## 3 Glottal Gap Segmentation and Facilitative Playbacks
### Facilitative Playbacks Representation
#### 3.3.2 Global-Dynamics Facilitative Playbacks
Then, the deflections among the vocal folds edges are computed by Equation (14), where \(\delta_{GVG}(pc,t_{k})\) represents the distance between the left \(\mathbf{c}_{pc}^{l}(tk)\) and right \(\mathbf{c}_{pc}^{r}(t_{k})\) fold at position \(pc\) and time \(t_{k}\):  
\[\delta_{GVG}(pc,t_{k})=\|\mathbf{c}_{pc}^{l}(t_{k})-\mathbf{c}_{pc}^{r}(t_{k}) \|_{2}\ ;\quad\forall k\text{ and }\forall pc \tag{14}\]  
Lastly, all the distances are stored in a matrix \(I_{GVG}\) (Equation (15)) and normalized within the interval \([0,1]\), with \(0\) corresponding to zero distance and \(1\) corresponding to maximal distance. For visualization purposes the matrix is coded with a grayscale map. \[I_{GVG}(x,y)=\left( \begin{array}{cccc}\delta_{GVG}(0,t_{1})&\delta_{GVG}(0,t _{2})&\cdots&\delta_{GVG}(0,t_{N})\\ \delta_{GVG}(1,t_{1})&\delta_{GVG}(1,t_{2})&\cdots&\delta_{GVG}(1,t_{N})\\ \vdots&\vdots&\ddots&\vdots\\ \delta_{GVG}(M,t_{1})&\delta_{GVG}(M,t_{2})&\cdots&\delta_{GVG}(M,t_{N}) \end{array} \right) \tag{15}\]The GVG depicts a well-shaped form of the vocal folds vibration even when detection errors occur during the segmentation or glottal main axis detection, providing a more intuitive representation of vocal folds dynamics (see Figure 14). However, GVG is not suitable to analyze symmetry between the vocal folds. There exist other global-dynamics FP that depend on vocal folds contour segmentation. The authors in [118] concatenate the vocal folds deflections \(\delta_{seg}^{l,r}(pc,t_{k})\) in order to obtain a 3D representation of the entire vocal folds dynamics. The representation is simple to implement but difficult to interpret. In [80] a playback named Vibration Profiles (VP) is introduced, where not only the edges but also the whole vocal folds were segmented. They computed the deflections between the medial axis in each vocal fold and the edges of each fold. The representation obtained keep relation with PVG playback, but attempts to preserve information of MW propagation. Empirical Orthogonal Eigenfunctions Analysis (EOF) is another playback that is based on the glottal edges computation. The authors extract independent spatio-temporal structures from the vocal-fold displacements analysis using empirical orthogonal functions. The technique appears to be an appropriate tool to extract principal glottal vibration modes from high-speed recordings. In [121], a Singular Value Descomposition (SVD) analysis of vocal folds edges was implemented to capture the spatial and temporal behavior of the vibrations over time and space respectively. The first spatial eigenfold captures the average shape of the vocal folds, the second spatial eigenfold captures the closing pattern of the folds, and the third spatial eigenfold captures the motion in the longitudinal direction of the folds. On the other hand, the first temporal eigenfold captures the amplitude differences between the left and the right folds, and the second temporal eigenfold captures the sample difference in phase  
Figure 13: Schematic representation of the PVG playback. (1) Segmentation; (2) Resampling of the extracted vocal folds edges; (3) Computation of vocal folds deflections \(\delta_{seg}^{l,r}(pc,t_{k})\); (4) Splitting of the glottal axis; (5) Virtual turning of the left fold; (6) Color coding of the vocal fold deflections; (7) \(I_{PVG}(x,y)\) representing a HSV with five glottal cycles. Adapted from [48]. between the left and right folds. The separation of spatial and temporal eigenfolds offers a compact visualization but does not allow localization of vibratory features in space and time. There are also representations that use information from other FP to compute new ones. For instance, the authors in [122] computed the Hilbert transformation of GAW to create an analytic phase plot that depicts the effects of jitter and shimmer as well as harmonic distortion in terms of perturbation and periodicity. Nevertheless, analyzing GAW signals does not differentiate between left and right vocal fold vibration and consequently, asymmetries remain unconsidered.