1. With Applications in Image Processing
The Story of Wavelets
Polikar, (C) 2007
With Applications in Image Processing
Guest Lecture
Dr. Robi Polikar

2. Outline Polikar, (C) 2007 Review of Fourier based methods
Analyzing non-stationary signals
Short-time Fourier transform
Continuous wavelet transform
Discrete wavelet transform
Applications
Discontinuity detection
Signal denoising
Compression
2-D wavelet transform
Image analysis and compression
Polikar, (C) 2007

3. What is a Transform and Why Do we Need One ?
Transform: A mathematical operation that takes a function or sequence and maps it into another one
Transforms are good things because…
The transform of a function may give additional /hidden information about the original function, which may not be available /obvious otherwise
The transform of an equation may be easier to solve than the original equation (recall Laplace transforms for “Defeques”)
The transform of a function/sequence may require less storage, hence provide data compression / reduction
An operation may be easier to apply on the transformed function, rather than the original function (recall convolution)
Polikar, (C) 2007

4. Properties of Transforms
Most useful transforms are:
Linear: we can pull out constants, and apply superposition
One-to-one: different functions have different transforms
Invertible:for each transform T, there is an inverse transform T-1 using which the original function f can be recovered (kind of – sort of the undo button…)
Continuous transform: map functions to functions
Discrete transform: map sequences to sequences
Semi-discrete transform: relate functions with sequences
Polikar, (C) 2007 T f F
T-1

5. What Does a Transform Look Like…?
Complex function representation through simple building blocks
Compressed representation through using only a few blocks (called basis functions / kernels)
Sinusoids as building blocks: Fourier transform
Frequency domain representation of the function
Polikar, (C) 2007

6. What Transforms Are Available?
Fourier series
Continuous Fourier transform
Discrete Fourier transform
Laplace transform
Z-transform
Polikar, (C) 2007

7. How Does FT Work Anyway? Polikar, (C) 2007
Recall that FT uses complex exponentials (sinusoids) as building blocks.
For each frequency of complex exponential, the sinusoid at that frequency is compared to the signal.
If the signal consists of that frequency, the correlation is high  large FT coefficients.
If the signal does not have any spectral component at a frequency, the correlation at that frequency is low / zero,  small / zero FT coefficient.
Polikar, (C) 2007

8. FT At Work
Polikar, (C) 2007

9. FT At Work F
Polikar, (C) 2007 F F

10. FT At Work
Polikar, (C) 2007 F

11. Stationary and Non-stationary Signals
FT identifies all spectral components present in the signal, however it does not provide any information regarding the temporal (time) localization of these components. Why?
Stationary signals consist of spectral components that do not change in time
all spectral components exist at all times
no need to know any time information
FT works well for stationary signals
However, non-stationary signals consists of time varying spectral components
How do we find out which spectral component appears when?
FT only provides what spectral components exist , not where in time they are located.
Need some other ways to determine time localization of spectral components
Polikar, (C) 2007

12. Stationary and Non-stationary Signals
Stationary signals’ spectral characteristics do not change with time
Non-stationary signals have time varying spectra
Polikar, (C) 2007
Concatenation

13. Non-stationary Signals
5 Hz
20 Hz
50 Hz
Polikar, (C) 2007
Perfect knowledge of what
frequencies exist, but no
information about where
these frequencies are
located in time

14. FT Shortcomings Polikar, (C) 2007
Complex exponentials stretch out to infinity in time
They analyze the signal globally, not locally
Hence, FT can only tell what frequencies exist in the entire signal, but cannot tell, at what time instances these frequencies occur
In order to obtain time localization of the spectral components, the signal need to be analyzed locally
HOW ?
Polikar, (C) 2007

15. Short Time Fourier Transform (STFT)
Choose a window function of finite length
Put the window on top of the signal at t=0
Truncate the signal using this window
Compute the FT of the truncated signal, save.
Slide the window to the right by a small amount
Go to step 3, until window reaches the end of the signal
For each time location where the window is centered, we obtain a different FT
Hence, each FT provides the spectral information of a separate time-slice of the signal, providing simultaneous time and frequency information
Polikar, (C) 2007

16. STFT Example
300 Hz Hz Hz Hz
Polikar, (C) 2007

17. STFT Example
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18. STFT Example
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a=0.01

