Biomedical signal processing: Wavelets Yevhen Hlushchuk, 11 November 2004

Usefull wavelets analyzing of transient and nonstationary signals analyzing of transient and nonstationary signals EP noise reduction = denoising EP noise reduction = denoising compression of large amounts of data (other basis functions can also be employed) compression of large amounts of data (other basis functions can also be employed)

Introduction Class of basis functons, known as wavelets, incorporate two parameters: Class of basis functons, known as wavelets, incorporate two parameters: 1. one for translation in time 2. another for scaling in time main point is to accomodate temporal information (crucial in evoked responses (EP) analysis) Another definition: A wavelet is an oscillating function whose energy is concentrated in time to better represent transient and nonstationary signals (illustration). illustration

Continuous wavelet transform (CWT) Example of continuous wavelet transform (here we see the scalogram)

Other ways to look at CWT The CWT can be interpreted as a linear filtering operation (convolution between the signal x(t) and a filter with impulse response ψ(-t/s)) The CWT can be viewed as a type of bandpass analysis where the scaling parameter (s) modifies the center frequency and the bandwidth of a bandpass filter (Fig 4.36) Fig 4.36Fig 4.36

Discrete wavelet transform CWT is highly redundant since 1-dimensional function x(t) is transformed into 2-dimensional function. Therefore, it is Ok to discretize them to some suitably chosen sample grid. The most popular is dyadic sampling: s=2 -j, τ = k2 -j s=2 -j, τ = k2 -j With this sampling it is still possible to reconstruct exactly the signal x(t).

Multiresolution analysis The signal can be viewed as the sum of: 1. a smooth (“coarse”) part – reflects main features of the signal (approximation signal); 2. a detailed (“fine”) part – faster fluctuations represent the details of the signal. The separation of the signal into 2 parts is determined by the resolution.

Scaling function and wavelet function The scaling function is introduced for efficiently representing the approximation signal x j (t) at different resolutions. The scaling function is introduced for efficiently representing the approximation signal x j (t) at different resolutions. This function has a unique wavelet function related to it. This function has a unique wavelet function related to it. The wavelet function complements the scaling function by accounting for the details of a signal (rather than its approximations) The wavelet function complements the scaling function by accounting for the details of a signal (rather than its approximations)

Classic example of multiresolution analysis

What should you want from the scaling and wavelet function? 1. Orthonormality and compact support (concentrated in time, to give time resolution) 2. Smooth, if modeling or analyzing physiological responses (e.g., by requiring vanishing moments at certain scale): Daubechies, Coiflets. 3. Symmetric (hard to get, only Haar or sinc, or switching to biorthogonality)

Scaling and wavelet functions Haar wavelet (square wave, limited in time, superior time localization) Haar wavelet (square wave, limited in time, superior time localization) Mexican hat (smooth) Mexican hat (smooth) Daubechies, Coiflet and others (Fig4.44) Daubechies, Coiflet and others (Fig4.44)Fig4.44

One more example but now with a smooth function Coiflet-4, you see, this one models the response somewhat better than Haar One more example but now with a smooth function Coiflet-4, you see, this one models the response somewhat better than Haar Haar

Denoising Truncation (denoising is done without sacrificing much of the fast changes in the signal, compared to linear techniques) Truncation (denoising is done without sacrificing much of the fast changes in the signal, compared to linear techniques) Hard thresholding (zeroing) Hard thresholding (zeroing) Soft thresholding (zeroing and shrinking the others above the threshold) Soft thresholding (zeroing and shrinking the others above the threshold) Scale-dependent thresholding Scale-dependent thresholding Time windowing Time windowing Scale-dependent time windowing Scale-dependent time windowing

Example: Daubechies-4. Noise – in finer scales!!! (as usually). Good reason for scale- dependent thresholding

When signal denoising is helpful? 1. Producing more accurate measurements of latency and time 2. Thus, of great value for single-trial analysis 3. Improves results of the Woody method (latency correction)

Application of scale- dependent thresholding

Summary The strongest point (as I see:) in the wavelets is flexibility (2-dimenionality) compared to other basis functions analysis we studied. Wavelet analysis useful in : analyzing of transient and nonstationary signals (single-trial EPs) analyzing of transient and nonstationary signals (single-trial EPs) EP noise reduction = denoising EP noise reduction = denoising compression of large amounts of data (other basis functions can also be employed) compression of large amounts of data (other basis functions can also be employed)

Happy end Oooooops………hu!

Non-covered issues (this and following slides :) Refinement equation Refinement equation Scaling and wavelet coefficients Scaling and wavelet coefficients

Calculating scaling and wavelet coefficients Analysis filter bank (top-down, fine-coarse) Analysis filter bank (top-down, fine-coarse) Synthesis filter bank (bottom- up, coarse-fine) Synthesis filter bank (bottom- up, coarse-fine)