Advisor : Jian-Jiun Ding, Ph. D.

Presenter : Ke-Jie Liao

NTU,GICE,DISP Lab,MD531

1

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

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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]

MRA To mimic human being’s perception characteristic

5[1]

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

An example -Using Mexican hat wavelet

7[1]

Image Pyramids Approximation pyramids

Predictive residual pyramids

8N*N

N/2*N/2

N/4*N/4

N/8*N/8

Image Pyramids Implementation

9

[1]

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

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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

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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

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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

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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

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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

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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]

FWT

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

( )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]

2D FWTs An example

LL LH

HL HH

29[1]

2D FWTs Splitting frequency characteristic

30

[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]

Edge detection

32

[1]

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

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Digital watermarking An embedding process

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?

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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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