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WELCOME
Introduction to Wavelet
Transform with
Applications to DSP
Hicham BERKOUK
Tarek Islam SADMI
E08/Computer Engineering
IGEE ...
OUTLINE
 Overview
 Historical Development
 Limitations of Fourier Transform
 Principle of Wavelet Transform
 Examples...
 Overview
 Historical Development
 Limitations of Fourier Transform
 Principle of Wavelet Transform
 Examples of Appl...
Overview:
What does Wavelet mean?
Oxford Dictionary: A wavelet is a small wave.
Wikipedia: A wavelet is a mathematical fun...
Advantages of using Wavelets:
 Provide a way for analysing waveforms in both frequency and
duration.
 Representation of ...
 Overview
 Historical Development
 Limitations of Fourier Transform
 Principle of Wavelet Transform
 Examples of Appl...
Historical Development:
 1909 : Alfred Haar – Dissertation “On the Orthogonal Function
Systems” for his Doctoral Degree. ...
 1985 : Yves Meyer : Construction of orthogonal wavelet basis
functions with very good time and frequency localization.
...
 Overview
 Historical Development
 Limitations of Fourier Transform
 Principle of Wavelet Transform
 Examples of Appl...
Limitations of Fourier Transform:
 To show the limitations of Fourier Transform,
we chose a well-known signal in SONAR an...
Fourier Transform of Chirp Signals:
 Different in time but same frequency
representation!!!
 Fourier Transform only gives what frequency
components exist in...
SOLUTION
?
WAVELET TRANSFORM
 Overview
 Historical Development
 Limitations of Fourier Transform
 Principle of Wavelet Transform
 Examples of Appl...
Principle of Wavelet Transform:
 The Continuous Wavelet Transform is
given by :
𝑋 𝑊𝑇 τ, 𝑠 =
1
𝑠 −∞
+∞
𝑥 𝑡 ѱ∗
𝑡 − τ
𝑠
𝑑𝑡
W...
 Here are some of the most popular mother
wavelets :
Each Mother Wavelet has its own equation :
 Haar’s basis equation :
 Daubechies’ basis equation:
 Morlet’s (Mexican Hat...
Steps to compute CWT of a given signal :
1. Take a wavelet and compare it to section at the start of
the original signal, ...
2. Shift the wavelet to the right and repeat
step 1 until the whole signal is covered.
3. Scale (stretch) the wavelet and repeat steps
1 through 2.
4. Repeat steps 1 through 3 for all scales.
Time-frequency representation of « up-chirp » signal
using CWT :
 The Discrete Wavelet Transform (DWT) is
given by:
𝑔 𝑡 =
𝑚∊ℤ 𝑛∊ℤ
𝑥, ѱ 𝑚,𝑛 . ѱ 𝑚,𝑛[𝑡]
Where : ѱ 𝑚,𝑛 𝑡 = 𝑠0
−𝑚/2
ѱ(𝑠0
−𝑚
𝑡 ...
 The Multi-Resolution Analysis :
Analyzing a signal at different frequencies
with different resolutions. By applying bank...
MRA time-frequency
representation:
 Good time resolution
for high frequencies.
 Good frequency
resolution for low
freque...
MRA time-frequency Representation
of Chirp Signal
 Overview
 Historical Development
 Limitations of Fourier Transform
 Principle of Wavelet Transform
 Examples of Appl...
FBI Fingerprints Compression:
o Since 1924, the FBI Collected about 200
Million cards of fingerprints.
o Each fingerprints...
 The FBI decided to adopt a wavelet-based
image coding algorithm as a national
standard for digitized fingerprint records...
JPEG 2000
 Image compression standard and coding system.
 Created by Joint Photographic Experts Group
committee in 2000....
Comparison between JPEG and
JPEG2000
Wavelets can be applied for many
different purposes :
 Audio compression.
 Speech recognition.
 Image and video compres...
 Overview
 Historical Development
 Limitations of Fourier Transform
 Principle of Wavelet Transform
 Examples of Appl...
Wavelet is a relatively new theory, it has enjoyed
a tremendous attention and success over the last
decade, and for a good...
 Overview
 Historical Development
 Limitations of Fourier Transform
 Principle of Wavelet Transform
 Examples of Appl...
