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Heart Sounds, ECG &
Fractals
The Heart
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
The heart is a 2-step mechanical
pump that is coordinated by
precisely timed electrical
impulses.
Fractal Results
The END
Lets
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The Heart
The Heart
ECG Wave
Heart sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
The heart is a pulsating pump that
composes of four chambers and
four heart valves.
The upper chambers are the right
atrium (RA) and left atrium (LA)
The lower chambers are the right
ventricle (RV) and left ventricle
(LV).
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Electrophysiology of cardiac conduction
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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4
Heart Valves
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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5
Events occurring during the cardiac cycle
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The cardiac cycle consists of two basic
components:
 A period of ventricular diastole during
which the ventricles are filled with
blood.
 A period of ventricular systole during
which blood is propelled out of the
heart.
The END
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Events occurring during the cardiac cycle
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
 Clinically, systole is taken as the
interval between the first and the
second heart sound.
 Diastole is considered to be the interval
between second heart sound and the
first heart sound.
The END
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7
The Electrical system
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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8
What is measured on the ECG
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Rate and rhythm of the heart.
Evidence of heart enlargement.
Evidence of damage to the heart
Fractal Dimension
Impaired blood flow to the heart
Sound Analysis
Heart rhythm problems
Fractal Results
Electrolyte imbalance
The END
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9
What are the limitations of the ECG
Electrophysiology
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The ECG is a static picture
Many heart attacks cannot be
detected by ECG.
Many abnormal patterns on an ECG
may be non-specific.
The ECG may be normal despite the
presence of a cardiac condition
The END
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10
ECG wave
Electrophysiology
ECG Wave
Heart Sounds
Abnormal Sounds
ECG tracings show a pattern of electrical
impulses that are generated by the heart.
Audicor’s Solution
P wave: the sequential activation
(depolarization) of the right and left atria
Fractal Dimension
QRS complex: right and left ventricular
depolarization
Sound Analysis
ST segmet: ventricular repolarization.
Fractal Results
The END
The T wave corresponds to electrical
relaxation and preparation for their next
muscle contraction.
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ECG wave
Electrophysiology
ECG Wave
Heart sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
ML Model
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12
Heart Sounds
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
The auscultation of the heart may reveal
different phenomena called heart sounds
and murmurs.
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
Heart sounds are a prolonged series of
vibrations of both high and low frequency
The murmurs are a longer series of
vibrations, mostly of either high or low
frequency.
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Heart Sounds
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
The sounds heard during auscultation are
called the first (S1) and second (S2) heart
sounds respectively,
with respect to their temporal relationship, and
are systolic sounds.
Phonocardiography often yields third (S3) and
fourth (S4) heart sounds especially in children
and in cases of heart disease. These are
diastolic sounds.
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14
Heart Sounds Genesis
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 Many hypotheses have been suggested to
explain the origin of these sounds.
 Some being controversial at the time.
 With the advent of echocardiography the
movement of intracardiac structures could
be monitored with virtually no time delay.
 Concerning S1, S2, and S4 these
controversies have largely been resolved.
However there still exists controversy
regarding S3.
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S1 & S2
Electrophysiology
ECG Measurements
Heart Sounds
Abnormal Sounds
Audicor’s Solution
S1 occurs when the mitral and
tricuspid valves close at the beginning of
systole.
Fractal Dimension
Sound Analysis
Fractal Results
S2 results from closure of the aortic and
pulmonic valves at the end of systole.
The END
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S3
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
low frequency sound: 0 - 70 Hz
50% in the 0-15 Hz band
Audicor’s Solution
occurring in early ventricular diastole
Fractal Dimension
due to over-distention of the ventricle
during the rapid early filling phase
Sound Analysis
Fractal Results
occurs 0.12 – 0.20 secs after S2
The END
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The physiological cause and effect of S3
Electrophysiology
ECG Measurements
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 The third heart sound (S3) occurs 0.12 to
0.20 seconds after S2 in early Diastole.
 Of the many proposed theories, the most
likely explanation is that excessive rapid
filling of the ventricle is suddenly halted,
causing vibrations that are audible as
S3.
 Pathologic states where an S3 is
encountered include anemia, thyrotoxicosis,
mitral regurgitation, hypertrophic
cardiomyopathy, aortic and tricuspid
regurgitation and left ventricular
dysfunction.
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18
S4
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
low frequency sound: 0 - 70 Hz
occuring at the late diastolic filling
phase at when the atria contract
Ventricles have decreased compliance,
or receive increased diastolic volume
occurs just before S1
70
ms after onset of ECG P wave
The END
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The physiological cause and effect of S4
Electrophysiology
ECG Measurements
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 The S4 occurs just before the first heart
sound in the cardiac cycle.
 It is produced in late diastole as a result of
atrial contraction causing vibrations of the
LV muscle, mitral valve apparatus, and LV
blood mass.
 Disease processes that produce an S4
include hypertension, aortic stenosis and
regurgitation, severe mitral regurgitation,
cardiomyopathy, and ischemic heart disease.
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The physiological cause and effect of
S3 & S4
Electrophysiology
ECG Measurements
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
In 1856, Potain first described "gallop rhythm"
as an audible phenomenon in which a tripling
or quadrupling of heart sounds resembles the
canter of a horse.
That term is still used to describe a third or
fourth heart sound.
The END
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The physiological cause and effect of
S3 & S4
Electrophysiology
ECG Measurements
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Gallops are diastolic events and seem to be
related to 2 periods of filling of the ventricles:
1. The rapid filling phase (ventricular diastolic
gallop or S3)
2. The presystolic filling phase related to
atrial systole (atrial gallop or S4)
Fractal Results
The END
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The physiological cause and effect of
S3 & S4
Electrophysiology
ECG Measurements
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 Experimental evidence in both humans and
animal models suggests that abnormal
compliance of the left ventricle is often
associated with an S4 and/or a pathological
S3.
 In the early diastolic phase of the cardiac
cycle, the left ventricle relaxes and the
intraventricular blood pressure falls below
that of the left atrium.
 Therefore, blood flows from the atrium into
the ventricle.
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The physiological cause and effect of
S3 & S4
Electrophysiology
ECG Measurements
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 This continues until the intraventricular
pressure equals the pressure in the atrium
and the flow of blood into the ventricle
therefore stops.
 This deceleration of the blood early in
diastole produces vibrations inside the
ventricle, which can result in an S3 if the
vibrations have sufficient energy.
 The steep left ventricular pressure increase
in early diastole causes a reversal of the
transmitral pressure gradient and hence a
more rapid deceleration of inflow.
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The physiological cause and effect of
S3 & S4
Electrophysiology
ECG Measurements
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 Since the vibrations of S3 occur during
deceleration of inflow, a conversion of
kinetic into vibratory energy is likely.
 These vibrations are audible if transmitted
with enough intensity.
 The higher the inflow rate (valve
regurgitation) and the steeper the rapid
filling wave (high filling rates), the greater
the deceleration and more likely an S3 will
occur.
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The physiological cause and effect of
S3 & S4
Electrophysiology
ECG Measurements
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
 S3 is produced when the rapidly distending
ventricle reaches a point when its distention
is checked by the resistance of its wall and
the ensuing vibrations are audible as the
third heart sound.
The END
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The physiological cause and effect of
S3 & S4
Electrophysiology
ECG Measurements
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 The remainder of the filling of the ventricle
occurs late in diastole because of active
contraction of the atrium.
 The deceleration of the blood later in diastole
also produces vibrations inside the ventricle.
 If the atrial contraction that produced the
late diastolic filling was sufficiently strong
and the ventricle is relatively stiff, these
vibrations may have enough energy to
produce an S4.
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Heart Sounds Characteristics
The Heart
http://www.cardiologysite.com/auscultation/html/s3_gallop.h
tml
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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28
Heart Sounds
The Heart
http://depts.washington.edu/physdx/heart/tech2.html
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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29
PCG against ECG
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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30
PCG against ECG
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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31
The relationship between heart sounds and
cardiac events
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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How do S3 & S4 Help?
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 Experimental evidence suggests that
abnormal compliance of the left ventricle is
often associated with an S4 and/or a
pathological S3.
 The S3 may be a normal finding in patients
less than 30 years old.
 However, in older patients, the S3 is usually
evidence of impaired ability of the ventricle to
contract during systole.
 The prevalence of the S4 increases with age
and usually indicates an abnormal increase in
ventricular stiffness
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Clinical significance of S3
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 The presence of S3 may be the earliest clue to
left ventricular failure.
 The presence of heart disease and may offer
valuable information about diagnosis,
prognosis, and treatment.
 The most useful clinical importance of S3 is in
detecting left-sided heart failure, especially in
the early stages when other signs may be
normal.
 More recently, S3 was the best predictor of
response to digoxin in CHF patients.
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Usefulness of S3, S4 & ECG in assisting
early detection
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 Several types of cardiac disease have
characteristic electrical and hemodynamic
manifestations.
 For example, acute myocardial ischemia is
typically associated both with displacement
of the ST segments of the ECG and with
alterations of the mechanical properties of
the left ventricle.
 The latter changes may produce pathological
heart sounds – S3 and/or S4.
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35
S3 In Children
The Heart
ECG Wave
 The genesis of S3 has been clearly associated
with the rapid filling phase of diastole.
Heart Sounds
 Present work has shown that S3 occurs
earlier in the cardiac cycle with increase in
Abnormal Sounds
age of child subjects.
Audicor’s Solution
Sound Analysis
 This supports the hypothesis that S3 is due
to L.V. reaching it’s elastic limit during
diastole.
Fractal Results

