Application of Wavelets in Signal Processing 1

8/6/2019 Application of Wavelets in Signal Processing 1

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Application of Wavelets in Signal

Processing

Special Emphasis : De-noising of 1

& 2 Dimensional Signals


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Outline of Presentation

Fourier Transform & STFT

Wavelets History

DWT vs. STFT

Continuous & Discrete Wavelets

Applications

Conclusion


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Signal Analysis

Most of the signals encountered in practice

are expressed in time ( space) domain.

M

any features of a signal are not explicit in timedomain

Analysis of these signals are carried in

frequency domain

Using Fourier Transform


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The Fourier Transform

AMathematical prism breaks a function into its constituent frequencies as a

prism breaks up light into colours

It transforms a function which depends on time ( or

space) into a new function which depends onfrequency.

A function and its Fourier transform are two faces ofthe same coin. The function displays the time ( or space) information and

hides information about frequencies The FT displays information about frequencies and hides

the information about time (or space) in phases.


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Important Features of Fourier

Transform

Hides information about time

Tells how much of each frequency a signal contains,

but is secretive about when these frequencies were

emitted

Note: FT hides information about time, but does

not destroy time information; otherwise we could

not reconstruct the signal from the transform. Time information is buried deep within the

phases.


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Important Features of Fourier

Transform

Computing time information from phases withenough precision is impossible.

Major drawback:

Information about one instant of a signal isdispersed throughout all the frequencies of theentire transform

A local characteristic of the signal becomes aglobal characteristic of the transform.

Example: a discontinuity is represented by asuperposition of all possible frequencies.


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Searching for Lost Time : Short Time

Fou

rierT

ransform Used for analyzing a signal both in time and

frequency

Divides the signal into different (fixed size) timesegments

Study the frequencies of a signal segment by

segment.

When one segment of the signal has been

analyzed, slide the window along the signal to

analyze another segment.


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Painful Compromises in STFT

Smaller the widow size,

better to locate sudden changes, such as peak or

discontinuities

But blinder to the lower frequency components of

the signal

Bigger the window size,

Worse at localization in time; difficult to locatesudden changes or discontinuities

See more of lower frequencies.


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Wavelet : A Mathematical Microscope

Automatically adapt to the different componentsof a signal (Multiresolution) uses small window to look at brief high frequency

components

Uses large window to look at long-lived low frequencycomponents

This procedure is called multiresolution; signal is studied at coarse resolution to get an overall

picture studied at higher and higher resolutions to see

increasingly fine details.


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History

First mention in appendix of the thesis of A. Haar(1909)

Cochlear transform, Zweig (1975)

Continuous Wavelet Transform (CWT), GrossmanandMorlet (1982) Geophysics

Discrete Wavelet Transform (DWT), Strmberg(1983)

Daubechies' orthogonal wavelets with compactsupport (1988)

Mallat's mult-iresolution framework (1989)


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DWT vs. STFT

Multi-ResolutionSTFT DWT


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Haar Wavelet

Change in scale and time


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Common Features of Different Types

of Wavelet Wavelet: a wave-like oscillation with an amplitude that starts out at zero, increases, and then

decreases back to zero

Wavelet-theory: wavelet with zero-integral and finite energy

Wavelet transform: projection on the sub-spaces associated with each wavelet

Different types of wavelets:

Ex1:-Morlet Wavelet Ex:-2Mexican Hat Ex:-3 Modified Haarwavelet

-4 -3 -2 -1 0 1 2 3 4-1

-0.8

-0.6

-0.4

-0.2

0

0. 2

0. 4

0. 6

0. 8

1

Morlet wavelet

0 5 10 15 20 25 30 35 40 -0.4

-0.2

0

0. 2

0. 4

0. 6

0. 8

1


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Discrete Wavelets and Filter Banks

0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1-70

-60

-50

-40

-30

-20

-10

0

10

[/T

Gain,

dB

Gain Response of 12 order Linear-

phase FIRSimple Filter bank consists of two Filters

The two main conditions for obtaining a perfect reconstruction and an alias-free filter bank respectively are :


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Orthogonal Haar Wavelet Scaling

Coefficients

Haar Wavelet Filters

Coefficients


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Example of Haar Transform for

1- Dimensional signal

02/10 !v2/32/10213 !vv2/7)2/13(2/14 !vv

ApproximatesCoefficients

Details

Coefficients

]2/3,2/3|2/3,2/5[


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Designing New Wavelet

To generate new wavelet from a given biorthogonal quadruplet, a finite

sequences of primal or dual elementary lifting steps (ELS) should be

applied. For a wavelet has four filters ,

the primal ELS can be defined as:

