Image processing and pattern recognition .ppt

CS589-04 Digital Image Processing
CS589-04 Digital Image Processing
Lecture 9. Wavelet Transform
Lecture 9. Wavelet Transform
Spring 2008
Spring 2008
New Mexico Tech
New Mexico Tech

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Wavelet Definition
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
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 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
Fourier transform:
Fourier transform:

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Wavelet Transform
Wavelet Transform
► Scale and shift original waveform
► Compare to a wavelet
► Assign a coefficient of similarity

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Scaling--
Scaling-- value of “stretch”
value of “stretch”
► Scaling a wavelet simply means stretching (or
compressing) it.
f(t) = sin(t)
scale factor1

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Scaling--
compressing) it.
f(t) = sin(2t)
scale factor 2

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Scaling--
compressing) it.
f(t) = sin(3t)
scale factor 3

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More on scaling
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a Compressed wavelet Rapidly changing detailsHigh
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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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
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
Shifting
C = 0.0004
C = 0.0034

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Five Easy Steps to a Continuous Wavelet
Transform
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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Transform
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
Coefficient Plots

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Discrete Wavelet Transform
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
Discrete Wavelet transform
signal
filters
Approximation
(a)
Details
(d)
lowpass highpass

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Results of wavelet transform
Results of wavelet transform
—
— approximation and details
approximation and details
► Low frequency:
 approximation (a)
► High frequency
 details (d)
► “Decomposition”
can be performed
iteratively

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Wavelet synthesis
Wavelet synthesis
•Re-creates signal from coefficients
•Up-sampling required

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Multi-level Wavelet Analysis
Multi-level Wavelet Analysis
Multi-level wavelet
decomposition tree Reassembling original signal

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Subband Coding
Subband Coding

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2-D 4-band filter bank
2-D 4-band filter bank
Approximation
Vertical detail
Horizontal detail
Diagonal details

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An Example of One-level Decomposition
An Example of One-level Decomposition

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An Example of Multi-level Decomposition
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 coefficients
j k

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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
 
 
 
 



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Wavelet Transforms in Two Dimensions
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 y
MN
 
 
 
 
 
1 1
, ,
0 0
1
( , , ) ( , ) ( , )
, ,
M N
i i
j m n
x y
W j m n f x y x y
MN
i H V D
 
 
 




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Inverse Wavelet Transforms in Two
Inverse Wavelet Transforms in Two
Dimensions
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 y
MN
W j m n x y
MN





 



  

Image processing and pattern recognition .ppt

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