WPD-CCA Method for Artifact Removal

• WPD-CCA is a hybrid method that combines wavelet packet decomposition with canonical correlation analysis to correct motion artifacts in single-channel EEG/fNIRS data.
• It generates pseudo-multichannel representations from single-channel recordings, enabling efficient separation and removal of artifact components.
• Performance metrics like ΔSNR and percentage artifact reduction illustrate its superiority over single-stage methods and traditional blind source separation techniques.

The WPD-CCA (Wavelet Packet Decomposition–Canonical Correlation Analysis) method offers a two-stage data-driven pipeline for correcting motion artifacts in single-channel electroencephalogram (EEG) and functional near-infrared spectroscopy (fNIRS) signals. Designed to address the non-stationarity and contamination issues inherent in wearable EEG/fNIRS measurements, WPD-CCA combines time–frequency localization provided by wavelet packet decomposition with the source separation capacity of canonical correlation analysis. This hybrid approach circumvents the requirement for multi-channel inputs or physically separate artifact references, instead generating pseudo-multichannel representations from a single sensor. WPD-CCA demonstrates superior denoising efficacy compared to single-stage WPD and other blind-source separation (BSS) methods, as evaluated by difference in signal-to-noise ratio (ΔSNR) and percentage reduction in motion artifacts (η) (Hossain et al., 2022).

1. Two-Stage Pipeline: Structure and Rationale

WPD-CCA is composed of two sequential operations:

1. Wavelet Packet Decomposition (WPD): The input signal $x(n)$ is decomposed into $2^j$ frequency-localized sub-bands via wavelet packet basis functions at level $j$. Each sub-band isolates distinct spectral content, exploiting the empirical observation that motion artifacts produce large-magnitude coefficients localized sparsely across few sub-bands. This process generates a pseudo-multichannel dataset from the original single-channel input.
2. Canonical Correlation Analysis (CCA): The WPD sub-bands are interpreted as channels. CCA is then applied between the (pseudo-multichannel) WPD sub-bands $X(t) \in \mathbb{R}^{M \times T}$ and their time-lagged versions $Y(t) = X(t-1) + X(t+1)$. By maximizing correlations between linear projections of $X$ and $Y$, CCA yields canonical variates ranked by autocorrelation strength. Artifact-dominated components typically possess the largest autocorrelation, and are identified for subsequent removal prior to reconstructing the artifact-suppressed signal (Hossain et al., 2022).

This two-stage approach is motivated by the observation that neither WPD nor CCA alone achieves optimal artifact suppression in single-channel settings. In WPD-CCA, WPD isolates artifact-rich sub-bands, while CCA further unmixed temporally autocorrelated artifact signals from the underlying neuronal or hemodynamic activity.

2. Mathematical Foundations

2.1 Wavelet Packet Decomposition

Given signal $x(n) = s(n) + v(n)$, with $s(n)$ the true underlying physiological signal and $v(n)$ the artifact, WPD at level %%%%10%%%% produces basis functions $\psi_{j,i}(n)$ and expansion coefficients

$$X_{j,i}(k) = \langle x(n), \psi_{j,i}(n-2^j k) \rangle$$

which decompose as $X_{j,i}(k) = S_{j,i}(k) + V_{j,i}(k)$. The basis recursion relations are

$$\psi_{j,2i}(n) = \sum_k h(k) \psi_{j-1,i}(n - 2^{j-1}k)$$

$$\psi_{j,2i+1}(n) = \sum_k g(k) \psi_{j-1,i}(n - 2^{j-1}k)$$

where $h$, $g$ are the low- and high-pass wavelet filters (e.g., Daubechies ‘db1’, ‘db2’, Fejér–Korovkin ‘fk4’, etc.).

2.2 Canonical Correlation Analysis

With $X(t)$ (size $M \times T$, $M=2^j$) and $Y(t) = X(t-1) + X(t+1)$, CCA finds $w_x$, $w_y$ to maximize

$\rho = \text{corr}(w_x^T X, w_y^T Y)$

Subject to constraints, this leads to the generalized eigenvalue problem

$$C_{XX}^{-1} C_{XY} C_{YY}^{-1} C_{YX} w_x = \rho^2 w_x$$

Canonical loadings $w_x$, $w_y$ unmix $X$ into variates $S=W X$ ordered by autocorrelation. Artifact components yield high canonical correlations.

2.3 Reconstruction

Artifacts are identified by evaluating whether zeroing a canonical variate increases the correlation with a ground-truth (reference) signal or surpasses an autocorrelation threshold. Excluding artifact-dominated canonical variates forms $\tilde{S}$, and the cleaned pseudo-channels are reconstructed by $\tilde{X} = W^{-1} \tilde{S}$. Summing across sub-bands yields the final artifact-corrected signal.

