## 3 The Stereo System and Its Underwater Calibration
### Off-Line Stereo Catadioptric System Calibration  
To reproduce a 3D scene from stereoscopic capturing images taken in different angle of view, the estimation of each model of the camera is essential. This process, called calibration, offers the intrinsic and extrinsic parameters that describe the physical functioning of the camera. In fact, it allows finding relationships between the pixel-wise dimensions on the image and the actual metric measurements. It can be achieved by having information either about the scene geometry or on the movement of the camera or simply by using calibration patterns. In this paper, we use the third possibility: the system of cameras and a checkerboard pattern of known geometry were placed underwater and a set of stereo images were taken for different orientations of the test pattern. The process of calibration consists on the definition of 3D corners of the pattern test and the extraction of the corresponding 2D projections in the image. The 3D and 2D corresponding coordinates constitute the data that will be processed in the calibration algorithm to determine an accurate model of each camera and the extrinsic measurements related to the stereo system and the laser pointing devices.  
We use the generalized parametric modelisation described and detailed by [PERSON] et al. in [45]. They propose a generalized approximation that relates 3D coordinates of a world vector and its 2D image projection. It consists on a polynomial approximation function denoted \"\(f\)\" which is defined through a set of coefficients \(\{a_{0};a_{1};a_{2};a_{3};\ldots\}\) as given in expression (Equation (1)). The calibration process is done once-time in off-line mode and the resulting coefficients will be used as variable inputs for the 3D reconstruction algorithm. The 2D coordinates \((u,v)\) represent the pixel coordinates of a feature point extracted from an image with respect to the principal point \((u_{0},v_{0})\). This latter is considered, at first, as the image center to generate a first parameter estimation. Then, by minimizing the re-projection error using the iterative RANdom SAmple Consensus (RANSAC) algorithm [46], the exact coordinates of the principal point is determined. This permits to highlight the misalignment between the optical axis of the camera and the axis of the mirror which allow refining the estimation results (Figure 2).  
\[f(\rho)=a_{0}+a_{1}\rho+a_{2}\rho^{2}+a_{3}\rho^{3}+a_{4}\rho^{4},\ \rho=\sqrt{u^{2}+v^{2}} \tag{1}\]  
Additional metric parameters of the stereoscopic vision system are measured such as the Baseline (the distance between the centers of both cameras) and the distance \"Zlaser\" of the laser belt with respect to each camera center. For underwater application, we use a plexi waterproof system of thickness \"\(e\)\" and radius \"\(d\)\". The estimated calibration parameters consider that all the system (stereo cameras + lasers + waterproof device) is a compact system, which means that the polynomial function \"\(f\)\" takes into consideration light refraction through the plexi. In fact, it permits to reconstruct a world point outside the waterproof system from its 2D image projection.  
Figure 2: Reprojection error before and after principal point estimation.