Elvir Causevic

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Fast Wavelet Estimation of Weak Biosignals
By Elvir Causevic Department of Applied
Mathematics Yale University Founder and
President Everest Biomedical Instruments
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Overview

• Introduction and Motivation
• Human auditory system
• Measurement of auditory function and difficulties
  in signal processing
• Introduction to wavelets and conventional wavelet
  denoising
• Novel wavelet denoising algorithm
• Frame recombination
• Denoising
• Variable threshold selection
• Estimation of rate of convergence
• Experimental results
• Future work
• Conclusion and summary

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Introduction

• Overall goal
• Creation of a fast estimator of weak biosignals
  based on wavelet signal processing. Application
  to auditory brainstem responses (ABRs) and other
  evoked potentials
• Specific objectives
• Reduce the length of time to acquire a valid ABR
  signal.
• Allow ABR signal acquisition in a noisy
  environment.
• Key obstacles
• Very large amount of acoustical and electrical
  noise present .
• Signals collected from ear and brain have very
  low SNR and require long averaging times

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Infant Hearing Screening

• Infant hearing screening is critically important
  in early intervention of treating deafness.
• Hearing loss affects 3 in 1,000 infants most
  commonly occurring birth defect.
• 25,000 hearing impaired babies born annually in
  the U.S. alone.
• Lack of early detection often leads to permanent
  loss of ability to acquire normal language
  skills.
• Early detection allows intervention that commonly
  results in development of normal speech by school
  age.
• Intervention involves hearing aids, cochlear
  implants and extensive parent and child education
  and training.
• 38 U.S. states mandate hearing screening, Europe,
  Australia, Asia following closely.

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Measurement of Hearing Function

• Auditory Brainstem Response (ABR) - neural test
• Response of the VIIIth nerve - auditory
  neuro-pathway to brain

VIIIth Nerve
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Auditory Brainstem Response (ABR)Signal
Processing Clinical Issuesfor Infant Hearing
Screening

• Stimulus 37 clicks per second, 65 dB SPL (30 dB
  nHL).
• Response scalp electrodes measure µV level
  signals.
• Noise completely buries the response (-35dB).
• Pass signal to noise ratio measure (called Fsp)
  greater than an experimentally determined value
  (NIH Multicenter study).
• With linear averaging, reliable results are
  obtained within 15 minutes of averaging of
  4000-8000 frames at a single level.
• We would like to test multiple levels (up to 10)
  , and with multiple tone pips (vs. clicks). This
  test normally takes over an hour, in a sound
  attenuated booth, manually administered by an
  expert.
• Currently only a single level response is tested
  and only a pass/fail result is provided, with
  over 5 false positive rate.
• Substantial improvement in rate of signal
  averaging is required to obtain a full diagnostic
  and reliable test.

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Auditory Brainstem Response example

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Infant Hearing Screening
Space Limitations
Time Constraints
Patient Tracking
Electrical Noise
Acoustic Noise
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Auditory Brainstem Response (ABR)Signal
Processing Clinical Issues
Frequency domain characteristics of a typical
ABR click stimulus as measured in the ear using
the ER-10C transducer
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Auditory Brainstem Response (ABR)Signal
Processing Clinical Issues

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Linear Averaging

• Linear averaging - sample mean estimate
• Linear averaging increases the amplitude SNR by a
  factor of N1/2
• Cramer Rao lower bound on variance

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Linear Averaging
Comparison of Fsp values with and without
stimulus presentation
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Wavelet Basics

• Traditional Fourier Transform
• Representation of signals in orthonormal basis
  using complex exponentials (real and imaginary
  sinusoidal components).
• Signal represented in frequency domain by a
  one-dimensional sequence.
• Loses time information.
• Features like transients, drifts, trends, etc.
  may be lost upon reconstruction.

• Wavelet Transform
• Representation of signals in unconditional
  orthonormal basis using waveforms of limited
  durations with average value of zero.
• Makes no assumption about length or periodicity
  of signals.
• Contains time information in coefficients
• Signal can be fully reconstructed using inverse
  transform, and local time features are
  preserved.

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Wavelet Transform

• Discrete wavelet transform (DWT)
• (a scale coefficient, ßtranslation
  coefficient)

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Example Wavelet Filters
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Wavelet Decomposition Example
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Conventional Wavelet Denoising

• Conventional denoising
• Perform wavelet transform.
• Set coefficients C(a,ß)ltd to zero, d
  threshold value. These coefficients are more
  likely to represent noise than signal.
• Perform inverse wavelet transform.
• Characteristics of conventional denoising
• Assumes that signal is smooth and coherent, noise
  rough and incoherent.
• Operation is performed on a single frame of data.
• Non-linear operation reduces the coefficients
  differently depending on their amplitude.

