A framework for ECG signal processing based on quadratic variation reduction

The analysis of the elctrocardiogram (ECG) is the standard diagnostic tool for the assessment of heart function and the diagnosis of cardiac diseases. Unfortunately, ECG signal is highly susceptible to several kinds of noise, such as thermal and electromyographic noise, power-line interference, motion artifacts, and baseline wander. In this dissertation, a framework for ECG signal processing based on the notion of quadratic variation reduction is proposed. The quadratic variation is a consistent measure of variability for vectors or sampled functions. In recent years, growing interest has been focused on the analysis of single waves of the ECG, like P-waves or T-waves. Being able to track beat-to-beat changes of such waves has proven to be helpful in predicting important heart diseases, such as atrial fibrillation (AF). Since single waves have low SNR, they are commonly analyzed via signal averaging. In this way information about beat-to- beat variability is unavoidably masked or possibly lost. To cope with this problem, in Chapter 3 a smoothing and denoising algorithm for ECG single waves is proposed, which is based on the notion of quadratic variation reduction. The algorithm is the closed-form solution to a constrained convex optimization problem, where the quadratic variation enters as a constraint. Numerical results show that the approach achieves considerable SNR gain over the whole range of practical input SNRs. In Chapter 3 the algorithm is successfully applied to the prediction of AF through the analysis of P-waves beat-to-beat variability. Denoising by quadratic variation reduction enters in the definition of a metric that measures P-waves variability. The study of the statistics of such a metric allows to discriminate between AF-prone and healthy subjects. Narrowband artifacts, such as power-line interference, are another common kind of noise affecting ECG waves. In Chapter 4 is proposed a joint approach to denoising and narrowband artifacts rejection. It is achieved by solving a constrained convex optimization problem, where the energy content of harmonic artifacts enters as an additional constraint. The improvement achieved by the approach based on quadratic variation reduction is notable, and this makes worth its extension to the smoothing of entire ECGs. However, this is effective if the local structure of the ECG is explicitly taken into account in the smoothing operation. In Chapter 5 a smoothing and denoising algorithm for entire ECGs is proposed. Smoothing is performed by locally reducing the variability, i.e., the quadratic variation, of the measured ECG, by an amount that is inversely related to the local SNR. Numerical results show that the smoothing performance is significant. The algorithm is favorable also in terms of complexity, which is linear in the size of the record to smooth. In the presence of noise and narrowband artifacts, a combined approach based on joint denoising and artifacts rejection is needed. This is pursued in Chapter 6, where smoothing and narrowband artifacts rejection are performed jointly combining local characteristics of the ECG and the spectral localization of the artifacts to reject. Simulation results confirm the effectiveness of the approach and highlight a notable ability to smooth and denoise ECG signals. Finally, in Chapter 7 the problem of baseline wander removal is tackled. This is an unavoidable step in any processing of ECG signals. Baseline wander and ECG have partially overlapping bands in the low-frequency region of the spectrum. This makes its removal difficult without affecting the ECG, in particular the ST segment, which is related to the diagnosis of acute coronary syndromes. Due to the in-band nature of this kind of noise, any technique for its removal that relies on spectral shaping, e.g., linear time-invariant filtering, will unavoidably alter the ST segment. To cope with this problem it is necessary to analyze baseline noise and ECG components contributing to the ST segment in a domain that is not the frequency domain. In this regard, the quadratic variation turns out to be a suitable measure of variability that is not directly related to the frequency domain. In Chapter 7 baseline wander is estimated as the component of measured ECG with the lowest variability. Comparative analyses show that the approach outperforms state-of-the-art algorithms and is superior in removing baseline wander, while preserving the ST segment. The approach compares favorably also in terms of computational complexity, which is linear in the size of the vector to detrend. This makes it suitable for real-time applications as well as for applications on devices with reduced computing power, e.g., handheld devices. All the techniques described for smoothing, rejection of power-line noise and, in general, narrowband artifacts, either for single waves or entire ECGs, and for baseline wander removal, share a common approach, although with different technicalities. This approach is based on the notion of quadratic variation reduction and constitutes the common framework for ECG signal processing, which is the object of this dissertation.