1. Applications in Signal and Image Processing
Wavelets
Applications in Signal and Image Processing

2. Motivation! The Fourier Transform Time domain -> frequency domain
Inner product of x(t) to a sin and cos function at frequency f
If X(f) is large for a given f the x(t) matches well and contains a lot of the frequency f.
If X(f) is small for a given f then x(t) doesn’t match well and does not contain frequency f. IMPULSE RESPONSE
In time domain we know the value of the signal at any time i.e. we have time resolution
In frequency domain we know exactly what frequencies exist

3. Problem
The FT of stationary and non stationary signals with the same frequency components are equivalent.
i.e. we are lacking time localization
Although FT tells us what frequencies appear in the signal it does not tell us at what time they appear!
Sationary waves : those where the frequency component exists at all times
Non stationary: certain frequencies appear at certain time

4. What has caused this?
e-2iπf is a function of infinite support / infinite window function
inner product of function to an infinite window looks for frequency f over all time
Nice as we get perfect frequency resolution in the frequency domain

5. Short Term Fourier Transform: STFT
Multiple FT over smaller windows translated in time
Compactly supported
We now have a time- frequency representation
YOU CAN ALL GO HOME 
W windowing function of width T

6. NOT!
Recall:
In the time domain we know exactly the value of the signal at any time (time resolution)
In the frequency domain we know exactly the frequencies in the signal (frequency resolution)
In STFT the kernel is compact … thus we can only see a band of frequencies based on the size of the kernel
We get perfect frequency resolution in the FT because the kernel is infinite and thus can represent all frequencies

7. Increasing window width
Consequence
Window size is application specific
Narrow window -> good time resolution, poor frequency resolution
Wide window -> good frequency resolution, poor time resolution
Increasing window width

8. Wavelets to the rescue
We would like to develop a method independent of the windowing function that gives us
Good time resolution and poor frequency resolution at high frequencies
Good frequency resolution but poor time resolution at low frequencies
Low frequency => Signal information
High frequency => Excess detail or noise in the signal
I have used FT and frequency analysis to motivate the use of WT however this isnt the only reason to use them there are many many other

9. Continuous Wavelet Transform: CWT
Ψ is the mother wavelet, the shape or choice of this depend on the properties of the signal we wish to analyze
Haar, 1st deriv gausian, mexican hat, morlet

10. Time localization Inspect the signal at different time steps
Introduce a translation parameter, t’, that controls the translation of the function:

11. Frequency Localization
Inspect the signal for different frequencies
Introduce a dilation parameter, s, that controls the scale of the function:
Smaller s the narrower the wavelet
1/sqrt(s) is for energy normalisation

12. Result Changing dilation parameter: Frequency Localization
Changing translation parameter: Time Localization

13. Result
+ve response
Low response
-ve response
0 response

14. Orthogonality / Orthonormal
i.e. 2 functions are, at no place the same or, are symmetric
Orthonormal:
So dilations and translations of a wavelet must be orthonormal to itself so as not to influence the construction of the coefficients
These allow for perfect reconstructions of the form
Orthogonality means that the information in the coefficient ψs,t’ is not repeated anywhere else. i.e removes redundancy.

15. Inverse Wavelet Transform: ICWT
Denoise by zeroing out coefficients

16. Frequency to time resolution
Low scales / high frequencies have good time resolution but poor frequency resolution.
High scales / low frequencies have good frequency resolution but poor time resolution.
STFT has constant time to frequency resolution as window size is fixed

17. Discrete Wavelet Transform: DWT
The Discrete Wavelet Transform is a sampled version of the Continuous case with discrete dilation and translation parameters
Filters or different cut of frequencies are used to analyze the signal at different scales or resolutions
We will be requiring a scaling filter/function and a wavelet filter/function in this case
Scaling function – low pass filter - approximation
Wavelet – high pass filter - details

18. Discrete wavelet Ψ Recall that the CW is defined as:
In a continuous transform we find the inner product over all scales S and translates t’. However now we must sample s and t’.
Logarithmic sampling of s means we need to move in discrete steps on t’ proportional to the scale s.

19. Dyadic scaling Dyadic scaling, choose s0=2 and t0’=1
Later this will lead to a nice down sampling routine
DWT to obtain detail coefficients becomes:
N.b. when we compute the CWT the the ICWT is accurate only up to the resolution of the samplings of s and t’.
With DWT the intergral is continuous but is only calculated on a discretized grid at m and n. thus we can reconstruct exactly the signal

20. Dyadic scaling

21. Discrete scaling function Φ
In the CWT we calculated the set of coefficients ψ over all scales s and translations t’ on the continuous signal x(t)
As we are sampling x(t) we cannot have these infinite coefficients. We need some way of keeping track of what the wavelet coefficients don’t express.
Therefore we must define how we sample the signal based on the current dilation, m, of the wavelet. This is done via a Scaling function
We can convolve the signal with the scaling function to get approximation coefficients
Scaling func is derived based on the wavelet or vicaversa

22. Discrete scaling function Φ

23. Approximation and detail
Approximation coefficients, ϕ, are produced by applying the scaling function to the sampled signal. They express the signal at a lower resolution as if the high frequencies had been removed
Detail coefficients, ψ, are produced by applying the wavelet to the sampled signal. They express the higher frequency components in the signal.
Thus a signal is represented as the sum of approximation and detail coefficients:
Approx tell you what lower frequency components of the signal are yet to be analysed.
Detail tels you the time-frequency representation of the current signal down to a certain band.
Original signal is now the combination of its approximation at scale m0 and the detail coefficients from lower levels

24. Multi-Resolution Analysis, MRA

25. Haar example

26. DWT via Filtering Filter convolution :
H (equivalent to wavelet) is high pass, stripping the signal of its lower band frequencies thus its coefficients represent high frequency components
G (equivalent to scaling function) is a low pass, stripping the signal of its higher frequencies thus is passed on to the next scale to remove the next band of high frequency

27. DWT via Lifting
Filters can be transformed in the time or frequency domain into distinct in-place processing steps on the signal rather than costly convolutions
Expressing a wavelet in terms of lifting steps is know as a Second Generation Wavelet
Here rather than low and high pass filters we perform a Prediction step and an Update step
Prediction – high pass filter – we predict what the signal is
Update – low pass filter – based on the prediction we update the signal

28. Lifting
Shift even to Up odd to Predict- down sample - predict / detail coefficients – update / approximation coefficients

29. Haar Lifting example
Take signal x(t) and split it into odd and even pairs
As a prediction step take the odd away from the even:
dj-1= oddj-P(evenj)
As an update step take the mean value of the odd and even parts
Sj-1=evenj+U(dj-1)

30. 2D DWT
Wavelets and scaling functions are orthogonal … hence separable.
We can apply the transform in one direction then the other

33. Z-transform Fourier Series: Z-Transform: Convolution Shift Left
Shift Right
Down sample
Up sample

34. Lifting to Polyphase
Split:
Prediction:
Update:

35. Filters to Z-transform

36. Filter Results = Polyphase Lifting
Lifting to Filters
Filter Results = Polyphase Lifting

37. Further reading Boundary problems! Vanishing Moments! Wavelet packets
Second generation wavelets
Multiwavelets
Curvelets, ridgelets …

38. Any Questions?