Biomedical Signal Processing Forum, August 30th 2005

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Biomedical Signal Processing Forum, August 30th
2005
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BioSens (2/2) wavelet based morphological ECG
analysis

• Joel Karel, Ralf Peeters, Ronald L. Westra
• Maastricht University
• Department of Mathematics

Biomedical Signal Processing Forum, August 30th
2005
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Biosens prehistory from 1994 onwards
TU Delft micro electronics
UM mathematics
Richard Houben Medtronic Bakken Research
Maastricht
Other companies
UM Fysiology Prof. M. Allessie
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STW project Biosens 2004-2008 Organization
research consortium
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Research topics studied so far

• 1. Efficient analog implementation of wavelets
• Analog implementation of wavelets allows
  low-power consuming wavelet transforms for e.g.
  implantable devices
• Wavelets cannot be implemented in analog circuits
  directly but need to be approximated A good
  approximation approach will allow reliable
  wavelet transforms
• 2. Epoch detection and segmentation
• Application of the Wavelet Transform Modulus
  Maxima method to T-wave detection in cardiac
  signals.
• 3. Optimal discrete wavelet design for cardiac
  signal processing
• What is the best wavelet relative to the data and
  pupose?

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• 1. Efficient implementation of analog wavelets
• Biosens team J.M.H. Karel (PhD-student),
• dr. R.L.M. Peeters, dr. R.L. Westra
• Accepted papers BMSC 2005 (Houffalize, Belgium),
  IFAC 2005 (Prague, Czech Republic), and CDC/EDC
  2005 (Sevilla, Spain)

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Implementation of wavelets

• Analog implementation of wavelets allows
  low-power consuming wavelet transforms for e.g.
  implantable devices
• Wavelets cannot be implemented in analog circuits
  directly but need to be approximated

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Wavelet approximation considerations

• A good approximation approach will allow reliable
  wavelet transforms
• will allow low-order implementation (low-power
  consuming) of wavelet transforms
• allows approximation of various types of wavelets
• is relatively easy applicable

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Wavelet approximation methodology
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Initial high-order discrete time MA-system

• Sampled wavelet function
• Required i.r.
• State-space realization in controllable companion
  form

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Model reduction with balance and truncate

• Balance Lyapunov-equations
• Note that P is identity matrix
• Reduce system based on Hankel singular values

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L2-approach

• Wavelet is approximated by impulse response of
  system
• The model class is determined by the system whos
  i.r. is used as a starting point

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State-space realization
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Morlet approximation
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Daubechies 3 wavelet
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Wavelet approximation methodology

• Allows approximation of wavelet function that
  previously could not be approximated
• Allows approximation of more generic functions
• Publications accepted for IFAC 2005 (Prague,
  Czech Republic), CDC/ECC 2005 (Sevilla, Spain),
  and BMSC 2005 (Houffalize, Belgium),

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• 2. ECG morphological analysis using designer
  wavelets
• Biosens team J.M.H. Karel (PhD-student), dr.
  R.L.M. Peeters, dr. R.L. Westra
• Graduation students Pieter Jouck, Kurt
  Moermans, Maarten Vaessen

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Wavelet based signal analysis
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Signal morphology and optimal wavelet design (1)

• Design wavelet such that they detect morphology
  in signal
• Wavelet transform can be seen as convolution
• Maximum values if wavelet resembles signal in an
  L2 sense
• Fit a wavelet to signal or create optimal wavelet
  in wavelet domain

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Signal morphology and optimal wavelet design (2)

• Can be used to detect epochs
• Detecting morphologies related to pathologies
• Computational efficient

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Example of optimized wavelet for R-peak detection
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Resulting transform
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• 2.1 Application of the wavelet
• Transform Modulus Maxima
• method to T-wave detection
• in cardiac signals
• J.M.H. Karel, P. Jouck, R.L. Westra

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T-wave detection in electrocardiograms

• Pilot study based on state-of-the-art approach
  (e.g. Li 1995, Butelli 2002)
• WTMM-based algorithm
• Approximation of singular value (Lipschitz
  coefficient) did not show to be particularly
  discriminating
• Approach successful on a variety of RT-wave
  morphologies
• Classification strategy rather ad hoc

