## 2 Contact Pressure Estimation from Laryngeal High-Speed Video
### Edge Tracking  
The tracking stage treats VF collision as a state estimation problem in the presence of noise and lost data. A Kalman Filter (KF) is implemented to predict the value and rate of change of the coefficients during the contact phase, wherein \(\Theta_{k}\) and \(\dot{\Theta}_{k}\) are observations for the filter. A dynamical model is needed for the KF process for which we assume a spring-mass-damper representation for each VF connected to their respective attachment lines (defined in the pre-processing stage). The mass in this system is normalized for simplicity. Thus, the structural VF model is given by  
\[X_{i,k+1} =AX_{i,k}+D_{k}\,, \tag{7}\] \[Y_{i,k} =CX_{i,k}+E_{k}\,,\] (8) \[Y_{i,k} = \begin{pmatrix}\theta_{i,k}\\ \dot{\theta}_{i,k}\end{pmatrix} \,,\quad A= \begin{pmatrix}1&\Delta t\\ -K\Delta t&1-b\Delta t\end{pmatrix} \,,\]  
where \(X_{i,k}\) is the state variable of a particular coefficient \(\theta_{i}\) in \(\Theta\) at the instant \(k\), \(Y_{i,k}\) are observations of the state (which we assume is available with \(C\) as the identity matrix), \(D_{k}\) and \(E_{k}\) are the process and measurement noise (uncorrelated and Gaussian distributed with variances \(\sigma_{d}\) and \(\sigma_{e}\), respectively), \(K\) is the coefficient stiffness, and \(b\) is the damping parameter of the model. The spring-mass-damper model needs to operate at a particular resonance \(w_{r}\), for which we define stiffness \(K\) and damping \(b\) values as functions of the control parameters \(w_{r}\) and \(\xi\) as  
\[K=\frac{w_{r}^{2}}{1-\xi^{2}}\,,\quad b=2\xi\sqrt{K}\,. \tag{9}\]  
Typical damping values associated with \(\xi\) are set between (0 to 0.03). The resonance \(w_{r}\) is calculated automatically from the kinematic information \((\theta_{i,k_{0}},\dot{\theta}_{i,k_{0}})\) at the previous instant of impact (\(k_{0}\)), using these as initial conditions. Then, the known analytic solution for the initial conditions is used to coincide the rebounding position \(\theta_{i,k_{1}}\) at the end of contact (\(k_{1}\)) with its analytic value. Thus, the proposed scheme adapts the elastic contact properties (stiffness and damping) to match the fundamental frequency of the VFs, which allows for the capture of pre-stress conditions. Thus, we need to find a particular resonance frequency \(w_{r}\) that satisfies  
\[w_{r}=\underset{w}{\text{argmin}}\quad\left\|F_{i,k_{1}}(w)- \theta_{i,k_{1}}\right\|\,, \tag{10}\] \[F_{i,k_{1}}(w)=\theta_{i,k_{0}}\cos\left(wt_{k_{1}}\right)+\frac {\dot{\theta}_{i,k_{0}}}{w}\sin\left(wt_{k_{1}}\right)\,, \tag{11}\]  
where \(t_{k_{1}}=\left(k_{1}-k_{0}\right)\Delta t\). The \(w_{r}\) parameter controls the required stiffness of the model to mimic a damped harmonic oscillation during collision. When the edges are visible, this contact model is not needed and, thus, is no longer used. To define when KF estimations are required, we calculate a \"non-detected points ratio\" \(\lambda\) and an uncertainty factor \(\rho\) as  
\[\lambda_{k}=\frac{D_{T}-D_{k}}{D_{T}}\,,\quad\rho_{k}=\frac{1}{1+e^{-\beta( \lambda_{k}-\gamma)}}\,, \tag{12}\]  
where \(D_{T}\) is the upper bound of all possible detected points at the edge, \(D_{k}\) is the actual detected points, \(\beta\) is a gain factor, and \(\gamma\) is an uncertainty threshold. When \(D_{k}\) is low, for example, \(\lambda_{k}\) tends to be larger than the uncertainly threshold and \(\rho_{k}\) goes to one; that is, there are too many lost data points, the polynomial fit is poor, and thus, KF estimations are needed.