

complex detection.







5.3.- Long-Term Evolution.Introduction to Wavelets

• Foundations:– Slow changes which take place in the signal, i.e.,

low frequency variations.– In the time domain: Wavelet transform has high

resolution in time at low frequencies.– In the frequency domain: the higher the

approximation level, the narrower the low pass filter is.

– Problem: depending on the baseline(trend) frequency, the approximation level may change.

– The baseline is considered as additive ‘noise’.


5.3.- Long-Term Evolution.

[ ] [ ] [ ] 10 −≤≤+= Nnnbnxny ,


Baseline

Signal

Observations of x + baseline

• Finite Length Signal with baseline variation:



[ ]( ) [ ]

[ ] [ ] [ ]nbnynx

nbnyWT

ˆˆ

ˆ

−=

=


• With the Wavelet Approximation of certain level, we estimate the baseline.

• It can be used as extra information about the signal or to reduce the baseline oscillation.


0 2 4 6 8 10 12 14 16x 104

-2000

-1900

-1800

-1700

-1600

-1500

-1400

-1300


• Application: Electrocardiogram baseline wandering reduction.








5.4.- Compression.Introduction to Wavelets

• Methods:– Quantization in the Wavelet domain. The set of

possible coefficients values is reduced:

jc

jc

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• Methods:– Discard those coefficients considered

‘insignificant’: Threshold.– Basis choice is very important.

Coefficients Histogram:

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• Application: ECG compression.

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6.- The Matlab Wavelet Toolbox.Introduction to Wavelets

Matlab: GUI (type wavemenu).

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Matlab: Command Line.

6.- The Matlab Wavelet Toolbox.

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Functions:DWT: Single-level 1-D wavelet decomposition.

[CA,CD] = DWT(X,'wname');

ApproximationCoefficients

DetailCoefficients

Signal MotherWavelet

Example:X=[1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1];

[CA,CD] = DWT(X,‘haar');

CA = 1.4142 0.7071 0 1.4142 0.7071 0 1.4142 1.4142

CD = 0 0.7071 0 0 0.7071 0 0 0


Prof_Fei_Hu

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Functions:IDWT: Single-level 1-D wavelet reconstruction.

X = IDWT(CA,CD,'wname');

WAVEINFO: Provides information about the wavelets in thetoolbox.

WAVEINFO(‘wname');

DWTMODE: Sets extension mode to handle border distortion.

DWTMODE(‘status');

sym,zpd,sp0,sp1,ppd


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Functions:WAVEDEC: Multiple-level 1-D wavelet decomposition.

[C,L] = WAVEDEC(X,N,'wname')


APPCOEF: Extracts 1-D approximation coefficients.

A = APPCOEF(C,L,'wname',N)

DETCOEF: Extracts 1-D detail coefficients.

D = DETCOEF(C,L,N)



Functions:WRCOEF: Reconstructs single branch from 1-D coefficients.

X = WRCOEF('type',C,L,'wname',N)


WDEN: Automatic 1-D denoising using wavelets.

[XD,CXD,LXD] = WDEN(X,TPTR,SORH,SCAL,N,'wname')

Denoisedsignal

Noisy signal

MotherWavelet

DecompositionLevel

Thresholdselection rule:rigrsureheursuresqtwolog

Soft ‘s’ or hard ‘h’ thresholding

Thresholdrescalingoneslnmln

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Example:Denoising an ECG signal using Wavelets (WaveletExample).








7.- Other Topics.Introduction to Wavelets

Wavelet Packets.



2D Wavelet Transform.Application to bidimensional signals: images. Noise reduction, compression, etc.Image is considered a matrix: Wavelet transform is applied on rows and columns.



Computer Graphics.Image illumination computation.Figure animation.Surface modelling.Image compression.Image query.Etc.



Self-Similarity Detection(Fractals).



Feature Extraction.