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Epoch detection and segmentation
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Example Application of the Wavelet Transform
Modulus Maxima method to T-wave detection in
cardiac signals
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Objectives

• ECG segmentation epoch detection
• Characterization of epoch

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Testcase T-wave detection

• Complications with T-wave detection
• low amplitude
• Wide variety of T-waves types
• fuzzy positioning

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Conventional Methodes

• 1st step Filter ? filtering of fluctuations and
  artifacts
• Different types of filters
• Differential filters
• Digital filters
• 2d step Signal comparsion using threshold

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Conventional Methods

• Advantages
• Simple and straightforward methodology
• Ease of implementation
• Disadvantages
• Sensitive for stochastic fluctuations
• Bad detection of complexes with low amplitudes

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Wavelet Transform

• Wavelet transform of signal f using wavelet ?
• Frequency and time domain
• Spectral analysis by scaling with a (dilation)
• Temporal analysis by translation with b

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Example WTMM
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WTMM-based QRS detection (1)

• Dyadic transform scales 21 22 23 24

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WTMM-based QRS detection (2)

• Identification of Modulus Maxima
• QRS-complex ? 2 modulus maxima (MM)
• Find all MM on all scales
• Delete redundant MMs
• 2 positive or 2 negative recurring MMs
• Proximate MM multiples (too close for comfort)

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WTMM-based QRS detection (3)

• Positioning of R-peak
• Zero-crossing between positive and negative MMs

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Adjustments for T-wave detection

• Transform to scale 10

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Adjustments for T-wave detection

• Search for Modulus Maxima
• Only MMs above a given threshold
• Position of T-wave peak
• Normal T-wave ? MM pare ? zero-crossing
• T-wave with single increase/decrease ? one MM
  near peak

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Results

• Sensitivity of WTMM-based methode to
• Low T-wave amplitudes
• Noise and stochastic variation
• Baseline-drift
• complex T-wave morphology

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Results
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Results

• Testcases Signals from i) MIT-BIH database, and
  ii) Cardiology department of Maastricht
  University.
• Performance
• Number of True Positive detections (TruePos)
• Number of False Positive detections (FalsePos)
• Number of True Negative detections (TrueNeg)
• Number of False Negative detections (FalseNeg)
• Total number of peaks (TotalPeak)
• Percentage of detected T-waves (Sensitivity)
• Percentage of correct detections (PercCorr)

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Testcase 1
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Testcase 2
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Testcase 3
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Testcase 4
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Discussion

• Problems
• Low amplitude high noise levels
• Extremely short ST-intervals
• Not all types of T-wave are detected
• Improvements
• Automatic scale adjuster
• More decision rules
• Learning algorithm

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Conclusion

• Reliable method
• Robust and noise-resistent
• Good performance in sense of sensitivity and
  percentage correct (typically gt 85)

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• 2.2 Optimal discrete wavelet
• design for cardiac signal
• processing
• J.M.H. Karel, R.L.M. Peeters and R.L. Westra

EMBC 2005 ( 27th Annual International Conference
of the IEEE Engineering in Medicine and Biology
Society), 1-4 September 2005, Shanghai,
Peoples Republic of China
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Optimal Wavelet Design

• What is a good wavelet for a given signal and a
  given purpose?
• Freedom in choice for analyzing wavelet ?(t)
• Best output ( wavelet coefficients ck) are well
  positioned in frequency-temporal space, i.e.
  sparse representation
• Essentials perfect reconstruction, orthogonal
  wavelet-multi resolution structure, vanishing
  moments of wavelets, flatness of filter, smooth
  wavelets
• Measure the performance of a given signal x(t)
  and a trial wavelet ?(t) with a criterion
  function V?

