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Wavelet Transform
4/14/2014 2
Wavelet Definition
“The wavelet transform is a tool that cuts up data, functions or operators into different frequency components, and then studies each component with a resolution matched to its scale”
Dr. Ingrid Daubechies, Lucent, Princeton U.
4/14/2014 3
Fourier vs. Wavelet
► FFT, basis functions: sinusoids
► Wavelet transforms: small waves, called wavelet
► FFT can only offer frequency information
► Wavelet: frequency + temporal information
► Fourier analysis doesn’t work well on discontinuous, “bursty” data
music, video, power, earthquakes,…
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Fourier vs. Wavelet
► Fourier
Loses time (location) coordinate completely
Analyses the whole signal
Short pieces lose “frequency” meaning
► Wavelets
Localized time-frequency analysis
Short signal pieces also have significance
Scale = Frequency band
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Fourier transform
Fourier transform:
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Wavelet Transform
► Scale and shift original waveform
► Compare to a wavelet
► Assign a coefficient of similarity
4/14/2014 7
Scaling-- value of “stretch”
► Scaling a wavelet simply means stretching (or
compressing) it.
f(t) = sin(t)
scale factor1
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Scaling-- value of “stretch”
► Scaling a wavelet simply means stretching (or
compressing) it.
f(t) = sin(2t)
scale factor 2
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Scaling-- value of “stretch”
► Scaling a wavelet simply means stretching (or
compressing) it.
f(t) = sin(3t)
scale factor 3
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More on scaling
► It lets you either narrow down the frequency band of interest, or determine the frequency content in a narrower time interval
► Scaling = frequency band
► Good for non-stationary data
► Low scalea Compressed wavelet Rapidly changing detailsHigh frequency
► High scale a Stretched wavelet Slowly changing, coarse features
Low frequency
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Scale is (sort of) like frequency
Small scale
-Rapidly changing details,
-Like high frequency
Large scale
-Slowly changing
details
-Like low frequency
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Scale is (sort of) like frequency
The scale factor works exactly the same with wavelets.
The smaller the scale factor, the more "compressed"
the wavelet.
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Shifting
Shifting a wavelet simply means delaying (or hastening) its
onset. Mathematically, delaying a function f(t) by k is
represented by f(t-k)
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Shifting
C = 0.0004
C = 0.0034
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Five Easy Steps to a Continuous Wavelet
Transform
1. Take a wavelet and compare it to a section at the start of
the original signal.
2. Calculate a correlation coefficient c
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Five Easy Steps to a Continuous Wavelet
Transform
3. Shift the wavelet to the right and repeat steps 1 and 2 until you've
covered the whole signal.
4. Scale (stretch) the wavelet and repeat steps 1 through 3.
5. Repeat steps 1 through 4 for all scales.
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Coefficient Plots
4/14/2014 18
Discrete Wavelet Transform
► “Subset” of scale and position based on power of two
rather than every “possible” set of scale and position in continuous wavelet transform
► Behaves like a filter bank: signal in, coefficients out
► Down-sampling necessary
(twice as much data as original signal)
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Discrete Wavelet transform
signal
filters
Approximation
(a) Details
(d)
lowpass highpass
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Results of wavelet transform — approximation and details
► Low frequency:
approximation (a)
► High frequency
details (d)
► “Decomposition”
can be performed
iteratively
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Wavelet synthesis
•Re-creates signal from coefficients
•Up-sampling required
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Multi-level Wavelet Analysis
Multi-level wavelet
decomposition tree Reassembling original signal
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Subband Coding
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2-D 4-band filter bank
Approximation
Vertical detail
Horizontal detail
Diagonal details
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An Example of One-level Decomposition
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An Example of Multi-level Decomposition
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Wavelet Series Expansions
0 0
0
0
2
, ,
Wavelet series expansion of function ( ) ( )
relative to wavelet ( ) and scaling function ( )
( ) ( ) ( ) ( ) ( )
where ,
( ) : approximation and/or scaling coefficients
j j k j j k
k j j k
j
f x L
x x
f x c k x d k x
c k
d
( ) : detail and/or wavelet coefficientsj k
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Wavelet Series Expansions
0 0 0, ,
, ,
( ) ( ), ( ) ( ) ( )
( ) ( ), ( ) ( ) ( )
j j k j k
j j k j k
c k f x x f x x dx
and
d k f x x f x x dx
4/14/2014 29
Wavelet Transforms in Two Dimensions
( , ) ( ) ( )
( , ) ( ) ( )
( , ) ( ) ( )
( , ) ( ) ( )
H
V
D
x y x y
x y x y
x y x y
x y x y
/2
, ,
/2
, ,
( , ) 2 (2 ,2 )
( , ) 2 (2 ,2 )
{ , , }
j j j
j m n
i j i j j
j m n
x y x m y n
x y x m y n
i H V D
0
1 1
0 , ,
0 0
1( , , ) ( , ) ( , )
M N
j m n
x y
W j m n f x y x yMN
1 1
, ,
0 0
1( , , ) ( , ) ( , )
, ,
M Ni i
j m n
x y
W j m n f x y x yMN
i H V D
4/14/2014 30
Inverse Wavelet Transforms in Two Dimensions
0
0
0 , ,
, ,
, ,
1( , ) ( , , ) ( , )
1 ( , , ) ( , )
j m n
m n
i i
j m n
i H V D j j m n
f x y W j m n x yMN
W j m n x yMN
