Wavelet Transform.ppt

Wavelet Transform
A very brief look

2
Wavelets vs. Fourier Transform
 In Fourier transform (FT) we represent a
signal in terms of sinusoids
 FT provides a signal which is localized
only in the frequency domain
 It does not give any information of the
signal in the time domain

3
 Basis functions of the wavelet transform
(WT) are small waves located in different
times
 They are obtained using scaling and
translation of a scaling function and
wavelet function
 Therefore, the WT is localized in both time
and frequency

4
 In addition, the WT provides a
multiresolution system
 Multiresolution is useful in several
applications
 For instance, image communications and
image data base are such applications

5
 If a signal has a discontinuity, FT produces
many coefficients with large magnitude
(significant coefficients)
 But WT generates a few significant
coefficients around the discontinuity
 Nonlinear approximation is a method to
benchmark the approximation power of a
transform

6
 In nonlinear approximation we keep only a few
significant coefficients of a signal and set the
rest to zero
 Then we reconstruct the signal using the
significant coefficients
 WT produces a few significant coefficients for
the signals with discontinuities
 Thus, we obtain better results for WT nonlinear
approximation when compared with the FT

7
 Most natural signals are smooth with a few
discontinuities (are piece-wise smooth)
 Speech and natural images are such signals
 Hence, WT has better capability for representing
these signal when compared with the FT
 Good nonlinear approximation results in
efficiency in several applications such as
compression and denoising

8
Series Expansion of Discrete-Time Signals
 Suppose that is a square-summable sequence, that
is
 Orthonormal expansion of is of the form
 Where
is the transform of
 The basis functions satisfy the orthonormality
constraint
[ ]
x n
2
[ ] ( )
x n  Z
[ ]
x n
[ ] [ ], [ ] [ ] [ ] [ ]
k k k
k k
x n l x l n X k n
  
 
 
 
Z Z
*
[ ] [ ], [ ] [ ] [ ]
k k
l
X k l x l n x l
 
  
[ ]
x n
k

[ ], [ ] [ ]
k l
n n k l
  
 
2 2
x X


9
 Haar expansion is a two-point avarage
and difference operation
 The basis functions are given as
 It follows that
Haar Basis
2
1 2 , 2 , 2 1
[ ]
0, otherwise
k
n k k
n

  
 

2 1
1 2 , 2
[ ] 1 2 , 2 1
0, otherwise
k
n k
n n k
 
 

   



2 0
[ ] [ 2 ],
k n n k
 
  2 1 1
[ ] [ 2 ]
k n n k
 
  

10
 The transform is
 The reconstruction is obtained from
Haar Basis
 
2
1
[2 ] , [2 ] [2 1] ,
2
k
X k x x k x k

   
[ ] [ ] [ ]
k
k
x n X k n


 
Z
 
2 1
1
[2 1] , [2 ] [2 1]
2
k
X k x x k x k
 
    

11
Two-Channel Filter Banks
 Filter bank is the building block of discrete-
time wavelet transform
 For 1-D signals, two-channel filter bank is
depicted below

12
 For perfect reconstruction filter banks we have
 In order to achieve perfect reconstruction the
filters should satisfy
 Thus if one filter is lowpass, the other one will be
highpass
x̂ x

0 0
1 1
[ ] [ ]
[ ] [ ]
g n h n
g n h n
  

  


14
 To have orthogonal wavelets, the filter bank
should be orthogonal
 The orthogonal condition for 1-D two-channel
filter banks is
 Given one of the filters of the orthogonal filter
bank, we can obtain the rest of the filters
1 0
[ ] ( 1) [ 1]
n
g n g n
   

15
Haar Filter Bank
 The simplest orthogonal filter bank is Haar
 The lowpass filter is
 And the highpass filter
0
1
, 0, 1
[ ] 2
0, otherwise
n
h n

 

 


1
1
, 0
2
1
[ ] , 1
2
0, otherwise
n
h n n





   






16
Haar Filter Bank
 The lowpass output is
 And the highpass output is
0 0 0
2
1 1
[ ] [ ]* [ ] [ ] [2 ] [2 ] [2 1]
2 2
n k
l
y k h n x n h l x k l x k x k


     

1 1 1
2
1 1
[ ] [ ]* [ ] [ ] [2 ] [2 ] [2 1]
2 2
n k
l
y k h n x n h l x k l x k x k


     


17
Haar Filter Bank
 Since and , the filter
bank implements Haar expansion
 Note that the analysis filters are time-reversed
versions of the basis functions
since convolution is an inner product followed by
time-reversal
0[ ] [2 ]
y k X k
 1[ ] [2 1]
y k X k
 
0 0
[ ] [ ]
h n n

  1 1
[ ] [ ]
h n n

 

18
Discrete Wavelet Transform
 We can construct discrete WT via iterated (octave-band) filter banks
 The analysis section is illustrated below
Level 1
Level 2
Level J

19
Discrete Wavelet Transform
 And the synthesis section is illustrated here
 If is an orthogonal filter and , then we have an
orthogonal wavelet transform
0
V
1
V
2
V
J
V
1
W
2
W
J
W
[ ]
i
h n [ ] [ ]
i i
g n h n
 

20
Multiresolution
 We say that is the space of all square-
summable sequences if
 Then a multiresolution analysis consists of
a sequence of embedded closed spaces
 It is obvious that
0
V
0 2( )
V 
2 1 0 2( )
J
V V V V
    
0 2
0
( )
J
j
j
V V

 

21
Multiresolution
 The orthogonal component of in will
be denoted by :
 If we split and repeat on , , …., ,
we have
1
j
V 
1 1
j j j
V V W
 
 
j
V
1
j
W 
1 1
j j
V W
 

0
V 1
V 2
V 1
J
V 
0 1 1 J J
V W W W V
    

22
2-D Separable WT
 For images we use separable WT
 First we apply a 1-D filter bank to the rows of the
image
 Then we apply same transform to the columns of
each channel of the result
 Therefore, we obtain 3 highpass channels
corresponding to vertical, horizontal, and
diagonal, and one approximation image
 We can iterate the above procedure on the
lowpass channel

23
2-D Analysis Filter Bank
1
h
0
h
1
h
1
h
0
h
0
h
x diagonal
vertical
horizontal
approximation

24
2-D Synthesis Filter Bank
x̂
diagonal
vertical
horizontal
approximation
1
g
1
g
1
g
0
g
0
g
0
g

25
2-D WT Example
Boats image WT in 3 levels

26
WT-Application in Denoising
Boats image Noisy image (additive Gaussian noise)

27
WT-Application in Denoising
Boats image Denoised image using hard thresholding

28
Reference
 Martin Vetterli and Jelena Kovacevic, Wavelets and
Subband Coding. Prentice Hall, 1995.

Wavelet Transform.ppt

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Wavelet Transform.ppt