Fractal Dimension
The END
This notion is supported further by the
finding of the spectral energy of S3 is
distributed more towards the high frequency
of the end of the spectrum with age.
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S3 In Children
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 This is consistent with an increase in
stiffness of the L.V. with age. The resonant
frequencies of L.V. increase with stiffness.
 higher frequencies are more attenuated by
passage through body tissue than lower
frequencies.
 As the frequency distribution of S3 is shifted
to higher frequencies as the child becomes
older, it would be expected that the energy in
S3 would decrease with age.
 Thus S3 usually disappears around
adulthood, but may reoccur with cardiac
pathology.
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Combining ECG & Heart Sounds
The Heart
Heart Sounds
ECG Wave
Audicor uses an advanced new technology
called correlated audioelectric
cardiography (COR).
This technology builds on the traditional
findings of the standard, 12-lead resting
Audicor’s Solution
ECG, augmenting it by simultaneously
Fractal Dimension acquiring acoustical signals from both the
V3 and V4 lead positions.
Sound Analysis
Abnormal Sounds
Fractal Results
The END
(Two acoustic sensors replace the V3 and
V4 ECG electrodes of a standard 12-Lead
ECG )
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Combining ECG & Heart Sounds
The Heart
Heart Sounds
Audicor CE combines the detection of
heart sounds with ECG data in order to
ECG Wave
provide physicians with additional
Abnormal Sounds
information that is valuable in
Audicor’s Solution assessing:
 S3 and S4 heart sounds that may be
Fractal Dimension
indicative of acute coronary
syndrome or heart failure
Sound Analysis
 Acute and prior (age-undetermined)
Fractal Results
myocardial infarction (MI)
The END
 Ischemia
 Left ventricular hypertrophy (LVH)
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Combining ECG & Heart Sounds
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
The S3 heart sound is often very difficult to
detect by auscultation due to its low
frequency and intensity.
Audicor’s Solution Noisy clinical environments further complicate
Fractal Dimension
Sound Analysis
Fractal Results
The END
this difficulty.
To improve the detection of the S3, Inovise
Medical, Inc. has developed AUDICOR®, a
device that records and algorithmically
interprets simultaneous 12-lead ECG and
electronic cardiac sound recording
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Audicor Heart Sound Algorithm
The Heart
ECG Wave
Heart Sounds
Heart Sounds
Audicor’s Solution The AUDICOR heart sounds algorithm receives
Fractal Dimension
Sound Analysis
three synchronous inputs:
1. A standard ECG signal
2. Two single-channel sound signals.
Fractal Results
The END
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41
Processing The Sound Data
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
The sound data for each channel is then
processed by removing offsets, prescaling, and
filtering it into narrow frequency bands to
optimize the detection of each S1 through S4
heart sound.
Audicor’s Solution Using the ECG as a reference, the S1 and S2
Fractal Dimension
Sound Analysis
Fractal Results
The END
detection time windows are identified for each
beat.
Utilizing a threshold adaptively computed from
a moving window root mean square for each
frequency band, the location of each S1 and S2
is determined within the computed detection
window.
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42
Detecting S3
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
The S3 detection time windows are located
using information within the ECG and the
computed position of the S2 offset.
The energy content is determined within the S3
detection time window.
Using a set of rules based on frequency and
amplitude measurements, possible S3s are
detected within the S3 windows.
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43
Detecting S4
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The S4 detection time windows are located
based on PQ intervals and Q-wave onset
positions.
Further processing on S4s is similar to that
described before for S3s.
The END
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44
ECG Diagnostic Algorithm
The Heart
ECG Wave
Heart Sounds
Superior diagnostic performance up to 84%
more sensitive than current systems in acute
MI detection, particularly in women was
achieved in the following ways :
Abnormal Sounds • Developed the computerized ECG diagnostic
Audicor’s Solution
algorithms using very large clinically
correlated databases of over 100,000 ECGs.
Fractal Dimension
Sound Analysis
Fractal Results
The END
• Divided the data into demographically
balanced learning and test sets to
help ensure that it was not overtraining the
algorithms using limited sets of data.
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45
ECG Diagnostic Algorithm
The Heart
ECG Wave
Heart sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
• Accounted for differences in gender and
age,
emphasizing features that discriminate
between prior and acute MI.
• Avoided circularity in developing and testing
the algorithms by selecting all cases and
non-cases of various diseases using criteria
independent of the ECG
The END
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46
ECG Diagnostic Algorithm
The Heart
ECG Wave
• Criteria for IMI is based upon
the relationships between portions of the
Heart Sounds
vectorcardiographic (VCG) QRS loop in the
frontal plane and the corresponding
Abnormal Sounds
portions
of the ECG QRS complexes recorded in leads
Audicor’s Solution
II and III.
Fractal Dimension
Sound Analysis
Fractal Results
The END
• Commercial ECG algorithms for detection of
prior myocardial infarction (MI)
predominantly rely on QRS criteria and on
established qualitative ST and T changes.
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47
ECG Diagnostic Algorithm
The Heart
ECG Wave
Heart Sounds
• Two distinct new approaches for quantifying
ST and T changes to assist with the
detection of prior MI.
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
1. The first method uses the mean axes of
vectorcardiographic T-loops taken from
the inverse Dower transform of the 12lead ECG to indicate ischemic regions of
the left ventricular wall.
2. The second method establishes regional
scores for residual ST elevation
supportive of ischemia or infarction.
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48
ECG Diagnostic Algorithm
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution • These 2 ST-T measures qualify borderline
Fractal Dimension
Sound Analysis
QRS infarct criteria, resulting in composite
criteria having higher sensitivities and
specificities than QRS criteria alone.
Fractal Results
The END
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49
ECG Diagnostic Algorithm
LVH is defined as values of the ECG left
ventricular mass index (LVMI) >116 g/m(2)
in men or >104 g/m(2) in women.
The Heart
ECG Wave
Heart Sounds
Univariate linear regression was performed
separately on the male and female subjects
Abnormal Sounds
in the Learning Set to find all the ECG
parameters that correlated significantly
Audicor’s Solution
with
Fractal Dimension
LVMI.
Sound Analysis
Fractal Results
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4
Multivariate linear regression (MLR) was
applied to these parameters to identify the
variables for each sex that discriminated
best between the subjects with and without
Lets
LVH.
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50
Diagnostic Performance of a Computerized Algorithm
for Augmenting the ECG with Acoustical Data
The Heart
ECG Wave
Heart Sounds
Results:
Abnormal Sounds
The following data show the ability of the
Audicor’s Solution computerized acoustical algorithm to detect an
S3 or an S4 in patients in a variety of clinical
settings.
Fractal Dimension
Sound Analysis
Fractal Results
The eND
The performance of the algorithm is compared
to a consensus of 2 experienced cardiologists
concerning the audibility of the recorded S3 or
the S4 in the each of the same patients.
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Diagnostic Performance of a Computerized Algorithm
for Augmenting the ECG with Acoustical Data
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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Audicor Analysis
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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53
Audicor Analysis
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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54
The AUDICOR Decision Pathway – EMS
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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ECG, Hemodynamic & Acoustical Findings:
Experimental Model of Myocardial Ischemia
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
Acute myocardial ischemia often displaces the
ST segments in the ECG.
However, since the specificity and sensitivity
for ischemia of ST segment displacement
are imperfect
Echocardiography and radionuclide studies are
often used to augment the ECG in
evaluating patients for ischemia.
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56
ECG, Hemodynamic & Acoustical Findings:
