Haar Wavelet Analysis

1. Haar Wavelet Analysis 吳育德 陽明大學放射醫學科學研究所 台北榮總整合性腦功能實驗室

2. A First Course in Wavelets with Fourier Analysis Albert Boggess Francis J. Narcowich Prentice-Hall, Inc., 2001

3. Outlines • Why Wavelet • Haar Wavelets • The Haar Scaling Function • Basic Properties of the Haar Scaling Function • The Haar Wavelet • Haar Decomposition and Reconstruction Algorithms • Decomposition • Reconstruction • Filters and Diagrams • Summary

4. 4.1 Why Wavelet • Wavelets were first applied in geophysics to analyze data from seismic surveys. • Seismic survey geophones seismic trace Sesimic trace Direct wave (along the surface) Subsequent waves (rock layers below ground)

5. Fourier Transform (FT) is not a good tool – gives no direct information about when an oscillation occurred. • Short-time FT : equal time interval, high- frequency bursts occur are hard to detect. • Wavelets can keep track of time and frequency information. They can be used to “zoom in” on the short bursts, or to “zoom out” to detect long, slow oscillations

6. frequency frequency + time (equal time intervals) frequency + time

7. 4.2 Haar Wavelets 4.2.1 The Haar Scaling Function • Wavelet functions • Scaling function Φ (father wavelet) • Wavelet Ψ (mother wavelet) • These two functions generate a family of functions that can be used to break up or reconstruct a signal • The Haar Scaling Function • Translation • Dilation

8. Using Haar blocks to approximate a signal • High-frequency noise shows up as tall, thin blocks. • Needs an algorithm that eliminates the noise and not distribute the rest of the signal. • Disadvantages of Harr wavelet: discontinuous and does not approximate continuous signals very well. Figure 2

9. Chap 6 Dubieties 3 Daubechies 4 Daubechies 8

10. 4.2.2 Basic Properties of the Haar Scaling Function • The Haar Scaling function is defined as • Φ(x-k) : same graph but translated by to the right (if k>0) by k units • Let V0 be the space of all functions of the form

11. V0 consists of all piecewise constant functions whose discontinuities are contained in the set of integers • V0 has compact support. Typical element in V0 Figure 5 Figure 6 has discontinuitiesat x=0,1,3, and 4

12. Let V1 be the space of piecewise constant functions of finite support with discontinuities at the half integers has discontinuities at x=0,1/2,3/2, and 2

13. Suppose j is any nonnegative integer. The space of step functions at level j, denoted by Vj , , is defined to be the space spanned by the set over the real numbers. • Vj is the space of piecewise constant functions of finite support whose discontinuities are contained in the set • means no information is lost as the resolution gets finer. Vj contains all relevant information up to a resolution scale order 2-j

14. A function f(x) belongs to V0 iff f(2jx) belongs to Vj

15. A function f(x) belongs to Vj iff f(2-jx) belongs to V0

16. How to decompose a signal into its Vj-components • When j is large, the graph of Φ(2j x) is similar to one of the spikes of a signal that we may wish to filter out. • One way is to construct an orthonormal basis for Vj using the L2 inner product

17. Theorem:

18. 4.2.4 The Haar Wavelet • We want to isolate the spikes that belong to Vj, but that are not members of Vj-1 • The way is to decompose Vj as an orthonormal sum of Vj-1 and its complement. • Start with V1, assume the orthonormal complement of Vo is generated by translates of some functions Ψ, we need:

19. Harr wavelet

21. Theorem 4.8 (extend to Vj)

23. Decomposing Vj

24. Theorem:

25. 4.3 Haar Decomposition and Reconstruction Algorithms

26. Implementation • Step 1 : Approximate the original signal f by a step function of the form

28. Example 4.11

29. General decomposition scheme Wj-1-component Vj-1-component

30. Theorem 4.12 (Haar Decomposition)

31. Example 4.13 V8-component V7-component V6-component V4-component W7-component

32. 4.3.2 Reconstruction

33. General reconstruction scheme

34. General reconstruction scheme

37. Theorem 4.14 (Haar Reconstruction)

38. Example 4.15 sample signal 80% compression 90% compression

39. 4.3.3 Filters and Diagrams • Decomposition algorithm • k=-1,0 • k=-1,0

40. downsampling operator

41. Reconstruction • k=0,1 • k=0,1

43. upsampling operator

44. Summary