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Wavelets An Introductory Exposition Features, Analysis Structures, and Selected Applications Entry of Wavelets into the Domain of SP Most significant event in SP after FT. Reference to Wavelet Transform: “Fourier Transform of 20th Century” Wavelets fills the missing link in signal processing: link between time and frequency studies “Little wave with big future” Groups Active in Wavelet Studies Wavelets Mathematics: Function Engineering, Applied Scientists Applications Analysis and Approximation In Signal Processing Industrial Groups Key Idea Alternative Signal Representations and Transformation Signal representation in a suitable domain for information extraction,…. Examples: Fourier Transform for spectral analysis Hilbert Transform in envelop detection KLT(PCA) for optimal function approx. Laguerre basis function Numerous other transforms( DCT,Radon,…) Two Domains for Signal Representation Projection space, Basis functions Signal Domain Transformed Domain Transformation Different projection spaces different signal representa- tions Fourier Transform Representation in Time Representation in Freq f(t) F(ω) Time Domain Studies Frequency Domain Studies Shortcomings of Fourier Transform signal f(t) with localized changes 0.04 0.02 f(t) 0 -0.02 0 50 100 150 200 time t 250 1 sin(t) 0 -1 0 200 400 600 800 1000 1 cos(t) 0 -1 0 200 400 600 800 1000 Sine and cosine as basis functions of Fourier Transform - Basis function vary within ± ∞, no localization - Only one Basis function: Sine and Cosine or its complex form - Fourier Coefficients: Projection of function f(t) onto sine and cosine bases functions - Information about the entire range of a function is contained in the coefficients, no localization Fourier Transform Loss of Local Information An Illustration f(x) x F(ω) freq ω Information about singularities and sharp changes spread across many frequencies and many basic functions Cost of computations for transients is high Starting Point of Wavelets Real-world Signals Dominance of transient and non- stationary signals Information often reside in transients, changes The need to develop tools for T-F analysis Shortcomings of STFT: Inability to model most of the nonstationary real- world signals What Are Wavelets Wavelets are wavelike oscilatory signals of finite bandwidth both in Time and in Frequency Wavelets are basis functions of spaces with certain properties Examples of Wavelets Db4 Db10 Db 4, Db 10 are from Daubechies family of wavelets. Db wavelets have no analytical expression, they are constructed numerically Examples of Wavelet Functions Coiflet 2 WP1,5 1 0 -1 0 20 40 60 80 100 120 140 160 180 2 WP2,5 0 -2 -4 0 20 40 60 80 100 120 140 2 WP3,5 0 -2 -4 0 20 40 60 80 100 120 140 160 180 Wavelet Functions Generated from Bior 3.9 bior3.9, wavelet packet W 30 10 5 0 -5 -10 0 50 100 150 200 250 300 350 Bior3.9, wavelet packet W 10 4 2 0 -2 -4 0 100 200 300 400 500 600 700 Shortcomings of Fourier Transform Representation in Time Representation in Freq f(t) F(ω) Time Domain Studies Frequency Domain Studies Loss of Time Information in FT SALAAM with switching the 1st 5000 samples with the tail segment 1 0.5 Original 0 -0.5 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 1 0.5 shifted 0 -0.5 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 Fourier Transform Loss of time information abs(fft) of SALAAM with shifting the 1st 5000 samples to the tail 4000 3000 2000 1000 0 0 1000 2000 3000 4000 5000 4000 3000 2000 1000 0 0 1000 2000 3000 4000 5000 An Important Property of Wavelets: Wavelets Filling the Gap Wavelets Representation in Time Representation in Freq f(t) F(ω) Time Domain Studies Frequency Domain Studies Transient Nature of Signals Most of the signals we deal are Nonstationary: Non-stationary refers to time variancy and spectral variation of the signals with time Unpredictable changes in statistics of the signals including changes in pdf function or statistical parameters The degree of unpredictability varies for different signals and applications An Example of a Transient Signal Speech Signal SALAAM 1 0 -1 0 0.5 1 1.5 2 2.5 3 3.5 0.5 4 x 10 sAlaam 0 -0.5 0 500 1000 1500 2000 2500 3000 1 saLaam 0 -1 0 500 1000 1500 1 saLaam 0 -1 0 500 1000 1500 2000 2500 3000 Three segments of SALAAM EEG Signals, Awake and Asleep EEG (EP) Examples of Transient Signals Engine Vibrations 2 Healthy 0 -2 500 1000 1500 2000 2500 3000 2 Faulty 0 -2 500 1000 1500 2000 2500 3000 Time (Samplings) An Example of Real-world Signals Engine Vibrations Wavelet Functions, WP Bior3.1, wavelet packet W2 40 20 0 -20 -40 0 20 40 60 80 100 db4, wavelet packet W 15 4 2 0 -2 -4 0 20 40 60 80 100 Main Stages in Signal Analysis Signal domain basis functions Transformed domain, Coeffs Signals Transformation Information Analysis extraction basis functions Recon Modified Signal Reconstruction Coefficients Examples of Processing in Coefficient Domain Coding for communication, transmission Compression and Data Storage Detection and Pattern Recognition Modification