an introduction to discrete wavelet transforms
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Advisor : Jian-Jiun Ding, Ph. D.Presenter : Ke-Jie Liao
NTU,GICE,DISP Lab,MD531
An Introduction to Discrete Wavelet
Transforms
1
IntroductionContinuous Wavelet TransformsMultiresolution Analysis Backgrounds Image PyramidsSubband Coding
MRADiscrete Wavelet TransformsThe Fast Wavelet Transform
Applications Image CompressionEdge DetectionDigital Watermarking
Conclusions
Outlines
2
Why WTs? F.T. totally lose time-information.
Comparison between F.T., S.T.F.T., and W.T.
Introduction(1)
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.
Introduction(2)Sampling
F.T.
,1( ) ( )s
xxss
0 0,
00
1( ) ( )j
s jj
x k sx
ss
Sampling
0, 0 0( ) exp ]( [ 2 ( )) js
jx A j ss fx k 4[5]
MRATo mimic human being’s perception
characteristic
Introduction(3)
5[1]
DefinitionsForward where
• Inverse exists only if admissibility criterion is satisfied.
CWT
,( , ) ( ) ( )sW s f x x dx
,1( ) ( )s
xxss
20
1 , xf x W s d dssC s s
2| ( ) || |fC dff
C
6
An example-Using Mexican hat wavelet
CWT
7[1]
Image PyramidsApproximation pyramidsPredictive residual pyramids
MRA Backgrounds(1)
8N*N
N/2*N/2
N/4*N/4
N/8*N/8
Image PyramidsImplementation
MRA Backgrounds(1)
9
[1]
Subband codingDecomposing into a set of bandlimited
componentsDesigning the filter coefficients s.t. perfectly
reconstruction
MRA Backgrounds(2)
10[1]
Subband codingCross-modulated condition
Biorthogonality condition
MRA Backgrounds(2)
0 1
11 0
( ) ( 1) ( )
( ) ( 1) ( )
n
n
g n h n
g n h n
10 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 codingOrthonormality for perfect reconstruction filter
Orthonormal filters
MRA Backgrounds(2)
( ), ( 2 ) ( ) ( )i jg n g n m i j m
1 0( ) ( 1) ( 1 )neveng n g K n
( ) ( 1 )i i evenh n g K n
12
The Haar Transform
MRA Backgrounds(2)
1 111 12
2H
01( ) 2 02
H k
11( ) 0 22
H k
DFT
Low pass
High pass 1
1( ) 1 12
h n
01( ) 1 12
h n
13[1]
Any square-integrable function can be represented byScaling functions – approximation partWavelet functions - detail part(predictive
residual) Scaling function Prototype Expansion functions
MRA
/2, ( ) 2 (2 )j jj k x x k
2( ) ( )x L R
,{ ( )}j j kV span x
14
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.
MRA
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.
MRA
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.
MRA
1j jV V
( 1)/2 1, ( ) ( )2 (2 )j jj k
n
x h n x n
, 1,( ) ( ) ( )j k j nn
x h n x
1/2( ) ( )2 (2 )n
x h n x n
Scaling vector/Scaling function coefficients 17
/2, ( ) 2 (2 )j jj k x x k
Wavelet functionFill up the gap of any two adjacent scaling
subspacesPrototype Expansion functions
MRA
( )x
/2, ( ) 2 (2 )j jj k x x k
,{ ( )}j j kW span x
1j j jV V W
0 0 0
21( ) j j jL V W W R
18
[1]
Wavelet function
Scaling and wavelet vectors are related by
MRA
1j jW V
, 1,( ) ( ) ( )j k j nn
x h n x
( 1)/2 1, ( ) ( )2 (2 )j jj 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
MRA
0 0
0
, ,
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
a d
j j k j j kk j j k
f x f x f x
f x c k x d k x
0 0 0
21( ) j j jL V W W R
( )f x
( )af x
( )df x0j
W
0jV
0 1jV
0
( ) 0jd k 0j j
20
Discrete wavelet transforms(1D)Forward
Inverse
DWT
00 ,1( , ) ( ) ( )j k
n
W j k f n nM
, 01( , ) ( ) ( ) ,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 TransformsExploits a surprising but fortune relationship
between the coefficients of the DWT at adjacent scales.
Derivations for
DWT
( ) ( ) 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 jm
h m k p m
2m k n
22
Fast Wavelet Transforms Derivations for
DWT
( , )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 TransformsWith a similar derivation for
An FWT analysis filter bank
DWT
( , )W j k
2 , 0( , ) ( ) ( 1, ) |n k kW j k h n W j n
24[1]
FWT
DWT
25[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
DWT
( )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 transformsOne separable scaling function
Three separable directionally sensitive wavelets
DWT
( , ) ( ) ( )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.
DWT
28[1]
2D FWTsAn example
DWT
LL LH
HL HH
29[1]
2D FWTsSplitting frequency characteristic
DWT
30
[1]
Image Compression have many near-zero coefficients JPEG : DCT-based JPEG2000 : FWT-based
Applications(1)
, ,H V DW W W
DCT-based FWT-based 31
[3]
Edge detection
Applications(2)
32
[1]
Digital watermarkingRobustnessNonperceptible(Transparency)Nonremovable
Applicatiosn(3)
Digital watermarking Watermark extracting
Channel/Signal
processing
WatermarkOriginal and/or Watermarked data
Secret/Public key Secret/Public key
Hostdata
Watermark orConfidencemeasure
33
Digital watermarkingAn embedding process
Applicatiosn(3)
34
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?
Conclusions&Future work
35
[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
References
36
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