Applications in Signal and Image Processing Wavelets Applications in Signal and Image Processing

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

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

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

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

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

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

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

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

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

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

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

Result +ve response Low response -ve response 0 response

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.

Inverse Wavelet Transform: ICWT Denoise by zeroing out coefficients

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

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

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.

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

Dyadic scaling

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

Discrete scaling function Φ

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

Multi-Resolution Analysis, MRA

Haar example

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

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

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

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)

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

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

Lifting to Polyphase Split: Prediction: Update:

Filters to Z-transform

Filter Results = Polyphase Lifting Lifting to Filters Filter Results = Polyphase Lifting

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

Any Questions?