Wavelets (Chapter 7)

1. Wavelets (Chapter 7) CS474/674 – Prof. Bebis

2. STFT - revisited • Time - Frequency localization depends on window size. • Wide window good frequency localization, poor time localization. • Narrow window good time localization, poor frequency localization.

3. Wavelet Transform • Uses a variable length window, e.g.: • Narrower windows are more appropriate at high frequencies • Wider windows are more appropriate at low frequencies

4. What is a wavelet? Sinusoid Wavelet • A function that “waves” above and below the x-axis with the following properties: • Varying frequency • Limited duration • Zero average value • This is in contrast to sinusoids, used by FT, which have infinite duration and constant frequency.

5. Types of Wavelets Morlet Daubechies Haar There are many different wavelets, for example:

6. Basis Functions Using Wavelets Like sin( ) and cos( ) functions in the Fourier Transform, wavelets can define a set of basis functions ψk(t): Spanof ψk(t): vector space S containing all functions f(t) that can be represented by ψk(t).

7. Basis Construction – “Mother” Wavelet The basis can be constructed by applying translations and scalings (stretch/compress) on the “mother” wavelet ψ(t): Example: scale ψ(t) translate

8. Basis Construction - Mother Wavelet (dyadic/octave grid) scale =1/2j (1/frequency) j k

9. Continuous Wavelet Transform (CWT) scale =1/2j (1/frequency) scale parameter (measure of frequency) translation parameter (measure of time) Forward CWT: mother wavelet (i.e., window function) normalization constant

10. Illustrating CWT • Take a wavelet and compare it to a section at the start of the original signal. • Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal. The higher C is, the more the similarity.

11. Illustrating CWT (cont’d) 3. Shift the wavelet to the right and repeat step 2 until you've covered the whole signal.

12. Illustrating CWT (cont’d) 4. Scale the wavelet and go to step 1. 5. Repeat steps 1 through 4 for all scales.

13. Visualize CTW Transform • Wavelet analysis produces a time-scale viewof the input signal or image.

14. Continuous Wavelet Transform (cont’d) Forward CWT: Inverse CWT: Note the double integral!

15. Fourier Transform vs Wavelet Transform weighted by F(u)

16. Fourier Transform vs Wavelet Transform weighted by C(τ,s)

17. Properties of Wavelets • Simultaneous localization in time and scale - The location of the wavelet allowsto explicitly represent the location of events in time. - The shape of the wavelet allowsto represent different detail or resolution.

18. Properties of Wavelets (cont’d) • Sparsity: for functions typically found in practice, many of the coefficients in a wavelet representation are either zero or very small.

19. Properties of Wavelets (cont’d) • Adaptability: Can represent functions with discontinuities or corners more efficiently. • Linear-time complexity: many wavelet transformations can be accomplished in O(N) time.

20. Discrete Wavelet Transform (DWT) (forward DWT) (inverse DWT) where

21. DFT vs DWT one parameter basis or two parameter basis DFT expansion: DWT expansion

22. Multiresolution Representation Using Wavelets fine details wider, large translations j coarse details

23. Multiresolution Representation Using Wavelets fine details j coarse details

24. Multiresolution Representation Using Wavelets fine details narrower, small translations j coarse details

25. Multiresolution Representation Using Wavelets high resolution (more details) j … low resolution (less details)

26. Pyramidal Coding - Revisited Approximation Pyramid (with sub-sampling)

27. Pyramidal Coding - Revisited (details) Prediction Residual Pyramid (details) (with sub-sampling) reconstruct Approximation Pyramid

28. Efficient Representation Using “Details” details D3 details D2 details D1 L0 (without sub-sampling)

29. Efficient Representation Using Details (cont’d) representation:L0 D1 D2 D3 in general: L0 D1 D2 D3…DJ • A wavelet representation of a function consists of • a coarse overall approximation • detail coefficients that influence the function at various scales

30. Reconstruction (synthesis) H3=H2 & D3 details D3 H2=H1 & D2 details D2 H1=L0 & D1 details D1 L0 (without sub-sampling)

31. Example - Haar Wavelets (with sub-sampling) L0 D1 D2 D3 • Suppose we are given a 1D "image" with a resolution of 4 pixels: [9 7 3 5] • The Haarwavelet transform is the following:

32. Example - Haar Wavelets (cont’d) [9 7 3 5] 1 • Start by averaging and subsampling the pixels together (pairwise) to get a new lower resolution image: • To recover the original four pixels from the two averaged pixels, store some detail coefficients.