Experimental Model of Myocardial Ischemia
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Ischemia also has hemodynamic effects that
include reduced left ventricular (LV)
contractility and compliance.
These hemodynamic changes are typically
associated with a third and fourth heart
sound (S3 and S4), respectively.
Fractal Results
The END
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57
ECG, Hemodynamic & Acoustical Findings:
Experimental Model of Myocardial Ischemia
The Heart
ECG Wave
Heart Sounds
Conclusion:
Abnormal Sounds
Detecting and recording heart sounds may
improve the identification of acute
myocardial ischemia as the cause of ST
segment abnormalities.
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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ECG & Acoustical Data in the detection of
Left Ventricular Enlargement
The Heart
ECG Wave
Heart Sounds
The presence of 3rd and 4th heart sounds was
Abnormal Sounds associated mainly with relative prolongation of
Audicor’s Solution the PR interval and with flattening or
negativity of T waves in multiple leads.
Fractal Dimension
Sound Analysis
Fractal Results
The END
Conversely these sounds were not associated
with the abnormalities of QRS voltage
traditionally attributed to increased left
ventricular mass.
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ECG & Acoustical Data in the detection of
Left Ventricular Enlargement
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
Conclusion:
ECG and acoustical data can detect
abnormalities of ventricular function that
the cardiac diseases responsible for LVE
produce.
The END
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Detecting Hemodynamic Abnormalities
Using ECG and Cardiac Acoustical Data
The Heart
ECG Wave
Heart Sounds
Background:
Hemodynamic abnormalities can produce ECG
changes.
Abnormal Sounds
For example, the ECG evidence of left
Audicor’s Solution ventricular hypertrophy (LVH) is a
consequence of the hemodynamic
Fractal Dimension abnormalities that produced the LVH.
Sound Analysis
Fractal Results
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However they hypothesized that abnormal
hemodynamics are more likely to predict
the presence of a third heart sound (S3)
than of ECG findings.
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Detecting Hemodynamic Abnormalities
Using ECG and Cardiac Acoustical Data
The Heart
ECG Wave
Heart Sounds
Conclusions:
The electronically recorded S3 is
associated with a wider range of
Audicor’s Solution hemodynamic abnormalities than is
Fractal Dimension ECG evidence of LVH,
ST-T or prior MI and can therefore
Sound Analysis
augment the diagnostic capabilities of
the ECG.
Fractal Results
Abnormal Sounds
The END
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Using Simultaneous ECG and Acoustical Data to
Evaluate and Monitor Patients with Cardiac Disease
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Background:
Acute myocardial ischemia is associated with
hemodynamic as well as ECG
abnormalities.
Audicor’s Solution For example, impaired left ventricular (LV)
Fractal Dimension
Sound Analysis
Fractal Results
systolic function can produce a third heart
sound (S3) that previous research, as
reflected in the ACC/AHA Practice
Guidelines, has shown to be associated
with increased clinical risk.
The END
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Using Simultaneous ECG and Acoustical Data to
Evaluate and Monitor Patients with Cardiac Disease
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
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Sound Analysis
Fractal Results
The END
Results:
In the 89 pre-cath patients, the recorded S3
had a sensitivity /specificity for detecting
an LV ejection fraction <50% and LV
enddiastolic pressure >15mmHg of 13/21
(sens, 62%); 60/68 (spec, 88%).
In the acute MI patient, acoustical changes
preceded ECG changes and a new S3
appeared shortly after the onset of the
acute MI.
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64
Using Simultaneous ECG and Acoustical Data to
Evaluate and Monitor Patients with Cardiac Disease
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
Conclusions:
Electronically recorded S3 identifies patients
with impaired LV systolic function and
recorded heart sounds can be added to
multi-parameter monitoring of patients with
suspected acute MI.
The END
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Automatic Segmentation of Heart Sound
Signals Using Hidden Markov Models
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
Fractal Dimension
Sound Analysis
Fractal Results
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Segmentation of heart sounds into their
component segments, using Hidden Markov
Models.
The heart sounds data is preprocessed into
feature vectors, where the feature vectors are
comprised of the average Shannon energy of
the heart sound signal, the delta Shannon
energy, and the delta-delta Shannon energy.
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Pre-processing
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
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The system filters the original heart sound
signal using a band-pass filter with cutoff
frequencies at 30 Hz and 200 Hz. Next, the
signal is normalized according to:
x( k )
xnorm (k ) 
max ( x(i) )
i
Then, it calculates the average Shannon energy
In continuous 0.04-second segments, with
0.02 seconds of overlap per segment.
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Pre-processing
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
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Sound Analysis
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Shannon energy emphasizes the medium
intensity signals and attenuates the high
intensity signals.
This tends to make medium
and high intensity signals similar in amplitude.
The system calculates the average Shannon
energy of each frame, where Xnorm is the
normalized heart signal, using:
N
2
2
E s  1 / N  x norm
(i )  log x norm
(i )
i 1
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Pre-processing
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
Fractal Dimension
Sound Analysis
Fractal Results
Then, the system normalizes the average
Shannon energy over all of the frames, where
E s (t ) is the average Shannon energy for
frame t
 ( E s (t )) the mean value
 ( E s (t )) the standard deviation
Es (t )   ( Es (t ))
Pa (t ) 
 ( Es (t ))
The END
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Mel-spaced filterbanks
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
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Sound Analysis
Fractal Results
The END
Next, the system extracts the spectral
Characteristics from the heart sound signal.
Since the average duration of the S1 sound is
0.16 seconds (empirical), the system divides
the signal into 0.15-second frames, with 0.02
seconds of overlap for each frame.
The frequency spectrum of S1 contains peaks
in the 10 to 50 Hz range and the 50 to 140
Hz range, while the frequency spectrum of S2
Contains peaks in the 10 to 80 Hz range, the 80
to 200 Hz range, and the 220 to 400 Hz range.
As a result, this study limits the spectral
feature extraction between the frequencies of
10 Hz and 430 Hz.
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Mel-spaced filterbanks
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
Fractal Dimension
Sound Analysis
Fractal Results
The END
Mel-Spaced filter banks provide a simple
method for extracting spectral characteristics
from an acoustic signal.
This method involves creating a set of
triangular filter banks across the spectrum.
The filterbanks are equally spaced along the
mel-scale, as defined in:
j
Mel ( f )  2595 log 10 (1 
)
700
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Mel-spaced filterbanks
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
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Sound Analysis
Fractal Results
The END
Equal spacing on the mel-scale provides nonLinear spacing on the normal frequency axis.
This non-linear spacing means that there are
numerous, small banks at the lower
frequencies and sparse, large banks at the
Higher frequencies.
Since most of the energy of the heart sounds is
in the lower frequency ranges, using a melscale matches the frequency spectrum of the
heart sounds.
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Mel-spaced filterbanks
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
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Sound Analysis
Fractal Results
Each triangular filter is multiplied by the
Discrete Fourier transfer of the heart sound
frame and summed.
This creates a set of frequency bins, where
each bin represents a portion of the frequency
spectrum.
The END
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Regression coefficients
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
Fractal Dimension
The final feature extraction step is to calculate
a set of regression coefficients. Regression
coefficients are used to represent the changes
in each feature over time.
The system computes the first order regression
(delta coefficients) and the second order
Coefficients (delta-delta coefficients) using the
following regression formula:
Sound Analysis
Fractal Results
The END