19. STFT Example
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a=0.001

20. STFT Example
Polikar, (C) 2007
a=0.0001

21. STFT Example
Polikar, (C) 2007 a=

22. Time – Frequency Resolution
Time – frequency resolution problem with STFT
Analysis window dictates both time and frequency resolutions, once and for all
Narrow window  Good time resolution
Narrow band (wide window)  Good frequency resolution
When do we need good time resolution, when do we need good frequency resolution?
Polikar, (C) 2007

23. Heisenberg Principle Polikar, (C) 2007
Time resolution: How well two spikes in time can be separated from each other in the transform domain
Frequency resolution: How well two spectral components can be separated from each other in the transform domain
Both time and frequency resolutions cannot be arbitrarily high!!!  We cannot precisely know at what time instance a frequency component is located. We can only know what interval of frequencies are present in which time intervals

24. The Wavelet Transform Polikar, (C) 2007
Overcomes the preset resolution problem of the STFT by using a variable length window
Analysis windows of different lengths are used for different frequencies:
Analysis of high frequencies Use narrower windows for better time resolution
Analysis of low frequencies  Use wider windows for better frequency resolution
This works well, if the signal to be analyzed mainly consists of slowly varying characteristics with occasional short high frequency bursts.
Heisenberg principle still holds!!!
The function used to window the signal is called the wavelet
Polikar, (C) 2007

25. Polikar, (C) 2007
1:44,500,000
1:2,500,000
1:375,500
1:62,500

26. Continuous Wavelet Transform
translation
Mother wavelet
Polikar, (C) 2007
Normalization factor
Scaling:
Changes the support of the wavelet based on the scale (frequency)
CWT of x(t) at scale
a and translation b
Note: low scale  high frequency

27. Why Wavelet? Polikar, (C) 2007
We require that the wavelet functions, at a minimum, satisfy the following:
Polikar, (C) 2007
Wave…
…let

28. Time-Frequency Grid in CWT
Scale
Polikar, (C) 2007
a=1/4
4Δf
Δt/4
a=1/2
Δt/2
2Δf
a=1 Δt Δf
a=2
2Δt
Δf/2
4Δt
Time
Δt * Δf ≥ ½

29. Matlab Demos on CWT
Polikar, (C) 2007

30. Discretization of Time & Scale Parameters
Recall that, if we use orthonormal basis functions as our mother wavelets, then we can reconstruct the original signal by
where W(a,b) is the CWT of x(t)
Q: Can we discretize the mother wavelet a,b(t) in such a way that a finite number of such discrete wavelets can still form an orthonormal basis (which in turn allows us to reconstruct the original signal)? If yes, how often do we need to sample the translation and scale parameters to be able to reconstruct the signal?
A: Yes, but it depends on the choice of the wavelet!
Polikar, (C) 2007

31. Dyadic Grid Polikar, (C) 2007
We do not have to use a uniform sampling rate for the translation parameters, since we do not need as high time sampling rate when the scale is high (low frequency).
Let’s consider the following sampling grid:
Polikar, (C) 2007 b
where a is sampled on a log scale, and b is sampled at a higher rate when a is small, that is,
where a0 and b0 are constants, and j,k are integers.
log a

32. An example on MRA Polikar, (C) 2007 f(t) f0(t) f2(t)
Let’s suppose we wish to approximate the function f(t) using a simpler function that has constant values over set intervals of 2j, such that the error ||f(t)-fj(t)|| is minimum for any j. The best approx. would be the average of the function over the interval length. In general, then…
f(t)
j=-
Polikar, (C) 2007
j=-2
j=-1
f0(t)
j=0
j=1
f2(t)
j=2
j=4
j=5
j=

33. An Example (cont.) Polikar, (C) 2007
As we go from a value of j to a higher value, we obtain a coarser approximation of the function, hence some detail is lost.
Let’s denote the detail lost at each approximation level with a function g(t):
Polikar, (C) 2007
Detail function
Approximation function
That is, the original function can be reconstructed simply by adding all the details (!)

34. Scaling & Wavelet Functions
The scaling function
Smoothed, scaling-function-dependent approximation of x(t) at scale j
Approximation coefficients at scale j
Polikar, (C) 2007
The wavelet function
Wavelet-function-dependent detail of x(t) at scale j
Detail coefficients at scale j
Furthermore:
&
Wavelet Reconstruction

35. Putting It All Together (cont.)
We can start reconstruction from any level (scale) in between:
Polikar, (C) 2007
Approximation coefficients at some arbitrary scale j0
Detail coefficients at scale j0 and below
Wavelet-function-dependent details of x(t) at scales j0 and below
Smoothed, scaling-function-dependent approximation of x(t) at the scale j0
The essence of wavelet transform: A signal can be obtained from (or decomposed into) an approximation at a specific level, and sum of details lost up to that level.

