## 3 Glottal Gap Segmentation and Facilitative Playbacks
### Facilitative Playbacks Representation
#### 3.3.2 Global-Dynamics Facilitative Playbacks
For doing this, it is assumed that the positions of the posterior and anterior glottal points do not change dramatically for all the intermediate images between the occurrences of two consecutive keyframes within a single oscillatory cycle. Therefore, such points are  
Figure 12: Illustration of the GAW playback. First row: six images of a particular glottal cycle length. Second row: GAW playback normalized within the interval [0,1] where 0 represents the minimum area and 1 the maximum area. computed approximately by linear interpolation via Equation (12) where \(t_{O}\) and \(t_{O+1}\) indicate two consecutive open-states. \[ \begin{array}{rcl}\mathbf{p}(t_{k})&=&\mathbf{p}(t_{O})+\frac{\mathbf{p}(t_{O +1})-\mathbf{p}(t_{O})}{t_{O+1}-t_{O}}\cdot(t_{k}-t_{O})\ ;&t_{O}<t_{k}<t_{O+1}\\ \\ \mathbf{a}(t_{k})&=&\mathbf{a}(t_{O})+\frac{\mathbf{a}(t_{O+1})-\mathbf{a}(t_ {O})}{t_{O+1}-t_{O}}\cdot(t_{k}-t_{O})\ ;&t_{O}<t_{k}<t_{O+1}\end{array}  \tag{12}\]  
By connecting the vocal folds edges \(\mathbf{C}^{l_{J}}(t_{k})\) to the approximated position of \(\mathbf{p}(t_{k})\) and \(\mathbf{a}(t_{k})\), a continuous representation of the vocal folds edges is obtained also in the parts that are undetected from the segmentation methods. Later, the glottal main axis \(\mathbf{G}(t_{k})\) and the vocal folds edges \(\mathbf{C}^{l_{J}}(t_{k})\) are equidistantly sampled with \(pc\in[0,M]\), where \(M\) corresponds to total percentage length of \(\mathbf{G}(t_{k})\) (100%). Then for each image the deflections of the vocal folds edges \(\delta_{seg}^{l_{J}}(pc,t_{k})\) are obtained \(\forall pc\in[0,M]\) and \(\forall t\in[1,N]\). \(\delta_{seg}^{l_{J}}(pc,t_{k})\) is positive, if the left/right fold contour is correctly located on the ipsilateral side of the glottal main axis. Contrariwise, if the vocal fold edges cross laterally, the glottal main axis, \(\delta_{seg}^{l_{J}}(pc,t_{k})\) becomes negative. Furthermore, the vocal folds are splitted longitudinally and the left vocal fold is turned \(180^{\circ}\) around the posterior commissure \(\mathbf{p}(t_{k})\). Lastly, all the computed \(\delta_{seg}^{l_{J}}(pc,t_{k})\) are stored in a matrix \(l_{PVG}(x,y)\in\mathbb{R}^{(2M+1)\times N}\) (Equation (13)). \[I_{PVG}(x,y)=\left( \begin{array}{cccc}\delta_{seg}^{l}(M,t_{1})&\delta_{seg }^{l}(M,t_{2})&\cdots&\delta_{seg}^{l}(M,t_{N})\\ \delta_{seg}^{l}(M-1,t_{1})&\delta_{seg}^{l}(M-1,t_{2})&\cdots&\delta_{seg}^{l }(M-1,t_{N})\\ \vdots&\vdots&\ddots&\vdots\\ \delta_{seg}^{l_{J}}(1,t_{1})&\delta_{seg}^{l}(1,t_{2})&\cdots&\delta_{seg}^{ l}(1,t_{N})\\ \delta_{seg}^{l_{J}}(0,t_{1})&\delta_{seg}^{l_{J}}(0,t_{2})&\cdots&\delta_{seg}^{ l_{J}}(0,t_{N})\\ \delta_{seg}^{r}(1,t_{1})&\delta_{seg}^{r}(1,t_{2})&\cdots&\delta_{seg}^{r}(1,t_{N})\\ \vdots&\vdots&\ddots&\vdots\\ \delta_{seg}^{r}(M,t_{1})&\delta_{seg}^{r}(M,t_{2})&\cdots&\delta_{seg}^{r}(M,t _{N})\end{array} \right) \tag{13}\]  
In order to visualize \(I_{PVG}(x,y)\), each element is represented by color tonalities: red represents positive deflections and blue represents negative deflections. PVG allows to distinguish left- and right-fold movements, and is thus more sensitive to the accuracy of glottal main-axis [90]. The complete procedure to compute PVG is illustrated in Figure 13 where five glottal cycles are depicted. The GVG playback was proposed in order to solve the difficulties to interpret the PVG and its strong dependence on the detection of the glottal main axis [90; 120]. The GVG synthesize the HSV in one single image and its computation uses a similar approach than the PVG formulation. However, unlike it, GVG computes the distance between the vocal folds edges themselves. Firstly, the vocal folds edges \(C^{l_{J}}(t)\) are equidistantly sampled with \(pc\in[0,M]\).