Multiresolution Analysis (MRA): Scaling Functions and Wavelets for Signal Processing

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

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

3. J=5 Antall samplinger: 2J = 32 Analysis /SynthesisExample

4. AnalysisSynthesisJ=5 Sampling: 25 = 32 j=5 j=4 j=3 j=2 j=1 j=0

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

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

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

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

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

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

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

12. 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

13. 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

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

15. 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

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

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

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

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

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

30. Analysis - From Fine Scale to Coarse Scale j=5 j=4

31. Analysis - From Fine Scale to Coarse Scale Synthesis - From Coarse Scale to Fine Scale Analysis Synthesis

32. Dirac Delta Function (Standard Time Domain Basis) f t

33. Fourier (Standard Frequency Domain Basis) f t

34. Two-band Wavelet Basis f t

35. J=5 Antall samplinger: 2J = 32 Analysis /SynthesisExample

36. AnalysisSynthesisJ=5 Sampling: 25 = 32 j=5 j=4 j=3 j=2 j=1 j=0

37. END