1. Wavelet Transform Fourier Transform Wavelet Transform
Gives only frequency component
No time information
We don’t know where the frequency change during the time
Good for stationary signal (no sudden change in frequency)
Wavelet Transform
Gives both frequency component + time
We know where the frequency change in time

2. Fourier Transform Weakness

3. Introduction to Wavelet Transform (1)
Objectives
Gives information about
what frequencies contain in the signal (frequency component)
Where the frequency change in time (time information)
Allow us to detect features that might go underdetected at one resolution but may be easy to spot at another
Basis function
Short-period wave (wavelet)
Mother wavelet
Scale and Translation wavelet
Example of Wavelet Application
Music example
Extract the frequency of notes and when to play that note

4. Time-Frequency Division

5. Introduction to Wavelet Transform (2)
is similar to Fourier Transform
Fourier Transform

6. Introduction to Wavelet Transform (3)
Example of Wavelet Basis Function
Mother wavelet
Scale & Translated wavelet

7. Introduction to Wavelet Transform (4)
Ex. Haar Wavelet
Mother wavelet
Scaled and Translated Wavelet

8. Forward Wavelet Transform
Scaling function (Approximation function)
Wavelet function (Detail function)

9. DWT Example
f (x) = [ ]
j0 = 0
Scaling function

10. DWT Example
j = j0 = 0; k = 0
Wavelet function

11. DWT Example
j = 1; k = 0
j = 1; k = 1
Wavelet function

12. Wavelet Coefficients
Scaling function
Wavelet function

13. Inverse DWT
Scaling function

14. Inverse DWT
Wavelet function

15. Generation of New Mother wavelet from simple rectangular pulse
Using two-scale relation
Mother wavelet
New Mother Wavelet

16. Wavelet (Proof for fast wavelet)
Mother wavelet
Wavelet

17. Wavelet Coefficients
n=2k, k>=0

18. Wavelet Blockdiagram 2

19. Approximation and Detail Wavelet2

20. Multi-resolution Wavelet Implementation

21. 2D Wavelet (Separable)
Row Wavelet
Column Wavelet
(LL)
(LH)
(HL)
(HH)

22. Example of Wavelet CoefficientsLL LH HL HH

23. END