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Wavelets Applications in Signal and Image Processing.

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Presentation on theme: "Wavelets Applications in Signal and Image Processing."— Presentation transcript:

1 Wavelets Applications in Signal and Image Processing

2  The Fourier Transform Motivation!

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!

4 What has caused this?  e -2iπf is a function of infinite support / infinite window function

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

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

7 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 a)Good time resolution and poor frequency resolution at high frequencies b)Good frequency resolution but poor time resolution at low frequencies  Low frequency => Signal information  High frequency => Excess detail or noise in the signal

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

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:

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

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

14 Orthogonality / Orthonormal  Orthogonal:  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

15 Inverse Wavelet Transform: ICWT Denoise by zeroing out coefficients

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

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 s 0 =2 and t 0 ’ =1  Later this will lead to a nice down sampling routine  DWT to obtain detail coefficients becomes:

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

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:

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

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:  d j-1 = odd j -P(even j )  As an update step take the mean value of the odd and even parts  S j-1 =even j +U(d j-1 )

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

31

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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 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?


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