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1. Wavelet Transform PSCI 702 December 7, 2005

2. Problem with Fourier · Fourier analysis -- breaks down a signal into constituent sinusoids of different frequencies. · A serious drawback in transforming to the frequency domain, time information is lost. When looking at a Fourier transform of a signal, it is impossible to tell when a particular event took place. Time information?

3. Gabor’s Proposal: Short Time Fourier Transform Requirements: Signal in time domain: require short time window to depict features of signal. Signal in frequency domain: require short frequency window (long time window) to depict features of signal.

4. Fourier  Gabor  Wavelet

5. FT At Work

6. FT At Work F F F

7. Stationary and Non-stationary Signals • FT identifies all spectral components present in the signal, however it does not provide any information regarding the temporal (time) localization of these components. Why? • Stationary signals consist of spectral components that do not change in time • all spectral components exist at all times • no need to know any time information • FT works well for stationary signals • However, non-stationary signals consists of time varying spectral components • How do we find out which spectral component appears when? • FT only provides what spectral components exist , not where in time they are located. • Need some other ways to determine time localization of spectral components

8. Stationary and Non-stationary Signals • Stationary signals’ spectral characteristics do not change with time • Non-stationary signals have time varying spectra Concatenation

9. Non-stationary Signals 50 Hz 20 Hz 5 Hz Perfect knowledge of what frequencies exist, but no information about where these frequencies are located in time

10. FT Shortcomings • Complex exponentials stretch out to infinity in time • They analyze the signal globally, not locally • Hence, FT can only tell what frequencies exist in the entire signal, but cannot tell, at what time instances these frequencies occur • In order to obtain time localization of the spectral components, the signal need to be analyzed locally • HOW ?

11. Short Time Fourier Transform (STFT) • Choose a window function of finite length • Put the window on top of the signal at t=0 • Truncate the signal using this window • Compute the FT of the truncated signal, save. • Slide the window to the right by a small amount • Go to step 3, until window reaches the end of the signal • For each time location where the window is centered, we obtain a different FT • Hence, each FT provides the spectral information of a separate time-slice of the signal, providing simultaneous time and frequency information

12. STFT

13. STFT Frequency parameter Time parameter Signal to be analyzed FT Kernel (basis function) STFT of signal x(t): Computed for each window centered at t=t’ Windowing function Windowing function centered at t=t’

14. STFT at Work Windowed sinusoid allows FT to be computed only through the support of the windowing function 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 0 100 200 300 0 100 200 300 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 0 100 200 300 0 100 200 300

15. STFT • STFT provides the time information by computing a different FTs for consecutive time intervals, and then putting them together • Time-Frequency Representation (TFR) • Maps 1-D time domain signals to 2-D time-frequency signals • Consecutive time intervals of the signal are obtained by truncating the signal using a sliding windowing function • How to choose the windowing function? • What shape? Rectangular, Gaussian, Elliptic…? • How wide? • Wider window require less time steps  low time resolution • Also, window should be narrow enough to make sure that the portion of the signal falling within the window is stationary • Can we choose an arbitrarily narrow window…?

16. Haar wavelet What are wavelets? · Wavelets are functions defined over a finite interval and having an average value  of zero.

17. What is wavelet transform? · The wavelet transform is a tool for carving up functions, operators, or data into components of different frequency, allowing one to study each component separately. · The basic idea of the wavelet transform is to represent any arbitrary function ƒ(t) as a superposition of a set of such wavelets or basis functions. · These basis functions or baby wavelets are obtained from a single prototype wavelet called the mother wavelet, by dilations or contractions (scaling) and translations (shifts).

18. The continuous wavelet transform (CWT) · Fourier Transform FT is the sum over all the time of signal f(t) multiplied by a complex exponential.

19. · Similarly, the Continuous Wavelet Transform (CWT) is defined as the sum over all time of the signal multiplied by scale , shifted version of the wavelet function : where * denotes complex conjugation. This equation shows how a function ƒ(t) is decomposed into a set of basis functions , called the wavelets. The variables s and tare the new dimensions, scale and translation (position), after the wavelet transform.

20. · The results of the CWT are many wavelet coefficients, which are a function of scale and position

21. · The wavelets are generated from a single basic wavelet , the so-called mother wavelet, by scaling and translation: s is the scale factor, tis the translation factor and the factor s-1/2 is for energy normalization across the different scales. · It is important to note that in the above transforms the wavelet basis functions are not specified. · This is a difference between the wavelet transform and the Fourier transform, or other transforms.

22. ·Scale · Scaling a wavelet simply means stretching (or compressing) it.

