1. 25/03/08

2. Outline Introduction The Fourier Transform
The Heisenberg Uncertainty Principle
The Windowed Fourier Transform
Wavelets History
Wavelets Basic Theory
The Continuous Wavelet Transform (CWT)
The Discrete Wavelet Transform (DWT) & MRA
Applications (1D & 2D)
Conclusion
Bibliography

3. Introduction

4. Introduction Music: a “time-scale game” Frequency of the notes Time
FR: Partition / US: Score
Time
Score Example

5. Introduction Fourier Analysis Wavelet Analysis Information :
Frequency of the notes
(Temporal information hidden)
Information :
Frequency of the notes
Time instants of the notes

6. A possible solution: use of the Wavelet Transform…
Introduction
Problem: how can we see the difference between two notes played simultaneously and two notes played one after the other ?
A possible solution: use of the Wavelet Transform…

7. Introduction

8. Magnitude of the Fourier Transform of s(t)
Introduction
Magnitude of the Fourier Transform of s(t)
points

9. Magnitude of the Wavelet Transform of s(t)
Introduction
Magnitude of the Wavelet Transform of s(t)

10. Introduction

11. Magnitude of the Fourier Transform of s(t) (“ringing” artifacts)
Introduction
Magnitude of the Fourier Transform of s(t)
Gibbs phenomenon
(“ringing” artifacts)
points

12. Magnitude of the Wavelet Transform of s(t)
Introduction
Magnitude of the Wavelet Transform of s(t)

13. Introduction

14. The Fourier Transform

15. The Fourier Transform
Fourier analysis (Fourier, J-B. J ) is one of the most known signal processing tool to study stationary signals.
It gives precise frequency information about a time-domain signal thanks to its decomposition basis along waves having precise frequency (sines and cosines).
stationary signals ~ continuous signals

16. The Fourier Transform
Its mathematical structure is very suitable to perform linear filtering operations (transfer function).
It has led to many algorithms (e.g. FFT) and many softwares, showing its actual notoriety of use.
But, Some limitations occur when it is about analyzing signals with local discontinuities such as peaks.
It is difficult to analyze high and low frequencies simultaneously…

17. The Fourier Transform
A “mathematical prism”: the Fourier Transform is a mathematical operation that decomposes a function according to its frequencies, just like the prism decomposes the light.
(direct)
S(f)
s(t)
(inverse)
The new function S(f) shows how many sines and cosines
are present in the original function s(t).

18. The Fourier Transform
A global transform: It cannot analyze the local frequency content or local regularity of a signal.
Phase: Temporal information is hidden in the phases (offsets between sinusoids).
In the Fourier Transform, it is difficult to calculate those phase-coefficients with sufficient accuracy to recover temporal information.
Non-causality: We must know the entire time-domain signal to be able to compute its Fourier Transform (FT).
“On-the-fly” computation and real-time analysis are impossible !

19. The Heisenberg Uncertainty Principle

20. The Heisenberg Uncertainty Principle
In quantum physics, the Heisenberg uncertainty principle states that:
This can be stated exactly as:
where is the uncertainty in position,
is the uncertainty in momentum,
h is the Planck's constant ( x J.s).
“ The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. ”
Heisenberg, W., uncertainty paper, 1927.
Werner Heisenberg

21. The Heisenberg Uncertainty Principle
In Schrödinger's wave mechanics, the quantum mechanical wave function contains information about both the position and the momentum of the particle. The position of the particle is where the wave is concentrated, while the momentum is the typical wavelength.
In signal processing, This is an exact counterpart to a well known result: the shorter a pulse in time, the less well defined the frequency. The width of a pulse in frequency space is inversely proportional to the width in time.

22. The Heisenberg Uncertainty Principle
The signal is well localized in time.
The FT is delocalized
in frequency.

23. The Heisenberg Uncertainty Principle
The signal is delocalized in time.
The FT is centered around zero, thus well localized in frequency.

24. The Heisenberg Uncertainty Principle
It is a fundamental result in Fourier analysis:
“The narrower the peak of a function, the broader the Fourier transform”.
Localizations of f and are related to the Heisenberg Uncertainty Principle which precise the link between the variances of f and .
It constraints the product of the dispersions and , in time and in frequency, respectively.

