2D Continuous Wavelet Transform

2D Continuous Wavelet Transform

Heejong Yoo(ECE)

April 26, 2001

Project Description

The idea of this project is from the software of Crit-tech.com(Psilets 3.0) (www.crit-tech.com)

Project Description– Understand 1D, 2D DWT(FWT_PO.m, FWT2_PO.m in

WaveLab)– Understand 1D CWT(CWT.m in WaveLab)– Develop 2D CWT algorithm for image processing at fixed

scale

CWT & DWT

CWT DWT

1. Scale At any scale Dyadic scales

2. Translation At any point Integer point

3. Wavelet Any wavelet that satisfies minimum criteria

Orthogonal, biorthogonal, …

4. Computation Large Small

5. Detection Easily detects direction, orientation

Cannot detect minute object if not finely tuned

6. Application Pattern Recognition

Feature extraction

Detection

Compression

De-noising

Transmission

Characterization

Continuous Wavelet Transform

),()( 2 RLxf

dxs

bxxf

sbscwt )()(

1),(

)()(),( swwFswsCWT

In one dimension,

(time domain)

(frequency domain)

In two dimension,

),(),( 2 RLyxf dydxs

by

s

axyxf

sbascwt

),(),(

1),,( (time domain)

)2,1()2,1()2,1,( swswwwFswwsCWT (frequency domain)

We want to calculate 2D CWT in frequency domain(just like WaveLab)

When scale s is fixed,

)2,1()2,1()2,1( wwwwFwwCWT

2D Mexican Hat wavelet

)(2

122

22

)2(),(yx

eyxyx

Time domain)21(

2

122

22

)21(2)2,1(ww

ewwww

Frequency domain

2D Mexican Hat wavelet (Movie)low frequency high frequency

<Time Domain Wavelet> <Fourier Domain Wavelet>

)},({),( 21 yxFww

)(2

122

21

2

1

2

1

2

1

2

122

2

1

2

121

2

1

2

1

2

1

2

122

1

2

12

2

1

2

1

2

1

2

122

1

2

12

)(2

122

21

22

21

22

21

21

22

22

21

2

2

1

2

1

2

2

2

2

2

1

2

2

2

1

2

1

2

2

2

2

2

1

2

21

22

)(2

)2)(2(2

)2)()1(2()2)()1(2(

)()(2

)()()()(

))((2

))(())((

])2[()},({),(

ww

ww

wwww

yjwyxjwx

xjwxyjwyyjwyxjwx

yjwyxjwx

xjwxyjwyyjwyxjwx

yjwxjwyx

eww

ee

eeweew

dyeedxee

dxeedyeeydyeedxeex

dxdyeeee

dydxeeeeydxdyeeeex

dxdyeeeyxyxFww

GUI based 2D CWT

In general, low scale means high frequency, high scale means low frequency

In WaveLab, low scale means low frequency, high scale means high frequency

This program follows the WaveLab

Output Image Display

Fourier Domain Wavelet

4 different Input Image

Scale(1~100)

Wavelet Select

Crit-tech Psilets 3.0 Output (1)

Scale = 38

Scale =2

Scale =1

Crit-tech Psilets 3.0 Output (2)

Scale =48

Scale =3