INTRODUCTION TO. Adapted from CS474/674 Prof. George Bebis Department of Computer Science & Engineering University of Nevada (UNR)

Transcription

1 INTRODUCTION TO WAVELETS Adapted from CS474/674 Prof. George Bebis Department of Computer Science & Engineering University of Nevada (UNR)

2 CRITICISM OF FOURIER SPECTRUM It gives us the spectrum of the whole time-series Which is OK if the time-series is stationary But what if its not? We need a technique that can march along a time series and that is capable of: Analyzing spectral content in different places Detecting sharp changes in spectral character

3 SHORT TIME FOURIER TRANSFORM STFT Time/Frequency localization depends on window size. Once you choose a particular window size, it will be the same for all frequencies. Many signals require a more flexible approach - vary the window size to determine more accurately either time or frequency.

4 THE WAVELET TRANSFORM Overcomes the preset resolution problem of the STFT by using a variable length window: Use narrower windows at high frequencies for better time resolution. Use wider windows at low frequencies for better frequency resolution.

5 The Wavelet Transform (cont d) Wide windows do not provide good localization at high frequencies.

6 The Wavelet Transform (cont d) Use narrower windows at high frequencies.

7 The Wavelet Transform (cont d) Narrow windows do not provide good localization at low frequencies.

8 The Wavelet Transform (cont d) Use wider windows at low frequencies.

9 The Wavelet Transform (cont d) STFT AND DWT BREAKDOWN OF A SIGNAL

10 WHAT ARE WAVELETS? Wavelets are functions that wave above and below the x-axis, have (1) varying frequency, (2) limited duration, and (3) an average value of zero. This is in contrast to sinusoids, used by FT, which have infinite energy. Sinusoid Wavelet

11 What are Wavelets? (cont d) Like sines and cosines in FT, wavelets are used as basis functions ψ k (t) in representing other functions f(t): f() t = akψ k() t k Span of ψ k (t): vector space S containing all functions f(t) that can be represented by ψ k (t).

12 What are Wavelets? (cont d) There are many different wavelets: Haar Morlet Daubechies

13 What are Wavelets? (cont d) = ψ jk (t)

14 What are Wavelets? (cont d) ψ ( j ) /2 () t = j jk 2 ψ 2 t k scale/frequency localization j time localization

15 MORE ABOUT WAVELETS normalization shift in time wavelet with scale, s and time, τ change in scale: big s means long wavelength Mother wavelet

16 SHANNON WAVELET Y(t) = 2 sinc(2t) sinc(t) mother wavelet τ=5, s=2 time

17 CONTINUOUS WAVELET TRANSFORM (CWT) translation parameter, measure of time scale parameter (measure of frequency) normalization constant 1 t C(,) τ s f ( t) dt s ψ τ = s t Forward CWT: Continuous Wavelet Transform of signal f(t) Mother wavelet (window) Scale = 1/j = 1/Frequency

18 CWT: MAIN STEPS 1. Take a wavelet and compare it to a section at the start of the original signal. 2. 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.

19 CWT: Main Steps (cont d) 3. Shift the wavelet to the right and repeat steps 1 and 2 until you've covered the whole signal.

20 CWT: Main Steps (cont d) 4. Scale the wavelet and repeat steps 1 through Repeat steps 1 through 4 for all scales.

21 COEFFICIENTS OF CTW TRANSFORM Wavelet analysis produces a time-scale view of the input signal or image. 1 t C(,) τ s f ( t) dt s ψ τ = s t

22 Continuous Wavelet Transform (cont d) Inverse CWT: 1 t τ f() t = C( τ, s) ψ( ) dτds s s τ s double integral!

23 FT VS WT weighted by F(u) weighted by C(τ,s)

24 PROPERTIES OF WAVELETS Simultaneous localization in time and scale - The location of the wavelet allows to explicitly represent the location of events in time. - The shape of the wavelet allows to represent different detail or resolution.

25 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. 1 t τ f() t = C( τ, s) ψ( ) dτds s s τ s Linear-time complexity: many wavelet transformations can be accomplished in O(N) time.

26 Properties of Wavelets (cont d) Adaptability: wavelets can be adapted to represent a wide variety of functions (e.g., functions with discontinuities, functions defined on bounded domains etc.). Well suited to problems involving images, open or closed curves, and surfaces of just about any variety. Can represent functions with discontinuities or corners more efficiently (i.e., some have sharp corners themselves).