dt 
 (c


1
t 

2  2
 1
74
 ct  )
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Regression coefficients
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
Fractal Dimension
Sound Analysis
Fractal Results
The system combines the Shannon energy, the
Spectral features, and the regression
coefficients into a single feature vector per
frame.
It stores these feature vectors for later use in
the training and testing of the heart sound
HMM.
The END
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Heart sound Hidden Markov Model
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
Fractal Dimension
Sound Analysis
Fractal Results
The END
One can model the phonocardiogram signal as
a four state HMM:

The first state corresponds to S1.

The second state corresponds to the silence
during the systolic period.

The third state corresponds to S2.

The fourth state corresponds to the silence
during the diastolic period.
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Heart sound Hidden Markov Model
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
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Sound Analysis
Fractal Results
The END
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Heart sound Hidden Markov Model
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
Fractal Dimension
Sound Analysis
This model ignores the possibility of the S3 and
S4 heart sounds, because these sounds are
difficult to hear and record; therefore, they are
most likely not noticeable in the heart sound
data.
Fractal Results
The END
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Heart sound Hidden Markov Model
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
Fractal Dimension
Sound Analysis
Fractal Results
The END
This four state HMM is useful for modeling the
sequence of symbols (or labels) of the
phonocardiogram;
However, it is too simple to accurately model
The transitions between sound and silence.
One solution is to embed another HMM inside
of each of the heart sound symbol states.
The embedded HMM models the signal as it
traverses a specific labeled region of the
signal.
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Heart sound Hidden Markov Model
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
Fractal Dimension
Sound Analysis
Fractal Results
The END
Using this combined approach, we can model
both the high-level state sequence of
our signal (S1-sil-S2-sil) and the continuous
transitions of the signal.
This type of model is similar to how a speech
processing system has a high-level
probabilistic grammar to model the transition
of words or phonemes, and an embedded HMM
for each Phoneme.
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Discussion
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
HMM Models
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Sound Analysis
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The END
Shannon Energy features with and without the
Melspaced filterbank features are nearly
identical in performance.
Shannon Energy features are better suited for
lowering the frame error rate while Mel-spaced
filterbanks are better suited for lowering the
model error rate.
Mel-spaced filterbanks are marginally better as
features for noisy PCGs than
Clean PCGs.
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FFT S3 Analysis
The Heart
ECG Wave

The PCG and ECG data were sampled at a
rate of 2042 Hz (this gave a Nyquist
frequency of about 1000 Hz).

The signals are then digitized by means of a
two channel, 8 bit analogue to digital
converter controlled by an Intel 8085
microprocessor based system (sdk85) with
8k memory.

Each sampled datum was represented as an
unsigned hexadecimal number.
These files are then simultaneously plotted
by means of a graphics terminal.
Heart Sounds
Abnormal Sounds
FFT
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Fractal Results
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FFT S3 Analysis
The Heart
ECG Wave
Heart Sounds

The ECG was used as a time reference for
the PCG plot, which aided in obtaining the
starting and end points of S3.

Each PCG file was multiplied by a file
containing a hamming window (0.54 +
0.46cosθ) co-positioned with the S3, but
zero everywhere else. This had the effect of
extracting the S3 from the PCG file and
multiplying it by a window function.

