Transcript 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 Crittech.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
In one dimension,
1
cwt
(
s
,
b
)

f ( x)  L2 ( R),
s

CWT ( s, w) 
f ( x) (
x b
)dx (time domain)
s
s F ( w) ( sw) (frequency domain)
In two dimension,
f ( x, y)  L2 ( R), cwt ( s, a, b)  1
f ( x, y ) (
s 
x a y b
,
)dxdy
s
s
CWT ( s, w1, w2)  s F ( w1, w2)( sw1, sw2)
(time domain)
(frequency domain)
We want to calculate 2D CWT in frequency domain(just like WaveLab)
When scale s is fixed,
CWT ( w1, w2)  F ( w1, w2)( w1, w2)
2D Mexican Hat wavelet
Time domain
 ( x, y )  ( x  y  2)e
2
2
1
 ( x2  y2 )
2
Frequency domain
(w1, w2)  2 (w1  w2 )e
2
2
1
 ( w12  w22 )
2
2D Mexican Hat wavelet (Movie)
low frequency  high frequency
<Time Domain Wavelet>
<Fourier Domain Wavelet>
(w1 , w2 )  F{( x, y)}
 ( w1 , w2 )  F { ( x, y )}    [( x  y  2)e
   (x e
2
1
 x2
2
 2   (e
  (x e
2
1
 x2
2
1
 x2
2
 2  (e
e
e
1
 x2
2
 jw1 x
e
 jw1 x
 jw1 x
e
 ( 2 (1  w )e
1
 w12
2
2
2
1
 y2
2
 jw2 y
)(e
e
1
 y2
2
e
1
 y2
2
1
 w2 2
2
1
 ( w12  w22 )
2
]e  jw1 x e  jw2 y dxdy
) dxdy    ( y e
2
1
 y2
2
e
 jw2 y
)(e
1
 x2
2
e  jw1 x ) dydx
 jw2 y
) dy   ( y e
2
1
 y2
2
e
 jw2 y
)dy  (e
1
 x2
2
e  jw1 x ) dx
e  jw2 y )dy
)( 2 e
)( 2 e
1
 ( x2  y2 )
2
e  jw2 y ) dxdy
1
 y2
2
) dx  (e
1
 w12
2
 2 ( w  w )e
2
1
2
)dx  (e
 jw1 x
2
1
 2( 2 e
)(e
2
1
 w2 2
2
)
)  ( 2 (1  w )e
2
2
1
 w2 2
2
)( 2 e
1
 w12
2
)
GUI based 2D CWT
Output Image Display
4 different Input Image
In general,
low scale means high frequency,
high scale means low frequency
In WaveLab,
low scale means low frequency,
high scale means high frequency
Wavelet Select
This program follows the WaveLab
Scale(1~100)
Fourier Domain Wavelet
Crit-tech Psilets 3.0 Output (1)
Scale = 38
Scale =2
Scale =1
Crit-tech Psilets 3.0 Output (2)
Scale =48
Scale =3