Where

The dual ELS can be defined as:


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Principle of Wavelet-based De-noising

Step-1: Obtain wavelet coefficients both

(approximates and details ) Approximates are the lowpass and downsampled output coefficients

Details are the highpass and downsampled output coefficients

Step-2: Determine the threshold type ( e.g. soft

and hard threshold) and value of the threshold

Step-3 :Eliminate details coefficients Step-4: Retain only those detail coefficients which

satisfy the threshold conditions and approximates


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Wavelet-based De-noising Procedure

Three main steps Compute wavelet transform of the signal (1D) or image (2D) by passing them through

Lowpass and Highpass filters

Threshold the coefficients using soft threshold and SURE to determine threshold value

0 50 100 -1

-0.8

-0.6

-0.4

-0.2

0

0. 2

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0. 8

1

the original signal

0 50 100 -1

-0.8

-0.6

-0.4

-0.2

0

0. 2

0. 4

0. 6

0. 8

1

Ha rd thresho ld of the signa l

0 50 100 -0.5

-0.4

-0.3

-0.2

-0.1

0

0. 1

0. 2

0. 3

0. 4

0. 5

soft threshold of the signal

Compute the inverse wavelet transform or performing reconstruction of the original signal by

passing the threshold coefficients through the synthesis filters to obtain the original

approximation coefficients.


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Advantages of Wavelet De-noising

Its possible to remove the noise with little

loss of details.

The idea of wavelet de-noising is based on theassumption that the amplitude, rather than

the location, of the spectra of the signal to be

as different as possible for that of the noise.


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Example1: -Duffings oscillator

0 200 400 600 -5

0

5x 10

-3 O riginal s ignal

0 200 400 600 -5

0

5

10x 10

-3 Noisy s ignal

0 200 400 600 -5

0

5x 10

-3 De-Noised signa


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Example-2:-Biomedical Data

0 10 20 30 40 50 60 70 80 0

50

100 Approximation cA4

0 200 400 600 -2

0

2Detail cD1

0 100 200 300 -2

0

2Detail cD2

0 50 100 150 -5

0

5Detail cD3

0 20 40 60 80 -5

0

5Detail cD4

-1000 -800 -600 -400 -200 0 200 400 600 800 1000 -1

0

1Correlation of Resudual

1 2 3 4 5 6 7 8 9 10 0

200

400Histogram O f Resudual

1 2 3 4 5 6 7 8 9 10 0

1000

2000

Cumulative Histogram Of Resudual

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 5

1 0

1 5

2 0t he o r i g i n a l s i g n a l

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 5

1 0

1 5

2 0t he d e - n o i s e d o r i g i n a l s i g n a l u s i n g wa v e l e t ba s e d


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Example 3:- De-noising Images

O riginal Ima ge

50 100 150 200 250

50

100

150

200

250

Noisy Image

P S N R = 1 0

50 100 150 200 250

50

10 0

15 0

20 0

25 0

De-N oised Image

P S N R = 2 7

50 100 150 200 250

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100

150

200

250

De -Noise d Image using Average F ilter 5X5

P S N R = 2 0

50 100 150 200 250

50

10 0

15 0

20 0

25 0

Original, Noisy (10db), Wavelet-based De-noised (27db), FT-based 5X5 AveragingFiltered (20db)


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Example 4:- De-noising of Seismic Data

0 5 1 0

0

50

10 0

15 0

20 0

25 0

TheoriginalSyntheticSeismogram

0 5 1 0

0

50

10 0

15 0

20 0

25 0

NoisySyntheticSeismogram

0 5 10

0

50

10 0

15 0

20 0

25 0De-N

oisedS

yntheticSeismogram

asanimage

0 5 1 0

0

50

100

150

200

250

A)The Noisy S ynthetic S e ism ogram

0 5 1 0

0

50

10 0

15 0

20 0

25 0

B)De-noised S e ism ic by MW

0 5 10

0

50

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150

200

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C)De-noised seism ic MW using PCA

Synthesised, Noisy(2db), 2D de-noised

seismic

Noisy(2d),Multivariate wavelet ,and Multivariate Wavelet Using PCA

1) For analysing seismic traces, both

oscillations and the time they occur are

important.

2) WT can be used to zoom in on the short

bursts and zoom out to detect long

oscillation

Application of Wavelets in Signal Processing 1

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