Introduction Continuous Wavelet Transforms Multiresolution Analysis Backgrounds
Image Pyramids Subband Coding
MRA Discrete Wavelet Transforms
The Fast Wavelet Transform
Applications Image Compression Edge Detection Digital Watermarking
Conclusions
2
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
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
xx
ss
0 0,
00
1( ) ( )
j
s jj
x k sx
ss
Sampling
0, 0 0( ) exp ]( [ 2 ( )) j
s
jx A j ss fx k 4[5]
Definitions Forward
where
• Inverse exists only if admissibility criterion is satisfied.
,( , ) ( ) ( )sW s f x x dx
,
1( ) ( )s
xx
ss
2
0
1,
xf x W s d ds
sC s s
2| ( ) |
| |
fC df
f
C
6
Subband coding Decomposing into a set of bandlimited components
Designing the filter coefficients s.t. perfectly reconstruction
10[1]
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]
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
The Haar Transform
1 11
1 12
2H
0
1( ) 2 0
2H k
1
1( ) 0 2
2H k
DFT
1
1( ) 1 1
2h n
0
1( ) 1 1
2h n
13[1]
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
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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]
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
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
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]
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 )nh n h n
19
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
Discrete wavelet transforms(1D) Forward
Inverse
00 ,
1( , ) ( ) ( )j k
n
W j k f n nM
, 0
1( , ) ( ) ( ) ,j k
n
W j k f n n for j jM
0
0
0 , ,
1 1( ) ( , ) ( ) ( , ) ( )j k j k
k j j k
f n W j k n W j k nM M
21
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 2 j
m
h m k p m
2m k n
22
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 kM
f n h m k n mM
h m k f n n mM
h m k W j k
,
1( , ) ( ) ( )j k
n
W j k f n nM
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
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]
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 12
h n
1
( ) 1 12
h n
1
( ) 2 02
H k
1
( ) 0 22
H k
26
[1]
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
2D fast wavelet transforms Due to the separable properties, we can apply 1D FWT
to do 2D DWTs.
28[1]
Image Compression have many near-zero coefficients
JPEG : DCT-based
JPEG2000 : FWT-based
, ,H V DW W W
DCT-based FWT-based 31
[3]
Digital watermarking Robustness
Nonperceptible(Transparency)
Nonremovable
Digital watermarking Watermark extracting
Channel/Signal
processing
Watermark
Original and/or Watermarked data
Secret/Public key Secret/Public key
Hostdata
Watermarkor
Confidencemeasure
33
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?
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[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
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