1. Wavelets: theory and applications
An introduction
GTDIR
Grupo de Investigación:
Tratamiento Digital de
Imágenes Radiológicas
Enrique Nava, University of Málaga (Spain)
Brasov, July 2006

2. What are wavelets? Wavelet theory is very recent (1980’s)
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

3. Overview Spectral analysis Continuous Wavelet Transform
Discrete Wavelet Transform
Applications
A wavelet tour of signal processing, S. Mallat, Academic Press 1998

4. Spectral analysis: frequency
Frequency (f) is the inverse of a period (T).
A signal is periodic if T>0 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)

5. Signals: examples

6. Signals: examples

7. 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’

8. Fourier series
1822

9. 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)

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

11. Fourier Transform The signal has a frequency point of view (spectrum)
Global representation
Lots of math properties
Linear operators

12. Discrete Fourier Transform
Practical implementation
Global representation
Lots of math properties
Linear operators
Easy discrete implementation (1965) (FFT)

13. Fourier transform

14. Random signals Stationary signals: Non-stationary signals:
Statistics don’t change with time
Frequency contents don’t change with time
Information doesn’t change with time
Non-stationary signals:
Statistics change with time
Frequencies change with time
Information quantity increases

15. Non-stationary signals
Time
Magnitude
Frequency (Hz)
2 Hz + 10 Hz + 20Hz
Stationary
Time
Magnitude
Frequency (Hz)
: 2 Hz +
: 10 Hz +
: 20Hz
Non-Stationary

16. Different in Time Domain
Chirp signal
Frequency: 2 Hz to 20 Hz
Frequency: 20 Hz to 2 Hz
Time
Magnitude
Frequency (Hz)
Different in Time Domain
Same in Frequency Domain

17. Fourier transform drawbacks
Global behaviour: we don’t 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

18. Short-time Fourier Transform (STFT)
Dennis Gabor (1946): “windowing the signal”
Signals are assumed to be stationally local
A 2D transform

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

20. Short-time Fourier Transform (STFT)

21. Short-time Fourier Transform (STFT)

22. Short-time Fourier Transform (STFT)
Narrow Window
Wide Window

23. 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

24. 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

25. Heisenberg Uncertainty Principle
Heisenberg Box

26. 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

27. Continuous Wavelet Transform
Definition:
Translation
(The location of the window)
Scale
Mother Wavelet

28. 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
S>1: dilated signal
S<1: compressed signal
Translation
Shifting of the signal

29. CWT practical computation
Energy normalization
Select s=1 and t=0.
Compute the integral and normalize by 1/
Shift the wavelet by t=Dt and repeat until wavelet reaches the end of signal
Increase s and repeat steps 1 to 3

30. 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

31. Comparison of transformations
From p.10

32. Mathematical view
CWT is the inner product of the signal and the basis function

33. Wavelet basis functions
2nd derivative of a Gaussian
is the Marr or Mexican hat wavelet

34. Wavelet basis functions
Frequency domain
Time domain

35. Wavelet basis properties

36. Discrete Wavelet Transform
Continuous Wavelet Transform
Discrete Wavelet Transform

37. Discrete CWT Sampling of time-scale (frequency) 2D space
Scale s is discretized in a logarithmic way
Scheme most used is dyadic: s=1,2,4,8,16,32
Time is also discretized in a logarithmic way
Sampling rate N is decreased so sN=k
Implemented like a filter bank

38. Discrete Wavelet Transform
Approximation
Details

39. Discrete Wavelet Transform

40. Discrete Wavelet Transform
Multi-level wavelet decomposition tree
Reassembling original signal

41. Discrete Wavelet Transform
Easy and fast to implement
Gives enough information for analysis and synthesis
Decompose the signal into coarse approximation and details
It’s not a true discrete transform S A1 A2 D2 A3 D3 D1

42. Examples fL Signal: 0.0-0.4: 20 Hz 0.4-0.7: 10 Hz 0.7-1.0: 2 Hz
Wavelet: db4
Level: 6
Signal:
: 20 Hz
: 10 Hz
: 2 Hz fH fL

43. Examples fL Signal: 0.0-0.4: 2 Hz 0.4-0.7: 10 Hz 0.7-1.0: 20Hz
Wavelet: db4
Level: 6
Signal:
: 2 Hz
: 10 Hz
: 20Hz fH fL

44. 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

45. Synthesis technique
Upsampling (insert zeros between samples)

46. Sub-band algorithm
Each step divides by 2 time resolution and doubles frequency resolution (by filtering)

47. Wavelet packets Generalization of wavelet decomposition
Very useful for signal analysis
Wavelet analysis: n+1 (at level n) different ways to reconstuct S

48. Wavelet packets We have a complete tree
Wavelet packets: a lot of new possibilities to reconstruct S:
i.e. S=A1+AD2+ADD3+DDD3

49. 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)

50. 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

51. Wavelet families: Daubechies
Compact support, orthonormal (DWT)

52. Other families

53. Matlab wavemenu command

54. 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)

55. De-noising signals Frequency is higher at the beginning
Details reduce with scale

56. De-noising images

57. Detecting discontinuities

58. Detecting discontinuities

59. Detecting self-similarity

60. Compressing images

61. 2-D Wavelet Transform

62. Wavelet Packets

63. 2-D Wavelets

64. 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

65. Practical use of wavelet
Wavelet software
Matlab Wavelet Toolbox
Free software
UviWave
Wavelab
Rice Tools

66. Useful Links to continue
Matlab wavelet tool using guide
waveletcourse/sig95.course.pdf