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Optimal Wavelet Design

• Orthogonal wavelets and filter banks

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Wavelet analysis and synthesis

• Low pass filter with transfer function C(z)
• High pass filter with transfer function D(z)
• Combination with down-sampling
• ? has compact support ? C, D are FIR

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Wavelets analysis and synthesis
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Filter bank
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Filter bank
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Polyphase decomposition
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Polyphase decomposition
the alternating flip construction relates
coefficients c and d
the remaining orthogal freedom in the 2n ck can
be expressed by a reparametrization in n new
parameters ?i the polyphase matrices R and ?
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Polyphase decomposition
The polyphase matrices R and ? in terms of
parameters ?i are defined as
Then define the 2x2 matrix H as
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Polyphase decomposition
Matrix H can be partitioned as
with
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Polyphase decomposition
Now the matrices C and D can be written as
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Design criteria on the wavelet put constraints on
C and D

• The polyphase decomposition handles the
  orthogonality of the filterbank
• Another desirable property are the vanishing
  moments
• This puts constraints on C and D, e.g. C(z) has
  zeros at z 1 0, ditto corresponding flatness
  of C and D at high/low frequencies
• The condition of vanishing moments translates to
  a linear set of constraints on the filter
  coefficients c and d

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Design criteria on the wavelet put constraints on
C and D

• For instance one vanishing moment states
• this translates to a constraint on c and d
• And so also to a constraint on the ?s

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Computing the wavelet and scaling function

• The scaling function ? relates to the wavelet ?
  via the dilation equation.
• Using the alternating flip construction this
  states

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Computing the wavelet and scaling function

• The resulting functions ? and ? may be
  discontinuous and fractal
• The dilation relation allows an iteration scheme
  for the coefficients

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Tree structure for dyadic scales
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Criteria to measure the quality of a wavelet for
given data

• The energy of the original signal x(t) is
• Because of the orthogonality this energy is
  preserved in the wavelet representation

• Therefore, the L2-norm is not a suitable
  criterion

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Wavelet design criteria

• Philosophy and algorithm
• Guiding principle proposed here is to aim for
  maximization of the variance.
• This is achieved by either
• maximization of the variance of the absolute
  values of the wavelet coefficients ? minimization
  of L1-norm
• maximization of the variance of the squared
  wavelet coefficients ? maximization of L4-norm

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Algorithm
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Wavelet Design Criteria

• Remarks
• The criteria min L1 and max L4 have been
  investigated to design wavelets for various given
  signals.
• When all wavelet coefficients are taken into
  account and no weighting is applied, both
  criteria tend to produce similar results.
• However, when only a few scales are taken into
  account (e.g. by weighting) the results may
  become different in case of minimization of the
  L1-norm energy tends to be placed in scales not
  taken into account, whereas in case of
  maximization of the L4-norm this does not happen.

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Experimentation

• I. Reconstruction of artificially generated noisy
  signals
• Make an artificial sparse signal x(t) by setting
  only a few wavelet-coefficients ck to non-zero
  values
• Note that this signal probably has a small
  L1-norm (sparse) and a large L4-norm (large
  variance)
• Add some white noise v(t) and apply the
  wavelet-design algorithm to this signal x(t)
  v(t)
• The reconstructed signal x(t) fits perfectly
  with x(t) up to a signal-to-noise ratio (snr) 1

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Experimentation

• II. Reconstruction of a reference signal
• Reference signal x(t) is obtained by averaging
  comparable episodes from ECG signals from the MIT
  Physionet normal sinus rhythm database
• Resulting smooth signal is upsampled

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Input signal averaged ECG
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Experimentation

• Local optima in the ?-parameter space
• Consider the situation with n3, i.e. three ?s
• Because ?1 ?2 ?3 ?/4 this situation has two
  degrees of freedom
• Now we can plot the L1-criterion in the (?2,
  ?3)-plane and study local optima

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Local Optima
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Experimentation

• Local optima in the ?-parameter space
• The coefficients of the local optima closely
  resemble the Daubechies 2 filter coefficients
• This observation gives a rationale for the use of
  the Daubechies 2 wavelet for cardiac signals
• When the sum of ?2 and ?3 already is close to ?/4
  only one degree of freedom is effectively used
• some of the minor local optima resemble the
  Daubechies 3 wavelet, however to lesser extent

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Experimentation

• The optimal number ?-parameters
• The filter size of the wavelet filter is
  determined by the number n of ?s used
• A large number of ?s gives freedom to fit the
  wavelet to the signal but also increases the
  complexity
• For n 1 to 25 the L1-criterion averaged over
  1000 random starting points is computed
• The graph is rather flat between n 5 and n
  20, and n 8 is a reasonable choice

