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Presents Robi Polikar Dept. of Electrical & Computer Engineering Rowan University The Story of Wavelets Technical Overview But…We cannot do that with Fourier Transform…. Time - frequency representation and the STFT Continuous wavelet transform Multiresolution analysis and discrete wavelet transform (DWT) Application Overview Conventional Applications: Data compression, denoising, solution of PDEs, biomedical signal analysis. Unconventional applications Yes…We can do that with wavelets too… Historical Overview 1807 ~ 1940s: The reign of the Fourier Transform 1940s ~ 1970s: STFT and Subband Coding 1980s & 1990s: The Wavelet Transform and MRA What is a Transform and Why Do we Need One ? Transform: A mathematical operation that takes a function or sequence and maps it into another one Transforms are good things because… The transform of a function may give additional /hidden information about the original function, which may not be available /obvious otherwise The transform of an equation may be easier to solve than the original equation (recall your fond memories of Laplace transforms in DFQs) The transform of a function/sequence may require less storage, hence provide data compression / reduction An operation may be easier to apply on the transformed function, rather than the original function (recall other fond memories on convolution). December, 21, 1807 “An arbitrary function, continuous or with discontinuities, defined in a finite interval by an arbitrarily capricious graph can always be expressed as a sum of sinusoids” J.B.J. Fourier Jean B. Joseph Fourier (1768-1830) Complex function representation through simple building blocks Basis functions Using only a few blocks Compressed representation Using sinusoids as building blocks Fourier transform Frequency domain representation of the function How Does FT Work Anyway? Recall that FT uses complex exponentials (sinusoids) as building blocks. For each frequency of complex exponential, the sinusoid at that frequency is compared to the signal. If the signal consists of that frequency, the correlation is high large FT coefficients. If the signal does not have any spectral component at a frequency, the correlation at that frequency is low / zero, small / zero FT coefficient. FT At Work FT At Work F F F FT At Work F FT At Work Complex exponentials (sinusoids) as basis functions: F An ultrasonic A-scan using 1.5 MHz transducer, sampled at 10 MHz 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 Stationary and Non-stationary Signals Stationary signals’ spectral characteristics do not change with time Non-stationary signals have time varying spectra Concatenation Stationary vs. Non-Stationary X4(ω) Perfect knowledge of what frequencies exist, but no information about where these frequencies are X5(ω) located in time Shortcomings of the FT Sinusoids and exponentials Stretch into infinity in time, no time localization Instantaneous in frequency, perfect spectral localization Global analysis does not allow analysis of non-stationary signals Need a local analysis scheme for a time-frequency representation (TFR) of nonstationary signals Windowed F.T. or Short Time F.T. (STFT) : Segmenting the signal into narrow time intervals, narrow enough to be considered stationary, and then take the Fourier transform of each segment, Gabor 1946. Followed by other TFRs, which differed from each other by the selection of the windowing function Short Time Fourier Transform (STFT) 1. Choose a window function of finite length 2. Place the window on top of the signal at t=0 3. Truncate the signal using this window 4. Compute the FT of the truncated signal, save. 5. Incrementally slide the window to the right 6. 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 STFT Time Frequency Signal to FT Kernel parameter parameter be analyzed (basis function) STFT of signal x(t): Windowing Windowing function Computed for each function centered at t=t’ window centered at t=t’ STFT t’=-8 t’=-2 t’=4 t’=8 STFT at Work STFT At Work STFT At Work 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…? Selection of STFT Window Two extreme cases: W(t) infinitely long: STFT turns into FT, providing excellent frequency information (good frequency resolution), but no time information W(t) infinitely short: STFT then gives the time signal back, with a phase factor. Excellent time information (good time resolution), but no frequency information Wide analysis window poor time resolution, good frequency resolution Narrow analysis windowgood time resolution, poor frequency resolution Once the window is chosen, the resolution is set for both time and frequency. Heisenberg Principle Time resolution: How well Frequency resolution: How two spikes in time can be well two spectral components separated from each other in can be separated from each the transform domain other in the transform domain Both time and frequency resolutions cannot be arbitrarily high!!! We cannot precisely know at what time instance a frequency component is located. We can only know what interval of frequencies are present in which time intervals The Wavelet Transform Overcomes the preset resolution problem of the STFT by using a variable length window Analysis windows of different lengths are used for different frequencies: Analysis of high frequencies Use narrower windows for better time resolution Analysis of low frequencies Use wider windows for better frequency resolution This works well, if the signal to be analyzed mainly consists of slowly varying characteristics with occasional short high frequency bursts. Heisenberg principle still holds!!! The function used to window the signal is called the wavelet The Wavelet Transform Translation parameter, Scale parameter, A normalization Signal to be measure of time measure of frequency constant analyzed Continuous wavelet transform The mother wavelet. All kernels are of the signal x(t) using the obtained by translating (shifting) and/or analysis wavelet (.) scaling the mother wavelet Scale = 1/frequency WT at Work Low frequency (large High frequency (small scale) scale) WT at Work WT at Work WT at Work Matlab