1Wavelet Transform

Wavelet Transform

2Wavelet Transform

Introduction

3Wavelet Transform

Why Transform

• To obtain further information that is not readily available in the raw signal

4Wavelet Transform

Limitation of Fourier Transform

• Loss of time/space information

5Wavelet Transform

Benefit of Wavelet Transform

• No Loss of time/space information

6Wavelet Transform

Non-Stationary Signal Analysis• Stationary Signal

– Properties do not evolve in time

– Fourier Transform is suitable

• Non-Stationary Signal– Properties evolve in

time– Time Frequency

Analysis• Short time Fourier

Transform (STFT)• Wavelet Transform

(WT)

7Wavelet Transform

• is a time domain windowing function• is the starting position of the window• STFT maps a function into 2-D plane• STFT uses sinusoidal wave as its basis function• Basis functions keep the same frequency over the

entire time interval• STFT uses a single analysis window

Short Time Fourier Transform (STFT)

( , ) ( )f t

( )w t

8Wavelet Transform

• A windowing technique with variable-sized regions• Long time intervals with more precise low-frequency

information• Short time intervals with high-frequency information

Wavelet Analysis

f

Time

Frequency

Wavelet Transform

9Wavelet Transform

Wavelet Analysis – A Contrast with other Methods

10Wavelet Transform

Wave vs Wavelet

• Wave: No compact support (extends to infinity)

• Transient signal (Anomaly, burst): Have compact support (non-zero only in a short interval)

• Many image features (e.g., edges) highly localized in spatial position.

0 100 200 300 400 500 600-8

-7

-6

-5

-4

-3

-2

-1

0

1

11Wavelet Transform

What is a Wavelet

• A wavelet is a waveform of effectively limited duration that has an average value of zero

Haar Wavelet

Finite Energy

12Wavelet Transform

What is a Wavelet (‘continued)

Basis of Fourier Analysis –unlimited duration sine waves

Smooth, predictable

Basis of Wavelet Analysis – limited duration wavelets

Irregular, asymmetric

• Fourier Analysis is breaking up of signal into sine waves of varying frequencies

• Wavelet Analysis is breaking up of signal into shifted and scaled version of mother wavelet

13Wavelet Transform

Continuous Fourier Transform (CFT)

( ) ( ) j tF f t e dt

14Wavelet Transform

Scaling a Sinusoid( ) sin

tf t

a

Time

Am

plit

ud

e

15Wavelet Transform

Scaling a Wavelet( )

tf t

a

Am

plit

ud

e

Time

16Wavelet Transform

Shifting a Wavelet

17Wavelet Transform

Continuous Wavelet Transform (CWT)

,( , ) ( ) ( )sW s f t t dt

,

1( )s

tt

ss

Wavelet Function

Scaling

Shifting

,2

0

( )1( ) ( , ) s tf t W s d ds

C s

CWT

Inverse CWT

2| ( ) |

| |

uC du

u

18Wavelet Transform

Computing Wavelet Transform

1

2

3

19Wavelet Transform

Wavelet Spectrum

20Wavelet Transform

Scale and Frequency

21Wavelet Transform

Discrete Wavelet Transform (DWT)DWT

Inverse DWT

,

1( , ) ( ) ( )

oo j kt

W j k f t tM

,

1( , ) ( ) ( )j k

t

W j k f t tM

Scaling Coefficients Wavelet Coefficients

, ,

1 1( ) ( , ) ( ) ( , ) ( )

o

o

o j k j kk j j k

f t W j k t W j k tM M

Scaling Function Wavelet Function

22Wavelet Transform

Some Scaling and Wavelet Functions

23Wavelet Transform

Two Dimensional Discrete Wavelet Transform

( , ) ( ) ( )x y x y

Separable Scaling function

( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

H

V

D

x y x y

x y x y

x y x y

Separable Wavelets

/ 2, , ( , ) 2 (2 ,2 )j j jj m n x y x m y n

Scaled and Translated Basis Functions

/ 2, , ( , ) 2 (2 ,2 ) { , , }i j i j jj m n x y x m y n i H V D

24Wavelet Transform

Two Dimensional Discrete Wavelet Transform(‘continued)

1 1

, ,0 0

1( , , ) ( , ) ( , )

o

M N

o j m nx y

W j m n f x y x yMN

2-D DWT

Scaling Coefficients

1 1

, ,0 0

1( , , ) ( , ) ( , ) { , , }

M Ni i

j m nx y

W j m n f x y x y i H V DMN

Wavelet Coefficients

Inverse 2-D DWT

, ,

, ,

1( , ) ( , , ) ( , )

1( , , ) ( , )

o

o

o j m nm n

i ij m n

i j j m n

f x y W j m n x yMN

W j m n x yMN

25Wavelet Transform

Some Applications of Wavelet Transform

• De-noising• Compression• Detection of Discontinuities• System Identification• Video Compression (MPEG-4)• Speech Recognition• Face Recognition

26Wavelet Transform

De-noising Application of Wavelet Transform

Noisy Image

De-Noised Image

De-Noised Image

Information removed in De-Noising Process

De-Noised Image

27Wavelet Transform

References

• Rafael C. Gonzalez, Richard E. Woods, Digital Image Processing, 2nd Edition, Prentice Hall, 2002

• Robi Polikar, The Wavelet Tutorial http://engineering.rowan.edu/~polikar/WAVELETS/WTtutorial.html

• Jan T. Bialasiewicz, Introduction to Wavelet Transform & its Applications, http://pelincec.isep.pw.edu.pl

• Wavelet Transform (WT) and JPEG 2000

http://missouri.edu/~llxm7/CS4670-7670/ Slides/Ch8%20Wavelet %20Transform%20for%20Image%20Coding.ppt

• Wavelet Toolbox – Matlab 7 Help