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Criterion versus number of ?s
n 8
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Experimentation

• Practical evaluation
• Next we design the best fitting wavelet for the
  test set episode 103 of the MIT-BIH arrythmia
  database. This is a 360 Hz annotated ECG signal.
• Two wavelets were designed using 8 ?s by
  minimizing a criterion function V, with
• for ?1 V L1-norm of the wavelet transform
• for ?2 V L4-norm of the wavelet transform

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L1-L4 wavelet
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Experimentation

• Quality of the wavelet transform of the
  reference averaged ECG-signal with the
  L4-criterion maximized wavelet
• The L4-wavelet has a fractal structure. The
  available degrees of freedom are not used to
  place poles at z -1 as with the Daubechies
  wavelets
• The fractal nature of the L4-wavelet is not
  relevant in processing, as only the coefficients
  are used in computations
• Moreover, the L4-wavelet is very effective in the
  wavelet decomposition of the reference signal.
  One single strong wavelet coefficient marks the
  location of largest correlation

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Wavelet transform of reference ECG signal with
the designed L4-wavelet
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Experimentation

• Comparison of the L1- and L4-wavelets with the
  Daubechies 2 wavelet
• The wavelet transform of the MIT-testset with was
  compared for the three wavelets (L1, L4, and
  Daubechies), on only a single level (scale)
• This level was selected for each wavelet
  individually to maximize performance
• A binary vector was constructed of all the
  wavelet coefficients of which the absolute value
  exceeded a certain threshold. These locations
  were related to the locations of the original
  signal
• The beat annotations in the testset were used as
  a reference to locate the QRS-complex.

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Experimentation

• Comparison of the L1- and L4-wavelets with the
  Daubechies 2 wavelet
• There are 1688 normal QRScomplexes in the
  MIT-testset. If the binary vector corresponding
  to the wavelet transform has detected a peak
  within 20 samples (56ms) of the marker, it is
  assumed that the QRS-complex is detected. If a
  peak is detected but no marker is within 20
  samples, a false detection is assumed
• This comparison yielded the following results

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Results
Comparison of the L1- and L4-wavelets with the
Daubechies 2 wavelet
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Conclusions

• Comparison of the L1- and L4-wavelets with the
  Daubechies 2 wavelet
• The table shows that the performance of the
  Daubechies 2 wavelet is quite good
• The L4 optimized wavelet however shows superior
  performance
• Furthermore the L4-wavelet is more robust with
  respect to the choice of threshold value, which
  may be a large advantage in practical
  applications

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References
1 Gilbert Strang,Truong Nguyen, Wavelets and
Filter Banks, Wellesley-Cambridge Press,
1996. 2 N. Neretti, N. Intrator, An adaptive
approach to wavelet filter design, Proc. IEEE
int. workshop on neural networks for signal
processing, 2002. 3 A.L. Goldberger et al..
Physiobank, physiotoolkit, and physionet Complex
physiologic signals. Circulation,
101(23)e215e220, June 2000. 4 Cuiwei Li et
al., Detection of ECG characteristic points using
wavelet transforms. IEEE Transactions on
Biomedical Engineering, 42(1)2128, January
1995. 5 Stephane Mallat. A Wavelet Tour of
Signal Processing. Ac. Press, 1999. 6 Ivo
Provaznýk, Ji Kozumplýk. Wavelet transform in
electrocardiography - data compression.
International Journal of Medical Informatics,
45(1-2)111128, June 1997. 7 M.P. Wachowiak et
al., Waveletbased noise removal for biomechanical
signals a comparative study. IEEE Transactions
on Biomedical Engineering, 47(3)360368, March
2000.
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Focus for further research

• Analysis of mathematical morphology of cardiac
  signals
• Characterization of cardiac signals (clustering)
• Expert input
• Design of optimal multiwavelets
• Development of online algorithm based on optimal
  wavelets and hardware implementation

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Discussion
Dr. Ronald L. Westra Department of
Mathematics Maastricht University westra_at_math.unim
aas.nl
Biomedical Signal Processing Forum, August 30th
2005