for Enhancement purposes Noise Reduction Watermarking Signal Separation Modeling Two Main Tasks of Wavelet Analysis Decomposition: Information Reside in Signal Constituents / Components Time-Frequency Representation Transformation of a signal into time- frequency representation Different basis and transformations result in different constituents and T-f information Key Concern in SP Resolution/Localization in Signal Processing Resolution: Information Extraction at a Narrow Band of Signal Span. Localization both in Time and in Frequency freq High freq Low freq Time/space Resolution/Localization in Time 3 3 3 Resolution/Localization in Frequency Power Spectrum, Vibration data at combustion zone, original and denoised 150 2 2 100 Original 50 0 0 2000 4000 6000 8000 10000 12000 14000 150 denoised, soft 100 50 0 0 2000 4000 6000 8000 10000 12000 14000 Freq. Hz Need for Joint Time-Frequency Analysis of High Resolution Why Joint T-F Analysis: Transient signal information reside in different bands of time and frequency domain The need for the study of signal at the limits of resolution determined by Heisenberg Uncertainty Limits Heisenberg Uncertainty Principle Heisenberg uncertainty principle defined in quantum physics and is applicable to signal processing problems as well. Position and momentum of a particle can not be determined simultaneously Sets a limit given as follows: Δt. Δf ≥ 1/4π Δx2=⌠{(x-μm )2 |f(x)|2 dx /⌠|f(x)|2 dx μm is center of mass of the function f(x): μm= ⌠{x| f(x)|2 dx /⌠|f(x)|2 dx Second Concern Alternative Time–Frequency Tiling Information about Different signal behavior reside at different T-F Bands Need to have alternative T-F tiling and cell structures Freq/scale Time/scale Third Concern Adaptive Multi-Resolution Study Different information reside at different resolutions. Redundancy in Signal Representation Freq/scale High Frequency Redundancy in tiling Low Frequency Time/scale Wavelets Tools for T-F Analysis Wavelets allow: High resolution and focused study of signals in time and in frequency Low resolution, Coarse and more general picture and trend analysis Comment: Wavelet analysis resembles human mental activity i.e capability for a detail focusing as well as general and a broadly-based data analysis Fourier vs Wavelet Transform Fourier Transform Wavelet Transform Stationary signal Analysis Nonstationary Transient signal Analysis Frequency Information only, Joint Time and Frequency Information Time/space information is lost Single Basis Function Many Basis Functions Computational Cost High Low computational costs Analysis Structures: Numerous Analysis structures: - FS( periodic functions only) CWT, - FT and DFT DWT(2 Band and M Band), WP DDWT, SWT, Adaptive Signal Transform, Numerous Best basis selection algorithms Frame structure Examples of Wavelets Coiflet 4 coif4 : phi coif4 : psi 1.2 1.5 1 1 0.8 0.5 0.6 0.4 0 0.2 -0.5 0 -0.2 -1 0 5 10 15 20 25 0 5 10 15 20 25 Scaling function Wavelet function Wavelet Functions Generated from HAAR function Haar, wavelet packet W 13 2 1 0 -1 -2 0 5 10 15 20 25 30 35 dmey, wavelet packet W 10 2 1 0 -1 -2 0 5 10 15 20 25 30 35 Gabor 2D Wavelet Gabor Wavelet at different scale and Orientation Laguerre Gaussian Wavelet Laguerre Gaussian Wavelet Wavelet Transform WaveletTransform is defined as in other transforms: W = <f(t),ψ(t)> = ∫f(t) ψ*(t)dt wjk = <f(t),ψj,k(t)> = ∫f(t) ψj,k*(t)dt ψj,k(t) is a shifted and scaled wavelet function, j is scaling and k translation parameters, J,K can be any scalars. In DWT they assume integer values. fj,k() t w , jk * jk Physical Interpretation of Wavelet Transform Correlation (extent of matching) of f(t) with the wavelet ψjk(t) at scale j and location k. It carries signal information at scale j, and location k. It gives localized information of a signal. Representation of a signal in a domain described by wavelet basis functions Wavelet Transform Given a function f(t) єL^2 Ψab(t)= Ψ(at-b) f(t) wab Function Domain Transformed Domain: single dimensional two dimensional, paramters a,b f(t) coeffs Translation b Time/space Scale a /freq Illustration of Translation by two different Wavelets f(t) t φ0k (t) s hifte d wave le ts t φ0 k (t) Haar s hifte d Haar wave le t t Illustrative Examples of WT Signal f(t) t φ0k(t) shifted wavelets t t φ0k(t) Haar φjk(t) shifted Haar wavelet scaled Haar wavelet t t t Wavelet Transform Window Function Interpretation Different information are extracted form the same signal using different wavelets. The need to access multitude of transforms and wavelet bases for extraction of different information of a given signal Wavelets as window functions may be considered as lenses having different resolutions. Different information are extracted by different lenses at different scales and different locations of a given signal STFT has only one window function at a given scale Wavelet Transform, Translation of Window Function Illustration of Wavelet Scaling/Dilation Common Wavelets Old Wavelets: Haar function Gabor function and wavelet(1D,2D) Morlet wavelets Shannon