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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
0. 4
0. 6
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
50
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
100
150
200
250
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
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