Transcript Wavelet Transform - Welcome to ECE at McMaster University

Wavelet Transform
A very brief look
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

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Wavelets vs. Fourier Transform
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
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Wavelets vs. Fourier Transform
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

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Wavelets vs. Fourier Transform
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
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Wavelets vs. Fourier Transform
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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
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Wavelets vs. Fourier Transform
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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
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Series Expansion of Discrete-Time Signals
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Suppose that x[n] is a square-summable sequence, that
is x[n]  2 (Z)
Orthonormal expansion of x[n] is of the form
x[n]   k [l ], x[l ] k [n]   X [k ]k [n]
kZ

Where
2
x  X
2
kZ
X [k ]  k [l ], x[l ]  k*[n] x[l ]
l

is the transform of x[n]
The basis functions  k satisfy the orthonormality
constraint
k [n], l [n]   [k  l ]
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Haar Basis
Haar expansion is a two-point avarage
and difference operation
 The basis functions are given as


2k [n]  1 2 , n  2k , 2k  1


0, otherwise
 1 2 , n  2k

2 k 1[n]  1 2 , n  2k  1
 0, otherwise

It follows that
2 k [n]  0 [n  2k ],
2 k 1[n]  1[n  2k ]
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Haar Basis

The transform is
X [2k ]  2 k , x 
X [2k  1]  2 k 1

1
 x[2k ]  x[2k  1] ,
2
1
,x 
 x[2k ]  x[2k  1]
2
The reconstruction is obtained from
x[n]   X [k ]k [n]
kZ
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Two-Channel Filter Banks
Filter bank is the building block of discretetime wavelet transform
 For 1-D signals, two-channel filter bank is
depicted below
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Two-Channel Filter Banks
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For perfect reconstruction filter banks we have
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In order to achieve perfect reconstruction the
filters should satisfy
 g0 [n]  h0 [n]
 g [ n ]  h [  n]
 1
1

Thus if one filter is lowpass, the other one will be
highpass
xˆ  x
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Two-Channel Filter Banks
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Two-Channel Filter Banks
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To have orthogonal wavelets, the filter bank
should be orthogonal
The orthogonal condition for 1-D two-channel
filter banks is
n
g1[n]  (1) g0 [n  1]

Given one of the filters of the orthogonal filter
bank, we can obtain the rest of the filters
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Haar Filter Bank
The simplest orthogonal filter bank is Haar
 The lowpass filter is
 1
, n  0, 1


h0 [n]   2

 0, otherwise

And the highpass filter
 1
 2, n0

 1
h1[n]  
, n  1
 2
 0, otherwise


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Haar Filter Bank
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The lowpass output is
y0 [k ]  h0 [n]* x[n] n  2k   h0 [l ]x[2k  l ] 
l

1
1
x[2k ] 
x[2k  1]
2
2
And the highpass output is
y1[k ]  h1[n]* x[n] n  2k   h1[l ]x[2k  l ] 
l
1
1
x[2k ] 
x[2k  1]
2
2
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Haar Filter Bank
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Since y0 [k ]  X [2k ] and y1[k ]  X [2k  1] , the filter
bank implements Haar expansion
Note that the analysis filters are time-reversed
versions of the basis functions
h0 [n]  0 [n]
h1[n]  1[n]
since convolution is an inner product followed by
time-reversal
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Discrete Wavelet Transform
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We can construct discrete WT via iterated (octave-band) filter banks
The analysis section is illustrated below
Level 1
Level 2
Level J
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Discrete Wavelet Transform
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And the synthesis section is illustrated here
If hi [ n ] is an orthogonal filter and gi [n]  hi [n] , then we have an
orthogonal wavelet transform
W1
V0
W2
V1
V2
WJ
VJ
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Multiresolution
We say that V0 is the space of all squaresummable sequences if V0  2 ( )
 Then a multiresolution analysis consists of
a sequence of embedded closed spaces

VJ 

 V2  V1  V0 
It is obvious that
J
j 0
2(
V j  V0 
)
2(
)
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Multiresolution

The orthogonal component of V j 1 in V j will
be denoted by W j 1 :
V j  V j 1  W j 1

V j 1  W j 1
If we split V0 and repeat on V1 , V 2 , …., VJ 1 ,
we have
V0  W1  W1 
 WJ  VJ
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2-D Separable WT
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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
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2-D Analysis Filter Bank
x
h1
h0
h1
diagonal
h0
vertical
h1
horizontal
h0
approximation
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2-D Synthesis Filter Bank
diagonal
g1
vertical
g0
horizontal
g1
approximation
g0
g1
xˆ
g0
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2-D WT Example
Boats image
WT in 3 levels
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WT-Application in Denoising
Boats image
Noisy image (additive Gaussian noise)
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WT-Application in Denoising
Boats image
Denoised image using hard thresholding
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Reference

Martin Vetterli and Jelena Kovacevic, Wavelets and
Subband Coding. Prentice Hall, 1995.
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