An introduction to

Wavelet Transform
Pao-Yen Lin
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering
National Taiwan University

1
Outlines
Introduction
Background
Time-frequency analysis
Windowed Fourier Transform
Wavelet Transform
Applications of Wavelet Transform

2
Introduction
Why Wavelet Transform?

Ans: Analysis signals which is a function of time and

frequency

Examples
Scores, images, economical data, etc.

3
Introduction

Conventional Fourier Transform

V.S.
Wavelet Transform

4
Conventional Fourier Transform

X( f )

5
Wavelet Transform

W{x(t)}

6
Background
Image pyramids
Subband coding

7
Image pyramids

Fig. 1 a J-level image pyramid[1]

8
Image pyramids

Fig. 2 Block diagram for creating image pyramids[1]

9
Subband coding

Fig. 3 Two-band filter bank for one-dimensional subband coding and

decoding system and the corresponding spectrum of the two bandpass
filters[1]
10
Subband coding
Conditions of the filters for error-free reconstruction

H 0   z  G0  z   H 1   z  G1  z   0
H 0  z  G0  z   H 1  z  G1  z   2
For FIR filter

g 0  n    1 h1  n 
n

g1  n    1 h0  n 
n +1

11
Time-frequency analysis
Fourier Transform

F  x t    x t e
 j 2 ft
dt  x  t  , e j 2 ft



Time-Frequency Transform

T  f      f  t  *  t  dt  f ,


  t  : time-frequency atoms   1
12
Heisenberg Boxes
 f ,  is represented in a time-frequency plane by
a region whose location and width depends on the tim
e-frequency spread of .

Center?
Spread?

13
Heisenberg Boxes
Recall that   1 ,that is:

||  ||2      dt  1
 2
| t |

Interpret as a PDF

Center : Mean
Spread : Variance

14
Heisenberg Boxes
Center (Mean) in time domain

      dt
 2
t | t |

Spread (Variance) in time domain

+
    =   t    |   t  |2 dt
2 2
t
-

15
Heisenberg Boxes
Plancherel formula


 ˆ   
2 2
d  2 

Center (Mean) in frequency domain
+
1
 ˆ    d
2
 = 
2 - 
Spread (Variance) in frequency domain

1
           ˆ    d
2 2 2

2 

16
Heisenberg Boxes

Fig. 4 Heisenberg box representing an atom  [1].

Heisenberg uncertainty
1
 t  
2
17
Windowed Fourier Transform
Window function
① Real
② Symmetric
 For a window function g  t 
① g  t   g  t 
② It is translated by μ and modulated by the frequency 
g  ,  t   e g  t   
i t

③ g  t  is normalized
g  t   1  g  ,  t 

18
Windowed Fourier Transform
Windowed Fourier Transform (WFT) is defined as

S  f   ,    f , g  ,   f  t  g  t    e i t dt


Also called Short time Fourier Transform (STFT)

Heisenberg box?

19
Heisenberg box of WFT
Center (Mean) in time domain
g  t  is real and symmetric, g  t  is centered at zero
g  ,  t  is centered at  in time domain

Spread (Variance) in time domain

 

  t   g  ,  t  dt   t g t
2 2 2
 
t
2 2
dt
 

independent of  and 

20
Heisenberg box of WFT
Center (Mean) in frequency domain
Similarly, g  ,  t  is centered at  in time domain

Spread (Variance) in frequency domain

By Parseval

theorem: 
1 1
    gˆ  ,      gˆ   
2 2 2
 
2
d  2
d
2 
2 

Both of them are independent of  and  .

