On the Removal of Artifacts of Known-Shape from Noisy Signals

1 Introduction

• Overlapping signal and disturbance spectra make linear filtering inadequate, motivating more advanced disturbance-rejection methods.The introduction notes that band-pass or stop filters work when signal and disturbance are spectrally separated.Source: §1 · p. 2
• Many blind-source-separation methods, including ICA, rely on multiple measurements, which may not be available.These methods aim to identify and separate the signal of interest from background noise and disturbances.Source: §1 · p. 2
• Prior LFP template-removal methods are described as having unclear implementations and performance analyses, and as ignoring artifact-instance amplitude.They typically subtract an averaged kernel from each identified artifact.Source: §1 · p. 2
• Autoencoders and other deep-learning methods require good training data, while synthetic training data postpones real-world performance verification.The introduction argues that insufficiently representative synthetic data can leave a model difficult to fix.Source: §1 · p. 2
• Artifact removal remains challenging when only one noisy measurement is available, spectra overlap, and no disturbance reference exists.The proposed approach targets signals affected by both noise and strong artifacts.Source: §1 · p. 2
• The paper introduces a method for removing known-shape artifacts from noisy signals using data-driven template matching.It also previews a sample modern-neurology use case with comparison to a machine-learning approach, and provides open-source Python implementations.Source: §1 · pp. 2–3

2 Methods

• The Python 3 implementation uses NumPy, SciPy, scikit-learn, and Matplotlib, and releases the signal-generation and artifact-removal code as open source.The cited libraries are used for numerical, scientific, machine-learning, and visualization tasks.Source: §2 · p. 3

2.1 Formalized Problem Statement

• The target is to reduce disturbance d[k] in noisy sampled signal x[k], with the measurement modeled as a linear mixture.The formulation considers measurement points 0 ≤ k < N − 1 and additional noise n[k].Source: §2.1 · p. 3
• The measurement is modeled as z[k] = x[k] + γ · d[k] + σ · n[k], with γ and σ positive.Source: §2.1 · p. 3
• The setting assumes one available measurement, partially overlapping signal and disturbance spectra, recurring pulses of known shape, and a known upper bound on pulse frequency.The frequency bound can also be interpreted as knowing the minimum time between artifact recurrences.Source: §2.1 · p. 3

2.2 Formal Process Description

• The disturbance is modeled as a concatenation of paused, scaled versions of an approximate dominant-artifact shape ψ[k].The process uses Dmin as the minimum number of samples between dominant artifacts.Source: §2.2 · pp. 3–4
• The method finds cross-correlation peaks above threshold rmin while enforcing minimum separation Dmin.The measurement can optionally be band-pass filtered to retain frequencies of interest, such as by removing slow drifts or high-frequency noise.Source: §2.2 · p. 4
• The artifact template τ[k] is formed by averaging measurement windows around detected peaks, then shifting and scaling it to start and end at zero with unit energy.Source: §2.2 · p. 4
• The estimated artifact signal is built by placing scaled copies of the template at the detected peak locations.The cleaning algorithm is called the wavelet template method, drawing inspiration from wavelet and shapelet analyses.Source: §2.2 · p. 4
• The clean-signal estimate is obtained by subtracting the estimated artifact signal from the measurement.Source: §2.2 · p. 4

2.3 Example Application

• The wavelet-template method extracts and averages measurement slices around cross-correlation peaks, then scales and concatenates templates to estimate the disturbance.The estimated disturbance is subtracted from the measurement to obtain the cleaned estimate.Source: Figure 1 · p. 5, §2.3 · p. 5
• The method targets ECG artifacts in LFP signals, not ECG analysis itself; the artifact can dominate 5–15 Hz while the LFP β-band spans 10–30 Hz.Only one measurement channel may be available in some cases.Source: §2.3 · p. 6

2.3.1 Ground-Truth Synthesis

• The synthetic ground truth uses superposed frequency-domain windowed white noise, while disturbance pulses are concatenated ECG-like pulses built from pseudo-random parameters.The measurement combines the ground-truth signal, disturbance, and generated noise; γ and σ are gain parameters for the respective addends.Source: §2.3.1 · p. 6, §2.3.2 · p. 6

2.3.2 ECG as Disturbance

• The ECG search template uses a Ricker wavelet because the QRS complex resembles its piecewise-linear shape.A 0.1 s search-template duration corresponds to a peak frequency of approximately 13.50 Hz, within the LFP β-band.Source: §2.3.2 · pp. 7–8

2.3.3 Machine-Learning Baseline

• The autoencoder is trained on synthetic data and evaluated against the template method on a separate, purpose-generated test set representing an idealized data-driven benchmark.Its training set contains approximately 83k overlapping 0.5 s slices; the training order varies disturbance and noise gains to encourage learning artifact recognition before noise rejection.Source: §2.3.3 · pp. 8–9

