1. A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001
Daubechies Wavelets
A first look
Ref: Walker (Ch.2)
Jyun-Ming Chen, Spring 2001

2. Introduction
A family of wavelet transforms discovered by Ingrid Daubechies
Concepts similar to Haar (trend and fluctuation)
Differs in how scaling functions and wavelets are defined
longer supports
Wavelets are building blocks that can quickly decorrelate data.

3. Haar Wavelets Revisited
The elements in the synthesis and analysis matrices are

4. Haar Revisited
Synthesis
Filter P3
Synthesis
Filter Q3

5. In Other Words

6. How we got the numbers Orthonormal; also lead to energy conservation
Averaging
Orthogonality
Differencing

7. How we got the numbers (cont)
See Haar.ppt (p.30)

8. Daubechies Wavelets How they look like: Translated copy dilation
Scaling functions
Wavelets

9. Daub4 Scaling Functions (n-1 level)
Obtained from natural basis
(n-1) level Scaling functions
wrap around at end due to periodicity
Each (n-1) level function
Support: 4
Translation: 2
Trend: average of 4 values
Discrete sampling of the fractal curve; I.e., if you give enough
Samples, you’ll see the complete “fractal” shape.

10. Daub4 Scaling Function (n-2 level)
Obtained from n-1 level scaling functions
Each (n-2) scaling function
Support: 10
Translation: 4
Trend: average of 10 values
This extends to lower levels

11. Daub4 Wavelets Similar “wrap-around” Obtained from natural basis
Support/translation:
Same as scaling functions
Extends to lower-levels

12. Numbers of Scaling Function and Wavelets (Daub4)

13. Property of Daub4
If a signal f is (approximately) linear over the support of a Daub4 wavelet, then the corresponding fluctuation value is (approximately) zero.
True for functions that have a continuous 2nd derivative
2nd derivative: curvature of f(x)

14. Property of Daub4 (cont)

15. MRA

16. Example (Daub4)

17. More on Scaling Functions (Daub4, N=8)
Synthesis
Filter P3

18. Scaling Function (Daub4, N=16)
Synthesis
Filter P3

19. Scaling Functions (Daub4)
Synthesis
Filter P2
Synthesis
Filter P1

20. More on Wavelets (Daub4)
Synthesis
Filter Q3
Synthesis
Filter Q2
Synthesis
Filter Q1

21. Summary Daub4 (N=32) j=5 j=4 j=3 j=2 In general N=2n support 1 4 10 22?
translation 2 8

22. Analysis and Synthesis
There is another set of matrices that are related to the computation of analysis/decomposition coefficient
In the Daubechies case, they are also the transpose of each other
Later we’ll show that this is a property unique to orthogonal wavelets

23. Analysis and Synthesisf

24. MRA (Daub4)

25. Energy Compaction (Haar vs. Daub4)

26. How we got the numbers Orthonormal; also lead to energy conservation
Orthogonality
Averaging
Differencing
Constant
Linear
4 unknowns; 4 eqns

27. Supplemental
Here we re-interpret “difference” as “constant correlation”.
We design D4 to capture both constant and linear correlation.

28. Conservation of Energy
Define
Therefore (Exercise: verify)

29. Energy Conservation
By definition:

30. Orthogonal Wavelets By construction Haar is also orthogonal
Not all wavelets are orthogonal!
Semiorthogonal, Biorthogonal

31. Other Wavelets (Daub6)
Wavelets are defined similarly.

32. Daub6 (cont) Constraints
If a signal f is (approximately) quadratic over the support of a Daub6 wavelet, then the corresponding fluctuation value is (approximately) zero.

33. DaubJ
Constraints
If a signal f is (approximately) equal to a polynomial of degree less than J/2 over the support of a DaubJ wavelet, then the corresponding fluctuation value is (approximately) zero.

34. Comparison (Daub20)

35. Supplemental on Daubechies Wavelets

37. Coiflets
Designed for maintaining a close match between the trend value and the original signal
Named after the inventor: R. R. Coifman

38. Ex: Coif6