A conventional FFT is then applied to these
files to produce the S3 spectra.
Abnormal Sounds
FFT
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Why FFT does not work for S3
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
FFT
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Sound Analysis
Fractal Results
The END
When applying it to the short duration S3, the
FFT suffers from a fundamental limitation in
frequency resolution determined by the
window size.
The FFT gives poor resolution for S3 spectral
analysis. The time duration of S3 is relatively
short (50 ms).
This short observation time, combined with the
spectral blurring effects of the window
function accounts for the poor resolution of the
FFT method.
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Why FFT does not work for S3
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
FFT
Fractal Dimension
Sound Analysis
Fractal Results
The major limitation of the FFT approach to
spectral analysis is that of frequency
resolution, i.e. the capability of distinguishing
between closely spaced spectral peaks.
The FFT resolution is about 1/T, where T is the
available data time. Hence, when dealing with
short data lengths the resolution is restricted.
The END
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Why FFT does not work for S3
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
FFT
Fractal Dimension
Sound Analysis
Fractal Results
The END
Another problem inherent to the FFT method is
the effect of spectral “leakage”.
In FFT analysis a real signal represents a
truncated function, which is equivalent to
multiplying it by a “window” function.
The resultant FFT spectrum contains energy
due to both the signal itself and the window
function.
The result is the spectrum of the convolution of
the signal and window functions. This leakage
can be reduced by appropriate design of the
window function, but this always results in
reduced frequency resolution.
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Maximum Entropy Spectral Analysis
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
MESA
Fractal Dimension
Sound Analysis
Fractal Results
The END
Classical spectral analysis requires the
assumptions, about the signal under analysis,
of long samples of data and of stationarity.
However in real applications of biomedical
spectral analysis both these assumptions are
violated.
In the case of the spectral analysis of S3, the
time duration is short enough to consider it
stationary; but the assumption of a long signal
history is obviously erroneous.
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Maximum Entropy Spectral Analysis
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
MESA
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88
The MESA technique has been demonstrated to
produce superior spectral resolution when
compared with more traditional methods,
especially for short data lengths [Burg 1967,
Kay 1981, Ulrych 1975].
Another advantage of MESA is that one can use
a simple rectangular window as there is no
spectral “leakage”.
Studies have shown that FFT is incapable of
satisfactorily resolving the frequency peaks of
in S3 and introduces unwanted leakage.
However the Maximum Entropy Method is
capable of satisfactory resolution with no Lets
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leakage.
Conclusions
The Heart
ECG Wave

Research has shown that S3 can
significantly enhance heart disease
analysis.

The normal range of human hearing lies
within the range of 20 Hz – 20000 Hz, with
maximum sensitivity lying in the speech
range; about 1000 Hz to 3000 Hz.

In order to be heard, low frequency sounds
such as S3, must attain energy levels
thousands of times greater than those
needed by vibrations within the speech
range.
Heart Sounds
Abnormal Sounds
Future Research
Fractal Dimension
Sound Analysis
Fractal Results
The END
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Conclusions
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds

Thus the extraction of S3 & S4 is almost
entirely dependant on the development of
accurate automated methods.

