## Materials and Methods
### Reconstruction Procedure  
#### 2.2.1 Overview  
In order to obtain the three-dimensional superior surface of the oscillating vocal folds from the two-dimensional high-speed footage, a series of steps has to be performed. The flowchart in Figure 7 displays the individual processing steps, subdivided in the calibration (green boxes), the reconstruction (blue boxes) and the input/output data (red ellipses).  
Figure 5: Projection of regular laser grid (\(31\times 31\) individual spots) at a working distance of \(50\,\mathrm{,}65\) and \(80\,\mathrm{mm}\) (from left to right) with a uniformity of \(10\%\) regarding the size and intensity of each point.  
Figure 6: Beam caustic comparison: average diameter of an individual laser point in the grid with respect to the position beneath the tip of the laserendoscope.  
Two different kinds of calibration targets as shown in Figure 8 with known metrical coordinates are used. In both cases, at least 15 pictures are taken under different angles that are acquired by tilting the target plane. The camera calibration target (white crosses on black background) is recorded without a laser projection and enables the calculation of the intrinsic parameters like focal length, chip resolution and possible skew. The fully automatic detection of the white crosses is threshold-based and does not require any user interaction. The laser calibration target displays a four-row checkerboard frame around empty white ground for the laser grid projection. Each point in the checkerboard pattern as well as the laser grid has to be detected and assigned to its position in the grid array. Based on the vanishing lines constraint, we are able to reconstruct the 3D coordinates of the projected laser points in each frame and therefore conclude the path of each laser beam in the array. By the use of the pinhole camera model [25], we can find the extrinsic parameters like rotation and translation between the camera and laser model. For each laser beam, the intrinsic and extrinsic parameters are combined to form the matrix transformation (homography) between the 3D coordinates and their 2D pixel image. A detailed description of this calibration process can be found in [20]. The calibration is only valid as long as the geometry (relative angle between camera and laser) and the optical system (zoom settings and focal adjustment) of the laserendoscope remain unchanged.  
The most time-consuming part of the 3D reconstruction is the determination of the 2D pixel position of each laser point in each frame of the high-speed footage. It turned out to be most effective to detect the initial laser grid in the first frame of the recording, assign the laser points to their absolute grid position and track the movement of each point in the following frames. By the concept of stereo triangulation, each 2D pixel point can be transformed into its 3D coordinates with the homography of the corresponding laser beam.  
Naturally, the reflections of the laser rays are only visible on the vocal folds, but not in the dark space in between, i.e., the glottis. The in-house software \"Glottis Analysis Tools\" [26] enables the semi-automatic segmentation of the empty and typically darker glottis area in between the vocal folds. The additional information of the two-dimensional glottis contour is incorporated in the 3D model by a projection on the interpolated, closed 3D surface. The vocal fold edge is then modelled by a G2-continuous Bezier curve towards the most medial point on the vocal fold, reflecting the natural tissue curvature. An in-depth description of the interpolation and edge modelling can be found in [22]. Various 3D parameters like mean and maximum amplitude and velocity in different phases of the oscillation cycle can be derived from the resulting 3D vocal fold surface model.  
Figure 7: Flowchart of the calibration and reconstruction process. White corners indicate new processing steps.