25. The Heisenberg Uncertainty Principle
This can be stated exactly as: , where: ,
They quantify the dispersions of and about their means µ and ξ, respectively, given by:

26. The Heisenberg Uncertainty Principle
Time-frequency Heisenberg Boxes in the Fourier Basis

27. ~ Problem ~ We wish to make Fourier local… but how ?

28. The Windowed Fourier Transform

29. The Windowed Fourier Transform
The Windowed Fourier Transform (Gabor, D ) is also known as Short-Time (or -Term) Fourier Transform (STFT):
Analysis window
Dennis Gabor
1971 Nobel prize for the invention of holography,
first suggested to make Fourier analysis local,
first introduced “time-frequency” wavelets (Gabor wavelets).

30. The Windowed Fourier Transform
The STFT makes it possible to analyze a signal x(t) in time and in frequency simultaneously, we talk about Time-Frequency Transform (TFT).
The idea is to perform a Fourier Transform inside a window that will be translated along the signal.
What is a window ?: a window (or envelope) is a function g(t), smooth, slowly variable and well localized in time. Its graphical representation is a portion of curve which delimitate an area containing oscillations.
In general, we choose the window g(t) even and real.

31. The Windowed Fourier Transform
When the window g(t) is a Gaussian function, we talk about the Gabor Transform.
time
Two “gaborettes” with different oscillations

32. The Windowed Fourier Transform
The size of the window (temporal support) is related to the size of the interval that will be analyzed. It doesn’t change during the process but it is filled with oscillations at different frequencies.
Like the Fourier Transform, it is possible to reconstruct the original signal x(t) with the help of the coefficients obtained during the analysis. The inversion formula (synthesis) is immediate and is given by:
where c>0 is a numerical constant
(for us, the value is not important here)

33. The Windowed Fourier Transform
Time-frequency Heisenberg Boxes in the STFT Basis.
Left: the window is narrow. Right: the window is broader.

34. ~ Problems ~ Since the size of the analysis window does not change during the STFT process, we have to make a compromise when analyzing different frequencies. A small window allows the analysis of transient components of a signal, i.e. high frequencies. A broader window allows the analysis of low frequencies. We cannot analyze high and low frequencies simultaneously !

35. ~ Solution ~ Find an analysis where the size of the window varies with the frequency… The Wavelets

36. Wavelets History

37. Wavelets History
We could go back in time and find origins of wavelets in At this time, in mathematics, wavelets were used under the name “atomic decompositions” to study different functional spaces. Some researchers have developed wavelets - under the name “autosimilar Gabor functions” - to model the human visual system.

38. Wavelets History
However, we usually take the year 1975 as the real starting point for the discovery of wavelets, and the works of the French geophysicist engineer Jean Morlet (from École Polytechnique), who worked for The French oil company Elf-Aquitaine and who invented time-scale wavelets to analyze the sound echoes used in oil prospecting.

39. Echoes : reflection signals
Wavelets History
Echoes : reflection signals
Pulse :
incoming signal
Ground
level 0
Interferences θ θ θ’ 1 2
θ’’
Oil layers of different thickness 3
θ’’’
Multiple Reflections

40. Wavelets History
The frequencies of those echoes are related to the thicknesses of oil layers.
High frequencies ≡ Thin layers
Problem: Difficulty to decorrelate all those reflection signals because they interfere a lot between them.
To extract information, J. Morlet first uses the STFT with windows of different sizes… Unsuccessfully ;-(
Thus, J. Morlet has a great idea: he fixes the number of oscillations inside the analysis window that he compresses or stretches like a accordion.
Wavelets were born !!!

42. Wavelets Basic Theory

43. Wavelets Basic Theory What is a wavelet ? (Graphical approach)
“wavelet” → small wave !
“The simplest transient signal we can imagine” (Y. Meyer)
A wavelet is a function that can be seen as:
FR: transitoire / US: transient
a fast-decaying
oscillating waveform
of finite-length
→ e.g. : Morlet Wavelet

44. Wavelets Basic Theory What is a wavelet ? (Mathematical context)
Let ψ be a carefully chosen function, regular and localized. This function * will be called wavelet if it verifies the following admissibility condition in the frequency space:
where is the Fourier Transform of .
→ The integral of the wavelet is null.
* L1 and L2 are the spaces of integrable functions and finite energy functions, respectively.

