1 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

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 1.Short-time Fourier Transform 2.Wigner-Ville Distribution

5 Classification Time-Frequency Distributions (TFDs) Linear STFT Wavelets Gabor Quadratic “Cohen class” (Shift invariant) Affine class (Scale invariant) Others Signal Dependent

6 Linear vs. Quadratic 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

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 Dimension nKernelName 1, 1 Wigner-Ville 1,Margenau-Hill 2Our work 3Born-Jordan 4Bessel 5Our work 6 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-JordanBessel Order-5 RadialOrder-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 – 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

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-JordanBessel Order-5 radialOrder-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.