By Elvir Causevic Department of Applied Mathematics Yale University Founder and President

1

ByBy

Elvir CausevicElvir Causevic

Department of Applied MathematicsDepartment of Applied Mathematics

Yale UniversityYale University

Founder and PresidentFounder and President

Everest Biomedical InstrumentsEverest Biomedical Instruments

Fast Wavelet Estimation of Fast Wavelet Estimation of Weak BiosignalsWeak Biosignals

2

OverviewOverview

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

3

IntroductionIntroduction Overall goalOverall goal

Creation of a fast estimator of weak biosignals based on Creation of a fast estimator of weak biosignals based on wavelet signal processing. Application to auditory wavelet signal processing. Application to auditory brainstem responses (ABRs) and other evoked potentialsbrainstem responses (ABRs) and other evoked potentials

Specific objectivesSpecific objectives Reduce the length of time to acquire a valid ABR signal.Reduce the length of time to acquire a valid ABR signal. Allow ABR signal acquisition in a noisy environment.Allow ABR signal acquisition in a noisy environment.

Key obstaclesKey obstacles Very large amount of acoustical and electrical noise Very large amount of acoustical and electrical noise

present .present . Signals collected from ear and brain have very low SNR Signals collected from ear and brain have very low SNR

and require long averaging timesand require long averaging times

4

• Infant hearing screening is critically important in early Infant hearing screening is critically important in early intervention of treating deafness.intervention of treating deafness.

• Hearing loss affects 3 in 1,000 infants: most commonly occurring Hearing loss affects 3 in 1,000 infants: most commonly occurring birth defect.birth defect.

• 25,000 hearing impaired babies born annually in the U.S. alone.25,000 hearing impaired babies born annually in the U.S. alone.• Lack of early detection often leads to permanent loss of ability to Lack of early detection often leads to permanent loss of ability to

acquire normal language skills.acquire normal language skills.• Early detection allows intervention that commonly results in Early detection allows intervention that commonly results in

development of normal speech by school age.development of normal speech by school age.• Intervention involves hearing aids, cochlear implants and Intervention involves hearing aids, cochlear implants and

extensive parent and child education and training.extensive parent and child education and training.• 38 U.S. states mandate hearing screening, Europe, Australia, 38 U.S. states mandate hearing screening, Europe, Australia,

Asia following closely.Asia following closely.

Infant Hearing ScreeningInfant Hearing Screening

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Measurement of Hearing Measurement of Hearing FunctionFunction

Auditory Brainstem Response (ABR) - Auditory Brainstem Response (ABR) - neural testneural test

– Response of the VIIIResponse of the VIIIthth nerve - auditory neuro- nerve - auditory neuro-pathway to brainpathway to brain

VIIIth Nerve

6

Auditory Brainstem Response Auditory Brainstem Response (ABR)(ABR)

Signal Processing & Clinical IssuesSignal Processing & Clinical Issuesfor Infant Hearing Screeningfor Infant Hearing Screening

Stimulus: 37 clicks per second, 65 dB SPL (30 dB nHL).Stimulus: 37 clicks per second, 65 dB SPL (30 dB nHL). Response: scalp electrodes measure μV level signals.Response: scalp electrodes measure μV level signals. Noise: completely buries the response (-35dB).Noise: completely buries the response (-35dB). Pass: signal to noise ratio measure (called Fsp) greater than an Pass: signal to noise ratio measure (called Fsp) greater than an

experimentally determined value (NIH Multicenter study).experimentally determined value (NIH Multicenter study). With linear averaging, reliable results are obtained within ~15 With linear averaging, reliable results are obtained within ~15

minutes of averaging of ~ 4000-8000 frames at a single level.minutes of averaging of ~ 4000-8000 frames at a single level. We would like to test multiple levels (up to 10) , and with multiple 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 tone pips (vs. clicks). This test normally takes over an hour, in a sound attenuated booth, manually administered by an expert.sound attenuated booth, manually administered by an expert.

Currently only a single level response is tested and only a pass/fail Currently only a single level response is tested and only a pass/fail result is provided, with over 5% false positive rate.result is provided, with over 5% false positive rate.

Substantial improvement in rate of signal averaging is required to Substantial improvement in rate of signal averaging is required to obtain a full diagnostic and reliable test.obtain a full diagnostic and reliable test.