function 1st and 2nd Derivatives of Gaussian function(mexican hat) Truncated and lapped Sine or Cosine functions Recent Wavelets Db Wavelets (orthogonal, biorthogonal), Coiflets, Symlets Biorthgonal wavelets Mallat Wavelet (1D,2D) Bathlets Curvelets Ridgelets Meyer wavelets Banana Wavelets Your Wavelets Second Generation Wavelets( wavelets constructed by lifting scheme) Commonly Used Numerical Wavelets Daubechies Wavelets Biorthogonal Wavelets Coiflets Symmlets Bathlets Dmeyer How Wavelets are Generated A few number of wavelets are constructed using known analog functions. Some are expressed in analog function form, e.g. Morlet wavelets Many others are designed and constructed numerically. Db wavelets are examples of these wavelets New wavelets are introduced by individuals each year Categorization of Wavelets Analytical or Numerical Real or Complex Symmetric, antisymmetric, asymmetric (e.g. db) Compactly supported or not compact Causal, Non-causal wavelets Do they have efficient computation algorithms such as FWT Wavelets with special features e.g. complex wavelet with real part (high freq) and complex part carry low frequency content Maximally flat waveletes (Flatness of spectrum, Rate of decay at ω=pi and origin) Have Orthogonal, biorthogonal analysis system Wavelets of high Resolution in time-frequency domain: ∆t ∆ω. Gabor: H=0.5, Haar: H=0.58 in normalized frequency (0,1). Balanced Wavelets (Bathlets) are designed under a criteria based on balancing ∆t and ∆ω. Two Common Analysis Structures 1- Standard Two Band Discrete Wavelet Transform High Freq Details Signal High Pass 2 High pass Low Pass 2 Low pass Low freq Approx Decimated Wavelet Transform-Analysis Stage Multiresolution Pyramidal Signal Analysis Structure Multiresolution as a subband coding of a signal, Signal components at different subbands are extracted Different analyzing wavelets results in different components at different bands An Illustration of Multiresolution Signal Decomposition Binary Tree of DWT Common Analysis Structures Wavelet Packets High freq details High pass 2 High Pass 2 V space of the signal Low pass 2 High pass 2 Low Pass 2 Low pass 2 Wavelet Packets -Analysis Stages Wavelet Packets Tree Standard Two Channel DWT and WP T-F Tiling Time-Freq. Tiling Standard DWT Standard WP Perfect Reconstruction Filterbank Filterbank implementation of DWT Wavelet Subspace W High Pass High pass Low Pass Low pass Perfect Reconstrution Filter bank Alternative Signal Analysis Architectures Wavelets are rich in analysis structures and algorithms Main Analysis Structures ORTHOGONAL BIORTHOGONAL REDUNDANT WAVELET TRANSFROM FRAME-BASED REDUNDANT TRANSFORM Beyond Orthogonality: Compression Requirements Coeffs 1 Coeffs 2 Slow rate of decay Fast rate of decay Orthogonal Expansion Non-orthogonal Expansion Primary Features of Wavelets used in Signal Processing 1. Time- frequency Localization (useful for transient data analysis) 2. Sparsity: Coefficients are often grouped into high and low amplitude 3. Function Approximation: near Optimal Function Approximation 4. Noise Reduction, No wavelet can model white noise leading to small amplitude coefficients 5. Decorrelation: Signal components( coefficients) in wavelet domain have a lower degree of correlation Categorization of Wavelet Applications Applications: Methodology/Algorithm Intensive Change Detection ( of all kinds) Compression Feature Extraction and Pattern Recognition with Numerous Applications Image Enhancement Noise reduction with Numerous Applications Diagnostics (such as NDT, NDE) Signal Separation, ICA using Wavelets Solution of Differential and Partial Differential Equations Analysis of Fractals System Identification in Control systems Data and Image Fusion Watermarking Applications, Industry Categorization Application Areas: Industry Intensive Power Systems, Transient data Analysis, Component Diagnosis, Signal detection NDT, Ultra sonic Flaw Detection, Diagnosis Communications, Coding, Transmission Medical Applications: Compression, Noise Reduction, signal separation, signal classification, image enhancement Machine Vision, Robotics, geometry and spatial edge detection Industrial Machinery and Machine Diagnosis, Remote Sensing, Satellite Image Analysis, Image segmentation, fusion Biometrics: Fingerprint, Iris and Facial Identification Nuclear Reactors, Boiler Pipe fault analysis, Oil Industry, Oil Exploration Transportation Industry (railroad engine diagnosis, Aero jet engine and Helicopter drive and gear diagnosis Seismic Data analysis Genetics Applications, Fields of Work Applications in different fields Physics, Geophysics, Mathematics, function approximation, signal processing, Harmonic analysis, Differential Equation, Frames, Function Space Analysis Electrical Engineering, numerous aplications Biomedical Engineering, EEG, ECG, MRI Finance and Financial Data Analysis Mechanical, Civil, Ocean Science and Engineering Remote Sensing, Forestry Nontechnical: Historial document retrieval Details of Applications To Be Presented in a Separate Slide Presentation