CSE489-02 &CSE589-02 Multimedia Processing
Lecture 8. Wavelet Transform
Spring 2009
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.
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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
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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 scaleCompressed wavelet Rapidly changing
detailsHigh frequency

High scale 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
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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
lowpass
highpass
filters
Approximation
(a)
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Details
(d)
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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
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Reassembling original signal
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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
Wavelet series expansion of function f ( x )  L2 ( )
relative to wavelet  ( x) and scaling function  ( x)

f ( x)   c j0 ( k ) j0 ,k ( x)    d j ( k ) j ,k ( x)
k
j  j0
k
where ,
c j0 (k ) : approximation and/or scaling coefficients
d j (k ) : detail and/or wavelet coefficients
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Wavelet Series Expansions
c j0 (k )  f ( x),  j0 ,k ( x)   f ( x) j0 ,k ( x)dx
and
d j (k )  f ( x), j ,k ( x)   f ( x) j ,k ( x)dx
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Wavelet Transforms in Two Dimensions
 ( x, y )   ( x) ( y )
 ( x, y )   ( x) ( y )
 j ,m,n ( x, y )  2 j /2  (2 j x  m, 2 j y  n)
 ( x, y )   ( x) ( y )
 i j ,m,n ( x, y )  2 j /2 i (2 j x  m, 2 j y  n)
 D ( x, y )   ( x) ( y )
i  {H ,V , D}
H
V
1
W ( j0 , m, n) 
MN
W i ( j , m, n) 
1
MN
M 1 N 1
 f ( x, y)
x 0 y 0
M 1 N 1

x 0 y 0
j0 , m, n
( x, y)
f ( x, y ) i j ,m,n ( x, y )
i   H ,V , D
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Inverse Wavelet Transforms in Two Dimensions
1
f ( x, y ) 
W ( j0 , m, n) j0 ,m,n ( x, y )

MN m n

1
i
i

W  ( j , m, n) j ,m,n ( x, y )



MN i  H ,V , D j  j0 m n
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