# Data-Driven Calibration Estimation for Robust Remote Pulse-Oximetry
## 2 Materials and Methods
### Apbv  
Instead of extracting amplitudes from the PPG waveforms and computing SpO\({}_{2}\) with ratio-of-ratios, APBV determines SpO\({}_{2}\) indirectly based on the signal quality of the pulse signals extracted with individual signature vectors \(\vec{P_{bv}}\) for each possible SpO\({}_{2}\) value [8]. This is attractive, as the optimum, indicating the SpO\({}_{2}\)-value, remains stable, even when the signal-to-noise ratio (SNR) worsens. The APBV-method can be mathematically summarized as:  
\[SpO_{2}^{*}=\underset{SpO_{2}\in SpO_{2}}{\arg\max}\;SNR\bigg{(}\underbrace{ \frac{\vec{W}_{pBV}}{k\vec{P_{bv}}(SpO_{2})[\mathbf{C_{n}C_{n}^{T}}]^{-1}} \mathbf{C_{n}}}_{\vec{P}}\bigg{)}, \tag{1}\]  
where \(P\) is the pulse signal, \(\mathbf{C_{n}}\) contains the DC-normalized color variations and scalar \(k\) is chosen such that \(\vec{W}_{PBV}\) has unit length. The calculation of the weights for extraction of the pulse signal, \(\vec{W}_{PBV}\), is formulated as a least squares problem using pulse signatures \(\vec{P_{bv}}\) for different SpO\({}_{2}\) values. The relation between the _direct_ ratio-of-ratios approach and _indirect_ APBV method can be described as:  
\[SpO_{2}=C_{1}+C_{2}\cdot\overbrace{\left(\frac{(\frac{AC}{DC})_{\lambda_{1}}} {(\frac{AC}{DC})_{\lambda_{2}}}\right)\overset{\text{Direct vs Indirect}}{\leq}\vec{P_{bv}}( SpO_{2})=\begin{bmatrix}(\frac{AC}{DC})_{\lambda_{1}}\\ (\frac{AC}{DC})_{\lambda_{2}}\end{bmatrix}(SpO_{2})}^{\text{Direct vs Indirect}}\; \vec{P_{bv}}(SpO_{2})}^{\text{Direct vs Indirect}}\;(SpO_{2})=\begin{bmatrix}( \frac{AC}{DC})_{\lambda_{1}}\\ (\frac{AC}{DC})_{\lambda_{2}}\end{bmatrix}(SpO_{2})}^{\text{Direct vs Indirect}}\;(SpO_{2})} ^{\text{Direct vs Indirect}}\;(SpO_{2})}, \tag{2}\]  
where \(R\) is the ratio of normalized pulse amplitudes and \(C_{1},C_{2}\) are calibration constants. The two most important implications of these formulations are that: (1) the use of ratio-of-ratios is limited to two wavelengths, whereas APBV can be executed with an arbitrary number of wavelengths, and (2) the calibration model of APBV, i.e., \(\vec{P_{bv}}(SpO_{2})\), should be determined a priori. The first implication is important as it has been shown that adding a third wavelength improves the robustness of the measurement during motion [8]. In this paper, we focus on the implication of the APBV calibration model, and more specifically on the accuracy of the calibration estimation.  
The SpO\({}_{2}\) signatures compiled in \(\vec{P_{bv}}\) for \(N\) wavelengths can be expressed as:  
\[\vec{P_{bv}}(SpO_{2})=\begin{bmatrix}(\frac{AC}{DC})_{\lambda_{1}}\\ (\frac{AC}{DC})_{\lambda_{2}}\\ \vdots\\ (\frac{AC}{DC})_{\lambda_{N}}\end{bmatrix}(SpO_{2})=\overbrace{\begin{pmatrix} \frac{AC}{DC})_{\lambda_{1}}\\ (\frac{AC}{DC})_{\lambda_{2}}\\ (\frac{AC}{DC})_{\lambda_{N}}\end{bmatrix}}^{\text{}}+(100-SpO_{2})\cdot\vec{P_ {bv}}^{\text{\#}}. \tag{3}\]  
Here, \(\vec{P_{bv}^{*}}\) denotes the (static) pulse signature vector for an oxygen saturation level of 100%, and \(\vec{P_{bv}^{\text{#}}}\) denotes the update vector consisting of \(N\) parameters which describe the change of the signature for decreasing SpO\({}_{2}\)-values. In order to apply this model for SpO\({}_{2}\) estimation, \(2(N-1)\) parameters have to be determined. The values for one wavelength can be set to a fixed value for both \(P_{bv}^{s}\) (e.g., 1) and \(P_{bv}^{\text{#}}\) (0), as the values in both vectors describe the ratio in pulse amplitude between the wavelengths. We will now discuss how the parameters can be determined based on opto-physiological modelling. Hereafter, we will present our proposed empirical approach.