Transcript Radial Kernel based Time-Frequency Distributions with

Radial Kernel based Time-Frequency
Distributions
with Applications to Atrial Fibrillation
Analysis
Sandun Kodituwakku
PhD Student
The Australian National University
Canberra, Australia.
Supervisors:
A/Prof. Thushara Abhayapala
Prof. Rod Kennedy
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Outline
• Background – Time-Frequency
Distributions (TFDs)
• Our work
1) Multi-D Fourier Transform based
framework for TFD kernel design
2) Unified kernel formula for generalizing
Wigner-Ville, Margenau-Hill, Born-Jordan
and Bessel
3) Applications to Atrial Fibrillation
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Motivation
• Real world signals -- speech, radar, biological
etc. -- are non-stationary in nature.
• Example: ECG Video
• Non-stationary – Period, Amplitudes,
Morphology changes in time.
• Limitations of Fourier Analysis – fails to locate
the time dependency of the spectrum.
• This motivates joint Time-Frequency
representation of a signal.
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Historical background
•
TFDs are a research topic for more than
half a century
• Famous two
1. Short-time Fourier Transform
2. Wigner-Ville Distribution
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Classification
Time-Frequency
Distributions
(TFDs)
Linear
•STFT
•Wavelets
•Gabor
Quadratic
•“Cohen class”
(Shift invariant)
•Affine class
(Scale invariant)
Others
•Signal Dependent
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Linear vs. Quadratic
Linear
Quadratic
Pros:
• Linear superposition
• No interference terms for
muti-component signals
Pros:
• Better time and frequency
resolutions than linear
• Shows the energy
distribution
Cons:
• Trade off between time
and frequency resolutions
Cons:
• Cross terms for multicomponent signals
Heisenberg inequality
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Cohen Generalization
• Breakthrough by L. Cohen in 1966
• All shift invariant TFDs are generalized to
a one class (Cohen class)
• Kernel function
a distribution
uniquely specifies
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Prominent members of Cohen
• Wigner-Ville (1948)
• Page (1952)
• Margenau-Hill (1961)
• Spectrogram – Mod squared of STFT
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Prominent members of Cohen
(cont.)
• Born-Jordan (1966)
• Choi-Williams (1989)
• Bessel (1994)
2-D time-frequency convolution of Wigner-Ville
will result others
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Kernel Questions?
• Why so many?
• Which one is the best?
• How to generate them?
• What are the applications?
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Our work
• Multi-D Fourier Transform based
framework for deriving Cohen kernels.
• Radial-δ kernel class generalizing WignerVille, Margenau-Hill, Born-Jordan, and
Bessel.
• Analysis of Atrial Fibrillation from surface
ECG.
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Multi-D Fourier Framework
Let
be a vector in n-D and
f
be a scalar-valued
multivariate function satisfying following
conditions.
C1:
C2:
C3:
ie. Radially symmetric
ie. Unit volume
ie. Finite support
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Multi-D Fourier Framework (cont.)
• Consider n-D Fourier Transform of
•
is radially symmetric as well.
Identify
by
radial kernel.
to obtain the order-n
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Realization based on δ function
• n-D radial δ function:
• It is radially symmetric (C1)
• It is normalised to give unit volume (C2)
• It has finite support for α ≤ ½ (C3)
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Realization based on δ function
(cont.)
• n-D Fourier transform of
• Thus order-n radial-δ kernel is given by,
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Lower dimensions simplified
Dimension n Kernel
1
1,
1,
2
3
4
5
6
and many more…..
Name
Wigner-Ville
Margenau-Hill
Our work
Born-Jordan
Bessel
Our work
Our work
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Kernel visualization
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TFD Properties
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TFD Properties (cont.)
• Realness
guaranteed by radial symmetry of
• Time and Frequency Shifting
guaranteed by independence of
and ω
from t
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TFD Properties (cont.)
• Time and Frequency marginals
guaranteed by unit volume condition
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TFD Properties (cont.)
• Instantaneous frequency and Group delay
guaranteed by radial symmetry of
unit volume condition together
and
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TFD Properties (cont.)
