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Wavelets Examples

Wavelets Examples. 王隆仁. Contents. Introduction Haar Wavelets General Order B-Spline Wavelets Linear B-Spline Wavelets Quadratic B-Spline Wavelets Cubic B-Spline Wavelets Daubechies Wavelets. I. Introduction. Wavelets are basis functions in continuous time.

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Wavelets Examples

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  1. WaveletsExamples 王隆仁

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

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

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

  5. (3) • 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,

  6. II. Haar Wavelets 2.1 Scaling functions • 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. • The two-scale relation for Haar scaling function is

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

  9. Fig. 5: Haar Wavelet  (x) .

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

  11. (4) (5) III. General Order B-Spline Wavelets 3.1 Scaling functions • The m-th order B-Splines Nm is defined by Note that the 1st order B-Spline N1(x) is the Haar scaling function.

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

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

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

  15. IV. Linear B-Spline Wavelets 4.1 Scaling functions • Linear B-Spline N2(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)

  16. Then the functions in V1 subspace are expressed explicitly as follows and is shown in Fig.7 . (7)

  17. Fig. 6: Linear B-Spline N2(x) .

  18. Fig. 7: Linear B-Spline N2(2x-k) .

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

  20. Fig. 8: Two-scale relation for N2 .

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

  22. The term N4(k) is cubic B-spline and the recursion relation for general order B-spline is given by This relation is useful to compute Nm(k)at some integer values. Non-zero values of Nm(k)for some small m are summarized in Table 1.

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

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

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

  26. 4.3 Decomposition relation • The decomposition sequences {ak} and {bk} are written for Linear B-Spline (m=2) as Noting that only three {pk} and five {qk} are non-zero, i.e., and

  27. V. Quadratic B-Spline Wavelets 5.1 Scaling functions • Quadratic B-spline N3(x) is shown in Fig.10 and given by

  28. Fig. 10: Quadratic B-Spline N3(x) .

  29. Functions in V1 space are expressed as

  30. The two-scale relation for quadratic B-Spline N3(x) is shown in Fig.11 and given as follow:

  31. Fig. 11: Two-scale relation for N3(x) .

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

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

  34. 5.3 Decomposition relation • The decomposition sequences {ak} and {bk} are written for Quadratic B-Spline (m=3) as Noting that only four {pk} and eight {qk} are non-zero, i.e., and

  35. VI. Cubic B-Spline Wavelets 6.1 Scaling functions • Cubic B-spline N4(x) shown in Fig.13 is given by

  36. Fig. 13: Cubic B-Spline N4(x) .

  37. The two-scale relation for cubic B-Spline N4(x) is and is shown in Fig.14.

  38. Fig. 14: Two-scale relation for N4(x) .

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

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

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

  42. That is, non-zero values of the two-scale sequence {pk} are : Note that the coefficients {pk} have properties p0 +p2 =p1 + p3 = 1 . Figure 16 and 17 show the Daubechies scaling functions, N is the length of the coefficients.

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

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

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

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

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

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

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