45. Wavelets Basic Theory What is a wavelet ? (Mathematical context)
We often want the wavelet to have (m+1) vanishing moments (oscillations):
A sufficient admissibility condition, easier to verify, can be written as follows:

46. Wavelets Basic Theory
There exist many different functions ψ, called mother-wavelets (prototypes).
Some of them have explicit mathematical formulas:
Morlet Wavelet
Mexican-hat Wavelet

47. Wavelets Basic Theory
Others are built upon more complex mathematical properties:
Meyer Wavelet
Daubechies Wavelet (‘db2’)

48. Wavelets Basic Theory Each wavelet has its own properties:
Symmetry: useful to avoid out-of-phase phenomenon,
Vanishing moments: useful for compression,
Regularity: useful to obtain smooth and regular reconstructed signals or images,
etc.

49. The Continuous Wavelet Transform (CWT)

50. CWT (1-D)
The (continuous) wavelet transform replaces the Fourier Transform's sinusoidal waves by a family (base atoms) generated by translations and dilatations of a mother-wavelet ψ.
where b is the translation parameter (time),
and a is the compression / dilatation parameter (scale).

51. Wavelet Time-Shifting (b parameter)
CWT (1-D)
Wavelet Time-Shifting (b parameter)
Amplitude
Time
-10 -5 +5
+10

52. CWT (1-D) Wavelet Compression / Dilatation (a parameter) Time
In order to have the same energy at each scale a, the wavelet is modified in amplitude ( ).
Amplitude 1
Dilatation -1
Base scale
Compression -2
Time

53. CWT (1-D)
The CWT of a function is defined by:

54. Example of Heisenberg boxes of wavelet atoms.
CWT
The previous function is centered around b. If the frequency center of ψ is η, then the frequency center of the dilated function is η /a.
Its time spread is proportional to a. Its frequency spread is proportional to the inverse of a.
Example of Heisenberg boxes of wavelet atoms.

55. CWT (1-D)
At the finer scales, more Heisenberg boxes can be placed side to side because there is a better time resolution.

56. CWT (1-D)
Properties
The wavelet transform has thus a time-frequency resolution which depends on the scale a. Under the (admissibility) condition:
It is a complete, stable and redundant representation of the signal; in particular, the wavelet transform is left invertible.

57. CWT (1-D)
Scalogram
If η denotes the frequency center of the base wavelet, then the frequency center of a dilated wavelet is ξ = η /a.
The scalogram of a signal is defined by:
The normalized scalogram is defined by:

58. CWT (1-D) What it looks like to perform the CWT in 1-D ?
Mother wavelet ψ: Morlet wavelet
Signal to be analyzed (e.g. 40 points):
The signal to be analyzed has voluntary been generated with a portion of curve very close to the shape of the mother-wavelet in order to underline the computation and the value of the wavelet coefficients.

59. CWT (1-D)
For the Morlet wavelet, the analysis window is Gaussian but to simplify the understanding of the animation, we will model the window by a simple rectangle.
The CWT is a continuous transform, thus we should perform the algorithm on each point of the signal. Of course, we will only show some steps !
Rectangular analysis window

60. ∫ π → one wavelet coefficient:
Signal to be analyzed
Time
The wavelet is centered at the beginning of the signal
π : signal x wavelet (inside the window)
The wavelet is shifted to the right
∫ π → one wavelet coefficient:
End of the algorithm for a=1 → 1st row of coefficients.

