1. Wavelet Transform Advanced Digital Signal Processing Lecture 12
SCHOOL of COMPUTING
UNIVERSITY OF EASTERn FINLAND
JOENSUU, FINLAND
Advanced Digital Signal Processing
Lecture 12
Wavelet Transform
Alexander Kolesnikov

2. Short-Term Fourier Transform again…

3. Spectrums of the test signals
Signals are different,
spectrums are similar
Why?

4. What is wrong with the Fourier Transform?
Two basis functions: sin(t) and (t)
Support region: In time In frequency
sin(t) 
(t) 
The basis function sin(t) is not localized in time,
The (t) (sample) is not localized in frequency.
Fourier Transform is good for stationary signals,
But it can not locate drift, trends, abrupt changes,
beginning and ends of events, etc.

5. Short-Term Fourier Transform (STFT)
The Gabor transform is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. The window function means that the signal near the time being analyzed will have higher weight. The mathematical definition is given below:
Short-Term Fourier Transform (STFT)
signal x(t)
window h(t-)
signal in window
Introduce basis functions which are compact in time and frequency domains.
Let us divide the input signal into time sub-intervals, and perform
DFT for every time sub-interval.

6. Gabor transfortm is a special case of STFT
Window is a Gaussian function:

7. STFT: Time-frequency plane
Equidistant frequencies:  t

8. Problems with STFT Uncertainty Principle:
We cannot localize events in time and frequency
simultaneously!
Improved time resolutionDegraded frequency resolution
Improved frequency resolution Degraded time resolution
Problem: the same  and t through the entire plane!

9. Continuous Wavelet Transform

10. The main idea
Use wider time window for lower frequencies!

11. Time-frequency plane for wavelets
Heisenberg cell

12. Our first wavelet transform:
Note that as alpha increases: the frequency decreases, and the window function expands.
Time shift now also in exponential. Can write:

13. Scale: Illustration Coast line Coarse scale: Flying in a jet at 5 km
Medium scale:
Bird flight at 100 m
Fine scale:
Beach walk

14. Wavelet mother & Baby function
Let we have a wavelet mother function (t).
How to obtain a set of wavelet functions?
Scaling (s) and Translation ()

15. Scaling (stretching or compressing), dilation

16. Translation (shift), 

17. Translation () and scaling (s)

18. Problems with Continuous Wavelet Transform
Redundancy of Continuous Wavelet Transform (CWT) .
The WT is calculated by continuously shifting of continuously
scalable function over a signal.
We still have an infinite number of wavelets in the wavelet
transform and we would like to reduce the number to a more
manageable count.
3) For the most functions the wavelet transform have no analytical
solution and they can be calculated only numerically. Fast algorithms
needed.

19. Discrete Wavelet Transform, DWT

20. Solution to the problems
To overcome the problems, Discrete Wavelets have been introduced.
Discrete Wavelets can be scaled and translated in discrete steps:
Here j and k are integer numbers, s0 is a fixed dilation step.
The translation factor depends on the dilation step.
The time-scale is sampled at discrete intervals.
We choose s0=1/2. This is natural choice for human ear, musics
and computers.
For the translation factor we choose 0=1.

21. Discrete Wavelets
...
...
k=0,1,...,

22. Discrete Wavelets

23. 23

24. Discrete Wavelets Normalization Orthogonality Expansion
Wavelet coefficients

25. Examples of wavelets

26. Haar function is the first wavelet (1909)
Scaling function

27. Approximation with Haar functions
Disadvantages of Haar wavelet: discontinuous and does not approximate continuous signals very well. k

28. Approximation with Haar functionsk

29. Direct approach: Example
Input data: x(n)=(2,5,8,9,7,4,-1,1)
0:
0,0:
1,0,1,1:
2,0, 2,1, 2,2 2,3 :

30. Forward Haar Transform: Analysis
Input data: X={2,5,8,9,7,4,-1,1}.
Haar wavelet transform: (a,b)(s,d) where:
s is for smooth component (LPF)
d is for details (HPF)
1st step:

31. Forward Haar Transform: Analysis
2nd step:
3rd step:

32. Forward Haar Transform: Analysis

33. Backward Haar Transform: Synthesis
1st step:

34. Backward Haar Transform: Synthes
2nd step:

35. Backward Haar Transform: Synthes
3rd step:

36. Haar Transform as subband filtering
LPF
HPF

37. Wavelet Transform and Filter Banks

38. Wavelet Transform and Filter Banks
Decimation in time means stretching in frequency domain

39. Wavelet Transform and Filter Banks
h0(n) is scaling function, low pass filter (LPF)
h1(n) is wavelet function, high pass filter (HPF)
is subsampling (decimation)

40. System of filters
Effectively, the DWT is nothing but a system of filters. There are two filters involved, one is the “wavelet filter”, and the other is the “scaling filter”. The wavelet filter, is a high pass filter, while the scaling filter is a low pass filter.
h0(n)
Level 1
h0(n)
h1(n)
h0(n)
h1(n)
Level 2
h1(n)
Level 3
Scaling Filter ~ Averaging Filter
Wavelet Filter ~ Details Filter
Pic from wikipedia.org

41. Inverse wavelet transform
is up-sampling (zeroes inserting)
In order to achieve perfect reconstruction the synthesis filters
should satisfy
g0(n) = h0(-n)
g1(n) = h1(-n)

42. Wavelet transform as Subband filtering
Discrete Wavelet Transform can be
Constructed via iterated
(octave-band) filter bank

43. Subspaces of DWT

44. Complexity of Discrete Wavelet Transform

45. Dilation equations or Two-scale relation
Scaling function:
Wavelet function:
The functions (t) and (t) are orthogonal:

46. Scaling (father) function and Wavelets
The most important property of the wavelets:
To obtain WT coefficients for level j we need to process
only WT coefficients for level j+1.

47. Haar: Scaling function and Wavelets

48. French Hat (piecewise linear spline) wavelet

49. French Hat wavelets

50. Daubechies wavelets D4
Haar=D1

51. Daubechies wavelets

52. 2-D Wavelet transform in JPEG2k
Horizontal filtering
Vertical filtering

53. 2-D wavelet transform Original Transform Coeff.
128, 129, 125, 64, 65, …
Transform Coeff.
4123, -12.4, -96.7, 4.5, …

54. 2-D wavelet transform
LL3
HH4
LH2
HH3
LH1
HL2
HH2
HL1
HH1

55. JPEG 2000 vs JPEG
DCT WT

56. JPEG 2000 vs JPEG: Quantization

57. JPEG 2000 vs JPEG: 0.3 bpp
JPEG
JPEG 2000

58. JPEG 2000 vs JPEG: Bitrate=0.3 bpp
MSE= MSE=73
PSNR=26.2 db PSNR=29.5 db

59. JPEG 2000 vs JPEG: Bitrate=0.2 bpp
MSE= MSE=113
PSNR=23.1 db PSNR=27.6 db

60. Examples of Wavelets

61. Examples of Wavelets

64. Denoising with Wavelets

65. Denoising with Wavelets

66. Wavelets with Matlab

67. Denoising with wavelets
See in Matlab: wavemenu ->SWT Denoising 1-D

68. Denoising with wavelets

69. Denoising with wavelets

70. Denoising with wavelets

71. Denoising with wavelets

72. History

73. History