## 2 Contact Pressure Estimation from Laryngeal High-Speed Video
### Curve Fitting  
In this step, a polynomial curve of order \(p\) is fitted to the \((x,y)\) points from the edge detection, defining the boundary of a given vocal fold with respect to the attachment line (the line defined by the attachment points \((x_{a},y_{a})\) and \((x_{b},y_{b})\)). A least squares (LS) estimator is used, taking into account the attachment points as root constraints for the solution. Letting \(u\) and \(v\) be the coordinates orthogonal to and aligned with the attachment line, respectively, the polynomial model \(M\) can be defined as:  
\[M(v)=\left(\sum_{i=0}^{p-2}\theta_{i}v^{i}\right)(v-v_{a})\left(v-v_{b}\right) +u_{a}\,. \tag{2}\]  
where \(\Theta=\left(\theta_{p-2}\quad\theta_{p-3}\quad\ldots\quad\theta_{0}\right)^{ \text{T}}\) is the vector of unknown model parameters with superscript T indicating vector transposition and \(u_{a}\), \(v_{a}\), and \(v_{b}\) are the attachment point locations in the \((u,v)\) coordinate system. Note that, assuming a rotation angle \(\varphi\) of the attachment line with respect to the vertical axis, the relationship between the \((x,y)\) and \((u,v)\) coordinate systems is given by  
\[\begin{pmatrix}u\\ v\end{pmatrix}=\begin{pmatrix}\cos\varphi&-\sin\varphi\\ \sin\varphi&\cos\varphi\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}\,. \tag{3}\]  
Figure 2: Edge detection example using a HSV recording. Representative steps of the detection process and a mid portion kymography of the glottis are shown. Gradient information is used (first row) to find left and right VF borders. VF edge points are missing during collision (second row). Edge-based kymogram (third row) and its temporally averaged version (fourth row) are also shown. Note that temporal averaging reduces the errors incurred by the edge detection, but it does not complete the trajectory in closing phases.  
Using an LS estimator solution, the polynomial parameters contained in \(\Theta\) are computed as  
\[\Theta=\left(V^{\mathrm{T}}V\right)^{-1}V^{\mathrm{T}}U \tag{4}\]  
where  
\[V=\begin{pmatrix}\tilde{v}_{1}v_{1}^{p-2}&\cdots&\tilde{v}_{1}v_{1}&\tilde{v}_{ 1}\\ \tilde{v}_{2}v_{2}^{p-2}&\cdots&\tilde{v}_{2}v_{2}&\tilde{v}_{2}\\ \vdots&\ddots&&\vdots\\ \tilde{v}_{D}v_{D}^{p-2}&\cdots&\tilde{v}_{D}v_{D}&\tilde{v}_{D}\end{pmatrix}; \quad U=\begin{pmatrix}\tilde{u}_{1}\\ \tilde{u}_{2}\\ \vdots\\ \tilde{u}_{D}\end{pmatrix} \tag{5}\]  
where \(\tilde{u}_{l}=u_{l}-u_{a}\), \(\tilde{v}_{l}=\left(v_{l}-v_{a}\right)\left(v_{l}-v_{b}\right)\), and \(l\) is an integer ranging from \(1\) to \(D\), with \(D\) representing the number of points defining the glottal edge detected in the previous stage. The velocity of the glottal edge is embedded in the rate of change of the coefficients \(\Theta\), which can be obtained via differentiating between frames as  
\[\dot{\Theta}_{k}=\frac{1}{\Delta t}\left(\Theta_{k}-\Theta_{k-1}\right)\,. \tag{6}\]  
where the subscript \(k\) is the frame index and \(\Delta t\) is the time lapse between successive HSV frames.  
The third frame in Figure 2 shows the output of the edge detection scheme. The second row of the figure shows how the edges are tracked from an open phase to a closed phase. Midline kymograms of the VF trajectories during the open phases are plotted at the bottom of the figure for both VFs.