23. High frequency low frequency ·Scale and Frequency ·Low scale a Compressed wavelet Rapidly changing details ·High scale a stretched wavelet slowly changing details ·Translation (shift) · Translating a wavelet simply means delaying (or hastening) its onset.

24. (1) admissibility condition: stands for the Fourier transform of • The admissibility condition implies that the Fourier transform of vanishes at the zero frequency, i.e. · A zero at the zero frequency ( DC component) also means that the average value of the wavelet in the time domain must be zero, must be a wave. Wavelet Properties · This means that wavelets must have a band-pass like spectrum. This is a very important observation, which we will use later on to build an efficient wavelet transform.

25. Haar wavelet

26. Function Representations –Desirable Properties • generality – approximate anything well • discontinuities, nonperiodicity, ... • adaptable to application • audio, pictures, flow field, terrain data, ... • compact – approximate function with few coefficients • facilitates compression, storage, transmission • fast to compute with • differential/integral operators are sparse in this basis • Convert n-sample function to representation in O(nlogn) or O(n) time

27. Wavelet History, Part 1 • 1805 Fourier analysis developed • 1965 Fast Fourier Transform (FFT) algorithm … • 1980’s beginnings of wavelets in physics, vision, speech processing (ad hoc) • … little theory … why/when do wavelets work? • 1986 Mallat unified the above work • 1985 Morlet & Grossman continuous wavelet transform … asking: how can you get perfect reconstruction without redundancy?

28. Wavelet History, Part 2 • 1985 Meyer tried to prove that no orthogonal wavelet other than Haar exists, found one by trial and error! • 1987 Mallat developed multiresolution theory, DWT, wavelet construction techniques (but still noncompact) • 1988 Daubechies added theory: found compact, orthogonal wavelets with arbitrary number of vanishing moments! • 1990’s: wavelets took off, attracting both theoreticians and engineers

29. If j=m and k=n others Discrete Wavelets ·Discrete wavelet is written as j and k are integers and s0 > 1 is a fixed dilation step. The translation factor t0 depends on the dilation step. The effect of discretizing the wavelet is that the time-scale space is now sampled at discrete intervals. We usually choose s0 = 2

30. · If we look at the scaling function as being just a signal with a low-pass spectrum, then we can decompose it in wavelet components and express it like · admissibility condition for scaling functions · Summarizing once more, if one wavelet can be seen as a band-pass filter and a scaling function is a low-pass filter, then a series of dilated wavelets together with a scaling function can be seen as a filter bank.

31. The Discrete Wavelet Transform · Calculating wavelet coefficients at every possible scale is a fair amount of work, and it generates an awful lot of data. What if we choose only a subset of scales and positions at which to make our calculations? · It turns out, rather remarkably, that if we choose scales and positions based on powers of two -- so-called dyadic scales and positions -- then our analysis will be much more efficient and just as accurate. We obtain just such an analysis from the discrete wavelet transform (DWT).

32. Approximations and Details · The approximations are the high-scale, low-frequency components of the signal. The details are the low-scale, high-frequency components. The filtering process, at its most basic level, looks like this: · The original signal, S, passes through two complementary filters and emerges as two signals .

33. Downsampling · Unfortunately, if we actually perform this operation on a real digital signal, we wind up with twice as much data as we started with. Suppose, for instance, that the original signal S consists of 1000 samples of data. Then the approximation and the detail will each have 1000 samples, for a total of 2000. · To correct this problem, we introduce the notion of downsampling. This simply means throwing away every second data point.

34. An example:

35. Reconstructing Approximation and Details Upsampling

36. Wavelet Decomposition Multiple-Level Decomposition The decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal is broken down into many lower-resolution components. This is called the wavelet decomposition tree.

37. · Scaling function (two-scale relation) · Wavelet · The signal f(t) can be expresses as DWT

38. Initialization: · How to calculate DWT given g(m), h(m) and a signal f(m)? Example: f={1,4,-3,0};

39. Wavelet Reconstruction (Synthesis) Perfect reconstruction :

40. HL LL HL L H LH HH LH HH original One scale two scales · 2-D Discrete Wavelet Transform · A 2-D DWT can be done as follows: Step 1: Replace each row with its 1-D DWT; Step 2: Replace each column with its 1-D DWT; Step 3: repeat steps (1) and (2) on the lowest subband for the next scale Step 4: repeat steps (3) until as many scales as desired have been completed

41. 2D Wavelets

42. Original Image

43. Wavelet Transformed

44. Reconstructed Image

45. Comparison

46. Discrete Wavelet Transform • Visual Comparison (a) (b) (c) (a) Original Image256x256Pixels, 24-BitRGB (b) JPEG (DCT) Compressed with compression ratio 43:1(c) JPEG2000 (DWT) Compressed with compression ratio 43:1