The document compares wavelet transforms and Fourier transforms. Wavelet transforms provide time-frequency localization while Fourier transforms only provide frequency localization. Wavelet transforms use small wave functions that are scaled and translated, allowing time-frequency localization. They also provide multiresolution analysis which is useful for applications like image processing. Wavelet transforms have better performance than Fourier transforms for signals that are piecewise smooth, like images and audio, due to producing fewer significant coefficients around discontinuities. The discrete wavelet transform is constructed via iterated filter banks and provides multiresolution analysis via embedded subspaces. Two-dimensional wavelet transforms separate into row and column filtering. Wavelet transforms are useful for applications like image denoising by thresholding coefficients
2. 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. 3
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
4. 4
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
5. 5
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
6. 6
Wavelets vs. Fourier Transform
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. 7
Wavelets vs. Fourier Transform
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. 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. 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. 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. 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. 12
Two-Channel Filter Banks
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. 14
Two-Channel Filter Banks
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. 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. 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. 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. 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. 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. 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. 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. 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. 23
2-D Analysis Filter Bank
1
h
0
h
1
h
1
h
0
h
0
h
x diagonal
vertical
horizontal
approximation
24. 24
2-D Synthesis Filter Bank
x̂
diagonal
vertical
horizontal
approximation
1
g
1
g
1
g
0
g
0
g
0
g