7

Auditory Brainstem ResponseAuditory Brainstem Response example example

8

9Acoustic Noise

Electrical Noise

Space Limitations

Time Constraints

Patient Tracking

Infant Hearing ScreeningInfant Hearing Screening

10


Signal Processing & Clinical IssuesSignal Processing & Clinical IssuesQuickTime™ and aGraphics decompressorare needed to see this picture. -100-90-80-70-60-50-40-30-20-100100100010000d

B V

Frequency in HzFrequency domain characteristics of a typical Frequency domain characteristics of a typical

ABR click stimulus as measured in the ear using the ER-10C ABR click stimulus as measured in the ear using the ER-10C transducer transducer

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Signal Processing & Clinical IssuesSignal Processing & Clinical Issues

0 1 2 3 4 5 6 7 8 9 10 11 12-20

-15

-10

-5

0

5

10

15

20

Typical single 512-sample frame with the final average ovelaid (Subject 3; right ear; 65 dB click)

Latency after click presentation (ms)

Am

plit

ude

( V

)

0 1 2 3 4 5 6 7 8 9 10 11 12-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Typical ABR waveform with manually labeled peak latencies(Subject 3; right ear; 65 dB click; 8,192 frame average, filtered)


Am

plit

ude

( V

)

peak V

12

Linear AveragingLinear Averaging

Linear averaging - sample mean estimate

Linear averaging increases the amplitude SNR by a factor of N1/2

Cramer Rao lower bound on variance

1

0

22

22

1

0

1]}[var{

1][

1var}ˆvar{

N

n

N

n NN

Nnx

Nnx

NA

1

0

][1ˆ

N

n

nxN

A

1

0

1

0

1]}[{

1][

1}ˆ{

N

n

N

n

ANAN

nxEN

nxN

EAE

.)];[(ln

1,

)];[(ln

1)ˆvar(

2

2

2

2const

AAnxp

E

cwhereN

c

AAnxp

EN

A

13

Linear AveragingLinear Averaging

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

3

6

9

12

15

18

21

24

Typical Fsp comparison for ABR recordingswith 65 dB stimulus vs no stimulus

Frame number

Fsp

val

ue

No stimulus 65 dB stimulus

Comparison of Fsp values with and without stimulus presentation Comparison of Fsp values with and without stimulus presentation

14

Wavelet BasicsWavelet Basics

Traditional Fourier Traditional Fourier TransformTransform

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 TransformWavelet 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.

knNjknN

N

n

knN eWwhereWnxkX )/2(

1

0

,][][

).2(2,][),( 2,,

1

0

knnwherennxkjC jj

kjkj

N

n

15

Wavelet TransformWavelet Transform

• Discrete wavelet transform (DWT)Discrete wavelet transform (DWT)

(α = scale coefficient, β=translation coefficient)(α = scale coefficient, β=translation coefficient)

ZkNjkfor

ngnfkjCC

jj

kjn

,,2,2

,),(, ,

Signal x[n] LP filter with H HP filter with G

Hx, Gx HHx HGx HHHx HHGx …. Final level

16

Example Wavelet FiltersExample Wavelet Filters

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1LP Decomposition filter H

0 1 2 3 4 5 6 7 8-0.5

0

0.5

1LP Reconstruction filter H'

0 1 2 3 4 5 6 7 8-1

-0.5

0

0.5HP Decomposition filter G

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5HP Reconstruction filter G'

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Wavelet Decomposition Wavelet Decomposition ExampleExample

18

Conventional Wavelet Conventional Wavelet DenoisingDenoising

Conventional denoising1. Perform wavelet transform.2. Set coefficients |C(α,β)|<α,β)|<δ to zero, δ – threshold value.

These coefficients are more likely to represent noise than signal.

3. 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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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)

.)(

functioncontinuousaisdt

tfdNn

n

n

.)(

0 functioncontinuousaisdt

tfddn

n

n

20


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.

21


22


0 0.5 1

-20

0

20

Noisy sinewave

-20

dB

0 0.5 1

-20

0

20

Simple low pass filter

0 0.5 1

-20

0

20

Conventional denoising

0 0.5 1-10

0

10

-10

dB

0 0.5 1-10

0

10

0 0.5 1-10

0

10

0 0.5 1-5

0

5

0 dB

0 0.5 1-5

0

5

0 0.5 1-5

0

5

0 0.5 1-2

0

2

+10

dB

0 0.5 1-2

0

2

0 0.5 1-2

0

2

0 0.5 1-2

0

2

+20

dB

0 0.5 1-2

0

2

0 0.5 1-2

0

2

23

Novel Wavelet DenoisingNovel Wavelet Denoising

Conventional denoising applied to weak biosignals• Setting coefficients |C(α,β)|<α,β)|< δ to zero, effectively removes all

the coefficients, including the ones that represent the signal.• SNR must be large (>20dB).

Novel Wavelet Denoising• Take advantage of multiple frames of data available.• Create new frames through recombination and denoising.

• Apply a different δk for each new set of recombined frames.