Following a review of the literature it is
apparent that a truly successful S3 & S4
detection algorithm has yet to be
developed.
Future Research
Fractal Dimension
Sound Analysis
Fractal Results
The END
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Future Research
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Future Research
Fractal Dimension
Sound Analysis
My future research may be focused on
developing an accurate S3 & S4 detection
algorithm.
Fractal Results
The END
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The End
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Thank You
Sound Analysis
Fractal Results
The END
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Fractal Definition
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
An object which appears self-similar under
varying degrees of magnification.
In effect, possessing symmetry across scale,
with each small part of the object replicating
the structure of the whole.
The END
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Self-similarity
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
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Fractal Dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
All fractals are characterized by their own
Audicor’s Solution
dimension, which is usually a non-integer
dimension,
Fractal Dimension that is greater than their topological dimension
and less than their Euclidean dimension.
Sound Analysis
Fractal Results
The END
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Fractal Dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
This definition of fractal dimension is often
used as an alternative definition of fractal
objects.
Fractal Dimension However, the fractal dimension may be
estimated in numerous ways, such as the boxSound Analysis
Fractal Results
counting dimension and the variance
dimension.
The END
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Chaotic dynamics
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
A chaotic system refers to a system that never
exactly repeats its behavior.
Fractal Dimension Regardless of how long we let the model run
for, we would never come across a repetition in
Sound Analysis
Fractal Results
the waveform due to its aperiodic feature.
The END
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Chaos
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
This behavior is known as chaos.
By plotting the models variables against each
other we receive a visualization of the
dynamics of the system.
Fractal Results
The END
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Phase Portrait
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The phase portrait will have the same form
although the model is started with different
initial conditions, the system will be attracted
to this type of final solution.
In two dimensions these plots are known as
phase portraits.
The END
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Strange Attractors
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
This plot is known as the attractor of the
system.
Audicor’s Solution
The attractors for chaotic systems are termed
Fractal Dimension strange attractors.
Sound Analysis
Fractal Results
The END
The fractal structures of these strange
attractors may be classified by calculating their
fractal dimensions.
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The Cantor Set
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
 The cantor set consists of an infinite set of
disappearing line segments in the unit
interval.
 The set is generated by iteratively removing
the middle third of line segments, resulting
in a collection of infinitely many
Fractal Dimension
disappearing line segments lying on the unit
Sound Analysis
interval.
Audicor’s Solution
Fractal Results
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 Both the line segments individual and
combined length approach zero as the
number of line segments approach infinity.
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The Cantor Set
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
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The Koch curve
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
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Sound Analysis
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 The iterative procedure used to construct the
Koch curve begins, similar to the Cantor set,
with the initiator of the set as the unit line
segment.
 The generator is constructed by removing
the middle third of the line segment and
then replace it with two equal segments
formed as two sides of a triangle.
 The process is repeated an infinite number of
times to produce the Koch curve.
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The Koch curve
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
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Regular Fractals
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
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Regular fractals are fractal objects that
possess exact self-similarity, objects with
structures comprising of exact copies of
themselves at all magnifications.
The most commonly known regular fractals are
possibly the Cantor set and the Koch curve,
both simply constructed using an iterative
procedure.
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Random Fractals
The Heart
ECG Wave
Heart Sounds
 Random fractals are statistically self-similar.
 Each small part of a random fractal has the
same statistical properties as the whole.
Abnormal sounds
 Random fractals may be constructed
mathematically by introducing a random
Audicor’s Solution
feature in the generating process of a regular
fractal.
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Sound Analysis
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106
 For instance, when generating the Cantor set
any third of the line segment is removed
instead of the middle third.
 Many properties of natural objects and
phenomena may be described using random
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fractals.
Fractal boundaries
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
In order for a fractal curve to be classified as a
fractal boundary it must meet two conditions:
1. The curve must be non-crossing,
meaning that the fractal curve does not
intersect itself
2.
As the fractal curve is zoomed in it
reveals more structure (details).
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Fractal boundaries
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
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Fractal boundaries
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Dimension measurements are suitable in order
to characterize and quantify the statistical selfsimilarity property of random fractal
boundaries.
Fractal Dimension As random fractals do not possess exact selfsimilarity the similarity dimension may not be
Sound Analysis
Fractal Results
The END
used. Instead we define estimates of the
fractal dimension of random fractals, which do
not require the exact self-similar property.
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The box counting dimension
The Heart
ECG Wave
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Abnormal sounds
Audicor’s Solution
The box counting dimension, DB enables noninteger dimensions to be found for fractal
curves.
DB covers the object in self-similar boxes.
In order to determine the box counting
Fractal Dimension dimension of a fractal object, the object is
covered with elements or boxes of side length
Sound Analysis
Fractal Results
The END
ε. The number of boxes, N, required to cover
the object together with the side length ε is
then used to determine the dimension.
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The box counting dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
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The box counting dimension
The Heart
ECG Wave
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Abnormal sounds
Audicor’s Solution
The straight line is covered with elements of
length ε, for simplicity we assume that the line
is of unit length.
Fractal Dimension In order to cover the line N elements are
required regardless of the dimension of the
Sound Analysis
Fractal Results
elements, here illustrated as cubes.
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The box counting dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
To cover the unit line segment any elements
with a dimension greater than or equal to the
dimension of the line itself may be used, and
still only require N of them.
This leads to the following expression:
L  N 1  1
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The box counting dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
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If, in stead, the same procedure would be
applied to a plane of unit area, the expression
received would be:
A  N   1
2
Similar reasoning with a 3-dimensional object
would lead to:
V  N  3  1
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The box counting dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
In general, in order to cover an object of unit
hypervolume the number of elements reuired
are:
N  1 /  DB
Sound Analysis
In logarithmic form:
Fractal Results
DB  log( N ) / log(1/  )
The END
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The box counting dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
By disregarding the assumption of unit
hypervolume, a general expression of the box
counting dimension may be received:
DB  (log( N )  log( V )) / log(1/  )
where V is the hypervolume of the fractal
object.
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The box counting dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
By rearranging the expression it is easy to see
that it is an equation of the straight line,
where the gradient of the line is the box
counting dimension of the object,
and by plotting log(N) against log(1/ε) for
various elements with different side lengths d
the box counting dimension may be
determined:
log( N )  DB  log(1/  )  log( V )
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The box counting dimension
The Heart
ECG Wave
Heart Sounds
To obtain a measure of the box counting
dimension there are different methods of
covering the fractal object. Three of them are
illustrated below.
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
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The box counting dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
 The first method illustrated is covering the
curve by placing boxes against each other in
a way that a minimum number of boxes are
being used.
Audicor’s Solution
 Another method is to cover the fractal object
with a grid of boxes and count the number of
Fractal Dimension
boxes that contain a part of the curve.
Sound Analysis
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 The last method illustrated is covering the
curve with circles instead of boxes, placed in
a similar way as with the boxes in the first
method.
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The box counting dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Regardless of which method being used, the
box counting dimension is still obtained from
the derivate of the plot of log(N) against
log(1/ε):
 log( N )
DB  lim
 0  log( 1 /  )
Fractal Results
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The box counting dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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121
The box counting dimension
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Normally, in practical applications, the box
counting dimension is estimated by selecting
two points at small values of ε in the plot,
resulting in an estimation given by:
DB  (log( N 2 )  log( N1 )) /(log( 1/  2 )  log(1/ 1 ))
Fractal Dimension
Sound Analysis
Fractal Results
The END
To receive a more accurate estimate of the box
counting dimension a best fitted line may be
drawn through the points at small values of ε.
The slope, and consequently the box counting
dimension, is then calculated from this best
fitted line.
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Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Regular Brownian motion, or Brownian motion,
is named after its discoverer Robert Brown.
He observed that small particles floating in
water underwent rapid irregular motions due
Fractal Dimension to their bombardment by water molecules.
Audicor’s Solution
Sound Analysis
Fractal Results
The END
If a group of particles is released at a certain
location the bombarding molecules will cause
the particles to spread out, diffuse, through
time.
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Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
The trajectories of particles undergoing
Brownian motion in the plane may cross over
themselves.
Hence, Brownian motion may not be classified
as a fractal boundary.
Furthermore, as the Brownian trajectory is
zoomed into, more structure is revealed,
indicating that the statistical self-similarity
features of the Brownian trajectory extends
over all scales of magnification
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Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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125
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
If the Brownian motion is sampled at intervals
Audicor’s Solution of t and the positions of the sampled points at
time t i are denoted by( xi , yi ) , then the observed
Fractal Dimension steps taken
in the two coordinate directions
xi  xi  xi 1 and yi  yi  yi 1 both follows a
Sound Analysis
Gaussian probability distribution.
Fractal Results
The END
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126
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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127
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Consequently, the step lengths between
observed points also follows a Gaussian
distribution:
ri  (xi ) 2  (yi ) 2
Sound Analysis
Fractal Results
The END
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128
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
The methods generally used to construct a
Brownian motion in the plane are derived from
these features.
Abnormal sounds
Hence, the motion is constructed by using
Audicor’s Solution steps in the two coordinate directions, ∆x and
∆y randomly selected from a Gaussian
Fractal Dimension distribution.
Sound Analysis
Fractal Results
The END
The step length r randomly selected from a
Gaussian distribution
T
he step angel is randomly selected from a
uniform distribution between 0 and .
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129
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
The time trace of Brownian motion, B(t), is
equal to the time history of the coordinates of
a Brownian trajectory, illustrating how the
coordinate values vary in time.
Audicor’s Solution
The construction of a Brownian motion trace is
derived from the property that successively
Fractal Dimension increments the trace following a Guassian
distribution:
Sound Analysis
Fractal Results
( B(t )  B(t  t ))
The END
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130
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Thus, by sampling the Brownian motion trace
at discrete times t i  i  t
a discrete approximation may be constructed,
by summarizing a series of random incremental
R (t j )
steps,
Fractal Dimension Thus B(t ,) is built up as an accumulated sum,
i
Sound Analysis
i
Fractal Results
The END
B(t i )   R(t j )  B(t i 1 )  R(t i )
j 1
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131
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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132
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
For a continuous Brownian motion trace, B(t),
the self-similar properties are apparent as
zooming into it.
Both the original trace and zoomed in traces
displays the similar irregularity, as they are
statistically self-similar.
However, in order to retain the self-similar
properties of the original trace the axes need
to be scaled differently.
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133
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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134
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
By considering pairs of points on a Brownian
motion trace separated by a time Ts
it is possible to state a relationship between
the mean absolute separation in B(t) between
these points, B  B(t  Ts )  B(t ) and the time
separation.
The obtained expression is given by
Fractal Dimension where the exponent, here equal to ½, is
denoted the Hurst exponent, H:
Sound Analysis
Fractal Results
B  TS1 / 2
The END
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135
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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136
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The Hurst exponent is the reason why a
Brownian motion trace only remains
statistically self-similar under scaling when the
axes B(t) and t are scaled differently.
Consequently, if the time is scaled by a factor
H
A, B(t) must be scaled by a factor A in order
to retain the similar relationship between and .
The END
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137
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
This is illustrated in the following example
where the time t is scaled by the factor A:
( A  t)  A  t
H
H
H
 A  B
H
This property of non-uniform scaling is known
as self-affinity, and is the reason for the two
scaling factors needed to retain the statistical
self-similar properties
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138
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
 All particles undergoing Brownian motion
diffuse through time in an average sense,
the traces start at t=0 where B(t)=0 and
continue to spread out from the origin as
time increases.
Audicor’s Solution
 If a large number of particles are spreading
out from the origin through time, the
Fractal Dimension
spreading process may be characterized
Sound Analysis
using averaged statistical properties.
Fractal Results
The END
 When considering diffusion related problems,
it is more natural to use the standard
deviation,  C as a measure of the
spreading.
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139
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
As the standard deviation and B are
proportional the scaling relationship is given
by :
 C  t 1/ 2
The expression is commonly re-expressed as
C  2 K t
where K is denoted the diffusion coefficient.
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140
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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141
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
It is possible to generate a Brownian motion
trace with a certain diffusion coefficient K, by
selecting the incremental steps, R (t j )
from a Gaussian distribution where the
standard deviation,
P is given by:

