Wavelets: theory and applications

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Wavelets theory and applications

• An introduction

Grupo de Investigación Tratamiento Digital de
Imágenes Radiológicas
Enrique Nava, University of Málaga (Spain)
Brasov, July 2006
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What are wavelets?

• Wavelet theory is very recent (1980s)
• There is a lot of books about wavelets
• Most of books and tutorials use strong
  mathematical background
• I will try to present an engineering version

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Overview

• Spectral analysis
• Continuous Wavelet Transform
• Discrete Wavelet Transform
• Applications

A wavelet tour of signal processing, S. Mallat,
Academic Press 1998
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Spectral analysis frequency

• Frequency (f) is the inverse of a period (T).
• A signal is periodic if Tgt0 and

• We need to know only information for 1 period
• Any signal (finite length) can be periodized.
• A signal is regular if the signal values and
  derivatives are equal at the left and right side
  of the interval (period)

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Signals examples
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Signals examples
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Why frequency is needed?

• To be able to understand signals and extract
  information from real world
• Electrical or telecommunication engineers tends
  
  to think in the frequency domain

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Fourier series
1822
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Fourier series difficulties

• Any periodic signal can be view as a sum of
  harmonically-related sinusoids
• Representation of signals with different periods
  is not efficient (speech, images)

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Fourier series drawbacks

• There are points where Fourier series does not
  converge
• Signals with different or not synchronized
  periods are not efficiently represented

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Fourier Transform

• The signal has a frequency point of view
  (spectrum)
• Global representation
• Lots of math properties
• Linear operators

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Discrete Fourier Transform

• Practical implementation
• Global representation
• Lots of math properties
• Linear operators
• Easy discrete implementation (1965) (FFT)

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Fourier transform
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Random signals

• Stationary signals
• Statistics dont change with time
• Frequency contents dont change with time
• Information doesnt change with time
• Non-stationary signals
• Statistics change with time
• Frequencies change with time
• Information quantity increases

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Non-stationary signals
2 Hz 10 Hz 20Hz
Stationary
0.0-0.4 2 Hz 0.4-0.7 10 Hz 0.7-1.0 20Hz
Non-Stationary
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Chirp signal

• Frequency 2 Hz to 20 Hz

• Frequency 20 Hz to 2 Hz

Same in Frequency Domain
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Fourier transform drawbacks

• Global behaviour we dont know what frequencies
  happens at a particular time
• Time and frequency are not seen together
• We need time and frequency at the same time
  time-frequency representation
• Biological or medical signals (ECG, EEG, EMG) are
  always non-stationary

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Short-time Fourier Transform (STFT)

• Dennis Gabor (1946) windowing the signal
• Signals are assumed to be stationally local

• A 2D transform

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Short-time Fourier Transform (STFT)
A function of time and frequency
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Short-time Fourier Transform (STFT)
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Short-time Fourier Transform (STFT)
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Short-time Fourier Transform (STFT)
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STFT drawbacks

• Fixed window with time/frequency
• Resolution
• Narrow window gives good time resolution but poor
  frequency resolution
• Wide windows gives good frequency resolution but
  poor time resolution

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Heisenberg Uncertainty Principle

• In signal processing
• You cannot know at the same time the time and
  frequency of a signal
• Signal processing approach is to search for what
  spectral components exist at a given time interval

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Heisenberg Uncertainty Principle

• Heisenberg Box

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Wavelet transform

• An improved version of the STFT, but similar
• Decompose a signal in a set of signals
• Capable of multiresolution analysis
• Different resolution at different frequencies

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Continuous Wavelet Transform

• Definition

Translation (The location of the window)
Scale
Mother Wavelet
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Continuous Wavelet Transform

• Wavelet small wave (ondelette)
• Windowed (finite length) signal
• Mother wavelet
• Prototype to build other wavelets with
  dilatation/compression and shifting operators
• Scale
• Sgt1 dilated signal
• Slt1 compressed signal
• Translation
• Shifting of the signal

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CWT practical computation
Energy normalization

1. Select s1 and t0.
2. Compute the integral and normalize by 1/
3. Shift the wavelet by tDt and repeat until
   wavelet reaches the end of signal
4. Increase s and repeat steps 1 to 3

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Time-frequency resolution
Better time resolution Poor frequency resolution
Frequency
Better frequency resolution Poor time resolution
Time