21
Heisenberg box of WFT

Fig. 5 Heisenberg boxes of two windowed Fourier atoms g  , and g ,

[1]

22
Wavelet Transform
Classification
① Continuous Wavelet Transform (CWT)
② Discrete Wavelet Transform (DWT)
③ Fast Wavelet Transform (FWT)

23
Continuous Wavelet Transform
Wavelet function
Define 

① Zero mean:    t  dt  0


② Normalized:   t 1

s :
③ Scaling by and translating it by
1 t  
  ,s  t    
s  s 

24
Continuous Wavelet Transform
Continuous Wavelet Transform (CWT) is defined as

1 t   
W  f   , s    f ,  ,s   f  t    dt
 s  s 

ˆ   
2

Define C  
0

d

It can be proved that C  


which is called Wavelet admissibility condition

25
Continuous Wavelet Transform
ˆ   
2

For C 

0

d

where C  
ˆ  0   0


   t  dt  0

Zero mean

26
Continuous Wavelet Transform
Inverse Continuous Wavelet Transform (ICWT)
 
1 1  t    ds
f  t 
C   W  f  , s  
0 

s  s  s
 du 2

27
Continuous Wavelet Transform
Recall the Continuous Wavelet Transform

1 t   
W  f   , s    f ,  ,s   f  t    dt
 s  s 

When W  f   , s   is known for s  s0 , to recover func

tion wef need a complement of information correspo
nding to W  for
f   , s   . s  s0

28
Continuous Wavelet Transform
Scaling function
Define that the scaling function is an aggregation of w
avelets at scales larger than 1.
Define
ˆ   
2
 
2 ds
ˆ      ˆ  s 
2

1
s
 


d

 1
2
lim ˆ     C Low pass filter
 0

29
Continuous Wavelet Transform
A function can therefore decompose into a low-freque
ncy approximation and a high-frequency detail
Low-frequency approximation of f at scale s :

1 t   
L  f   , s    f ,  , s   f  t    dt
 s  s 

30
Continuous Wavelet Transform
The Inverse Continuous Wavelet Transform can be re
written as:

s0
1 ds 1
f  t  W f  .,s  ,s  t 2  L f  .,s0  ,s  t
C 0 s Cs 0

31
Heisenberg box of Wavelet atoms
Recall the Continuous Wavelet Transform

1 t   
W  f   , s    f ,  , s   f  t    dt
 s  s 
The time-frequency resolution depends on the time-fr
equency spread of the wavelet atoms  .,s

32
Heisenberg box of Wavelet atoms
Center in time domain
Suppose that  is centered at zero,
which implies that   ,s is centered at  .
Spread in time domain
 

  t     ,s  t  dt  s  
2 2

2 2
v 2
v dv  s t
2 2

 
t
v
s

33
Heisenberg box of Wavelet atoms
Center in frequency domain
for ̂    , it is centered at

1
  ˆ   
2
 d
2 

and ˆ ,s     sˆ  s  exp  i 

 
1 1
  ˆ     s ˆ  s 
2 2
c  d  d
2 2
,s
 
 
1 1 1 1 
 v ˆ  v   v ˆ  v  dv 
2 2
 dv 
2 
s s 2 
s
34
Heisenberg box of Wavelet atoms
Spread in frequency domain
Similarly,
 2  2
1   1  
    s   ,s    d  2     s    s  sd
2 2
ˆ ˆ
2
 
1 1 1 1  2
  s    ˆ  s    v    ˆ  v 
2 2 2 2
 d  d  2
2 
s 2 
s 2
s

35
Heisenberg box of Wavelet atoms
Center in time domain: 
Spread in time domain: s 2 t2
Center in frequency domain: 
s

Spread in frequency domain: 
2

s2
Note that they are function of s ,
but the multiplication of spread remains the same.

36
Heisenberg box of Wavelet atoms

Fig. 6 Heisenberg boxes of two wavelets. Smaller scales decrease the

time spread but increase the frequency support and vice versa.[1]

37
Examples of continuous wavelet
Mexican hat wavelet
Morlet wavelet
Shannon wavelet

38
Mexican hat wavelet

Also called the second derivative of

the Gaussian function

1  
t 2
 t2
 t  e 2 2
 2  1 
2  
3
 
Fig. 7 The Mexican hat wavelet[5]

39
Morlet wavelet
 t  1 4 imt  t 2 2
e e
ˆ      U  e
   m 
2
1 4 2
U(ω): step function

Fig. 8 Morlet wavelet with m equals to 3[4]

