## 3 Glottal Gap Segmentation and Facilitative Playbacks
### Facilitative Playbacks Representation
#### 3.3.2 Global-Dynamics Facilitative Playbacks  
The global-dynamics FP analyse the vocal folds behaviour along the whole glottal length. Most of them are focused on vocal folds edge motion by means of glottal segmentation algorithms. The most widespread are: Glottal Area Waveform (GAW), Phonovibrogram (PVG) and Glottoviprogram (GVG). GAW uses the glottal segmentation to compute a glottal gap area function along time from which several parameters can be estimated [8; 39]. Let us consider \(I_{seg}(\textbf{x},t)\) as a segmented HSV, having the same size of the original video \(I(\textbf{x},t)\). The segmented HSV is a set of binary images, where 1 is assigned to pixels belonging to the glottal gap area (foreground) and 0 is assigned to pixels belonging to the other laryngeal structures (background). Equation (8) computes \(I(\textbf{x},t_{k})\) in time \(t_{k}\):  
\[I_{seg}(\textbf{x},t_{k})= \begin{cases}1&\text{pixels}\in\text{ glottal gap}\\ 0&\text{background}\end{cases}  \tag{8}\]  
Therefore, the glottal gap area at \(t_{k}\) is computed via Equation (9) as follows:  
\[A_{o}(t_{k})=\sum_{i=1}^{n}\sum_{j=1}^{m}I_{seg}(\textbf{x}_{ij},t_{k})\ ;\quad k =\{1,2,\cdots,N\} \tag{9}\]  
then GAW can be expressed in vector notation by Equation (10), where each of its elements represents the area of the glottal gap at a particular instant of time. \[\textbf{GAW}(t)=[A_{o}(t_{1})\ A_{o}(t_{2})\ \cdots\ A_{o}(t_{k})\ \cdots\ A_{o}(t_{N})] \tag{10}\]  
The GAW playback measures the glottal area function throughout the glottal cycle, being possible to compute some features as: opening and close phase of the vocal folds oscillations, maximum and minimum glottal areas, Open Quotient (OQ), closed quotient and speed quotient, among others. Figure 12 illustrates a GAW normalized within the interval [0,1]. The peaks represent the open-states of the vibratory cycles, meanwhile the valleys represent the closed-states of the vibratory cycles. The maximum and minimum amplitudes of the whole vibratory cycles can be computed by finding the maximum and minimum glottal area respectively. The period of the GAW playback is equivalent to the duration of the glottal cycles, and also to the sum of the opening and closing phase duration. On the other hand, PVG and GVG FP are 2-D representations of the vocal folds vibratory pattern as a function of time, for which the movements of the vocal folds edges along the anterior-posterior axis are summarized into a time-varying line of the image. PVG, \(I_{p\epsilon g}(x,y)\), is a further development of spatiotemporal plots of vocal folds vibrations presented in [117] and of the glottal shape representation proposed by [118]. PVG is a 2-D diagram introduced in [119] where a set of segmented contours of the moving vocal folds are unambiguously transformed into a set of geometric objects that represents the entire HSV sequence. Let us consider that the video sequence \(I_{seg}(\mathbf{x},t)\) was computed in advance by any segmentation algorithm. Then, the set of frames with the maximal glottal gap area are identified and named as keyframes \(I_{key}(\mathbf{x},t)\) (Equation (11)). \[I_{key}(\mathbf{x},t)=\operatorname*{arg\,max}_{k=1\dots N}I(\mathbf{x},t_{k}) \tag{11}\]  
For each keyframe, \(I_{key}(\mathbf{x},t)\), a linear regression line is computed to identify the main orientation of the glottal gap area. The regression line intersects with \(\mathbf{C}^{l,r}(t_{k})\) at the points \(\mathbf{p}(t_{k})\) and \(\mathbf{a}(t_{k})\). Such points are used to split the vocal folds edges into the left \(\mathbf{C}^{l}(t_{k})\) and right folds \(\mathbf{C}^{r}(t_{k})\). Since the vocal folds contours were computed independently, it is necessary to derive a continuous representation of the vocal folds vibrations that links the posterior and anterior point of all images within the HSV sequence.