•
Time and Frequency support
guaranteed by finite support condition
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Simulation of FM + Chirp signals
• Time-frequency analysis of the sum of FM
and chirp signal.
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Simulation of FM + Chirp signals (cont.)
Born-Jordan
Order-5 Radial
Bessel
Order-6 Radial
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Simulation of FM + Chirp signals (cont.)
Order-7 Radial
Order-5 radial-δ kernel works best.
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Summary so far……..
• A unified kernel formula
which contains 4 of the famous kernels
(Wigner-Ville, Margenau-Hill, Born-Jordan
and Bessel).
• Formula derived from n-dimensional FT of
a radially symmetric δ function.
• Superiority of high order radial-δ kernels.
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An application of novel TFDs
Atrial Fibrillation Analysis
from surface ECG
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What is ECG?
• ECG – Electrocardiogram
• ECG is a time signal which shows the
changes in body surface potentials due to
the electrical activity of the heart.
• Gold standard for diagnosing
cardiovascular disorders.
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Typical healthy ECG
Source: Wikipedia
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What is AF?
AF
healthy
• AF – Atrial Fibrillation
• Cardiac arrhythmia condition
• Consistent P waves are replaced by rapid
oscillations.
• Fibrillatory waves vary in amplitude, frequency
and shape.
• Associates with an irregular ventricular
response.
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Why AF important?
• AF is the most common sustained cardiac
arrhythmia condition.
• Increases in prevalence with age.
• Affects approx. 8% of the population over
age of 80.
• Accounts for 1/3 of hospitalizations for
cardiac rhythm disturbances.
• Associated with an increased risk of
stroke.
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Motivation
• Spectrum of Atrial activity of ECG under
AF has a dominant peak (AF frequency ).
• AF frequency gives insight to spontaneous
or drug induced termination of AF.
• Thus, importance of accurately tracking AF
frequency in time.
• TFDs are a good tool for this task.
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Previous work
• Stridh[01] used STFT and cross WignerVille distributions for estimating the AF
frequency.
• Sandberg[08] used HMM based method
for AF frequency tracking.
• We obtained better results using higher
order radial-δ kernels.
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System model
• Atrial fibrillation is modelled by a sum of
frequency modulated sinusoidals with time
varying amplitudes, and its harmonics
[Stridh & Sornmo 01]
where,
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Synthetic ECG with AF
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Objective
• AF frequency given by,
• Accurately estimate
, especially
when
is higher compared to
.
• Approximation to the real AF.
• Can be used to compare performance of
different algorithms.
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Born-Jordan
Order-5 radial
Bessel
Order-6 radial
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Simulation Results (cont.)
Order-7 radial
Order-6 radial-δ kernel works best.
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Performance measure
• Maximise ratio between auto term energy
and interference term energy.
• Find the order (n) with maximum ratio
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Performance measure (cont.)
(Auto term)/(Cross term) dB
6.75
6.7
Bessel
6.65
6.6
Born-Jordan
Wigner-Ville
6.55
Margenau-Hill
6.5
0
1
2
3
4
5
6
7
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Kernel Order
• Best results for the AF model obtained by order6 radial-δ kernel
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Comparison with Choi-Williams
6th Order Radial
5
5
10
10
15
15
20
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Time (s)
Time (s)
Choi-Williams
25
30
25
30
35
35
40
40
45
45
0
2
4
6
8
10
12
Frequency (Hz)
14
16
18
20
0
2
4
6
8
10
12
14
16
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Frequency (Hz)
• Less interference in order-6 radial-δ kernel.
• Choi-Williams does not satisfy time and
frequency support properties.
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PhysioBank data
• AF termination challenge database- ECG
record n02
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Future directions
• Parameterizing TFD for paroxysmal and
persistent AF conditions.
• Pharmacological therapy and DC
cardioversion influence on TFD.
• Generalization for other supraventricular
tachyarrhythmias – Atrial Flutter.
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Summary
• A unified kernel formula for Cohen class
of TFDs
based on n-dimensional Fourier Transform
of a radially symmetric δ function.
• Atrial Fibrillation cardiac arrhythmia
condition analysis using TFDs with higher
order radial-δ kernels.
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