## 3 Glottal Gap Segmentation and Facilitative Playbacks
### Facilitative Playbacks Representation
#### 3.3.1 Local-Dynamics Facilitative Playbacks
The current position of \(\mathbf{g}_{pc}(t_{k})\) is updated at every frame to compensate for the relative movement of the endoscope, of the larynx, or of the vocal folds length changes via Equation (2). \[\mathbf{g}_{pc}(t_{k})=\mathbf{p}(t_{k})+(\mathbf{p}(t_{k})-\mathbf{a}(t_{k})) \left(\frac{pc(\%)}{100\%}\right)\;\in\;\mathbf{G}(t_{k}) \tag{2}\]  
Later, the intersection between the vocal folds edges \(\mathbf{C}^{l,r}(t_{k})\) and the trajectory line \(\mathbf{L}(t_{k})\) is computed by Equation (3):  
\[\mathbf{c}^{l,r}_{pc}(t_{k}):\mathbf{c}^{l,r}_{pc}(t_{k})\in\mathbf{L}(t_{k})\; \wedge\;\mathbf{c}^{l,r}_{pc}(t_{k})\in\mathbf{C}^{l,r}(t_{k}) \tag{3}\]  
Figure 10: Schematical drawing of the successive phases of a glottal cycle in three views. First row: frontal section of the vocal folds. Second row: larynxogoscopy (top view of the vocal folds). Third row: DKG at the line \(y_{j}\). Adapted from [116]. The vocal folds trajectory is obtained by Equation (4) as:  
\[\delta^{l,r}_{seg}(pc,t_{k})=\|\mathbf{g}_{pc}(t_{k})-\mathbf{c}^{l,r}_{pc}(t_{k}) \|_{2}\;;\quad k=\{1,2,\ldots,N\} \tag{4}\]  
where \(\delta^{l,r}_{seg}(pc,t_{k})\) are the deflections of the vocal folds edges at the point \(\mathbf{c}^{l,r}_{pc}(t_{k})\), and \(pc\) indicates the position of \(\mathbf{g}_{pc}(t_{k})\) in the glottal main axis. The VFT is illustrated in Figure 11 and expressed in vector notation at Equation (5). \[\delta^{l,r}_{seg}(pc,t)=[\delta^{l,r}_{seg}(pc,t_{1})\ \delta^{l,r}_{seg}(pc,t_{2})\ \cdots\ \delta^{l,r}_{seg}(pc,t_{k})\ \cdots\ \delta^{l,r}_{seg}(pc,t_{N})] \tag{5}\]  
There are other local FP that have been less explored in literature such as Mucosal Wave Kymography (MKG), Optical Flow Based Waveform (OFW) and Optical Flow Kymogram (OFKG). These FP compute motion vectors to assess the fine detail of MW including the propagation of the mucosal edges during the opening and closing glottal phases. They are computed based on motion estimation techniques where the motion field \(\mathbf{\mathcal{W}}(\mathbf{x},t_{k})\) at time \(t_{k}\) is obtained using the intensity variation of consecutive frames (\(t_{k}\) and \(t_{k}+1\)). The matrix representation of a total motion field at time \(t_{k}\) is depicted in Equation (6). \[\mathbf{\mathcal{W}}(\mathbf{x},t_{k})= \begin{pmatrix}\widetilde{\mathbf{w}}( \mathbf{x}_{11},t_{k})&\widetilde{\mathbf{w}}(\mathbf{x}_{12},t_{k})&\cdots& \widetilde{\mathbf{w}}(\mathbf{x}_{1n},t_{k})\\ \widetilde{\mathbf{w}}(\mathbf{x}_{21},t_{k})&\widetilde{\mathbf{w}}( \mathbf{x}_{22},t_{k})&\cdots&\widetilde{\mathbf{w}}(\mathbf{x}_{2n},t_{k})\\ \vdots&\vdots&\ddots&\vdots\\ \widetilde{\mathbf{w}}(\mathbf{x}_{m1},t_{k})&\widetilde{\mathbf{w}}( \mathbf{x}_{m2},t_{k})&\cdots&\widetilde{\mathbf{w}}(\mathbf{x}_{mn},t_{k}) \end{pmatrix}  \tag{6}\]  
where \(\widetilde{\mathbf{w}}(\mathbf{x}_{ij},t_{k})=(u(\mathbf{x}_{ij},t_{k}),v( \mathbf{x}_{ij},t_{k}))\) is the vector displacement of \(\mathbf{x}_{ij}\) at time \(t_{k}\). The vector displacement \(\widetilde{\mathbf{w}}(\mathbf{x}_{ij},t_{k})\) has two components: one in the \(x\)-axis direction (\(u(\mathbf{x}_{ij},t_{k})\)) and another in the \(y\)-axis direction (\(v(\mathbf{x}_{ij},t_{k})\)). For instance, in OFKG, a row vector \(\mathbf{OFKG}(t_{k})=[u(\mathbf{x}_{inj},t_{k})\ u(\mathbf{x}_{n-1j},t_{k})\ u( \mathbf{x}_{n-2j},t_{k})\cdots\ u(\mathbf{x}_{1j},t_{k})]\) is selected from a horizontal line at time \(t_{k}\) and position \(y_{j}\). Then, the OFKG matrix \(I^{R,G,R}_{OFKG}(x,y)\) is  
Figure 11: Illustration of the VFT playback. First row: the image sequence of one glottal cycle. Second row: vocal folds trajectories, DKG and DKG+VFT playbacks of five glottal cycles. constructed with a set of \(N\) vectors \(\{\textbf{OFKG}(t_{k})\in\mathbb{R}^{n}\), \(k=1,\cdots,N\}\), where each \(\textbf{OFKG}(t_{k})\) is a column vector of \(I_{OFKG}^{R,G,B}(x,y)\):  
\[I_{OFKG}^{R,G,B}(x,y)=