• Each box represents a equal portion
• Resolution in STFT is selected once for entire
  analysis

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Comparison of transformations
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Mathematical view

• CWT is the inner product of the signal and the
  basis function

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Wavelet basis functions
2nd derivative of a Gaussian is the Marr or
Mexican hat wavelet
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Wavelet basis functions
Frequency domain
Time domain
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Wavelet basis properties
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Discrete Wavelet Transform

• Continuous Wavelet Transform
• Discrete Wavelet Transform

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Discrete CWT

• Sampling of time-scale (frequency) 2D space
• Scale s is discretized in a logarithmic way
• Scheme most used is dyadic s1,2,4,8,16,32
• Time is also discretized in a logarithmic way
• Sampling rate N is decreased so s?Nk
• Implemented like a filter bank

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Discrete Wavelet Transform
Approximation
Details
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Discrete Wavelet Transform
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Discrete Wavelet Transform
Multi-level wavelet decomposition tree
Reassembling original signal
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Discrete Wavelet Transform

• Easy and fast to implement
• Gives enough information for analysis and
  synthesis
• Decompose the signal into coarse approximation
  and details
• Its not a true discrete transform

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Examples
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Examples
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Signal synthesis

• A signal can be decomposed into different scale
  components (analysis)
• The components (wavelet coefficients) can be
  combined to obtain the original signal
  (synthesis)
• If wavelet analysis is performed with filtering
  and downsampling, synthesis consists of filtering
  and upsampling

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Synthesis technique

• Upsampling (insert zeros between samples)

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Sub-band algorithm

• Each step divides by 2 time resolution and
  doubles frequency resolution (by filtering)

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Wavelet packets

• Generalization of wavelet decomposition
• Very useful for signal analysis

Wavelet analysis n1 (at level n) different ways
to reconstuct S
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Wavelet packets

• We have a complete tree

Wavelet packets a lot of new possibilities to
reconstruct S i.e. SA1AD2ADD3DDD3
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Wavelet packets

• A new problem arise how to select the best
  decomposition of a signal x(t)?
• Posible solution
• Compute information at each node of the tree
• (entropy-based criterium)

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Wavelet family types

• Five diferent types
• Orthogonal wavelets with FIR filters
• Haar, Daubechies, Symlets, Coiflets
• Biorthogonal wavelets with FIR filters
• Biorsplines
• Orthogonal wavelets without FIR filters and with
  scaling function
• Meyer
• Wavelets without FIR filters and scaling function
• Morlet, Mexican Hat
• Complex wavelets without FIR filters and scaling
  function
• Shannon

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Wavelet families Daubechies

• Compact support, orthonormal (DWT)

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Other families
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Matlab wavemenu command
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Wavelet application

• Physics (acoustics, astronomy, geophysics)
• Telecommunication Engineering (signal processing,
  subband coding, speech recognition, image
  processing, image analysis)
• Mecanical engineering (turbulence)
• Medical (digital radiology, computer aided
  diagnosis, human vision perception)
• Applied and Pure Mathematics (fractals)

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De-noising signals
Frequency is higher at the beginning
Details reduce with scale
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De-noising images
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Detecting discontinuities
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Detecting discontinuities
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Detecting self-similarity
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Compressing images
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2-D Wavelet Transform
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Wavelet Packets
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2-D Wavelets
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Applications of wavelets

• Pattern recognition
• Biotech to distinguish the normal from the
  pathological membranes
• Biometrics facial/corneal/fingerprint
  recognition
• Feature extraction
• Metallurgy characterization of rough surfaces
• Trend detection
• Finance exploring variation of stock prices
• Perfect reconstruction
• Communications wireless channel signals
• Video compression JPEG 2000

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Practical use of wavelet

• Wavelet software
• Matlab Wavelet Toolbox
• Free software
• UviWave http//www.tsc.uvigo.es/wavelets/uvi_wave
  .html
• Wavelab http//playfair.stanford.edu/wavelab/
• Rice Tools http//jazz.rice.edu/RWT/

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Useful Links to continue

• Matlab wavelet tool using guide
• http//www.wavelet.org
• http//www.multires.caltech.edu/teaching/
• http//www-dsp.rice.edu/software/RWT/
• www.multires.caltech.edu/teaching/courses/
  waveletcourse/sig95.course.pdf
• http//www.amara.com/current/wavelet.html