Wavelets
description
Transcript of Wavelets
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WaveletsApplications in Signal and Image Processing
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The Fourier Transform
Motivation!
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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!
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What has caused this?
e-2iπf is a function of infinite support / infinite window function
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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
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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
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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
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Wavelets to the rescue
We would like to develop a method independent of the windowing function that gives usa) Good time resolution and poor frequency resolution at
high frequenciesb) Good frequency resolution but poor time resolution at low
frequencies
Low frequency => Signal information High frequency => Excess detail or noise in the signal
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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
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Time localization
Inspect the signal at different time steps Introduce a translation parameter, t’, that controls the
translation of the function:
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Frequency Localization Inspect the signal for different frequencies Introduce a dilation parameter, s, that controls the
scale of the function:
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Result
Changing translation parameter: Time Localization
Changing dilation
parameter: Frequency
Localization
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Result
+ve response
-ve response
0 response
Low response
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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
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Inverse Wavelet Transform: ICWT
Denoise by zeroing out coefficients
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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.
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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
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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.
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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:
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Dyadic scaling
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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
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Discrete scaling function Φ
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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:
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Multi-Resolution Analysis, MRA
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Haar example
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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
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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
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Lifting
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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)
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2D DWT Wavelets and scaling functions are orthogonal …
hence separable. We can apply the transform in one direction then the
other
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Z-transform
Fourier Series: Z-Transform:
Convolution Shift Left Shift Right Down sample Up sample
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Lifting to Polyphase
Split: Prediction: Update:
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Filters to Z-transform
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Lifting to Filters
Filter Results = Polyphase Lifting
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Further reading
Boundary problems! Vanishing Moments! Wavelet packets Second generation wavelets Multiwavelets Curvelets, ridgelets …
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Any Questions?