3. Why WTs?
F.T. totally lose time-information.
Comparison between F.T., S.T.F.T., and W.T.
f f f
t t t
F.T. S.T.F.T. W.T.
3
4. Difficulties when CWT DWT?
Continuous WTs Discrete WTs
need infinitely scaled wavelets to represent a given
function Not possible in real world
Another function called scaling functions are used to
span the low frequency parts (approximation parts)of
the given signal.
Sampling
F.T.
,
1
( ) ( )s
x
x
ss
0 0
,
00
1
( ) ( )
j
s jj
x k s
x
ss
Sampling
0, 0 0( ) exp ]( [ 2 ( )) j
s
j
x A j ss fx k 4[5]
5. MRA
To mimic human being’s perception characteristic
5
[1]
6. Definitions
Forward
where
• Inverse exists only if admissibility criterion is satisfied.
,( , ) ( ) ( )sW s f x x dx
,
1
( ) ( )s
x
x
ss
2
0
1
,
x
f x W s d ds
sC s s
2
| ( ) |
| |
f
C df
f
C
6
10. Subband coding
Decomposing into a set of bandlimited components
Designing the filter coefficients s.t. perfectly
reconstruction
10[1]
11. Subband coding
Cross-modulated condition
Biorthogonality condition
0 1
1
1 0
( ) ( 1) ( )
( ) ( 1) ( )
n
n
g n h n
g n h n
1
0 1
1 0
( ) ( 1) ( )
( ) ( 1) ( )
n
n
g n h n
g n h n
(2 ), ( ) ( )i jh n k g k i j
11
or
[1]
12. Subband coding
Orthonormality for perfect reconstruction filter
Orthonormal filters
( ), ( 2 ) ( ) ( )i jg n g n m i j m
1 0( ) ( 1) ( 1 )n
eveng n g K n
( ) ( 1 )i i evenh n g K n
12
13. The Haar Transform
1 11
1 12
2H
0
1
( ) 2 0
2
H k
1
1
( ) 0 2
2
H k
DFT
1
1
( ) 1 1
2
h n
0
1
( ) 1 1
2
h n
13
[1]
14. Any square-integrable function can be represented by
Scaling functions – approximation part
Wavelet functions - detail part(predictive residual)
Scaling function
Prototype
Expansion functions
/2
, ( ) 2 (2 )j j
j k x x k
2
( ) ( )x L R
,{ ( )}j j kV span x
14
15. MRA Requirement
[1] The scaling function is orthogonal to its integer
translates.
[2] The subspaces spanned by the scaling function at low
scales are nested within those spanned at higher scales.
1 0 1 2V V V V V V
15
[1]
16. MRA Requirement
[3] The only function that is common to all is .
[4] Any function can be represented with arbitrary
precision.
jV ( ) 0f x
{0}V
2
{ ( )}V L R
16
17. Refinement equation
the expansion function of any subspace can be built
from double-resolution copies of themselves.
1j jV V
( 1)/2 1
, ( ) ( )2 (2 )j j
j k
n
x h n x n
, 1,( ) ( ) ( )j k j n
n
x h n x
1/2
( ) ( )2 (2 )
n
x h n x n
Scaling vector/Scaling function coefficients 17
/2
, ( ) 2 (2 )j j
j k x x k
18. Wavelet function
Fill up the gap of any two adjacent scaling subspaces
Prototype
Expansion functions
( )x
/2
, ( ) 2 (2 )j j
j k x x k
,{ ( )}j j kW span x
1j j jV V W
0 0 0
2
1( ) j j jL V W W R
18
[1]
19. Wavelet function
Scaling and wavelet vectors are related by
1j jW V
, 1,( ) ( ) ( )j k j n
n
x h n x
( 1)/2 1
, ( ) ( )2 (2 )j j
j k
n
x h n x n
1/2
( ) ( )2 (2 )
n
x h n x n
Wavelet vector/wavelet function coefficients
( ) ( 1) (1 )n
h n h n
19
20. Wavelet series expansion
0 0
0
, ,
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
a d
j j k j j k
k j j k
f x f x f x
f x c k x d k x
0 0 0
2
1( ) j j jL V W W R
( )f x
( )af x
( )df x
0jW
0jV
0 1jV
0
( ) 0jd k 0j j
20
21. Discrete wavelet transforms(1D)
Forward
Inverse
00 ,
1
( , ) ( ) ( )j k
n
W j k f n n
M
, 0
1
( , ) ( ) ( ) ,j k
n
W j k f n n for j j
M
0
0
0 , ,
1 1
( ) ( , ) ( ) ( , ) ( )j k j k
k j j k
f n W j k n W j k n
M M
21
22. Fast Wavelet Transforms
Exploits a surprising but fortune relationship between
the coefficients of the DWT at adjacent scales.