3. Detailed Algorithmic Steps

3.1 Preprocessing

• EEG: Downsample from 2048 Hz to 256 Hz, apply a 50 Hz notch filter, remove polynomial baseline drift.
• fNIRS: Use 25 Hz sampling rate, apply similar notch/baseline removal.
• Entire signals (~9 minutes duration) are processed as a single segment.

3.2 WPD Stage

• Decomposition level $j=4$ ($16$ sub-bands).
• Wavelet packet options: db1, db2, db3, sym4–6, coif1–3, fk4, fk6, fk8 (12 total).
• Compute $x_i(n)$ for $i = 1, \dots, 16$.

3.3 WPD-Based Artifact Detection

• In single-stage WPD, each sub-band $x_i(n)$ is dropped if its exclusion increases the sum’s correlation with a reference channel.
• Artifact-reduced signal is obtained by summing the undropped sub-bands.

3.4 CCA Stage (WPD-CCA)

• Stack the 16 sub-bands as $X(t)$.
• Compute $Y(t) = X(t-1) + X(t+1)$.
• Estimate covariance matrices $C_{XX}, C_{YY}, C_{XY}$.
• Solve for canonical weights and project $S = W X$.
• Identify and zero artifact-related canonical variate columns in $S$.
• Reconstruct $\tilde{X} = W^{-1}\tilde{S}$, sum sub-bands to yield the final signal.

Parameter selection is driven by trade-offs between frequency resolution and computational load (suggested $j=4$), and best performance is empirically observed for db1–3 and fk4–8 wavelets.

4. Quantitative Performance and Evaluation Metrics

Performance is quantified using:

• Difference in Signal-to-Noise Ratio (ΔSNR):

$$\Delta \mathrm{SNR} = 10 \log_{10}(\sigma^2/\delta_{after}^2) - 10 \log_{10}(\sigma^2/\delta_{before}^2) = 10 \log_{10}(\delta_{before}^2/\delta_{after}^2)$$

$\sigma^2$ denotes variance of the clean signal, $\delta_{before}^2$ and $\delta_{after}^2$ those of the corrupted and corrected signals, respectively.

• Percentage Reduction in Motion Artifacts (η):

$$\eta = 100 \cdot (\rho_{after} - \rho_{before}) / (1 - \rho_{before})$$

with $\rho_{before}$ and $\rho_{after}$ the correlations of the clean signal with the corrupted and corrected outputs.

Summarized results (average across subjects):

	Single-stage WPD (best)	WPD-CCA (best)	Relative Improvement
EEG	ΔSNR=29.44 dB (db2) η=51.4%	ΔSNR=30.76 dB (db1) η=59.51%	↑η by 11.3%
fNIRS	ΔSNR=16.11 dB (fk4) η=26.4%	ΔSNR=12.41 dB (fk8) η=41.40%	↑η by 56.8%

The WPD-CCA method provides higher percentage reduction in motion artifacts ($\eta$) and, in EEG, also higher ΔSNR than its single-stage counterpart (Hossain et al., 2022).

5. Comparative Analysis: Advantages and Limitations

Compared to earlier single-channel motion artifact removal techniques, including DWT, EMD/EEMD, VMD, ICA, or CCA alone, WPD-CCA demonstrates several notable advantages:

• Operates on single-channel recordings by generating pseudo-channels via WPD.
• Jointly exploits the frequency localization properties of wavelet packets and the multivariate separation capabilities of CCA.
• Does not require a physically separate uncorrupted reference at runtime.
• Artifact bands and components are identified in a data-adaptive manner.
• Robust performance across multiple wavelet bases.
• Outperforms single-stage WPD, as well as several BSS strategies, in denoising effectiveness.

However, certain limitations persist:

• Artifact component selection in CCA presently relies on a reference-correlation or autocorrelation threshold; universal, fully automatic thresholds remain undetermined.
• As the WPD level and sub-band count grow, so does computational complexity.
• In the absence of a reference signal, surrogate criteria (such as autocorrelation drops) must be used to detect artifact-related components.

6. Context, Applicability, and Future Considerations

WPD-CCA is particularly suited to biological signal modalities (EEG, fNIRS) where motion artifacts are spectrally sparse yet temporally autocorrelated, and multi-channel acquisition is impractical or infeasible. Its reliance on pseudo-multichannel analysis supports broader application in single-sensor wearable systems. A plausible implication is that further advances may be realized by integrating more sophisticated or fully unsupervised artifact identification schemes, potentially reducing dependence on reference correlation or hand-tuned thresholds. Optimization of decomposition level and wavelet basis, and parallel implementation strategies, could further enhance practical deployment (Hossain et al., 2022).