1. 1 Wavelets Examples 王隆仁

2. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets o Cubic B-Spline Wavelets o Daubechies Wavelets

3. 3 I. Introduction o Wavelets are basis functions in continuous time. o A basis is a set of linearly independent functions that can be used to produce all admissible functions : o The special feature of the wavelet basis is that all functions are constructed from a single mother wavelet. (1)

4. 4 o A typical wavelet is compressed times and shifted times. Its formula is o The remarkable property that is achieved by many wavelets is orthogonality. The wavelets are orthogonal when their “inner products” are zero : o Orthogonality leads to a simple formula for each coefficient in the expansion for. (2)

5. 5 o Multiply the expansion displayed in equation (1) by and integrate : All other terms in the sum disappear because of orthogonality. Equation (2) eliminates all integrals of times, except the one term that has j=J and k=K. That term produces. Then is the ratio of the two integrals in equation (3). That is, (3)

6. 6 II. Haar Wavelets 2.1 Scaling functions o Haar scaling function is defined by and is shown in Fig. 1. Some examples of its translated and scaled versions are shown in Fig. 2-4. o The two-scale relation for Haar scaling function is

7. 7 Fig.1: Haar scaling function  (x).Fig.2: Haar scaling function  (x-1). Fig.3: Haar scaling function  (2x).Fig.4: Haar scaling function  (2x-1).

8. 8 2.2 Wavelets o The Haar wavelet  (x) is given by and is shown in Fig. 5. o The two-scale relation for Haar wavelet is

9. 9 Fig. 5: Haar Wavelet  (x).

10. 10 2.3 Decomposition relation o Both of the two-scale relation together are called the reconstruction relation. o The decomposition relation can be derived as follows.

11. 11 3.1 Scaling functions o The m-th order B-Splines N m is defined by Note that the 1st order B-Spline N 1 (x) is the Haar scaling function. (4) (5) III. General Order B-Spline Wavelets

12. 12 o The two-scale relation for B-spline scaling functions of general order m is where the two-scale sequence {p k } for B-spline scaling functions are given by :

13. 13 3.2 Wavelets o The two-scale relation for B-spline wavelets for general order m is given by where

14. 14 3.3 Decomposition relation o The decomposition relation for m-th order B- Spline is where

15. 15 4.1 Scaling functions o Linear B-Spline N 2 (x) is derived from the recurrence (4) and (5) as the case m=2 for general B-Splines as follows and is shown in Fig.6. (6) IV. Linear B-Spline Wavelets

16. 16 o Then the functions in V 1 subspace are expressed explicitly as follows and is shown in Fig.7. (7)

17. 17 Fig. 6: Linear B-Spline N 2 (x).

18. 18 Fig. 7: Linear B-Spline N 2 (2x-k).

19. 19 o Since the support of is [0, 2], its two-scale relation is in the form o By substituting the expressions (6) and (7) for each 1/2 interval between [0, 2] into (8), the coefficients {p k } are obtained and the two scale relation for Linear B-Spline is shown in Fig.8 and is given by (8)

20. 20 Fig. 8: Two-scale relation for N 2.

21. 21 4.2 Wavelets o The two-scale relation for Linear B-Spline wavelets for general order m=2 is where

22. 22 o The term N 4 (k) is cubic B-spline and the recursion relation for general order B-spline is given by This relation is useful to compute N m (k) at some integer values. Non-zero values of N m (k) for some small m are summarized in Table 1.

23. 23 Table 1: Non-zero N m (k) values for m = 2,…, 6.

24. 24 o Then the two-scale sequence {q k } for is computed as follows: o Thus the Linear B-Spline wavelets is

25. 25 Fig. 9: Linear B-Spline wavelet.

26. 26 4.3 Decomposition relation o The decomposition sequences {a k } and {b k } are written for Linear B-Spline (m=2) as Noting that only three {p k } and five {q k } are non-zero, i.e., and

27. 27 5.1 Scaling functions o Quadratic B-spline N 3 (x) is shown in Fig.10 and given by V. Quadratic B-Spline Wavelets

28. 28 Fig. 10: Quadratic B-Spline N 3 (x).

29. 29 o Functions in V 1 space are expressed as

30. 30 o The two-scale relation for quadratic B-Spline N 3 (x) is shown in Fig.11 and given as follow:

31. 31 Fig. 11: Two-scale relation for N 3 (x).

32. 32 5.2 Wavelets o The quadratic B-spline wavelet is shown in Fig.12 and the two-scale relation is given by

33. 33 Fig. 12: Quadratic B-Spline wavelet.

34. 34 5.3 Decomposition relation o The decomposition sequences {a k } and {b k } are written for Quadratic B-Spline (m=3) as Noting that only four {p k } and eight {q k } are non-zero, i.e., and

35. 35 6.1 Scaling functions o Cubic B-spline N 4 (x) shown in Fig.13 is given by VI. Cubic B-Spline Wavelets

36. 36 Fig. 13: Cubic B-Spline N 4 (x).

37. 37 o The two-scale relation for cubic B-Spline N 4 (x) is and is shown in Fig.14.

38. 38 Fig. 14: Two-scale relation for N 4 (x).

39. 39 6.2 Wavelets o The Cubic B-Spline wavelet is shown in Fig.15. 6.3 Decomposition relation o The decomposition sequences for Cubic B- Spline are :

40. 40 Fig. 15: Cubic B-Spline wavelet.

41. 41 7.1 Scaling functions o Daubechies scaling function is defined by the following two-scale relation : VII. Daubechies Wavelets

42. 42 o That is, non-zero values of the two-scale sequence {p k } are : Note that the coefficients {p k } have properties p 0 + p 2 = p 1 + p 3 = 1. Figure 16 and 17 show the Daubechies scaling functions, N is the length of the coefficients.

43. 43 Fig. 16: Daubechies Scaling Functions, N=4,6,8,10.

44. 44 Fig. 17: Daubechies Scaling Functions, N=12,16,20,40.

45. 45 7.2 Wavelets o The two-scale relation for the Daubechies wavelets is in the following form :

46. 46 o Therefore the non-zero values of the two-scale sequence {q k } are : Figure 18 and 19 show the Daubechies wavelets, N is the length of the coefficients.

47. 47 Fig. 18: Daubechies Wavelets, N=4,6,8,10.

48. 48 Fig. 19: Daubechies Wavelets, N=12,16,20,40.