40
Shannon wavelet

  t   sinc  t 2  cos  3 t 2 

1 0.5  f  1
ˆ  f   
0 otherwise

Fig. 9 The Shannon wavelet in time and frequency domains[5]

41
Discrete Wavelet Transform (DWT)
Let f  t   f  Nt 

 
W f   , s   N 1 2W  f  N  , Ns  

Usually we choose N  2 j
discrete wavelet set:
 j ,k  x   2 j 2   2 j x  k 
discrete scaling set:
 j ,k  x   2 j 2   2 j x  k 

42
Discrete Wavelet Transform
Define
V j  Span   j ,k  x  
k
V j can be increased by increasing j .

There are four fundamental requirements of multireso

lution analysis (MRA) that scaling function and wavele
t function must follow.

43
Discrete Wavelet Transform
MRA(1/2)
① The scaling function is orthogonal to its integer transla
tes.
② The subspaces spanned by the scaling function at low r
esolutions are contained within those spanned at high
er resolutions:
V    V1  V0  V1  V2    V
③ The only function that is common to all is . That is
V   0

44
Discrete Wavelet Transform
MRA(2/2)
④ Any function can be represented with arbitrary precisi
on. As the level of the expansion function approaches i
nfinity, the expansion function space V contains all the
subspaces.
V   L2  R  

45
Discrete Wavelet Transform
subspace V j can be expressed as a weighted sum of the
expansion functions of subspace V j 1 .
 j ,k  x    n j 1,k  x 
n
  x    h  n  2  2 x  n 
n

scaling function coefficients

46
Discrete Wavelet Transform
Similarly,
Define

W j  span  j ,k  x 
k

The discrete wavelet set  j ,k  x  spans the difference bet
ween any two adjacent scaling subspaces, and
Vj
V j 1 .

V j 1  V j  W j

47
Discrete Wavelet Transform

Fig. 10 the relationship between scaling and wavelet function space[1]

48
Discrete Wavelet Transform
Any wavelet function can be expressed as a weighted s
um of shifted, double-resolution scaling functions

  x    h  n  2  2 x  n 
n

wavelet function coefficients

49
Discrete Wavelet Transform
By applying the principle of series expansion, the DW
T coefficients of f  x  defined as:
are

1
W  j0 , k  
M
 f  x   x
x
j0 , k

Arbitrary scale
1
W  j, k  
M
 f  x   x
x
j ,k

Normalizing factor

50
Discrete Wavelet Transform
 f  x  can be expressed as:

1 1
f  x 
M
W 
k
 j0 , k  j0 ,k  x  
M
 W  j , k    x 
j  j0 k
 j ,k

51
Fast Wavelet Transform (FWT)
Consider the multiresolution refinement equation
  x    h  n  2  2 x  n 
n

By a scaling of x by 2 j , translation of x by k units:


  2 j x  k    h  n  2 2  2 j x  k   n
n

  h  m  2k  2  2 j 1 x  m 
m

m  2k  n
52
Fast Wavelet Transform
Similarly,
  2 j x  k    h  m  2k  2  2 j 1 x  m 
m

Now consider the DWT coefficient functions

1
W  j, k  
M
  
f x 2 j 2
  x k
2 j

1  
W  j, k    f  x 2   h  m  2k  2  2 x  m  
j 2 j 1

M x m 

53
Fast Wavelet Transform
Rearranging the terms:
 1 
W  j , k    h  m  2k    f  x 2 
2  2 x  m 
j 1 2 j 1

m  M x 

W  j0 , k  with j0  j  1

54
Fast Wavelet Transform
Thus, we can write:

W  j , k    h  m - 2k  W  j  1, k 
m
Similarly,

W  j , k    h  m  2k  W  j  1, k 
m

55
Fast Wavelet Transform

Fig. 11 the FWT analysis filter bank[1]

56
Fast Wavelet Transform

Fig. 12 the IFWT synthesis filter bank[1]

