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Extraction of f Waves

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Atrial Fibrillation from an Engineering Perspective

Part of the book series: Series in BioEngineering ((SERBIOENG))

Abstract

This chapter provides a comprehensive overview of methods for f wave extraction, divided into the following categories: average beat subtraction and variants, interpolation, extended Kalman filtering, adaptive filtering, principal component analysis, singular spectral analysis, autoregressive modeling and prediction error analysis, and independent component analysis. Different performance measures are described, used either for real or simulated ECG signals.

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Notes

  1. 1.

    This signal model was also considered for determining a QRST template using a maximum likelihood (ML) approach. Since the performance of the ML-based method was found to be inferior to that of the method based on the MSE criterion described in this section, the interested reader is referred to [21] for further details.

  2. 2.

    For the single-lead case, i.e., \(L=1\), STC simplifies to ABS, except for that misalignment in time can still be handled by \(\mathbf {J}_{\tau }\) and amplitude mismatch to \(\hat{\mathbf {S}}_t\) by the scaling factor \(a_1\); the rotation matrix reduces to a scalar equal to one. The single-lead version of STC is closely related to the extraction method based on singular value decomposition, as described in Sect. 5.6.1, which also involves amplitude scaling.

  3. 3.

    Time-dependent scaling and rotation have been considered when evaluating a method for vectorcardiographic loop alignment [38], see also [39]. A mathematical model was proposed in which the time-dependent, angular variation associated with the rotation matrix is assumed to be proportional to the amount of air in the lungs during a respiratory cycle—a property modeled by the product of two sigmoidal functions reflecting inhalation and exhalation, respectively, cf. Sect. 3.4.4.

  4. 4.

    In general, it is desirable to use an orthogonal set of basis functions for signal representation, so that the signal component associated with a certain basis function do not interfere with the components associated with the other basis functions. For example, the Hermite functions are well-suited for modeling of the QRST complex [61, 62]. However, orthogonality is of less importance to the described simulation model, and, therefore, Gaussian functions, being nonorthogonal basis functions, are considered.

  5. 5.

    An early precursor to the idea of using multiple state-space models was explored for the identification of certain persistent ECG rhythms [70], although none of them were AF. In that study, the proposed rhythm models were linear in nature, and, therefore, the discrete-time Kalman filter could be used.

  6. 6.

    For the eigenvector \(\pmb {\varphi }_1\) to be virtually identical to a scaled version of the ensemble average, the ensemble with similar-shaped beats should be reasonably well-aligned in time and the noise level should not be so high that f waves are completely obscured, i.e., two conditions which are easily met in practice.

  7. 7.

    The observation model in (5.125) is also central to the methods exploring higher-order statistics for independent component analysis, see Sect. 5.9. The mixing matrix \(\mathbf {A}\) is here constrained to be orthogonal, whereas not so in Sect. 5.9.

  8. 8.

    Alternatively, the mean cross prediction error \(E[e_d(n) e_d(n-q)]\) in (5.133) may be minimized by introducing the cost function \(J^{\prime }(\mathbf {w};q) = \mathbf {w}^T (\mathbf {E}_x(q) + \mathbf {E}_x^T(q)) \mathbf {w}\). The eigenvector corresponding to the smallest eigenvalue of the symmetric matrix \(\mathbf {E}_x(q) + \mathbf {E}_x^T(q)\) minimizes \(J^{\prime }(\mathbf {w};q)\), and therefore taken as an estimate of \(\mathbf {w}\).

  9. 9.

    An overview of methods for AR model parameter estimation, as well as for model order estimation, can be found in, e.g., [4, 118].

  10. 10.

    The analysis of lagged covariance matrices for the purpose of separating different signal sources with a temporal structure was first explored in a nonparametric setting, leading to the AMUSE algorithm [119], see also [95, Chap. 18]. Using that algorithm, the time lag is usually chosen to be \(q=1\), just like in [113].

  11. 11.

    The wavelet coefficients are obtained by correlating the observed signal with the selected mother wavelet at different dilations and time shifts.

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Authors and Affiliations

  1. Department of Biomedical Engineering and Center for Integrative Electrocardiology, Lund University, Lund, Sweden

    Leif Sörnmo

  2. Biomedical Engineering Institute, Kaunas University of Technology, Kaunas, Lithuania

    Andrius Petrėnas & Vaidotas Marozas

  3. Biomedical Signal Interpretation and Computational Simulation (BSICoS), Aragón Institute of Engineering Research (I3A), Centro de Investigacíon Biomédica en Red de Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Zaragoza University, Zaragoza, Spain

    Pablo Laguna

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Sörnmo, L., Petrėnas, A., Laguna, P., Marozas, V. (2018). Extraction of f Waves. In: Sörnmo, L. (eds) Atrial Fibrillation from an Engineering Perspective. Series in BioEngineering. Springer, Cham. https://doi.org/10.1007/978-3-319-68515-1_5

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