References :
 [1] :https://www. en.wikipedia.org/wiki/Wavelet.
 [2] : Introduction to Wavelet : Bhushan D Patil PhD Rese...
Introduction to Wavelet Transform with Applications to DSP
Introduction to Wavelet Transform with Applications to DSP
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Introduction to Wavelet Transform with Applications to DSP

Introduction to wavelet transforms.
History,
Mathematical development,
Applications in digital signal processing.

Introduction to Wavelet Transform with Applications to DSP

  1. 1. WELCOME
  2. 2. Introduction to Wavelet Transform with Applications to DSP Hicham BERKOUK Tarek Islam SADMI E08/Computer Engineering IGEE – Boumerdes.
  3. 3. OUTLINE  Overview  Historical Development  Limitations of Fourier Transform  Principle of Wavelet Transform  Examples of Applications  Conclusion  References
  4. 4.  Overview  Historical Development  Limitations of Fourier Transform  Principle of Wavelet Transform  Examples of Applications  Conclusion  References
  5. 5. Overview: What does Wavelet mean? Oxford Dictionary: A wavelet is a small wave. Wikipedia: A wavelet is a mathematical function used to divide a given function or continuous-time signal into different scale components. A Wavelet Transform is the representation of a function by wavelets.
  6. 6. Advantages of using Wavelets:  Provide a way for analysing waveforms in both frequency and duration.  Representation of functions that have discontinuities and sharp peaks.  Accurately deconstructing and reconstructing finite, non-periodic and/or non-stationary signals.  Allow signals to be stored more efficiently than by Fourier transform.
  7. 7.  Overview  Historical Development  Limitations of Fourier Transform  Principle of Wavelet Transform  Examples of Applications  Conclusion  References
  8. 8. Historical Development:  1909 : Alfred Haar – Dissertation “On the Orthogonal Function Systems” for his Doctoral Degree. The first wavelet related theory .  1910 : Alfred Haar : Development of a set of rectangular basis functions.  1930s : - Paul Levy investigated “The Brownian Motion”. - Littlewood and Paley worked on localizing the contributing energies of a function.  1946 : Dennis Gabor : Used Short Time Fourier Transform .  1975 : George Zweig : The first Continuous Wavelet Transform CWT.
  9. 9.  1985 : Yves Meyer : Construction of orthogonal wavelet basis functions with very good time and frequency localization.  1986 : Stephane Mallat : Developing the Idea of Multiresolution Analysis “MRA” for Discrete Wavelet Transform “DWT”.  1988 : The Modern Wavelet Theory with Daubechies and Mallat.  1992 : Albert Cohen, Jean fauveaux and Daubechies constructed the compactly supported biorthogonal wavelets.
  10. 10.  Overview  Historical Development  Limitations of Fourier Transform  Principle of Wavelet Transform  Examples of Applications  Conclusion  References
  11. 11. Limitations of Fourier Transform:  To show the limitations of Fourier Transform, we chose a well-known signal in SONAR and RADAR applications, called the Chirp.  A Chirp is a signal in which the frequency increases (‘up-chirp’) or decreases (‘down- chirp’).
  12. 12. Fourier Transform of Chirp Signals:
  13. 13.  Different in time but same frequency representation!!!  Fourier Transform only gives what frequency components exist in a signal.  Fourier Transform cannot tell at what time the frequency components occur.  However, Time-Frequency representation is needed in most cases.
  14. 14. SOLUTION ?
  15. 15. WAVELET TRANSFORM
  16. 16.  Overview  Historical Development  Limitations of Fourier Transform  Principle of Wavelet Transform  Examples of Applications  Conclusion  References
  17. 17. Principle of Wavelet Transform:  The Continuous Wavelet Transform is given by : 𝑋 𝑊𝑇 τ, 𝑠 = 1 𝑠 −∞ +∞ 𝑥 𝑡 ѱ∗ 𝑡 − τ 𝑠 𝑑𝑡 Where: • τ is the translation parameter. • s is the scale parameter. • ψ is the Mother Wavelet.
