# Data-Driven Calibration Estimation for Robust Remote Pulse-Oximetry
## 2 Materials and Methods
### Apbv
#### 2.2.2 Empirical  
In the current research, we aim at providing an empirical, data-driven alternative for the modeling approach, which shall be compared with the earlier modeling-based approach to show it is more accurate and more generic.  
Framework  
The general framework of our calibration approach is visualized in Figure 2 and consists of three steps. The first step is to extract the raw PPG signals from the videos. Hereafter, in the second step, an exhaustive search is executed over a range of possible pulse signatures of normalized amplitudes, \(\vec{P_{bv}}\), where the signature corresponding with the best pulse-signal quality is selected as optimum. Finally, based on the collection of estimated signature vectors, a _relative_ calibration model is fitted where the reference data are used to arrive at the APBV calibration model with vectors \(\vec{P_{bv}^{s}}\) and \(\vec{P_{bv}^{u}}\) which can be used for SpO\({}_{2}\) estimation. The three main differences between this approach and the previous calibration method for camera SpO\({}_{2}\)[9] are that: (1) the method can be applied for the indirect APBV method for an arbitrary number of wavelengths, (2) it can be applied on short time windows which allows the capturing of dynamics in SpO\({}_{2}\), and (3) the camera measurements are not directly linked to SpO\({}_{2}\) values from the reference. In the next paragraphs, we discuss the steps of our proposed calibration approach in more detail.  
Pre-Processing  
We extracted the PPG signals from the forehead region-of-interest (ROI) using triangulation of facial landmark points. We spatially averaged the intensities of the pixels within the ROI for all wavelengths and concatenated the values for each video frame. For each time-window of 10 s, we divided the raw PPG signals for each wavelength (\(\mathbf{C^{i}}\)) by its quasi-DC signal obtained by low-pass filtering (LPF), and bandpass filtered (BPF) the resulting signal to obtain the DC-normalized PPG waveforms:  
\[\mathbf{C_{n}^{i}}=BPF\left(\frac{\mathbf{C^{i}}}{LPF(\mathbf{C^{i}})}\right) \qquad\text{for i = 1,...,N}, \tag{6}\]  
where the \(LPF\) is a first-order Butterworth filter with a cut-off frequency of 0.7 Hz, and the \(BPF\) is a fourth-order Butterworth filter with a passband in the range 0.7-4 Hz, the typical range of pulse rates for healthy adults. The DC-normalized PPG waveforms \(\mathbf{C_{n}}\) are calculated for each step-size of 1 s and are used as the input for the exhaustive search.  
Exhaustive Search  
As mentioned in Section 2.2, the signature vector \(\vec{P_{bv}}\) comprises the _relative_ pulse amplitudes for each wavelength. Therefore, for \(N\) wavelengths, the search has to be performed over \(N-1\) dimensions, arbitrarily setting one of the wavelength pulse-amplitude to unity (1). For a three-wavelength system, this can be formulated as:  
Figure 2: The general framework of our empirical calibration approach can be described by three steps: (1) extracting the raw PPG signals from the videos, (2) executing an exhaustive search over a range of pulse signatures using pulse quality as optimization criterion, and (3) fitting of a calibration model to the data.  
\[\vec{P_{b_{0}}}(SpO_{2})=\begin{bmatrix}\alpha_{\lambda_{1}}\\ 1_{\lambda_{2}}\\ \beta_{\lambda_{3}}\end{bmatrix}, \tag{7}\]  
where the exhaustive search is performed over a range of \(\alpha\) and \(\beta\) relative to the unit amplitude of \(\lambda_{2}\). For our wavelengths present in the dataset, we set the evaluation range for \(\alpha\) and \(\beta\) to 0-2 with a sampling resolution of 0.001. An example of the optimization for the wavelengths \([\lambda_{1},\lambda_{2},\lambda_{3}]=[675,800,905]\) nm is visualized in Figure 3. We are aware that exhaustive search is computationally expense and that more efficient approaches exist, e.g., stochastic gradient descent or other simulated annealing approaches. We, however, do not worry too much that computational efficiency of what we describe could be further improved, as long as the accuracy is guaranteed.