36. Discrete Wavelet Transform ImplementationG H 2 +
x[n]
Decomposition
Reconstruction ~
Polikar, (C) 2007 2
Down-sampling
Up-sampling G
Half band high pass filter
Half band low pass filter H 2
2-level DWT decomposition. The decomposition can be continues as long as there are enough samples for down-sampling.

37. DWT - Demystified Polikar, (C) 2007 g[n] h[n] g[n] h[n] g[n] h[n]
Length: 512
B: 0 ~ 
|H(jw)| w
/2
-/2
g[n]
h[n]
Length: 256
B: 0 ~ /2 Hz
Polikar, (C) 2007
Length: 256
B: /2 ~  Hz 2 2 a1
d1: Level 1 DWT
Coeff.
|G(jw)| w  -
/2
-/2
g[n]
h[n]
Length: 128
B: 0 ~  /4 Hz
Length: 128
B: /4 ~ /2 Hz 2 2 a2
d2: Level 2 DWT
Coeff.
g[n]
h[n] 2 2
Length: 64
B: 0 ~ /8 Hz
Length: 64
B: /8 ~ /4 Hz
…a3….
d3: Level 3 DWT
Coeff.
Level 3 approximation
Coefficients

38. Matlab Implementation Command Line
Single level decomposition and reconstruction
[cA1 cD1]=dwt(x, ‘db4’) single level decomposition
x=idwt(cA1, cD1, ‘db4’):3 single level inverse transform
Direct reconstruction of approximation and detail signals
D1=upcoef(‘d’, cD1, ‘db4’, 1, N): one-level reconstruction of detail signal from detail coefficient
A1=upcoef(‘a’, cA1, ‘db4’, 1, N): one level reconstruction of approximation signal from approximation coefficients
Multi-level decomposition & reconstruction
[C L]=wavedec(x, K, ‘db4’): k-level decomposition
A0=waverec(C, L, ‘db4’): reconstruct the signal
Extracting coefficients
cA3=appcoef(C, L, ‘db4’, 3);
cD5=detcoef(C, L, 5);
Single branch reconstruction
A3=wrcoef(‘a’, C, L, ‘db4’, 3); Reconstruct level 3 approx. signal
D2=wrcoef(‘d’, C, L, ‘db4’, 2); Reconstruct level 2 detail signal
Polikar, (C) 2007
L=[lA3 lD3 lD2 lD1 lsignal] C= A3 D3 D2 D1
lA3
lD3
lD2
lD1

39. Quadrature Mirror Filters
It can be shown that
that is, h[] and g[] filters are related to each other:
in fact, that is, h[] and g[] are mirrors of each other, with every other coefficient negated. Such filters are called quadrature mirror filters. For example, Daubechies wavelets with 4 vanishing moments…..
Polikar, (C) 2007

40. DB-4 Wavelets Polikar, (C) 2007 ~ ~ L: filter length (8, in this case)

41. Implementation of DWT on MATLAB
Choose wavelet
and number
of levels
Load signal
Hit Analyze
button
s=a5+d5+…+d1
Polikar, (C) 2007
Approx. coef.
at level 5
Level 1 coeff.
Highest freq.
(Wavedemo_signal1)

42. Details vs. Approximations
Polikar, (C) 2007

43. Applications
Polikar, (C) 2007
Detect
discontinuities

44. Applications
Polikar, (C) 2007
Detect hidden
discontinuities

45. Applications
Polikar, (C) 2007
Simple
denoising

46. Compression Polikar, (C) 2007
DWT is commonly used for compression, since most DWT are very small, can be zeroed-out!
Polikar, (C) 2007

47. Compression
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48. 1D-DWT2D-DWT Polikar, (C) 2007 Recall fundamental concepts of 1D-DWT
Provides time-scale (frequency) representation of non-stationary signals
Based on multiresolution approximation (MRA)
Approximate a function at various resolutions using a scaling function, (t)
Keep track of details lost using wavelet functions, (t)
Reconstruct the original signal by adding approximation and detail coeff.
Implemented by using a series of lowpass and highpass filters
Lowpass filters are associated with the scaling function and provide approximation
Highpass filters are associated with the wavelet function and provide detail lost in approximating the signal
Polikar, (C) 2007