33. Example - Haar Wavelets (cont’d) 1 Harr decomposition: • Repeating this process on the averages (i.e., low resolution image) gives the full decomposition:

34. Example - Haar Wavelets (cont’d) L0 D1 D2 D3 1 -1 2 [6] • The original image can be reconstructed by adding or subtracting the detail coefficients from the lower-resolution representations.

35. Example - Haar Wavelets (cont’d) Detail coefficients become smaller and smaller scale decreases. Dj How should we compute the detail coefficients Dj ? Dj-1 D1 L0

36. Multiresolution Conditions • If a set of functions V can be represented by a weighted sum of ψ(2jt - k), then a larger set, including V, can be represented by a weighted sum of ψ(2j+1t - k). high resolution ψ(2j+1t - k) j ψ(2jt - k) low resolution

37. Multiresolution Conditions (cont’d) Vj+1: span of ψ(2j+1t - k): Vj: span of ψ(2jt - k):

38. Multiresolution Conditions (cont’d) Nested Spaces ψ(t - k) V0 j=0 j=1 j ψ(2t - k) V1 … Vj ψ(2jt - k) if f(t) ϵV jthen f(t) ϵ V j+1

39. How to compute Dj ? IDEA: Define a set of basis functions that span the difference between Vj+1 and Vj

40. How to compute Dj? (cont’d) • Let Wj be the orthogonal complement of Vj in Vj+1 Vj+1 = Vj + Wj

41. How to compute Dj ? (cont’d) If f(t) ϵVj+1, then f(t) can be represented using basis functions φ(t) from Vj+1: Vj+1 Alternatively, f(t) can be represented using two sets of basis functions, φ(t) from Vj and ψ(t) fromWj: Vj+1 = Vj + Wj

42. How to compute Dj ? (cont’d) Think of Wj as a means to represent the parts of a function in Vj+1 that cannot be represented in Vj Vj+1 differences between Vj and Vj+1 Vj Wj

43. How to compute Dj ? (cont’d) •  using recursion on Vj: Vj+1 = Vj + Wj Vj+1 = Vj-1+Wj-1+Wj = …= V0 + W0 + W1 + W2 + … + Wj if f(t) ϵ Vj+1 , then: V0 W0, W1, W2, … basis functions basis functions

44. Summary: wavelet expansion (Section 7.2) • Wavelet decompositions involve a pair of waveforms: encodes low resolution info encodes details or high resolution info φ(t) ψ(t) Terminology: scaling function wavelet function

45. 1D Haar Wavelets • Haarscaling and wavelet functions: φ(t) ψ(t) computes details (high pass) computes average (low pass)

46. 1D Haar Wavelets (cont’d) Let’s consider the spaces corresponding to different resolution 1D images: . .. …. V0 (j=0) 1-pixel (j=1) V1 2-pixel (j=2) V2 4-pixel etc.

47. 1D Haar Wavelets (cont’d) j=0 • V0 representsthe space of 1-pixel (20-pixel) images • Think of a 1-pixel image as a function that is constant over [0,1) Example: width: 1 0 1

48. 1D Haar Wavelets (cont’d) j=1 • V1represents the spaceof all 2-pixel(21-pixel) images • Think of a 2-pixel image as a function having 21 equal-sized constant pieces over the interval [0, 1). 0 ½ 1 width: 1/2 Example: Note that: e.g., + =

49. 1D Haar Wavelets (cont’d) • V j represents all the 2j-pixel images • Functions having 2jequal-sized constant pieces over interval [0,1). Vj-1 Example: width: 1/2j Note that: V1 ϵ Vj ϵ Vj width: 1/2j-1 width: 1/2

50. 1D Haar Wavelets (cont’d) V0, V1, ..., Vjare nested i.e., Vj … V1 V0 fine details coarse info