27 Properties of Wavelets (cont d) Admissibility condition: Implies that Ψ(ω) 0 both as ω 0 and ω, so Ψ(ω) must be bandlimited

28 Fourier spectrum of Shannon Wavelet frequency, ω ω Spectrum of higher scale wavelets

29 DISCRETE WAVELET TRANSFORM (DWT) CWT computes all scales and positions in a given range DWT scales and positions are only computed in powers of 2 (dyadic scales) This subset can be shown to have the same accuracy as DWT Dyadic scales allow for tree decompositions

30 DISCRETE WAVELET TRANSFORM (DWT) ajk = f t jk t t * () ψ () (forward DWT) f() t a ψ () t = k j jk jk (inverse DWT) where ψ ( j ) /2 () t = j jk 2 ψ 2 t k

31 DFT VS DWT DFT expansion: one parameter basis or f() t = alψ l() t l DWT expansion two parameter basis f() t a ψ () t = k j jk jk

32 MULTIRESOLUTION REPRESENTATION USING ψ jk () t f() t wider, large translations f() t a ψ () t = k j jk jk fine details j coarse details

33 MULTIRESOLUTION REPRESENTATION USING ψ jk () t f() t fine details f() t a ψ () t = k j jk jk j coarse details

34 MULTIRESOLUTION REPRESENTATION USING ψ jk () t f() t narrower, small translations f() t a ψ () t = k j jk jk fine details j coarse details

35 MULTIRESOLUTION REPRESENTATION USING ψ jk () t f() t high resolution (more details) f ˆ 1 () t f ˆ () t 2 fˆ s () t f() t a ψ () t = k j jk jk j low resolution (less details)

36 Mallat* Filter Scheme n Mallat was the first to implement this scheme, using a well known filter design called two channel sub band coder, yielding a Fast Wavelet Transform

37 Approximations and Details: n Approximations: High-scale, low-frequency components of the signal n Details: low-scale, high-frequency components LPF Input Signal HPF

38 Decimation n The former process produces twice the data it began with: N input samples produce N approximations coefficients and N detail coefficients. n To correct this, we Down sample (or: Decimate) the filter output by two, by simply throwing away every second coefficient.

39 Decimation (cont d) So, a complete one stage block looks like: LPF A* Input Signal HPF D*

40 Multi-level Decomposition n Iterating the decomposition process, breaks the input signal into many lower-resolution components: Wavelet decomposition tree:

41 Wavelet reconstruction n Reconstruction (or synthesis) is the process in which we assemble all components back Up sampling (or interpolation) is done by zero inserting between every two coefficients

42 WAVELETS LIKE FILTERS Relationship of Filters to Wavelet Shape - Choosing the correct filter is most important. - The choice of the filter determines the shape of the wavelet we use to perform the analysis.

43 FILTER BANK REPRESENTATION OF THE DWT DILATIONS

44 WAVELET PACKET ANALYSIS (DWPA) TREE DECOMPOSITION

45 Prediction Residual Pyramid In the absence of quantization errors, the approximation pyramid can be reconstructed from the prediction residual pyramid. Prediction residual pyramid can be represented more efficiently. (with sub-sampling)

46 Efficient Representation Using Details details D 3 details D 2 details D 1 L 0 (no sub-sampling)

47 Efficient Representation Using Details (cont d) representation: L 0 D 1 D 2 D 3 in general: L 0 D 1 D 2 D 3 D J (decomposition or analysis) A wavelet representation of a function consists of (1) a coarse overall approximation (2) detail coefficients that influence the function at various scales.

48 RECONSTRUCTION (SYNTHESIS) H 3 =L 2 +D 3 details D 3 H 2 =L 1 +D 2 details D 2 H 1 =L 0 +D 1 details D 1 L 0 (no sub-sampling)

49 EXAMPLE - HAAR WAVELETS Suppose we are given a 1D "image" with a resolution of 4 pixels: [ ] The Haar wavelet transform is the following: L 0 D 1 D 2 D 3 (with sub-sampling)

50 Example - Haar Wavelets (cont d) Start by averaging 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. 1

51 Example - Haar Wavelets (cont d) Repeating this process on the averages gives the full decomposition: 1

52 Example - Haar Wavelets (cont d) The Harr decomposition of the original four-pixel image is: We can reconstruct the original image to a resolution by adding or subtracting the detail coefficients from the lowerresolution versions

53 Example - Haar Wavelets (cont d) Note small magnitude detail coefficients! D j D j-1 How to compute D i? L 0 D 1

54 MULTIRESOLUTION CONDITIONS If a set of functions can be represented by a weighted sum of ψ(2 j t - k), then a larger set, including the original, can be represented by a weighted sum of ψ(2 j+1 t - k): high resolution scale/frequency localization j time localization low resolution

55 Multiresolution Conditions (cont d) If a set of functions can be represented by a weighted sum of ψ(2 j t - k), then a larger set, including the original, can be represented by a weighted sum of ψ(2 j+1 t - k): V j : span of ψ(2 j t - k): f () t = aψ () t j k jk k V j+1 : span of ψ(2 j+1 t - k): fj+ 1 () t = bkψ ( j+ 1) k() t k V V j j + 1