 P  2  K  t
Where is the time interval between each
sample.
Hence, after number of time steps, the time t
equals:
t  i  t
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142
Regular Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Combining the above equations will result in
that the standard deviation of diffusing
particles may be expressed as:
 c   P  i1 / 2
Fractal Results
The END
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143
Fractional Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
The fractional Brownian motions, abbreviated
fBms, are a generalisation of the regular
Brownian motion.
The Hurst exponents for fBms range from
0<H<1
where the special case of H=0.5 results in
regular Brownian motion.
Normally, fBms are denoted BH (t ) where the
subscript H equals the Hurst exponent that is
classifying the motion.
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144
Fractional Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Similar to regular Brownian motion traces the
fBm traces are self-affine processes.
In addition, a scaled up part of an fBm requires
Fractal Dimension different scaling factors for the t and
BH (t )
axes in order to retain its statistical self-similar
Sound Analysis
Fractal Results
properties.
The END
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145
Fractional Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
The scaling relationship between the mean
absolute separation along the fBm trace and
the time of the separation is expressed as:
BH  TSH
and similarly the standard deviation of
Fractal Dimension diffusing particles scales as:
Sound Analysis
H
H
Fractal Results
The END
C  t
Where
~
 C  (2  K F  t )
K F is the fractional diffusion coefficient.
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146
Fractional Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
The fractional diffusion coefficient is obtained
by plotting ( C )1 / H against time since release.
This results in a linear relationship, where the
slope of the plot corresponds to twice the
fractional diffusion coefficient.
Fractal Dimension However, this requires that is known, which is
not always the case in practical applications.
Sound Analysis
Fractal Results
The END
Instead a logarithmic plot of against time may
be used, where the gradient of the best-fitted
line through the experimental data equals and
the crossing point on the  C axis equals
H  log( 2K F )
147
Lets
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Fractional Brownian motion
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
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148
Heart Sound Analysis with Time
Dependent Fractal Dimensions
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 First I will explain the construction and use
of fractal dimension trajectories, and how
the selection of windows affects the
appearance of the trajectory.
 Thereafter follows a description of the
different methods used to calculate the
dimension trajectories.
 The description consists of two parts, a brief
introduction to the methods containing the
necessary theory, followed by some practical
considerations that have to be accounted for
when applying them to discrete signals.
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149
Heart Sound Analysis with Time
Dependent Fractal Dimensions
The Heart
ECG Wave
Heart Sounds
Abnormal sounds
Audicor’s Solution
Fractal Dimension
Sound Analysis
Fractal Results
The END
 First I will explain the construction and use
of fractal dimension trajectories, and how
the selection of windows affects the
appearance of the trajectory.
 Thereafter follows a description of the
different methods used to calculate the
dimension trajectories.
 The description consists of two parts, a brief
introduction to the methods containing the
necessary theory, followed by some practical
considerations that have to be accounted for
when applying them to discrete signals.
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150
The End
The Heart
ECG Wave
Heart Sounds
Abnormal Sounds
Audicor’s Solution
Fractal Dimension
Thank You
Sound Analysis
Fractal Results
The END
Lets
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151