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Explanation of Wavelet Transform
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1Wavelet Transform
Wavelet Transform
2Wavelet Transform
Introduction
3Wavelet Transform
Why Transform
• To obtain further information that is not readily available in the raw signal
4Wavelet Transform
Limitation of Fourier Transform
• Loss of time/space information
5Wavelet Transform
Benefit of Wavelet Transform
• No Loss of time/space information
6Wavelet Transform
Non-Stationary Signal Analysis• Stationary Signal
– Properties do not evolve in time
– Fourier Transform is suitable
• Non-Stationary Signal– Properties evolve in
time– Time Frequency
Analysis• Short time Fourier
Transform (STFT)• Wavelet Transform
(WT)
7Wavelet Transform
• is a time domain windowing function• is the starting position of the window• STFT maps a function into 2-D plane• STFT uses sinusoidal wave as its basis function• Basis functions keep the same frequency over the
entire time interval• STFT uses a single analysis window
Short Time Fourier Transform (STFT)
( , ) ( )f t
( )w t
8Wavelet Transform
• A windowing technique with variable-sized regions• Long time intervals with more precise low-frequency
information• Short time intervals with high-frequency information
Wavelet Analysis
f
Time
Frequency
Wavelet Transform
9Wavelet Transform
Wavelet Analysis – A Contrast with other Methods
10Wavelet Transform
Wave vs Wavelet
• Wave: No compact support (extends to infinity)
• Transient signal (Anomaly, burst): Have compact support (non-zero only in a short interval)
• Many image features (e.g., edges) highly localized in spatial position.
0 100 200 300 400 500 600-8
-7
-6
-5
-4
-3
-2
-1
0
1
11Wavelet Transform
What is a Wavelet
• A wavelet is a waveform of effectively limited duration that has an average value of zero
Haar Wavelet
Finite Energy
12Wavelet Transform
What is a Wavelet (‘continued)
Basis of Fourier Analysis –unlimited duration sine waves
Smooth, predictable
Basis of Wavelet Analysis – limited duration wavelets
Irregular, asymmetric
• Fourier Analysis is breaking up of signal into sine waves of varying frequencies
• Wavelet Analysis is breaking up of signal into shifted and scaled version of mother wavelet
13Wavelet Transform
Continuous Fourier Transform (CFT)
( ) ( ) j tF f t e dt
14Wavelet Transform
Scaling a Sinusoid( ) sin
tf t
a
Time
Am
plit
ud
e
15Wavelet Transform
Scaling a Wavelet( )
tf t
a
Am
plit
ud
e
Time
16Wavelet Transform
Shifting a Wavelet
17Wavelet Transform
Continuous Wavelet Transform (CWT)
,( , ) ( ) ( )sW s f t t dt
,
1( )s
tt
ss
Wavelet Function
Scaling
Shifting
,2
0
( )1( ) ( , ) s tf t W s d ds
C s
CWT
Inverse CWT
2| ( ) |
| |
uC du
u
18Wavelet Transform
Computing Wavelet Transform
1
2
3
19Wavelet Transform
Wavelet Spectrum
20Wavelet Transform
Scale and Frequency
21Wavelet Transform
Discrete Wavelet Transform (DWT)DWT
Inverse DWT
,
1( , ) ( ) ( )
oo j kt
W j k f t tM
,
1( , ) ( ) ( )j k
t
W j k f t tM
Scaling Coefficients Wavelet Coefficients
, ,
1 1( ) ( , ) ( ) ( , ) ( )
o
o
o j k j kk j j k
f t W j k t W j k tM M
Scaling Function Wavelet Function
22Wavelet Transform
Some Scaling and Wavelet Functions
23Wavelet Transform
Two Dimensional Discrete Wavelet Transform
( , ) ( ) ( )x y x y
Separable Scaling function
( , ) ( ) ( )
( , ) ( ) ( )
( , ) ( ) ( )
H
V
D
x y x y
x y x y
x y x y
Separable Wavelets
/ 2, , ( , ) 2 (2 ,2 )j j jj m n x y x m y n
Scaled and Translated Basis Functions
/ 2, , ( , ) 2 (2 ,2 ) { , , }i j i j jj m n x y x m y n i H V D
24Wavelet Transform
Two Dimensional Discrete Wavelet Transform(‘continued)
1 1
, ,0 0
1( , , ) ( , ) ( , )
o
M N
o j m nx y
W j m n f x y x yMN
2-D DWT
Scaling Coefficients
1 1
, ,0 0
1( , , ) ( , ) ( , ) { , , }
M Ni i
j m nx y
W j m n f x y x y i H V DMN
Wavelet Coefficients
Inverse 2-D DWT
, ,
, ,
1( , ) ( , , ) ( , )
1( , , ) ( , )
o
o
o j m nm n
i ij m n
i j j m n
f x y W j m n x yMN
W j m n x yMN
25Wavelet Transform
Some Applications of Wavelet Transform
• De-noising• Compression• Detection of Discontinuities• System Identification• Video Compression (MPEG-4)• Speech Recognition• Face Recognition
26Wavelet Transform
De-noising Application of Wavelet Transform
Noisy Image
De-Noised Image
De-Noised Image
Information removed in De-Noising Process
De-Noised Image
27Wavelet Transform
References
• Rafael C. Gonzalez, Richard E. Woods, Digital Image Processing, 2nd Edition, Prentice Hall, 2002
• Robi Polikar, The Wavelet Tutorial http://engineering.rowan.edu/~polikar/WAVELETS/WTtutorial.html
• Jan T. Bialasiewicz, Introduction to Wavelet Transform & its Applications, http://pelincec.isep.pw.edu.pl
• Wavelet Transform (WT) and JPEG 2000
http://missouri.edu/~llxm7/CS4670-7670/ Slides/Ch8%20Wavelet %20Transform%20for%20Image%20Coding.ppt
• Wavelet Toolbox – Matlab 7 Help