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Wavelets: theory and applications

GTDIR. Wavelets: theory and applications. An introduction. Grupo de Investigación: Tratamiento Digital de Imágenes Radiológicas. Enrique Nava, University of Málaga (Spain). Brasov, July 2006. What are wavelets?. Wavelet theory is very recent (1980’s)

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Wavelets: theory and applications

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  1. GTDIR Wavelets: theory and applications An introduction 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: • 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. Magnitude Magnitude Time Frequency (Hz) Magnitude Magnitude Time Frequency (Hz) 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

  16. Different in Time Domain Magnitude Magnitude Magnitude Magnitude Time Frequency (Hz) Time Frequency (Hz) Chirp signal • Frequency: 20 Hz to 2 Hz • Frequency: 2 Hz to 20 Hz 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. Narrow Window Wide Window Short-time Fourier Transform (STFT)

  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 • 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 Energy normalization

  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. From http://www.cerm.unifi.it/EUcourse2001/Gunther_lecturenotes.pdf, p.10 Comparison of transformations

  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. S S D1 A1 D2 A2 A3 D3 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

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

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

  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

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