Proprietary confidential information

24

Tree DenoisingTree Denoising

Create a tree1. Collect a set of N frames of original data [f1, f2, …, fN]

2. Take the first two frames of the signal, f1 and f2, and average together, f12=

(f1+f2)/2

3. Denoise this average f12 using a threshold δk , fd12=den(f12 ,δ1).

4. Linearly average together two more frames of the signal, f34 ,and denoise that

average, fd34=den(f34 ,δ1). Continue this process for all N frames

5. Create a new level of frames consisting of [fd12, fd34, …, fdN-1,N].

6. Linearly average each two adjacent new frames to create f1234=(fd12 +fd34), and

denoise that average to create fd1234=den(f1234 ,δ2).

7. Continue to apply in a tree like fashion.

8. Apply a different δk for denoising frames at each new level .


25

Tree Denoising GraphTree Denoising Graph


1. Original signal x[n] consisting of N=8 frames of data of n signal samples in each frame

f1 f 2 f 3 f 4 f 5 f 6 f 7 f 8

N

2. Create a signal x1[n] at level k=1 by averaging frames of x[n] (f12=(f1+f2)/2 and denoising by δ1

fd12 fd 34 fd56 fd 78

N/2

3. Create a signal x2[n] at level k=2 by averaging frames of x1[n] and denoising by δ2

fd1234 fd 5678

N/4

4. Create a signal x3[n] at level k=3 by averaging frames of x2[n] and denoising by δ3

fd12345678

N/8

26

Cyclic Shift Tree Denoising Cyclic Shift Tree Denoising (CSTD)(CSTD)


1. Original signal x[n] consisting of N=8 frames of data of n signal samples in each frame

f1 f 2 f 3 f 4 f 5 f 6 f 7 f 8

N

2. Create a signal x1[n] at level k=1 by averaging frames of x[n] (f12=(f1+f2)/2), then cyclic shifting

frames x[n] to create new cyclic shift averages (f23=(f2+f3)/2), and denoising

fd12 fd 34 fd56 fd 78 fd 23 fd45 fd 67 fd 81

N/2 N/2

3. Create a signal x2[n] at level k=2 by averaging frames of x1[n], then cyclic shifting frames x1[n]

to create new cyclic shift averages, and denoising


N/4 N/4 N/4 N/4

4. Create a signal x3[n] at level k=3 by averaging frames of x2[n], then cyclic shifting frames x2[n]

to create new cyclic shift averages, and denoising


N/8 N/8 N/8 N/8 N/8 N/8 N/8 N/8

27

Cyclic Shift Tree Denoising Cyclic Shift Tree Denoising (CSTD)(CSTD)


Original signal Denoise with δ1

k=1 Denoise with δ2

k=2… Denoise with δ3

… … Denoise with δk

Final level

28

Frame PermutationsFrame Permutations


- Create new arrangements of original frames prior to CSTD

- xnew=(p*xold) mod N

- Increase total number of new frames by a factor of 0.5*N*log2(N)

p Frame 0 Frame 1 Frame 2 Frame 3 Frame 4 Frame 5 Frame 6 Frame 71 0 1 2 3 4 5 6 73 0 3 6 1 4 7 2 55 0 5 2 7 4 1 6 37 0 7 6 5 4 3 2 1

29

Threshold SelectionThreshold Selection


.44

0,)

4cos(

1)

4cos(

,2

12

,e

1

,log

1log

,1

1

k

22

K

k

Kk

andK

k

and

ande

(k)and(k)

kandk

kandk

k

k

kk

Threshold function selection

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.5 1 1.5 2 2.5

Initial threshold value

1 / sqrt(2)^k

1 / exp ^k

1 / k

constant

1/ k^2

1 / log(k)

cos (2*Pi*k)

sqrt (2) ^k

30

Estimated Rate of Estimated Rate of ConvergenceConvergence

Linear averaging - sample mean estimate

CSTD Creates M=log2(N)*N new frames.

Permutations prior to CSTD create at most M=0.5*(N2 * log2(N) new frames.

CSTD can improve the Cramer-Rao lower bound by at most a factor of 0.5*N*log2(N).

The new frames are not linearly dependent, but also not all statistically independent.

