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7/17/2019 Evoked Potentials in Human Body
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EVOKEDPOTENTIALS
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Evoked Potentials (EPs)
Event!related brain activity where the stimulus isusually of sensory origin.
Acquired with conventional EEG electrodes.
Time!synchronized = time interval from stimulus toresponse is usually constant.
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EP = A Transient Waveform
Evoked potentials are usually ”hidden" in the EEGsignal.
Their amplitude ranges from 0.1 % 10 µV, to becompared with 10 % 100 µV of the EEG.
Their duration is 25 % 500 milliseconds.
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Examples of Evoked Potentials
Note the widely di" erent amplitudes and time scales.
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EP – Definitions
Time forstimulus
Latency
Amplitude
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Auditive Evoked Potentials–
AEPs
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Visual Evoked Potentials–
VEPs
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Somatosensory Evoked
Potentials–SEPs
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SEPs during Spinal Surgery
electrode #1 electrode #2
stimulation
recording electrodes
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EP Scalp Distribution
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A. Evoked potentials
resulting from a colortask in which red andblue flashed checker!
boards were presentedin a rapid, randomizedsequence at the center
of the screen.
B. Scalp voltagedistributions evokedpotentials at di" erent
latency ranges.
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Brainstem Auditive EP
(BAEPs) in Newborns wave III
stimulus
2 months
8 months
1 ms
I
IIIII
IV
V
VI VII
IIII
IVV
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BAEPs of Healthy Children
2 % 8 years
0 % 1 years
newborn
premature
latency # ms $
IIII IV V II wave
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Cognitive EPs
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Ensemble Formation
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Formation of an EP Ensemblestimulus#
EEG signal
E P
e n s e m b l e
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10 Superimposed EPs
latency # ms $
a m p l i t u d e
#
m
i c r o V $
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Model for Ensemble Averaging
fixed shape
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Noise Assumptions
I.
II.
III.
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Ensemble Averaging
The ensemble average is defined by
The more familiar # scalar $expression for ensemble
averaging is given by
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Ensemble Averaging
evoked potentials
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Noise Variance
The variance of the ensemble average is inverselyproportional to the the number of averaged potentials,
that is:
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Reduction of Noise Level
#potentials
The noise estimatebefore division by the
reduction factor
1/√ M
Reduction in noise levelof the ensemble average
as a function of#potentials
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Exponential averaging
The ensemble average can becomputed recursively because:
assuming
Exponential averaging results fromreplacing the weight 1/M with alpha:
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Exponential averaging
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Noise Reduction of EPs with
Varying Noise Level
Assumption: all evoked potentials have
identical shapes s# n $ but with
varying noise level.
Such an heterogenous ensemble is processed by weighted averaging .
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Weighted Averaging
The weighted average is obtained by weighting eachpotential x i# n $ with its inverse noise variance:
where each
weight w ithus is
This expression reduces to the ensemble average when
the noise variance is identical in all potentials.
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Weighted averaging: An Example
Weighteda vera ge
Ensemblea vera ge
T h e e n s e m b l e c o n s i s t s o f 8 0 E P s
w i t h v a r i a n c e 1 a n d 2 0 E P s w i t h v a r i a n c e 2 0 # h e t e r o g e n o u s $
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Robust Waveform Averaging
Gaussiannoise
Laplaciannoise
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The Effect of Latency Variations
xi(n) = s(n−!i) + vi(n)
Signal model:
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Lowpass Filtering of the Signal
The expected value of the ensemble average, inthe presence of latency variations, is given by:
or, equivalently, in the frequency domain:
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Latency Variation andLowpass Filtering
Gaussian PDF Uniform PDF
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Techniques for Correction ofLatency Variations
Synchronize with respect to a peak of the signal or
similar property.Crosscorrelation between two EPs.
Woody’s method for iterative synchronization of all
responses of the ensemble. The method terminates when no further latency corrections are done.
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Estimation of LatencyAn Illustration
Input signal
Template waveform
Correlationfunction
Latency estimate
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Woody’s Method
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Woody’s Method: Different SNRs
good SNR
not so good SNR
bad SNR
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the ensemble average. The optimal “filter” is
SNR-based Weighting
E (s(n)− sa(n)w(n))2
Design a weight function which minimizesw(n)
where denotes the desired signal ands(n) sa(n)
w(n) = !
2
s(n)
!2s(n) +
!2
v
M
= 1
1+ !