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Applications in electrocardiology:

• ECG Compression.• ECG Pattern Recognition: detection and classification

of ECG waves, discriminating normal and abnormal cardiac patterns.

• ST segment analysis. Change detection.• Heart rate variability: ECG is assumed to be

nonstationary.• High resolution electrocardiography: detection of late

potentials.







8.- Summary.Introduction to Wavelets

• Fourier transform is not suitable to obtain time and frequency information: Wavelet Transform.

• Multiresolution analysis. It is possible to change time and scale.• Discrete Wavelet Transform. Pyramidal Algorithm: Efficient

algorithm to calculate the Wavelet Transform in a computer.• Applications in signal processing: noise reduction, long-term

evolution, compression, abrupt changes.• Very important to the analysis of biological signals since most of

the statistical characteristics of such signals are nonstationary-• Matlab Wavelet Toolbox: definitions and functions to work with

Wavelets.• Many other topics: Wavelet packets, Computer graphics, etc.







9.- Bibliography.Introduction to Wavelets

1. C.S.Burrus et al. “Introduction to Wavelets and Wavelet Transforms”. Prentice Hall (1998).2. The Wavelet Tutorial. Robi Polikar. http://www.public.iastate.edu/~rpolikar/wavelets3. T. Edwards. “Discrete Wavelet Transforms: Theory and Implementation”. (1991)4. Michael L. Hilton, "Wavelet and Wavelet Packet Compression of Electrocardiograms", IEEE

Trans. on Biomedical Engineering, Vol. 44, pp.394—402 , (1997).5. Matlab Wavelet Toolbox User’s Manual.6. I.Provaznik et al., “Wavelet Transform in ECG signal Processing”. Proceedings of the 15th

Biennial Eurasip Conference BIOSIGNAL 2000. pp. 21-25, (2000).7. M. Unser, “Wavelets, statistics, and biomedical applications”. Proceedings of 8th IEEE

Signal Processing Workshop on Statistical Signal and Array Processing. pp. 244-249, (1996)

8. Cuiwei Liyet al., “Detection of ECG Characteristic Points Using Wavelet Transforms”, IEEE Transactions on Biomedical Engineering, vol. 42, no.1 , pp. 21-28. (1995)

9. A. Bezarienas et al. “ Selective Noise filtering of High Resolution ECG Through Wavelet Transform “, Computers in Cardiology pp 637-640, (1996)

10. H. Inoue; A. Miyazaki, “A Noise Reduction Method for ECG Signals Using the Dyadic Wavelet Transform”, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, pp. 1001-1007, (1998)

11. Donoho D., “De-noising by soft-thresholding”, IEEE Trans. Information Theory, vol. 41, no. 3, pp.612-627, (1995)


9.- Bibliography.Introduction to Wavelets

12. B.Vidakovic and P. Müller, “Wavelets for Kids. A Tutorial Introduction”. Duke University.13. J.P. Couderc and W. Zareba, “Contribution of the Wavelet Analysis to the Non-Invasive

Electrocardiology”. University of Rochester.14. M.V. Wickerhauser, “Lectures on Wavelet Packet Algorithms”. Department of Mathematics,

Washington University (1991).15. K.Berkner and R.O.Wells, “Wavelet Transforms and Denoising Algorithms”. Department of

Mathematics, Rice University. IEEE (1998).16. L.V.Novikov, “Adaptive Wavelet Analysis of Signals”.17. M. Kanapathipillai et al.,”Adaptive Wavelet Representation and Classification of ECG

Signals”. IEEE (1994).18. B.Jawerth and W.Sweldens, “An Overview of Wavelet Based Multiresolution Analysis”.

Department of Mathematics, University of South Carolina.19. R.R.Coifman and M.V. Wickerhauser, “Entropy-Based Algorithms for Best Basis Selection”.

Department of Mathematics, Yale University.


Signal ProcessingSignal Processing


•Laboratory Exercises

•Assignments

Documents

Pattern Recognition Techniques Applied to Biomedical