61. Scalogram
Scale (a)
Amplitude 10 25 40 1 +
Time (b)

62. Same operations → Scale Change → Wavelet Dilatation.
Time
Same operations → Scale Change → Wavelet Dilatation.

63. Scalogram
Scale (a)
Amplitude 10 25 40 1 + 2
Time (b)

64. Same operations → Scale Change → Wavelet Compression
Time
Same operations → Scale Change → Wavelet Compression
ETC…

65. The Discrete Wavelet Transform (DWT)

66. DWT (1-D)
The DWT has been designed to perform fast algorithms by discretizing the continuous form.
For convenience, in the discretization, we restrict a and b to the following dyadic values:
where are the sampling steps,
where (resolution: 2j, frequency: 2-j).
Thus, the set of functions constitutes an orthonormal basis of :

67. DWT (1-D)
The DWT is a spatial-frequency decomposition that provides a flexible multiresolution analysis of the image. In one dimension, the aim of the wavelet transform is to represent the signal as a superposition of wavelets.
Let f(x) be a discrete signal, its wavelet decomposition is then:
where j and k are integers. This ensures that the signal is decomposed into normalized wavelets at octave scales (when an octave is reached, the frequency doubles).

68. DWT (1-D)
For an iterated wavelet transform, additional coefficients aj,k are required at each scale.
At each scale aj,k and aj-1,k describe the approximations of the function f at resolution 2j and at the coarser resolution 2j-1, respectively.
At each scale, the coefficients cj,k describe the difference between one approximation and the other.
In order to obtain the coefficients cj,k and aj,k at each scale, and position, a scaling function is needed that is similarly defined to the previous equation.

69. A single stage wavelet analysis and synthesis in one dimension.
DWT (1-D)
A single stage wavelet analysis and synthesis in one dimension.
h: low-pass analysis filter, : low-pass synthesis filter
g: high-pass analysis filter, : high-pass synthesis filter

70. DWT (2-D)
To extend the wavelet transform to two dimensions, it is just necessary to separate filter and downsample in the horizontal and vertical directions.
This produces four subbands at each scale. Denoting the horizontal frequency first and then the vertical frequency second, this produces low-low (LL), low-high (LH), high-low (HL) and high-high (HH) image subbands.
By recursively applying the same scheme to the LL subband, a multiresolution decomposition can be achieved.

71. DWT (2-D): Multiresolution Analysis (MRA)
One stage of the 2-D DWT decomposition.

72. DWT (2-D): Multiresolution Analysis (MRA)
2nd resolution level
Approximation
at level 1
Horizontal details
Vertical details
Diagonal details
1st resolution level
Normal layout of the 2D-DWT.
At each scale, the subbands are sensitive to frequencies at that scale
and the LH, HL and HH subbands are sensitive to horizontal, vertical and diagonal frequencies respectively. The sizes of frequency bands will decrease as the decomposition goes on.

74. Applications

75. Applications Analysis Denoising Compression
Wavelets are very efficient to solve 3 classical problems in signal processing
Analysis
Denoising
Compression

76. Applications 1-D Analysis (linear chirp)
“Linear chirp”: signal which frequency varies linearly with time.
sig = fmlin(256,0.1,0.4)
(Shannon normalized)
ESD
Linear evolution of the frequency with time,
by a simple observation of the scalogram

77. Applications 1-D Denoising
Definition: Recover a signal from observations corrupted by an additive noise (N samples, standard deviation: ).
Principle: We transform the signal into the wavelet domain then we select, by a thresholding method (‘hard’ or ‘soft’), coefficients from which the final signal is reconstructed in the time-domain, using an inverse wavelet transform.
Universal Threshold (Donoho, D., Stanford University)
Example: Electric consumption signal denoising

78. Applications 1-D Denoising (Electric consumption signal) Noised signal
Linear evolution of the frequency with the time,
by a simple observation of the scalogram

79. Wavelet denoised signal
Applications
1-D Denoising (Electric consumption signal)
Wavelet denoised signal
Linear evolution of the frequency with the time,
by a simple observation of the scalogram

80. Applications 2-D Compression
Definition: Reduce the size of data while maintaining their integrity and quality as high as possible.
Principle: Close to the denoising principle. Wavelets are, in general, able to concentrate in few non-null coefficients, the most important part of the energy of a signal, hence the compression phenomena.
Example: Image Compression (JPEG vs. JPEG 2000)

81. Applications 2-D Compression (JPEG vs. JPEG 2000)
JPEG: based on the Discrete Cosine Transform (DCT)
8x8 blocks analysis of the image,
“Pixellization” effects ( more and more visible as the compression ratio increases !).
JPEG 2000: based on the Wavelet Transform
Global analysis of the image,
Very good quality even at a high compression ratio,
Possibility to calculate the size of the compressed image,
Progressive display of the image during the reconstruction.

