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

xx

ss

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

xx

ss

2

0

1,

xf x W s d ds

sC s s

2| ( ) || |f

C 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 11

1 12

2H

0

1( ) 2 0

2H k

1

1( ) 0 2

2H k

DFT

Low pass

High pass 1

1( ) 1 1

2h n

0

1( ) 1 1

2h 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 x

0jW

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

, 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 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 j

m

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 1

2h n

1( ) 1 1

2h n

1( ) 2 0

2H k

1( ) 0 2

2H 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

Watermark

Original 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

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