Wavelets

Applications in Signal and Image Processing

The Fourier Transform

Motivation!

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!

What has caused this?

e-2iπf is a function of infinite support / infinite window function

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

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

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 usa) 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

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

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:

Result

Changing translation parameter: Time Localization

Changing dilation

parameter: Frequency

Localization

Result

+ve response

-ve response

0 response

Low response

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

Inverse Wavelet Transform: ICWT

Denoise by zeroing out coefficients

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.

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:

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

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:

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

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

Lifting to Filters

Filter Results = Polyphase Lifting

Further reading

Boundary problems!

Vanishing Moments!

Wavelet packets

Second generation wavelets

Multiwavelets

Curvelets, ridgelets …

Any Questions?