2
v
M !2
s(n)
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SNR-based Weighting
Noise!free signal
Ensemble average
Weight function
Weight function multiplied with ensemble average
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Noise Reduction by Filtering
Estimate the signal and noise power spectra from theensemble of signals.
Design a linear, time!invariant, linear filter such thatthe mean square error is minimized, i.e., design a Wiener filter.
Apply the Wiener filter to the ensemble average toimprove its SNR.
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Wiener Filtering
S s(e j !)
S v(e j !)
: signal power spectrum
: noise power spectrum
Wiener filter:
H (e j !) = S s(e
j !)
S s(e j !
) + (e j !
)for one potential
H (e j !) = S s(e
j !)
S s(e j !
) + v(e j !
)
for M potentials
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i
n
e
n
S
R
Filtering of Evoked Potentials
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Limitations of Wiener filtering
Assumes that the observed signal is stationary # which
in practice it is not... $.
Filtering causes the EP peak amplitudes to be severelyunderestimated at low SNRs.
As a result, this technique is rarely used in practice.
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Tracking of EP Morphology
So far, noise reduction has been based on the entireensemble, e.g., weighted or exponential averaging
We will now track changes in EP morphology by so!called single!sweep analysis. More a priori informationis introduced by describing each EP by a set of basis
functions.
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Selection of Basis Functions
Orthonormality is an important function property
of basis functions.Sines/cosines are well!known basis functions, but itis often better to use...
...functions especially determined for optimal # MSE $representation of di" erent waveform morphologies# the Karhunen!Loève representation $.
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Orthogonal Expansions
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Basis Functions: An Example
Linear combinationsof two basis functions
model a variety of signal morphologies
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Calculation of the Weights
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Mean-Square Weight Estimation
i.e. identical to theprevious expression
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Truncated Expansion
The underlying idea of signal estimation through atruncated series expansion is that a subset of basisfunctions can provide an adequate representation ofthe signal part.
Decomposition into”signal” and ”noise” parts:
The estimate of the signalis obtained from:
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Truncated Expansion, cont’
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Examples of Basis Functions
Sine/ Cosine
Walsh
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Sine/Cosine Modeling
K = 3
VEP without noise
#basis functions
K = 7
K = 12
K = 500
Si /C i M d li A lit d
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Sine/Cosine Modeling: Amplitude
Estimate and MSE Error
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MSE Basis Functions
How should the basis functions be designed so that thesignal part is e&ciently represented with
a small number of functions?
We start our derivation by decomposing the seriesexpansion of the signal into two sums, that is,
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Karhunen–Loève Basis Functions
The Karhunene % Loève # KL $ basis functions, minimizingthe MSE, are obtained as the solution of the ordinary
eigenvalue problem, and equals the eigenvectorscorresponding to the largest eigenvalues:
The MSE equals the sum of the # N!K $
smallest eigenvalues
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KL Performance Index
Example of theperformance index
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How to get R x?
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Example: KL Basis FunctionsBasis functions Signals
Observed
signal: xi
si
Signalestimate:
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Time-Varying Filter Interpretation
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Modeling with Damped Sinusoids
The original Prony method
The least!squares Prony method
Variations
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Adaptive Estimation of Weights
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Adaptive Estimation of Weights
The instantaneous LMS algorithm, in which the weights of the series expansion are adapted at every
time instant, thereby producing a weight vector w # n $
The block LMS algorithm, in which the weights areadapted only once for each EP # “block” $, thereby
producing a weight vector w i that corresponds to thei:th potential.
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Estimation Using Sine/Cosine
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Estimation Using KL Functions
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Limitations
Sines/cosines and the KL basis functions lack theflexibility to e&ciently track changes in latency of
evoked potentials, i.e., changes in waveform width.
The KL basis functions are not associated with anyalgorithm for fast computations since the functions are
signal!dependent.
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Wavelet AnalysisWavelets is a very general and powerful class of basisfunctions which involve two parameters: one fortranslation in time and another for scaling in time.
The purpose is to characterize the signal with good
localization in both time and frequency.
These two operations makes it possible to analyze thejoint presence of global waveforms # “large scale” $ as
well as fine structures # “small scale” $ in a signal.
Signals analyzed at di" erent scales, with an increasinglevel of detail resolution, is referred to as a multi!resolution analysis.
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Wavelet Applications
signal characterization
signal denoising
data compression
detecting transient waveforms
and much more!