56 The factor of two scaling means that the spectra of the wavelets divide up the frequency scale into octaves (frequency doubling intervals) ω 1 / 8 ω ny ¼ω ny ½ω ny ω ny

57 Ψ 1 is the wavelet, now viewed as a bandpass filter. This suggests a recursion. Replace: ω 1 / 8 ω ny ¼ω ny ½ω ny ω ny with low-pass filter ω ½ω ny ω ny

58 And then repeat the processes, recursively

59 CHOOSING THE LOW-PASS FILTER The low-pass filter, f lp (w) must match wavelet filter, Ψ(ω). A reasonable requirement is: f lp (ω) 2 + Ψ(ω) 2 = 1 That is, the spectra of the two filters add up to unity. A pair of such filters are called Quadature Mirror Filters. They are known to have filter coefficients that satisfy the relationship: Ψ N-1-k = (-1) k f lp k Furthermore, it s known that these filters allows perfect reconstruction of a time-series by summing its low-pass and high-pass versions

60 time-series of length N HP LP 2 2 Recursion for wavelet coefficients γ(s 1,t) HP LP γ(s 1,t): N/2 coefficients 2 γ(s 2,t) HP 2 LP γ(s 2,t): N/4 coefficients γ(s 2,t): N/8 coefficients Total: N coefficients 2 γ(s 3,t) 2

61 Coiflet low pass filter Coiflet high-pass filter time, t time, t

62 Spectrum of low pass filter Spectrum of wavelet frequency, ω frequency, ω

63 SUMMARY: WAVELET EXPANSION Wavelet decompositions involve a pair of waveforms (mother wavelets): encode low resolution info φ(t) ψ(t) encode details or high resolution info The two shapes are translated and scaled to produce wavelets (wavelet basis) at different locations and on different scales. φ(t-k) ψ(2 j t-k)

64 Summary: wavelet expansion (cont d) f(t) is written as a linear combination of φ(t-k) and ψ(2 j t-k) : No se puede mostrar la imagen. Puede que su equipo no tenga suficiente memoria para abrir la imagen o que ésta esté dañada. Reinicie el equipo y, a continuación, abra el archivo de nuevo. Si sigue apareciendo la x roja, puede que tenga que borrar la imagen e insertarla de nuevo. scaling function wavelet function

65 1D HAAR WAVELETS Haar scaling and wavelet functions: φ(t) ψ(t) computes average computes details

66 1D Haar Wavelets (cont d) V j represents all the 2 j -pixel images Functions having constant pieces over 2 j equal-sized intervals on [0,1). width: 1/2 j Note that Examples: No se puede mostrar la imagen. Puede que su equipo no tenga suficiente memoria para abrir la imagen o que ésta esté dañada. Reinicie el equipo y, a continuación, abra el archivo de nuevo. Si sigue apareciendo la x roja, puede que tenga que borrar la imagen e insertarla de nuevo. No se puede mostrar la imagen. Puede que su equipo no tenga suficiente memoria para abrir la imagen o que ésta esté dañada. Reinicie el equipo y, a continuación, abra el archivo de nuevo. Si sigue apareciendo la x roja, puede que tenga que borrar la imagen e insertarla de nuevo. ϵ V j ϵ V j

67 1D Haar Wavelets (cont d) V 0, V 1,..., V j are nested i.e., Vj V j + 1 V J fine details V 2 V 1 coarse details

68 1D Haar Wavelets (cont d)

69 EXAMPLE

70 Example (cont d)

71 2D HAAR BASIS FOR STANDARD DECOMPOSITION To construct the standard 2D Haar wavelet basis, consider all possible outer products of 1D basis functions. φ 0,0 (x) Example: V 2 =V 0 +W 0 +W 1 ψ 0,0 (x) ψ 0,1 (x) ψ 1,1 (x)

72 2D HAAR BASIS FOR STANDARD DECOMPOSITION To construct the standard 2D Haar wavelet basis, consider all possible outer products of 1D basis functions. φ 00 (x), φ 00 (x) ψ 00 (x), φ 00 (x) ψ 01 (x), φ 00 (x) j j ϕ ( x) ϕ ( x) ψ ( x) ψ ( x) i ji i ji

73 2D HAAR BASIS OF STANDARD DECOMPOSITION V 2 j ϕ ( x) ϕ ( x) i ji ψ ( x) ψ ( x) j i ji

74 WAVELETS APPLICATIONS Noise filtering Image compression Fingerprint compression Image fusion Recognition Image matching and retrieval

75

76 Original Image Compressed Image Threshold: 3.5 Zeros: 42% Retained energy: 99.95%