2.4 Performance Evaluation

• Performance is assessed by comparing the cleaned signal and estimated disturbance with their respective synthetic ground truths using time- and frequency-domain metrics.RMSE measures time-domain error; RMSLE measures log-power error across the spectrum, based on estimated power spectra.Source: §2.4 · p. 9
• Negative ΔRMSE and ΔRMSLE indicate improvement over the raw measurement, with each delta comparing cleaned-signal error against unfiltered error.This comparison addresses the possibility that cleaning could distort measurements that are scarcely affected by a disturbance.Source: §2.4 · p. 10
• Artifact detection is evaluated with TPR and FPR, defined from matching peaks in the estimated disturbance and ground truth.A matching peak occurs at the same index in both signals; the rates use the number of true peaks and the total detected peaks, respectively.Source: §2.4 · p. 10

2.5 Parametric Sweep

• The sweep evaluates the cleaning algorithm over a grid of mixing parameters (γ, σ), with other values held constant.The setup anticipates that larger γ eases artifact detection, while larger σ lowers accuracy; these are expectations, not reported outcomes.Source: §2.5 · p. 10
• The full parameter grid is evaluated on N randomly generated datasets, varying the number, locations, and widths of peak frequencies.The passage does not specify the resulting performance trend across the grid.Source: §2.5 · p. 10
• Table 1 reports artifact-estimation and removal performance for 100 simulations with the same configuration, using median and 95th-percentile values.The supplied caption identifies the table’s scope and summary statistics but does not provide its cell values.Source: Table 1 · p. 11

• The single-configuration test used 100 synthetic signals with overlapping signal and disturbance frequencies and significant noise (SNR = 4).Source: §3 · p. 11
• RMSLE was evaluated across the full frequency range and was dominated by noise differences, making even small RMSLE changes important.Source: §3 · p. 11
• 33% median signal-estimate RMSE improvement for WT versus 29% for AE; Figure 3 shows both methods approximating the disturbance spectrum near its peak.Source: §3 · p. 11, Figure 3 · p. 12
• AE had smaller median-to-95th-percentile score gaps, suggesting greater consistency, while WT performed better on average for almost all metrics except TPR.Source: §3 · p. 11
• The 100-experiment parameter sweep varied noise and disturbance gains across a grid, using randomly generated signal frequencies, widths, disturbances, and noise.Source: §3 · p. 11
• Across the parameter sweep, AE showed greater consistency, but WT could yield better results on individual metrics and runs.Source: §3 · p. 13

4 Discussion

• Both WT and AE perform best when the disturbance dominates the measurement, i.e. γ > 1 ∧ γ > σ.The presented algorithm has shown effective artifact removal, particularly for dominant artifacts.Source: §4 · p. 13
• The WT needs an initial disturbance-shape guess, but extracts the actual template from the measurement, partially decoupling that choice from performance.A reasonable initial guess has been shown to yield satisfactory results.Source: §4 · p. 13
• The algorithm always removes artifacts whether or not they are present, and performance worsens when artifacts are comparatively small, near γ → 1+.The authors say artifact-removal need should therefore be assessed before applying the results.Source: §4 · p. 13
• The WT can outperform the AE on some metrics and runs, although the AE is more consistent overall and performs better in on-off artifact detection.Figure 4 also reports WT gains in RMSE for larger σ; its thin lines show individual runs and thick lines show medians.Source: §4 · p. 13, Figure 4 · p. 14

5 Conclusions

• The data-augmented WT is presented as a valid alternative to the autoencoder for removing artifact-like disturbances from signals.The comparison included a parametric sweep over measurement parameters.Source: §5 · p. 15
• WT avoids generalization uncertainty and can be adapted to different scenarios without expensive retraining, though it requires an initial artifact-shape guess.The paper contrasts this guess with designing and retraining an AE for a specific task.Source: §5 · p. 15
• The proposed method and the methods for generating evaluation signals have been released publicly, with source code freely available.The authors hope this will prompt further development and investigation.Source: §5 · p. 15

Author Contributions (CRediT)

• AS and HM are credited with conceptualization, software, and formal analysis; AS also led investigation and visualization.AS and HM contributed to methodology and writing; AS drafted the original manuscript.Source: Author Contributions (CRediT) · p. 15
• GC is credited with funding acquisition and project administration, while HT is credited with writing review and editing.GC also contributed to writing review and editing.Source: Author Contributions (CRediT) · p. 15
• All authors read and agreed to the submitted version of the manuscript.Source: Author Contributions (CRediT) · p. 15