1

0

][1ˆ

N

n

nxN

A .)];[(ln

1,

)];[(ln

1)ˆvar(

2

2

2

2const

AAnxp

E

cwhereN

c

AAnxp

EN

A

31

Experimental ResultsExperimental ResultsNoisy SinewavesNoisy Sinewaves


0 2 4 6 8 10 12-1

0

1

2

3

4

5

6

Linear Average and CSTD for Sinewave data at -20 dB512 frames with

1 =1

Time (ms)

Ma

gni

tud

e (

with

plo

tting

offs

et)

Linear Avg.CSTD Original

102

103

10-2

10-1

100

Variance of Linear Avgerage and CSTD for Sinewave at -20 dB512 frames with

1 =1

Number of frames averagedE

stim

ato

r V

aria

nce

Linear AverageCSTD

32

Experimental ResultsExperimental ResultsABR DataABR Data

0 2 4 6 8 10 12-0.5

0

0.5

1

1.5

2

2.5

3

Linear Avgerage and CSTD for ABR Data (Subject 3)512 frames with

1 =1


Mag

nitu

de

(with

plo

ttin

g of

fset

)

Linear AverageCSTD Final Average

102

103

10-2

10-1

100

Variance of Linear Avgerage and CSTD for ABR Data (Subject 3)512 frames with

1 =1

Number of frames averaged

Est

imat

or V

aria

nce

Linear AverageCSTD

Frames σdenoised δav eraged Ratio

2 5.2457 10.1841 1.944 2.5953 3.9456 1.528 2.6032 3.0031 1.15

16 1.5141 1.8796 1.2432 0.5419 0.9337 1.7264 0.077 0.2533 3.29

128 0.0448 0.1143 2.55256 0.0379 0.075 1.98512 0.0139 0.0357 2.57

33

Experimental ResultsExperimental ResultsABR DataABR Data

0 2 4 6 8 10 12-1

0

1

2

3

4

5

6

7

Linear Average and CSTD for ABR Data (Subject 3)128 frames with

1 =1


Mag

nitu

de

(w

ith p

lotti

ng o

ffse

t)

Linear Avg.CSTD Final Avg.

0 2 4 6 8 10 12-1

0

1

2

3

4

5


1 =1


Mag

nitu

de

(w

ith p

lotti

ng o

ffse

t)


0 2 4 6 8 10 12-0.5

0

0.5

1

1.5

2

2.5

3


1 =1


Mag

nitu

de

(w

ith p

lotti

ng o

ffse

t)


34

Experimental ResultsExperimental ResultsAMLR DataAMLR Data

V

Nb

PbPa

Time (ms)

Na

(e)

(a)

(b) (c)

(d)

Performance of CSTD algorithm compared to linear averaging 256 data frames. (a): Template 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 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; (c): Single frame consisting of AMLR model plus WGN; (d): Linear average of 256 frames; (e): Result of CSTD algorithm frames; (e): Result of CSTD algorithm

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The Final ProductThe Final Product

36

Future Work & Other Future Work & Other applicationsapplications

Wavelet denoising using wavelet packets EEG/EP Recording and Monitoring

• Use in ambulances and emergency roomsUse in ambulances and emergency rooms

• At-home patient monitoringAt-home patient monitoring

Depth of Anesthesia Monitoring• Monitor brain stem and cortex activity during surgeryMonitor brain stem and cortex activity during surgery

• Use in all operating roomsUse in all operating rooms

Oto-toxic drug administration• Certain strong antibiotics cause hearing loss - ototoxicCertain strong antibiotics cause hearing loss - ototoxic

• Dosage can be monitored on-lineDosage can be monitored on-line

• Use in intensive care unitsUse in intensive care units

37

ED Bedside in minutes

Non-patient care Environment-hours

38

39HLB PRELIMINARY CONCEPT

40HLB PRELIMINARY CONCEPT

41

Thank you!Thank you!

Questions?Questions?

42

Experimental ResultsExperimental ResultsNoisy SinewavesNoisy Sinewaves


FramesLinear

varianceDenoised variance

Variance ratio

Linear SNR (dB)

Denoise SNR (dB)

SNRdB

Improvement2 23.6363 2.0081 11.77 -16.74 -6.03 10.714 12.6286 0.9706 13.01 -14.02 -2.87 11.148 5.967 0.6821 8.75 -10.76 -1.34 9.42

16 2.967 0.3341 8.88 -7.72 1.76 9.4832 1.6339 0.161 10.15 -5.13 4.93 10.0664 0.8 0.0827 9.67 -2.03 7.82 9.86

128 0.4052 0.0479 8.46 0.92 10.2 9.28256 0.1902 0.0271 7.02 4.21 12.68 8.47512 0.1023 0.0184 5.56 6.9 14.35 7.45

43

Example Wavelet FiltersExample Wavelet Filters

An additional property of a basis is being unconditional. A basis {φn} is an unconditional basis for a

normed space if there is some constant C<∞ such that

00 n

nnn

nnn cCc

for coefficients cn, and any sequence {εn} of zeros and ones. This means that if some coefficients cn are set to zero by

the sequence {εn}, the norm of the remaining series is always bounded. Sines and cosines are NOT unconditional

bases.

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By Elvir Causevic Department of Applied Mathematics Yale University Founder and President

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