Wavelets for bio signal lprocessing

8/13/2019 Wavelets for bio signal lprocessing

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Biomedical signal processing:

Wavelets

Yevhen Hlushchuk,

11 November 2004


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Usefull wavelets

analyzing of transient and nonstationary

signals

EP noise reduction = denoising

compression of large amounts of data(other

basis functions can also be employed)


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Introduction

Class of basis functons, known as wavelets,

incorporate two parameters:

1. one for translationin time

2. another forscalingin timemain 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).
http://localhost/var/www/apps/conversion/tmp/scratch_5/wavelet3scales.tifhttp://localhost/var/www/apps/conversion/tmp/scratch_5/wavelet3scales.tif


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Continuous wavelet transform

(CWT)Example of

continuous

wavelet transform

(here we see thescalogram)


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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 (-t/s))

The CWT can be viewed as a type of bandpass

analysis where the scaling parameter (s)

modifies thecenter frequencyand the

bandwidth of a bandpass filter(Fig 4.36)
http://localhost/var/www/apps/conversion/tmp/scratch_5/wavelet3scales.tifhttp://localhost/var/www/apps/conversion/tmp/scratch_5/wavelet3scales.tif


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Discrete wavelet transform

CWT is highly redundant since 1-dimensional

functionx(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,= k2-j

With this sampling it is still possible to

reconstruct exactly the signalx(t).


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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 isdetermined by the resolution.


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Scaling function and wavelet

function

Thescaling functionis introduced for

efficiently representing the approximation

signalxj(t)at different resolutions.

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)


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Classic example

of multiresolution

analysis


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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 tobiorthogonality)


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Scaling and wavelet functions

Haar wavelet

(square wave,

limited in time,

superior timelocalization)

Mexican hat

(smooth)

Daubechies, Coiflet

and others (Fig4.44)
http://localhost/var/www/apps/conversion/tmp/scratch_5/scalingandwavelet_functions.bmphttp://localhost/var/www/apps/conversion/tmp/scratch_5/scalingandwavelet_functions.bmp


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One more example but

now with a smooth

function Coiflet-4, you

see, this one models theresponse somewhat

better than Haar


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Denoising

Truncation(denoising is done withoutsacrificing much of the fast changes in thesignal, compared to linear techniques)

Hard thresholding(zeroing) Soft thresholding(zeroing and shrinking the

others above the threshold)

Scale-dependent thresholding Time windowing

Scale-dependent time windowing


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Example:

Daubechies-4.

Noisein finer

scales!!! (as

usually). Good

reason for

scale-

dependent

thresholding


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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)


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Application of

scale-

dependentthresholding


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Summary

The strongest point (as I see:) in the wavelets isflexibility (2-dimenionality) compared to otherbasis functions analysis we studied.

Wavelet analysis useful in : analyzing of transient and nonstationary

signals (single-trial EPs)

EP noise reduction = denoising compression of large amounts of data(other

basis functions can also be employed)


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Happy end

Oooooopshu!


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Non-covered issues (this and

following slides :)

Refinement equation

Scaling and wavelet coefficients


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Calculating scaling and wavelet

coefficients

Analysis filter

bank (top-down,

fine-coarse)

Synthesis filterbank (bottom-

up, coarse-fine)

Wavelets for bio signal lprocessing

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