Demos on CWT Discrete Wavelet Transform CWT computed by computers is really not CWT, it is a discretized version of the CWT. The resolution of the time-frequency grid can be controlled (within Heisenberg’s inequality), can be controlled by time and scale step sizes. Often this results in a very redundant representation How to discretize the continuous time-frequency plane, so that the representation is non-redundant? Sample the time-frequency plane on a dyadic (octave) grid Discrete Wavelet Transform Dyadic sampling of the time –frequency plane results in a very efficient algorithm for computing DWT: Subband coding using multiresolution analysis Dyadic sampling and multiresolution is achieved through a series of filtering and up/down sampling operations x[n] H y[n] Discrete Wavelet Transform Implementation x[n] x[n] ~ G G + 2 2 ~ 2 ~ 2 H G 2 2 G + H ~ 2 2 H H Decomposition Reconstruction G Half band high pass filter 2 Down-sampling H Half band low pass filter 2 Up-sampling 2-level DWT decomposition. The decomposition can be continues as long as there are enough samples for down-sampling. DWT - Demystified Length: 512 |H(jw)| B: 0 ~ g[n] h[n] w Length: 256 -/2 /2 Length: 256 2 2 B: 0 ~ /2 Hz B: /2 ~ Hz a1 |G(jw)| d1: Level 1 DWT Coeff. g[n] h[n] Length: 128 w Length: 128 2 2 B: 0 ~ /4 Hz - -/2 /2 B: /4 ~ /2 Hz a2 d2: Level 2 DWT g[n] h[n] Coeff. 2 2 Length: 64 Length: 64 B: 0 ~ /8 Hz B: /8 ~ /4 Hz d3: Level 3 …a3….Level 3 approximation DWT Coefficients Coeff. Implementation of DWT on MATLAB Choose wavelet and number Load signal of levels Hit Analyze s=a5+d5+…+d1 button Approx. coef. at level 5 Level 1 coeff. (Wavedemo_signal1) Highest freq. Applications of Wavelets Compression De-noising Feature Extraction Discontinuity Detection Distribution Estimation Data analysis Biological data NDE data Financial data Compression DWT is commonly used for compression, since most DWT are very small, can be zeroed-out! Compression Compression ECG- Compression Denoising Implementation in Matlab First, analyze the signal with appropriate wavelets Hit Denoise (Noisy Doppler) Denoising Using Matlab Choose thresholding method Choose noise type Choose thrsholds Hit Denoise Denosing Using Matlab Discontinuity Detection (microdisc.mat) Discontinuity Detection with CWT (microdisc.mat) Application Overview Data Compression Wavelet Shrinkage Denoising Source and Channel Coding Biomedical Engineering EEG, ECG, EMG, etc analysis MRI Nondestructive Evaluation Ultrasonic data analysis for nuclear power plant pipe inspections Eddy current analysis for gas pipeline inspections Numerical Solution of PDEs Study of Distant Universes Galaxies form hierarchical structures at different scales Application Overview Wavelet Networks Real time learning of unknown functions Learning from sparse data Turbulence Analysis Analysis of turbulent flow of low viscosity fluids flowing at high speeds Topographic Data Analysis Analysis of geo-topographic data for reconnaissance / object identification Fractals Daubechies wavelets: Perfect fit for analyzing fractals Financial Analysis Time series analysis for stock market predictions History Repeats Itself… 1807, J.B. Fourier: All periodic functions can be expressed as a weighted sum of trigonometric function Denied publication by Lagrange, Legendre and Laplace 1822: Fourier’s work is finally published … … 143 years … … 1965, Cooley & Tukey: Fast Fourier Transform History Repeats Itself: Morlet’s Story 1946, Gabor: STFT analysis: high frequency components using a narrow window, or low frequency components using a wide window, but not both Late 1970s, Morlet’s (geophysical engineer) problem: Time - frequency analysis of signals with high frequency components for short time spans and low frequency components with long time spans STFT can do one or the other, but not both Solution: Use different windowing functions for sections of the signal with different frequency content Windows to be generated from dilation / compression of prototype small, oscillatory signals wavelets Criticism for lack of mathematical rigor !!! Early 1980s, Grossman (theoretical physicist): Formalize the transform and devise the inverse transformation First wavelet transform ! Rediscovery of Alberto Calderon’s 1964 work on harmonic analysis 1980s 1984, Yeves Meyer : Similarity between Morlet’s and Colderon’s work, 1984 Redundancy in Morlet’s choice of basis functions 1985, Orthogonal wavelet basis functions with better time and frequency localization Rediscovery of J.O. Stromberg’s 1980 work the same basis functions (also a harmonic analyst) Yet re-rediscovery of Alfred Haar’s work on orthogonal basis functions, 1909 (!). Simplest known orthonormal wavelets Transition to the Discrete Signal Analysis Ingrid Daubechies: Discretization of time and scale parameters of the wavelet transform Wavelet frames, 1986 Orthonormal bases of compactly supported wavelets (Daubechies wavelets), 1988 Liberty in the choice of basis functions at the expense of redundancy Stephane Mallat: Multiresolution analysis w/ Meyer, 1986 Ph.D. dissertation, 1988 Discrete wavelet transform Cascade algorithm for computing DWT …However… Decomposition of a discrete into dyadic frequencies (MRA) , known to EEs under the name of “Quadrature Mirror Filters”, Croisier, Esteban and Galand, 1976 (!) Transition to the Discrete Signal Analysis Martin Vetterli & Jelena Kovacevic Wavelets and filter banks, 1986 Perfect reconstruction of signals using FIR filter banks, 1988 Subband coding Multidimensional filter banks, 1992 1990s Equivalence of QMF and MRA, Albert Cohen, 1990 Compactly supported biorthogonal wavelets, Cohen, Daubechies, J. Feauveau, 1993 Wavelet packets, Coifman, Meyer, and Wickerhauser, 1996 Zero Tree Coding, Schapiro 1993 ~ 1999 Search for new wavelets with better time and frequency localization properties. Super-wavelets Matching Pursuit, Mallat, 1993 ~ 1999 New & Noteworthy Zero crossing representation signal classification computer vision data compression denoising Super wavelet Linear combination of known basic wavelets Zero Tree Coding, Schapiro Matching Pursuit , Mallat Using a library of basis functions for decomposition New MPEG standard