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The Correlation Operation
Recall the fundamental operation in orthonormal basisfunction analysis: in discrete!time, the correlation betweenthe observed signal x # n $ and the basis functions !k# n $:
In wavelet analysis, the two operations of scaling and
translation in time are most simply introduced when thecontinuous!time description is adopted:
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The Mother Wavelet
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The Wavelet Transform
CW T
ICW T
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The Scalogram
Compositesignal
Scalogram
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The Discrete Wavelet Transform
The discrete wavelet
transform # DWT $
The inverse discrete wavelet transform # IDWT $
The CWT w # s, ! $ is highly redundant
and needs to be sampled
Dyadic sampling
The discretized waveletfunction
l i l i A l i
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Multiresolution Analysis
A signal can be viewed as the sum of a smooth # “coarse” $part and a detailed # “fine” $ part.
The smooth part reflects the main features of thesignal, therefore called the approximation signal.
The faster fluctuations represent the signal details.
The separation of a signal into two parts is determinedby the resolution with which the signal is analyzed, i.e.,by the scale below which no details can be discerned.
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Multiresolution Analysis Exemplified
M l i l i A l i ’
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Multiresolution Analysis, cont’
In mathematical terms this is expressed as:
Th S li F i
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The Scaling Function
The scaling function !# t $ is introduced for the purposeof e&ciently representing the approximation signal x j# t $at di" erent resolution.
This function, being related to a unique waveletfunction "# t $, can be used to generate a set of scalingfunctions defined by di" erent translations:
where the index “0” indicates that no time scaling isperformed.
Th S li F i ’
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The Scaling Function, cont’
The design of a scaling function !# t $ must be such thattranslations of !# t $ constitute an orthonormal set offunctions, i.e.,
Its design is not considered in this course, but someexisting scaling functions are applied.
Th A i i Si l
( )
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The Approximation Signal x0(t)
Th A i i Si l
( )
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The Approximation Signal x j(t)
# dyadic sampling $
Th M l i l i P
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The Multiresolution Property
Th R fi E i
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The Refinement Equation
h!# n $ is a sequence of scaling coe&cients
Th W l F i
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The Wavelet Function
It is desirable to introduce the function "# t $ whichcomplements the scaling function by accounting forthe details of a signal rather than its approximations.
For this purpose, a set of orthonormal basis functionsat scale j is given by
which spans the di" erence between the two subspacesVj and Vj+1.
S li d W l t F ti
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Scaling and Wavelet Functions
O th l C l t
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Orthogonal Complements
Th W l t S i E i
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The Wavelet Series Expansion
Compare this expansion with the orthogonal expansionsmentioned earlier such as the one with sine/cosine basis
functions, i.e., the Fourier series. The wavelet/scalingcoe&cients do not have a similar simple interpretation.
Multiresolution Signal Analysis:
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Multiresolution Signal Analysis:A Classical Example
The Haarscaling function
The Haar wavelet function
These functions are individually and mutually orthonormal
Th H S li F ti
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The Haar Scaling Function
H M lti l ti A l i
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Haar Multiresolution Analysis
Approximationsignals Detail signals
Haar Scaling and
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Haar Scaling andWavelet Functions
Comp tation of Coefficients
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Computation of Coefficients
The scaling and wavelet coe&
cients can computedrecursively by exploring the refinement equation
so that, for example, the scaling coe&cients are
computed with
see derivationon page 300
Filter Bank Implementation
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Filter Bank Implementation
DWT Calculation
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DWT Calculation
Inverse DWT Calculation
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Inverse DWT Calculation
Scaling Function Examples
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Scaling Function Examples
Coiflet Multiresolution Analysis
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Coiflet Multiresolution Analysis
Scaling Coefficients in Noise
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Scaling Coefficients in Noise
Denoising of Evoked Potentials
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Denoising of Evoked Potentials
EP Wavelet Analysis
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EP Wavelet Analysis
coe&cientsof W 3
reconstructed waveform
coe&cients
of V 3
reconstructed waveform
Visual EP
from Ademoglu et al., 1997
EP Wavelet Analysis cont’
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EP Wavelet Analysis, cont
Normal
Dement
Waveforms reconstructed from V 3 and superimposed for24 normal subjects # upper panel $ and for 16 patients withdementia # lower panel $.
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EMGIS NOT
COVEREDIN THISCOURSE