82. Applications 2-D Compression (JPEG vs. JPEG 2000) Original image
256*256
24-Bit RGB
JPEG
(Ratio 43:1)
JPEG 2000
(Ratio 43:1)

83. Applications 2-D Compression - Performance Original image 512*512
24-Bit RGB
786 Ko
1st compression
(Ratio 75:1)
10.6 Ko
2nd compression
(Ratio 150:1)
5.3 Ko
3rd compression
(Ratio 300:1)
2.6 Ko

84. Applications Others Fields Geophysics (seism detection),
Medicine (ECG, EEG,…),
Satellite imagery,
Video encoding (divX 4/5),
Internet traffic modeling,
Cryptography,
Optoelectronics,
Biometrics, …

85. Conclusion

86. Conclusion: What we have to remember
New Technique in signal processing (Birth : 1975, Morlet, J.),
Local analysis of a signal at different scales:
« Multiresolution analysis (MRA) »
Adaptability to the different components of a signal:
« Wavelets = Mathematical Microscope »
Wavelet Transform → Time-scale analysis (time:b, scale:a)
Transformation of a signal into numerical coefficients:
« Wavelet Coefficients: Cf (a,b) »

87. Conclusion: What we have to remember
There exist many wavelets and one can construct his own wavelet,
Many possible applications in various fields,
Nowadays, several variants of the wavelets (ridgelets, curvelets, contourlets, bandelets, etc.)
JPEG octets
(wavelets)
Original Image
(~ 100 Ko)
Let It Wave octets
(bandelets)

88. Conclusion: Wavelets’ Hall of Fame
Jean Morlet
Alex Grossmann
Ecole Polytechnique (X-1954)
French Geophysicist Engineer (Elf-Aquitaine)
Inventor of the Time-scale Wavelets
Mathematician
1984: proved that the inverse wavelet transform is exact (ε=0)
Wavelets ≡ Fourier local

89. Conclusion: Wavelets’ Hall of Fame
Yves Meyer
Ingrid Daubechies
Fellow Professor at École Normale Supérieure de Cachan (ENS-CMLA)
French Academy of Sciences Member since 1993
Discovered orthogonal wavelets (Meyer wavelet)
Professor at Princeton University, Department of Mathematics
Discovered a compactly-supported wavelets family
(Daubechies wavelets, ‘dbX’)

90. Conclusion: Wavelets’ Hall of Fame
Stéphane Mallat
Professor at École Polytechnique (CMLA)
Invented a fast transform algorithm (with Y. Meyer)
Start-up: “Let It Wave” → Wavelets Applications in imaging

91. Bibliography: Books & Papers
[1] : Barbara Burke Hubbard, « Ondes et ondelettes – La saga d’un outil mathématique »,
Belin Pour la Science – 2000.
[2] : Michel Misiti, Yves Misiti, Georges Oppenheim, Jean-Michel Poggi,
« Les ondelettes et leurs applications », Hermes Science Publications – 2003.
[3] : Ingrid Daubechies, « Ten Lectures on Wavelets », SIAM – 1999.
[4] : Stéphane Mallat, « A Wavelet Tour of Signal Processing, 2nd Edition »,
Academic Press – 1999.
[4] : Stéphane Jaffard, Yves Meyer, Robert D. Ryan, « Wavelets – Tools for Science & Technology », SIAM – 2001.
[5] : Nicolas Morizet, « Initiation aux ondelettes », Revue de l’Electricité et de l’Electronique (REE) – REE 2006 Best Paper Award.

92. Bibliography: Websites
French Websites
[1] : La Recherche: – édition du 01/02/2005.
[3] : Première Start-Up sur les « bandelettes » :
[4] : Forum sur les ondelettes:
[5] : Un enseignement sur les ondelettes:
English Websites
[6] : Wavelet Digest:
[7] : An excellent MATLAB Toolbox to practice wavelet analysis:
[8] : The Wavelet Tutorial – An introduction to wavelet analysis:
[9] : JPEG – JPEG 2000:

93. End of the Lecture
Thank You Questions ?
Short Answers Test ?...