## 2 Materials and Methods
### Normalized Least Squares Filter and Ensemble Empirical Mode Decomposition  
Because the driving environment is dynamic, \(HR_{comp}\) cannot be used directly as a measure of the HR. In order to refine the extracted features via post-processing, an adaptive filter was applied.  
Figure 2: Plots of the correlation and correlation coefficients (\(\rho\)) between the face (horizontal axis) and background (vertical axis) signals in each channel. Each dot denotes samples of the joint blind source separation (JBSS) results at every frame: (**a**) red channel; (**b**) green channel; (**c**) blue channel.  
The normalized least mean squares (NLMS) filter is one of the most useful adaptive filters used in noise cancellation [24]. The reasonable performance of an NLMS filter in a remote PPG experiment was previously reported [4]. Although [PERSON] et al. used NLMS to subtract the noisy background signal from the face signal, our method employed it to refine each \(HR_{comp}\) channel. NLMS can be represented as follows:  
\[d_{i}=x_{i}^{t}w_{i}+n_{i}, \tag{3}\]  
where \(x_{i}^{t}\) and \(n_{i}\) are the input vector and white Gaussian noise with \(\mathcal{N}\Big{(}0,\sigma_{v}^{2}\Big{)}\), which is independent from \(x_{i}^{t}\), at the \(i_{th}\) iteration, respectively, and \(d_{i}\) is the signal to be refined. For each iteration, the optimal weight coefficient is updated and \(d_{i}\) is estimated as  
\[w_{i}=w_{i-1}-\mu\Big{[}\
abla^{2}J[w_{i-1}]\Big{]}^{-1}\big{[}\
abla J[w_{i-1} ]\big{]}^{*}, \tag{4}\]  
where * and \(\
abla\) denote the conjugate operation and derivation, respectively, and \(J(w_{i-1})\) is the cost function with respect to the weight coefficient. The NLMS algorithm operated iteratively to minimize the error \(n_{i}\), which was the difference between the desired signal \(d_{i}\) and \(x_{i}^{t}w_{i}\). The weight coefficient \(w\) was initialized as zero at the beginning of the iterations.  
[PERSON] et al. proposed several HR feature signals using a linear combination of RGB channels, and [PERSON] et al. analyzed the optimal feature signals in a dataset for various environments [8; 15]. Based on previous results, the signal for each channel refined using NLMS was transformed into a _RoverG_ signal, which was the ratio of the red and blue channels.  
Ensemble empirical mode decomposition (EEMD) is a noise-assisted method for extracting the intrinsic mode function (IMF) from observed data. EEMD operates by iteratively averaging a series of trials in which white noise is added to the observed signal. As demonstrated in Reference [25], IMFs representing the source components of complicated observations can be decomposed after several iterations. Therefore, EEMD was employed on the _RoverG_ signals to extract the IMF corresponding to the HR. In Reference [25], [PERSON] et al. discovered that the fourth IMF of facial observation signals is the closest to the HR; thus, our approach also used the fourth IMF from _RoverG_ via EEMD.  
An example of these steps is presented in Figure 3. While the other approaches fluctuated over time, HR estimation using the NLMS filter followed the ground-truth in a relatively stable manner.