Derivations for
( ) ( ) 2 (2 )
n
p h n p n
( , )W j k
(2 ) ( ) 2 2(2 )j j
n
p k h n p k n
1
( 2 ) 2 2j
m
h m k p m
2m k n
22
23. Fast Wavelet Transforms
Derivations for ( , )W j k
/2
/2 1
( 1)/2 1
1
( , ) ( )2 (2 )
1
( )2 ( 2 ) 2 (2 )
1
( 2 ) ( )2 (2 )
( 2 ) ( 1, )
j j
n
j j
n m
j j
m n
m
W j k f n n k
M
f n h m k n m
M
h m k f n n m
M
h m k W j k
,
1
( , ) ( ) ( )j k
n
W j k f n n
M
1
(2 ) ( 2 ) 2 2j j
m
n k h m k n m
2 , 0( , ) ( ) ( 1, ) |n k kW j k h n W j n 23
24. Fast Wavelet Transforms
With a similar derivation for
An FWT analysis filter bank
( , )W j k
2 , 0( , ) ( ) ( 1, ) |n k kW j k h n W j n
24[1]
26. Inverse of FWT
Applying subband coding theory to implement.
acts like a low pass filter.
acts like a high pass filter.
ex. Haar wavelet and scaling vector
( )h n
( )h n
DFT
1
( ) 1 1
2
h n
1
( ) 1 1
2
h n
1
( ) 2 0
2
H k
1
( ) 0 2
2
H k
26
[1]
27. 2D discrete wavelet transforms
One separable scaling function
Three separable directionally sensitive wavelets
( , ) ( ) ( )x y x y
( , ) ( ) ( )H
x y x y
( , ) ( ) ( )V
x y y x
( , ) ( ) ( )D
x y x y
x
y
27
28. 2D fast wavelet transforms
Due to the separable properties, we can apply 1D FWT
to do 2D DWTs.
28
[1]
33. Digital watermarking
Robustness
Nonperceptible(Transparency)
Nonremovable
Digital watermarking Watermark extracting
Channel/
Signal
processin
g
Watermark
Original and/or
Watermarked data
Secret/Public key Secret/Public key
H
o
s
t
d
a
t
a
Watermark
or
Confidence
measure
33
35. Wavelet transforms has been successfully applied to
many applications.
Traditional 2D DWTs are only capable of detecting
horizontal, vertical, or diagonal details.
Bandlet?, curvelet?, contourlet?
35
36. [1] R. C. Gonzalez, R. E. Woods, "Digital Image
Processing third edition", Prentice Hall, 2008.
[2] J. J. Ding and N. C. Shen, “Sectioned Convolution
for Discrete Wavelet Transform,” June, 2008.
[3] J. J. Ding and J. D. Huang, “The Discrete Wavelet
Transform for Image Compression,”,2007.
[4] J. J. Ding and Y. S. Zhang, “Multiresolution Analysis
for Image by Generalized 2-D Wavelets,” June, 2008.
[5] C. Valens, “A Really Friendly Guide to Wavelets,”
available in http://pagesperso-
orange.fr/polyvalens/clemens/wavelets/wavelets.html
36