  18. 18.  Here are some of the most popular mother wavelets :
  19. 19. Each Mother Wavelet has its own equation :  Haar’s basis equation :  Daubechies’ basis equation:  Morlet’s (Mexican Hat) basis equation:
  20. 20. Steps to compute CWT of a given signal : 1. Take a wavelet and compare it to section at the start of the original signal, and calculate a correlation coefficient C.
  21. 21. 2. Shift the wavelet to the right and repeat step 1 until the whole signal is covered.
  22. 22. 3. Scale (stretch) the wavelet and repeat steps 1 through 2. 4. Repeat steps 1 through 3 for all scales.
  23. 23. Time-frequency representation of « up-chirp » signal using CWT :
  24. 24.  The Discrete Wavelet Transform (DWT) is given by: 𝑔 𝑡 = 𝑚∊ℤ 𝑛∊ℤ 𝑥, ѱ 𝑚,𝑛 . ѱ 𝑚,𝑛[𝑡] Where : ѱ 𝑚,𝑛 𝑡 = 𝑠0 −𝑚/2 ѱ(𝑠0 −𝑚 𝑡 − 𝑛τ0).
  25. 25.  The Multi-Resolution Analysis : Analyzing a signal at different frequencies with different resolutions. By applying bank filters :
  26. 26. MRA time-frequency representation:  Good time resolution for high frequencies.  Good frequency resolution for low frequencies.
  27. 27. MRA time-frequency Representation of Chirp Signal
  28. 28.  Overview  Historical Development  Limitations of Fourier Transform  Principle of Wavelet Transform  Examples of Applications  Conclusion  References
  29. 29. FBI Fingerprints Compression: o Since 1924, the FBI Collected about 200 Million cards of fingerprints. o Each fingerprints card turns into about 10 MB, which makes 2,000 TB for the whole collection. Thus, automatic fingerprints identification takes a huge amount of time to identify individuals during criminal investigations.
  30. 30.  The FBI decided to adopt a wavelet-based image coding algorithm as a national standard for digitized fingerprint records.  The WSQ (Wavelet/Scalar Quantization) developed and maintained by the FBI, Los Alamos National Lab, and the National Institute for Standards and Technology involves: • 2-dimensional discrete wavelet transform DWT. • Uniform scalar quantization. • Huffman entropy coding.
  31. 31. JPEG 2000  Image compression standard and coding system.  Created by Joint Photographic Experts Group committee in 2000.  Wavelet based compression method.  1:200 compression ratio Mother Wavelet used in JPEG2000 compression
  32. 32. Comparison between JPEG and JPEG2000
  33. 33. Wavelets can be applied for many different purposes :  Audio compression.  Speech recognition.  Image and video compression  Denoising Signals  Motion Detection and tracking
  34. 34.  Overview  Historical Development  Limitations of Fourier Transform  Principle of Wavelet Transform  Examples of Applications  Conclusion  References
  35. 35. Wavelet is a relatively new theory, it has enjoyed a tremendous attention and success over the last decade, and for a good reason. Almost all signals encountred in practice call for a time-frequency analysis, and wavelets provide a very simple and efficient way to perform such an analysis. Still, there’s a lot to discover in this new theory, due to the infinite variety of non-stationary signals encountred in real life. Conclusion :
  36. 36.  Overview  Historical Development  Limitations of Fourier Transform  Principle of Wavelet Transform  Examples of Applications  Conclusion  References
  37. 37. References :  [1] :https://www. en.wikipedia.org/wiki/Wavelet.  [2] : Introduction to Wavelet : Bhushan D Patil PhD Research Scholar.  [3] : Introduction to Wavelets : Amara Graps : http:www.ieee.org.  [4] : The FBI Wavelet : Los Alamos National Laboratory – TOM Hopper –.  [5] : Some Applications of Wavelets : Anna Rapoport.  [6] : https://www.en.wikipedia.org/wiki/Daubechies_wavelet  [7] : http://en.wikipedia.org/wiki/Chirp  [8]: http://en.wikipedia.org/wiki/Haar_wavelet  [9]: http://en.wikipedia.org/wiki/Discrete_wavelet_transform  [10] : Chapter 2, Wavelets Theory and applications for manufacturing, by Gao, R.X.; Yan, R.

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