Chapter 03 Multiresolution Analysis (MRA)

1. Chapter 03Multiresolution Analysis (MRA) V0  V1  V2

2. Multiresolution Gjennomsnitt V0 V1 V2 V3 V4 Differens W0 W1 W2 W3

3. Multiresolution analysis (MRA) The multiresolution analysis (MRA) of L2(R) is an increasing sequence of closed, nested subspaces (linear vector space) {Vj}j  Z such that:

4. Scaling functionExample 1 1 1 2 1 3 n n+1

5. Scaling function that span V0 Scaling function V0  L2(R)

6. Scaling Function that span V0Example 1 1 2 3 4 5 5 1 1 2 3 4 5

7. Scaling Function (unnormalized) that span Vj 1 1 Dilation Translation 1 1 1 1 1 1 1 1 1 1 1 1

8. Scaling Function (normalized) that span Vj 1 1 Dilation Translation 2 1/2 2 1/2 2 2 2 2

9. Scaling functions (normalized) Scaling function V0  V1  V2

10. Normalization of scaling functions Scaling function Inner product Norm Scaling functions (Orthonormal)

11. Haar Scaling Functions (unnormalized) that span Vj k 0 1 2 3 j 0 1 1 For hver j: Basisfunksjoner: (2jt-k) k = 0,…2 j-1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1

12. Haar Scaling Functions (normalized) that span Vj k 0 1 2 3 j 0 1 1 For hver j: Basisfunksjoner: j,k(t) k = 0,…2 j-1 2 1/2 2 1/2 1 1 1 2 2 2 2 2 1 1 1 1 2 3/2 2 3/2 2 3/2 2 3/2 3 1 1 1 1

13. V1  V2 Scaling Function that span V1 and V2 1 1

14. Haar Scaling Functions that span Vj j = 0,1,2,3 j 0 1 2 3 k 0 1 2 3 4 5 6 7

15. V0  V1 Relation between V0 and V1Haar Wavelet - Triangle Wavelet Scaling function

16. Properties of the h-coefficients (1/5) V0  V1

17. Properties of the h-coefficients (2/5) V0  V1

18. Properties of the h-coefficients (3/5)

19. Properties of the h-coefficients (4/5)

20. Properties of the h-coefficients (5/5)

21. Examples of h-coefficients n=2 D2 Haar scaling function n=3 n odd --> one coefficient = 0 n=4 D4 Daubechies four-tap solution One degree of freedom

22. Examples of h-coefficients n=4 One degree of freedom D2 D4

23. Examples of h-coefficients n=6

24. Examples of h-coefficients D6 D8

25. DaubechiesVanishing moments The continuous wavelet transform (CWT) Taylor series at t=0 until order n (b=0 for simplicity) Moments of the Wavelet

26. DaubechiesVanishing moments Wavelet until Daubechies: - Haar Compact support, but discontinuous - Shannon Smooth, but extend the whole real line - Linear spline Continuous, but infinite support Daubechies: Hierarchy of Wavelets: n = 2 : Haar Compact support, discontinuous M0 = 0 n = 4 : D4 Compact support, continuous, not diff. Mi = 0 i=0..n/2-1 n = 6 : D6 Compact support, continuous, 1 diff. Mi = 0 i=0..n/2-1 n = 8 : D8 Compact support, continuous, 2 diff. Mi = 0 i=0..n/2-1 ...

27. Wavelet functions Scaling function V0  V1  V2 W0 W1 Wavelet function

28. Properties of the g-coefficients Scaling function V0  V1  V2 W0 Wavelet function W1

29. Relation between V0 and V1Haar Wavelet and Triangle Wavelet Wavelet function

30. Constructing the Scaling FunctionIterative Procedure

31. Constructing Scaling Function / Wavelet FunctionLength-4 Scaling Coefficient Vector

32. Constructing Scaling Function / Wavelet FunctionScaling Function / Wavelet Function

33. Constructing Scaling Function / Wavelet FunctionD4 Scaling Function / D4 Wavelet Function

34. Series of Scaling and Wavelet functions V0  V1  V2 W0 W1

35. Series of Scaling and Wavelet functions J j0

36. Series of Scaling and Wavelet functions

37. Display of the Discrete Wavelet Transform and the Wavelet Expansion 1. The signal itself (or samples of the signal) in the time-domain. No frequency or scale information. 2. Plot of the expansion coefficients or DWT values. Cj,k and dj,k over the j,k plane. 3. Generating time functions fj(t) at each scale by summing over k. Illustrates the components of the signal at each scale. 4. Generating time localization of the wavelet expansion by time functions fk(t) at each translation by summing over j. Illustrates the components of the signal at each integer translation. 5. Tiling the time-frequency plane. Based on a partitioning of the time-scale plane as if the time translation index and scale were continuous variables.

38. Display - Time functions at each scale 3. Generating time functions fj(t) at each scale by summing over k. Illustrates the components of the signal at each scale. Vj-1  Vj  Vj+1 Wj+1 Wj

39. Display - Time functions at each translation 4. Generating time localization of the wavelet expansion by time functions fk(t) at each translation by summing over j. Illustrates the components of the signal at each integer translation. Vj-1  Vj  Vj+1 Wj+1 Wj

40. Display - Time functions at each scale/translation j j t t

41. Haar Scaling Functions that span Vj j = 0,1,2,3 j 0 1 2 3 k 0 1 2 3 4 5 6 7

42. Decomposition of V1 = V0 + W0 j 0 1 2 3 =

43. Decomposition of V2 = V1 + W1 j 0 1 2 3 =

44. Decomposition of V3 = V2 + W2 j 0 1 2 3 =

45. Decomposition of V3 = V0 + W0 + W1 + W2 j 0 1 2 3 =

46. Haar Scaling Functions that span V2 - Example 5 5 1 1 2 x 3 x 6 x 1 x 2 x - 2 x 2 x 0 x

47. Haar Scaling Functions that span V3 - Example 5 5 1 1 4 x 4 x ….. 4 x 5 x 2 x 3 x 3 x 1 x

48. Decomposition of V3 = V0 + W0 + W1 + W2

49. Analysis - From Fine Scale to Coarse Scale

50. Analysis - From Fine Scale to Coarse Scale