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Radial Kernel based Time-Frequency Distributions with Applications to Atrial Fibrillation Analysis

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. Outline.

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Radial Kernel based Time-Frequency Distributions with Applications to Atrial Fibrillation Analysis

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  1. Radial Kernel based Time-Frequency Distributionswith Applications to Atrial Fibrillation Analysis Sandun Kodituwakku PhD Student The Australian National University Canberra, Australia. Supervisors: A/Prof. Thushara Abhayapala Prof. Rod Kennedy

  2. 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

  3. 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.

  4. Historical background • TFDs are a research topic for more than half a century • Famous two • Short-time Fourier Transform • Wigner-Ville Distribution

  5. Classification

  6. Linear 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 Linear vs. Quadratic

  7. 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

  8. Prominent members of Cohen • Wigner-Ville (1948) • Page (1952) • Margenau-Hill (1961) • Spectrogram – Mod squared of STFT

  9. Prominent members of Cohen (cont.) • Born-Jordan (1966) • Choi-Williams (1989) • Bessel (1994) 2-D time-frequency convolution of Wigner-Ville will result others

  10. Kernel Questions? • Why so many? • Which one is the best? • How to generate them? • What are the applications?

  11. 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.

  12. 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

  13. 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.

  14. 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)

  15. Realization based on δ function (cont.) • n-D Fourier transform of • Thus order-n radial-δ kernel is given by,

  16. Lower dimensions simplified and many more…..

  17. Kernel visualization

  18. TFD Properties

  19. TFD Properties (cont.) • Realness guaranteed by radial symmetry of • Time and Frequency Shifting guaranteed by independence of from t and ω

  20. TFD Properties (cont.) • Time and Frequency marginals guaranteed by unit volume condition

  21. TFD Properties (cont.) • Instantaneous frequency and Group delay guaranteed by radial symmetry of and unit volume condition together

  22. TFD Properties (cont.) • Time and Frequency support guaranteed by finite support condition

  23. Simulation of FM + Chirp signals • Time-frequency analysis of the sum of FM and chirp signal.

  24. Simulation of FM + Chirp signals (cont.) Born-Jordan Bessel Order-5 Radial Order-6 Radial

  25. Simulation of FM + Chirp signals (cont.) Order-7 Radial Order-5 radial-δ kernel works best.

  26. 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.

  27. An application of novel TFDs Atrial Fibrillation Analysis from surface ECG

  28. 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.

  29. Typical healthy ECG Source: Wikipedia

  30. What is AF? 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

  31. 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.

  32. 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.

  33. 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.

  34. System model • Atrial fibrillation is modelled by a sum of frequency modulated sinusoidals with time varying amplitudes, and its harmonics [Stridh & Sornmo 01] where,

  35. Synthetic ECG with AF

  36. 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.

  37. Born-Jordan Bessel Order-5 radial Order-6 radial

  38. Simulation Results (cont.) Order-7 radial Order-6 radial-δ kernel works best.

  39. Performance measure • Maximise ratio between auto term energy and interference term energy. • Find the order (n) with maximum ratio

  40. Performance measure (cont.) • Best results for the AF model obtained by order-6 radial-δ kernel

  41. Comparison with Choi-Williams • Less interference in order-6 radial-δ kernel. • Choi-Williams does not satisfy time and frequency support properties.

  42. PhysioBank data • AF termination challenge database- ECG record n02

  43. 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.

  44. 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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