1 radial kernel based time-frequency distributions with applications to atrial fibrillation analysis...
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Radial Kernel based Time-Frequency Distributions
with Applications to Atrial Fibrillation Analysis
Sandun KodituwakkuPhD Student
The Australian National UniversityCanberra, Australia.
Supervisors: A/Prof. Thushara Abhayapala
Prof. Rod Kennedy
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Outline
• Background – Time-Frequency Distributions (TFDs)
• Our work1) Multi-D Fourier Transform based framework for TFD kernel design2) Unified kernel formula for generalizing Wigner-Ville, Margenau-Hill, Born-Jordan and Bessel3) 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. QuadraticLinear
Pros:• Linear superposition• No interference terms for
muti-component signals
Cons:• Trade off between time
and frequency resolutions
Heisenberg inequality
Quadratic
Pros:• Better time and frequency
resolutions than linear• Shows the energy
distribution
Cons:• Cross terms for multi-
component signals
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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 uniquely specifies a distribution
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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 Wigner-Ville, 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: ie. Radially symmetric
C2: ie. Unit volume
C3: 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 to obtain the order-n radial kernel.
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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 Name
1 , 1 Wigner-Ville
1 , Margenau-Hill
2 Our work
3 Born-Jordan
4 Bessel
5 Our work
6 Our work
and many more…..
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TFD Properties (cont.)
• Realness
guaranteed by radial symmetry of
• Time and Frequency Shifting
guaranteed by independence of from t and ω
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TFD Properties (cont.)
• Instantaneous frequency and Group delay
guaranteed by radial symmetry of and unit volume condition together
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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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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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What is AF?
• 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.
healthy
AF
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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 Wigner-
Ville 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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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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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.)
• Best results for the AF model obtained by order-6 radial-δ kernel
0 1 2 3 4 5 6 7 86.5
6.55
6.6
6.65
6.7
6.75
Kernel Order
(Aut
o te
rm)/
(Cro
ss te
rm)
dB
Bessel
Born-Jordan
Wigner-Ville
Margenau-Hill
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Comparison with Choi-Williams
• Less interference in order-6 radial-δ kernel.• Choi-Williams does not satisfy time and
frequency support properties.
Frequency (Hz)
Tim
e (s
)
Choi-Williams
0 2 4 6 8 10 12 14 16 18 20
5
10
15
20
25
30
35
40
45
6th Order Radial
Frequency (Hz)
Tim
e (s
)
0 2 4 6 8 10 12 14 16 18 20
5
10
15
20
25
30
35
40
45
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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.