57
2-D DWT
Two-dimensional scaling function

  x, y     x    y 
Two-dimensional wavelet functions
 H  x, y     x    y 

 V  x, y     x    y 

 D  x, y     x    y 

58
2-D DWT
  H  x, y  : variations along columns

  V  x, y  : variations along rows

  D  x, y  : variations along diagonals

59
2-D DWT
Basis

j ,m,n  x, y   2   2 x  m, 2 y  n 
j 2 j j

 ij ,m,n  x, y    j 2  i  2 j x  m, 2 j y  n  , i   H , V , D

60
2-D DWT
The discrete wavelet transform of function f  x, y  of s
ize M  N:
M 1 N 1
1
W  j0 , m, n  
MN
  f  x, y 
x 0 y 0
j0 , m , n  x, y 
M 1 N 1
1
Wi  j , m, n  
MN

x 0 y 0
f  x, y  ij ,m,n  x, y  i   H ,V , D

61
2-D DWT
Two-dimensional IDWT

1
f  x, y   
MN m n
W  j0 , m, n j0 ,m, n  x, y 

1
   
MN i  H ,V , D j  j0 m n
W 
i
 j , m , n  j , m , n  x, y 
i

62
2-D DWT

Fig. 13 the resulting decomposition of 2-D DWT[1]

63
2-D FWT

Fig. 14 the two-dimensional FWT analysis filter bank[1]

64
2-D FWT

Fig. 15 the two-dimensional IFWT synthesis filter bank[1]

65
2-D DWT

Fig. 16 A three-scale FWT[1]

66
Comparison
Resolution
Complexity
Given function f  t 

 sin  2 100t  0  t<0.5


f  t   sin  2 200t  0.5  t<1
sin  2 400t  1  t<1.5


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Comparison of resolution
Fourier Transform

Fig. 17 the result using Fourier Transform

68
Comparison of resolution
Windowed Fourier Transform

Fig. 18 the result using Windowed Fourier Transform

69
Comparison of resolution
Discrete Wavelet Transform

Fig. 19 the result using Discrete Wavelet Transform

70
Comparison of resolution

Fig. 20 Time-frequency tilings for Fourier Transform[1]

71
Comparison of resolution

Fig. 21 Time-frequency tilings for Windowed Fourier Transform

with different window size[1]

72
Comparison of resolution

Fig. 22 Time-frequency tilings for Wavelet Transform[1]

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Comparison of complexity

FFT WFT FWT

Complexity O  N log 2 N  O  N 2 log 2 N  O  N log 2 N 

Table. 1 Comparison of complexity between FFT, WFT and FWT

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Applications of Wavelet Transform
Image compression
Edge detection
Noise removal
Pattern recognition
Fingerprint verification
Etc.

75
Applications of Wavelet Transform
Image compression

Fig. 23 Input image Fig. 24 Output image with

compression ratio 30%

76
Applications of Wavelet Transform
Edge detection

Fig. 25 example of edge detection using Discrete Wavelet Transform[1]

77
Applications of Wavelet Transform
Noise removal

Fig. 26 example of noise removal using Discrete Wavelet Transform[1]

78
Conclusion

79
Reference
1. R. C. Gonzalez and R. E. Woods, Digital Image Processing 2/E. Upp
er Saddle River, NJ: Prentice-Hall, 2002, pp. 349-404.
2. S. Mallat, Academic press - A Wavelet Tour of Signal Processing 2/E.
San Diego, Ca: Academic Press, 1999, pp. 2-121.
3. J. J. Ding and N. C. Shen, “Sectioned Convolution for Discrete Wav
elet Transform,” June, 2008.
4. Clecom Software Ltd., “Continuous Wavelet Transform,” available i
n
http://www.clecom.co.uk/science/autosignal/help/Continuous_W
avelet_Transfor.htm
.
5. W. J. Phillips, “Time-Scale Analysis,” available in
http://www.engmath.dal.ca/courses/engm6610/notes/node4.html .
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