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Conventional Wavelet Denoising

• Why does wavelet denoising work?
• The underlying signal is smooth and coherent,
  while the noise is rough and incoherent
• A function f(t) is smooth if
•  
• A function f(t) is smooth to a degree d, if
• Bandlimited functions are smooth
• Measured biologic functions are smooth (such as
  ABR)

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Conventional Wavelet Denoising

• Coherent vs. incoherent
• A signal is coherent if its energy is
  concentrated in both time and frequency domains.
• A reasonable measure of coherence is the
  percentage of wavelet coefficients required to
  represent 99 of signal energy.
• An example well-concentrated signal may require
  5 of coefficients to represent 99 of its
  energy.
• Completely incoherent noise requires 99 of
  coefficients to represent 99 of its energy.

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Conventional Wavelet Denoising
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Conventional Wavelet Denoising
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Novel Wavelet Denoising

• Conventional denoising applied to weak biosignals
• Setting coefficients C(a,ß)lt d to zero,
  effectively removes all the coefficients,
  including the ones that represent the signal.
• SNR must be large (gt20dB).
• Novel Wavelet Denoising
• Take advantage of multiple frames of data
  available.
• Create new frames through recombination and
  denoising.
• Apply a different dk for each new set of
  recombined frames.

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Tree Denoising

• Create a tree
• Collect a set of N frames of original data f1,
  f2, , fN
• Take the first two frames of the signal, f1 and
  f2, and average together, f12 (f1f2)/2
• Denoise this average f12 using a threshold dk ,
  fd12den(f12 ,d1).
• Linearly average together two more frames of the
  signal, f34 ,and denoise that average,
  fd34den(f34 ,d1). Continue this process for all
  N frames
• Create a new level of frames consisting of fd12,
  fd34, , fdN-1,N.
• Linearly average each two adjacent new frames to
  create f1234(fd12 fd34), and denoise that
  average to create fd1234den(f1234 ,d2).
• Continue to apply in a tree like fashion.
• Apply a different dk for denoising frames at each
  new level .

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Tree Denoising Graph
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Cyclic Shift Tree Denoising (CSTD)
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Cyclic Shift Tree Denoising (CSTD)
  Original signal ?Denoise with
d1 k1 ? Denoise with d2 k2 ?
Denoise with d3     ? Denoise with
dk Final level  
 
 
 
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Frame Permutations

• - Create new arrangements of original frames
  prior to CSTD
• xnew(pxold) mod N
• Increase total number of new frames by a factor
  of 0.5Nlog2(N)

 
 
 
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Threshold Selection
 
 
 
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Estimated Rate of Convergence

• Linear averaging - sample mean estimate
• CSTD Creates Mlog2(N)N new frames.
• Permutations prior to CSTD create at most
  M0.5(N2 log2(N) new frames.
• CSTD can improve the Cramer-Rao lower bound by at
  most a factor of 0.5Nlog2(N).
• The new frames are not linearly dependent, but
  also not all statistically independent.

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Experimental ResultsNoisy Sinewaves
 
 
 
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Experimental ResultsABR Data
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Experimental ResultsABR Data
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Experimental ResultsAMLR Data
Performance of CSTD algorithm compared to linear
averaging 256 data frames. (a) Template of AMLR
evoked potential waveform from Spehlmann (b)
linear average of 8192 AMLR frames (c) Single
frame consisting of AMLR model plus WGN (d)
Linear average of 256 frames (e) Result of
CSTD algorithm
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The Final Product
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Future Work Other applications

• Wavelet denoising using wavelet packets
• EEG/EP Recording and Monitoring
• Use in ambulances and emergency rooms
• At-home patient monitoring
• Depth of Anesthesia Monitoring
• Monitor brain stem and cortex activity during
  surgery
• Use in all operating rooms
• Oto-toxic drug administration
• Certain strong antibiotics cause hearing loss -
  ototoxic
• Dosage can be monitored on-line
• Use in intensive care units

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ED Bedside in minutes
Non-patient care Environment-hours
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HLB PRELIMINARY CONCEPT
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HLB PRELIMINARY CONCEPT
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Thank you!

• Questions?

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Experimental ResultsNoisy Sinewaves
 
 
 
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Example Wavelet Filters
An additional property of a basis is being
unconditional. A basis fn is an unconditional
basis for a normed space if there is some
constant Clt8 such that
for coefficients cn, and any sequence en
of zeros and ones. This means that if some
coefficients cn are set to zero by the sequence
en, the norm of the remaining series is always
bounded. Sines and cosines are NOT
unconditional bases.