Wavelets An Introduction by flwifnVD

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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

								
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