A low-rank matrix factorization approach for joint harmonic and baseline noise suppression in biopotential signals
Introduction
Noise removal is very often a crucial step in processing and interpreting biopotential data. The noise artifacts most commonly encountered in such signals are harmonic (a.k.a. powerline) and baseline noise. Harmonic noise is characterized by a few time-variant harmonic components centered at integer multiples of the fundamental frequency (50 Hz/60 Hz). It typically arises as a combination of several unwanted effects: capacitive coupling between the subject and power lines, differences in the electrode impedances, separation between the electrodes, measurement interconnections, to name a few [1], [2], [3], [4]. Baseline noise is defined as a long-term drift (slowly-varying DC component) whose frequency range usually falls below 1 Hz [5]. It is mainly caused by subject's movements during the measurements, spatial-variant electrode-skin impedance and breathing [6], [7], [8]. Both noise sources spectrally overlap with frequency bands that contain clinically relevant information. Moreover, the useful signal instantaneous amplitude is often very close to the noise level, giving rise to low signal-to-noise ratio (SNR) measurements. Accordingly, to ensure minimal signal distortion while suppressing the artifacts is a very difficult task.
Related work encompasses a number of approaches where the problem of harmonic and baseline noise removal was usually treated separately. Amongst the approaches that aim at suppressing the harmonic noise the most salient are notch filtering [9], adaptive filtering [10], [11], [12], [13], [14], [15], outlier detection [16], component subtraction [17], [18], intrinsic mode decomposition [19], wavelet analysis [20]. Regarding baseline removal problem, the research focus was set on high-pass filtering [21], adaptive filtering [22], curve fitting [23], [24], empirical mode decomposition [25], to name the most cited. To the best of our knowledge, only a few approaches deal jointly with both noise sources: FIR filtering [26], fractal and empirical mode decomposition [27] and sinusoidal modeling [28].
Herein we focus our attention on matrix sparsity and non-negative rank, which are one of the most prominent concepts of the low-rank matrix decomposition theory [29], [30]. Briefly, it states that a large matrix with a small number of significant non-negative entries can be replaced by a lower rank matrix, providing a more efficient representation of the relationship between data elements. Such a matrix brings out the most relevant components of the data while, at the same time, mitigating the effect of possible disturbances. Harmonic and baseline noise are time-variant narrow-band signals which are almost sparse in the time-frequency (TF) domain i.e. their individual spectrograms can be compactly described by non-negative vectors/matrices of a very small rank. However, in presence of a biopotential signal, this convenient description often does not hold for two reasons. On the one hand, some biopotential signals are wide-band signals (e.g. EMG) which spectrally encompass the harmonic and baseline noise. As a biopotential signal is usually the dominating source in the data, the sparsity condition of the harmonic and baseline noise might be seriously degraded. On the other hand, we cannot uniquely determine the non-negative rank of individual sources in the data if no additional information on the mixing system is available. A usual shortcut to this problem is to assign a user-defined rank to each source and then tune them up by some iterative trial-and-error input matrix decomposition procedure. Such an approach is computationally costly and is not guaranteed to converge to optimal individual ranks.
Recently there have been a few attempts which aimed at separating ECG–EMG signal mixtures by making use of sparsity and unsupervised learning algorithms based on NMF [31], [32]. The former focused on enhancing the sparsity of the ECG spectrogram by filtering and downsampling the input signal, thus bringing out the QRS complexes against the EMG background. The latter processed input data by the wavelet transform with adequately chosen basis functions (Cauchy-type non-linearly scaled wavelets) which captured the basic ECG waveforms shapes with a relatively small number of wavelet coefficients. Both methods provided a sparse ECG TF representation which was used as the initialization for the NMF algorithms and they both achieved a very good ECG–EMG separation compared to the existing state-of-the-art techniques. However, both methods have important drawbacks, namely: (1) they do not work with data containing more than two signal sources, (2) their performance largely depends on the analysis window size used to generate TF representations, and (3) they always consider unitary rank matrix description independent of the signal source dynamics.
The present application deals with separating more than two sources from the input mixture; therefore, the above methods could not be used in the current form. Adopting the general approach based on sparsity and unsupervised learning, we propose a novel method which poses the harmonic and baseline noise removal task as a source separation problem (the preliminary research in the context of EMG denoising was discussed in [33]). The proposed method overcomes the aforementioned drawbacks by introducing new algorithm features, namely: unambiguous non-negative rank estimation of individual sources in the data and phase-preserving spectrogram segmentation. Optimal non-negative rank is a key factor in any non-negative matrix decomposition because: (1) it provides a way to drastically reduce problem dimensionality without losing any relevant characteristic of the underlying source; (2) it allows for fast convergence of the NMF algorithms. However, existing algorithms for computing the non-negative rank introduce a prohibitively high computational cost, which impedes its use in on-line applications. Herein we develop a study which (1) discusses a general relationship between standard and non-negative matrix rank, and (2) shows under which circumstances those ranks can be used interchangeably in the context of the present application. A major benefit of this study is that the estimation of the non-negative rank can be carried out by an economy-size singular value decomposition (SVD), which drastically relaxes the algorithm's overall computational cost. Furthermore, we introduce a spectrogram segmentation procedure which aim at coarse separation of underlying signal sources in the TF domain by means of a set of data-driven spectrogram shaping vectors. Such a segmentation scheme ensures that no spectral phase modification occurs; accordingly, the estimated signal component waveforms are virtually distortionless.
