Transcript Biomedical signal processing: Wavelets

Biomedical signal processing:
Wavelets
Yevhen Hlushchuk,
11 November 2004
Usefull wavelets
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analyzing of transient and nonstationary
signals
EP noise reduction = denoising
compression of large amounts of data (other
basis functions can also be employed)
Introduction
Class of basis functons, known as wavelets,
incorporate two parameters:
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1.
2.
one for translation in time
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).
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)
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
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
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The scaling function is introduced for
efficiently representing the approximation
signal xj(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)
Classic example
of multiresolution
analysis
What should you want from the
scaling and wavelet function?
1.
2.
3.
Orthonormality and compact support
(concentrated in time, to give time
resolution)
Smooth, if modeling or analyzing
physiological responses (e.g., by requiring
vanishing moments at certain scale):
Daubechies, Coiflets.
Symmetric (hard to get, only Haar or sinc, or switching to
biorthogonality)
Scaling and wavelet functions
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Haar wavelet
(square wave,
limited in time,
superior time
localization)
Mexican hat
(smooth)
Daubechies, Coiflet
and others (Fig4.44)
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One more example but
now with a smooth
function Coiflet-4, you
see, this one models the
response somewhat
better than Haar 
Denoising
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Truncation (denoising is done without
sacrificing much of the fast changes in the
signal, 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
Example:
Daubechies-4.
Noise – in finer
scales!!! (as
usually). Good
reason for
scaledependent
thresholding
When signal denoising is helpful?
1.
2.
3.
Producing more accurate measurements of
latency and time
Thus, of great value for single-trial analysis
Improves results of the Woody method
(latency correction)
Application of
scaledependent
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)
 EP noise reduction = denoising
 compression of large amounts of data (other
basis functions can also be employed)
Happy end
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Oooooops………hu!
Non-covered issues (this and
following slides :)
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Refinement equation
Scaling and wavelet coefficients
Calculating scaling and wavelet
coefficients
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Analysis filter
bank (top-down,
fine-coarse)
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Synthesis filter
bank (bottomup, coarse-fine)