49. 2-D DWT
How do we generalize these concepts to 2D?
2D functions  images f(x,y)  I[m,n] intensity function
What does it mean to take 2D-DWT of an image? How do we interpret?
How can we represent an image as a function?
How do we define low frequency / high frequency in an image?
How to we compute it?
Why would we want to take 2D-DWT of an image anyway?
Compression
Denoising
Feature extraction
Polikar, (C) 2007

50. 2D Scaling/Wavelet Functions
We start by defining a two-dimensional scaling and wavelet functions:
If (t) is orthogonal to its own translates,
is also orthogonal to its own translates. Then, if fo(x,y) is the projection of f(x,y) on the space Vo generated by s(x,y):
Polikar, (C) 2007

51. 2D-DWT
Just like in 1D we generated an approximation of the 2D function f(x,y). Now, how do we compute the detail lost in approximating this function?
Unlike 1D case there will be three functions representing the details lost:
Details lost along the horizontal direction
Details lost along the vertical direction
Details lost along the diagonal direction
1D  Two sets of coeff.; a(k,n) & d (k,n)
2D Four sets of coefficients: a(k,n), b(k, n), c(k, n) & d(k,n)
Polikar, (C) 2007

52. Four Faces of 2D-DWT Polikar, (C) 2007
One level of 2D DWT reconstruction:
Polikar, (C) 2007
Approximation coefficients
Detail coefficients along
the horizontal direction
Detail coefficients along
the vertical direction
Detail coefficients along
the diagonal direction

53. Implementation of 2D-DWT
INPUT
IMAGE ……
ROWS
COLUMNS H ~
2 1 G
1 2 LL LH HL HH
LHH
LLH
LHL
LLL
Polikar, (C) 2007

54. Up and Down … Up and Down Polikar, (C) 2007 2 1 1 2 2 1 1 2
Downsample columns along the rows: For each row, keep the even indexed columns, discard the odd indexed columns
2 1
Polikar, (C) 2007
Downsample columns along the rows: For each column, keep the even indexed rows, discard the odd indexed rows
1 2
Upsample columns along the rows: For each row, insert zeros at between every other sample (column)
2 1
Upsample rows along the columns: For each column, insert zeros at between every other sample (row)
1 2

55. Implementing 2D-DWT
Decomposition
COLUMN j
ROW i
Polikar, (C) 2007

56. Reconstruction Polikar, (C) 2007 LL H 1 2 2 1 H LH G 1 2 HL H 1 2 G
1 2 H
2 1 H
Polikar, (C) 2007 LH
1 2 G
ORIGINAL
IMAGE HL
1 2 H
2 1 G HH
1 2 G

57. 2-D DWT ON MATLAB Polikar, (C) 2007 Load Image Choose (must be
.mat file)
Choose
wavelet type
Hit
Analyze
Polikar, (C) 2007
Choose
display
options

58. 2D Wavelets
Other uses for Wavelets in image processing (as seen in later slides)
Image Compression
The Discrete wavelet transform decomposes an image into a set of successfully smaller orthogonal images. Often it is possible to coarsely quantize or eliminate low valued coefficients without sacrificing the integrity of the image
Polikar, (C) 2007
5MB
5MB
2.5MB

59. 2D Wavelets Polikar, (C) 2007 2) Image Enhancement
Since the DWT decomposes the image into components of different size, position and orientation, you can alter the coefficients before reconstruction so that you can attenuate certain characteristics of the image.
Polikar, (C) 2007

60. 2D Wavelets Polikar, (C) 2007 3) Image Fusion
Image fusion combines 2 or more registered images of the same object into a single object, that in most cases is better than the original. This is helpful in the medical imaging field where multiple images from different machines. The images are combined in the transform domain by taking the highest amplitude coefficient then performing the inverse on the new fused image.
Polikar, (C) 2007
Run Image Fusion GUI

61. Image Denoising
Polikar, (C) 2007

62. Project Idea Polikar, (C) 2007 Implement JPEG 2000
A much improved form of JPEG compression that primarily uses the wavelet based compression idea we just discussed.
Polikar, (C) 2007