By combining these novel features with NMF, our algorithm achieves an unambiguous and physically meaningful signal-noise separation from the spectrogram of the input data in only a few computational iterations. Moreover, such a separation strategy was proven experimentally to be especially efficient in data acquisitions with very low SNR.
Section snippets
Methods
As we deal with matrices with non-negative entries, it is assumed that the signal to be processed is in the form of the Short-Time Fourier Transform: , and , where m equals the number of frequency bins in [0, π) rad/sample, n is given by the number of overlapped short-time segments, whilst the operator ‘◦’ stands for the Schur–Hadamard entry-wise multiplication. Thus, X contains the amplitudes whilst Θ encodes the phases in the form of unitary-magnitude complex
Data
We used two sets of surface EMG signals, recorded under different experimental and measurement conditions. In both cases the experimental procedure was conducted in accordance with the Declaration of Helsinki and was approved by the Local Ethics Committee. Each subject provided an informed written consent before participation in the study. The first set of EMG signals (EMG set I) was obtained from the vastus medialis obliquus during a sustained fatiguing contraction at 30% of the maximal
Discussion
In the present study, we addressed a problem of simultaneous harmonic and baseline noise removal from single-channel biopotential recordings without an a priori knowledge of the mixing matrix. In this analysis scenario, which often arises in biomedical signal processing applications, noise removal always leads to significant distortions of the biopotential signal.
The noise reduction method described in this study was specially designed to operate on data characterized by low input
Conclusions
We presented a novel approach to joint harmonic and baseline noise removal from biopotential signals in the framework of the low-rank matrix factorization. We showed that suitable biopotential spectrogram modifications generate sparse non-negative representations, which admit decomposition in low-rank matrices. Such matrices convey the information about concurrent sources in the input data and can easily yield the corresponding time waveforms.
The principal contributions of the present work are
References (46)
- et al.
ECG baseline wander and powerline interference reduction using nonlinear filter bank
Signal Process.
(2005) - et al.
Removing harmonic power line interference from biopotential signals in low cost acquisition systems
Comput. Biol. Med.
(2009) - et al.
Filtering electrocardiographic signals using an unbiased and normalized adaptive noise reduction system
Med. Eng. Phys.
(2009) A frequency domain Hampel filter for blind rejection of sinusoidal interference from electromyograms
J. Neurosci. Methods
(2009)- et al.
Power line interference filtering on surface electromyography based on the stationary wavelet packet transform
Comput. Methods Programs Biomed.
(2013) - et al.
The low-frequency response of electrocardiographs: a frequent source of recording errors
Am. Heart J.
(1966) - et al.
Electrocardiogram baseline noise estimation and removal using cubic splines and state-space computation techniques
Comput. Biomed. Res.
(1977) - et al.
ECG signal denoising and baseline wander correction based on the empirical mode decomposition
Comput. Biol. Med.
(2008) - et al.
Fractal and EMD based removal of baseline wander and powerline interference from ECG signals
Comput. Biol. Med.
(2013) - et al.
Simultaneous powerline interference and baseline wander removal from ECG and EMG signals by sinusoidal modeling
Med. Eng. Phys.
(2013)
Applications of structured low-rank approximation
Wavelet-based unsupervised learning method for electrocardiogram suppression in surface electromyograms
Med. Eng. Phys.
On the nonnegative rank of Euclidean distance matrices
Linear Algebra Appl.
Overlap-add methods for time-scaling of speech
Speech Commun.
60-Hz interference in electrocardiography
IEEE Trans. Biomed. Eng.
High-quality recording of bioelectric events. Part 1. Interference reduction, theory and practice
Med. Biol. Eng. Comput.
Bioelectrical signal processing in cardiac and neurological applications
A comprehensive model for power line interference in biopotential measurements
IEEE Trans. Instrum. Meas.
Multiadaptive bionic wavelet transform: application to ECG denoising and baseline wandering reduction
EURASIP J. Adv. Signal Process.
Recommendations for measurement standards in quantitative electrocardiography: the CSE working party
Eur. Heart J.
Baseline reconstruction for localization of rapid ventricular tachycardia from body surface potential maps
Physiol. Meas.
Elimination of AC interference in electrocardiogram using HR notch filter with transient suppression
IEEE Trans. Biomed. Eng.
Adaptive noise cancelling: principles and applications
Proc. IEEE
Cited by (5)
EWT-IIT: a surface electromyography denoising method
2022, Medical and Biological Engineering and ComputingVMD-based denoising methods for surface electromyography signals
2019, Journal of Neural Engineering