# Technological Development and Application of Photo and Video Theodolites
## 3 Measurement and Processing Approaches
### IATS System and Camera Calibration
\[\tan(\alpha)=1/f. \tag{2}\]To precisely and accurately determine displacements, the camera's angular resolution must also be precisely and accurately determined, as the object displacements on the image given in pixels must be multiped by the value of angular resolution to be presented in angular units, i.e., in linear units. The object displacements can be up to several pixels; thus, small uncertainty in the value of angular resolution can lead to a large error in the computed angle [59]. Manufacturers support academic research projects covering different calibration methods of integrated cameras. For example, [PERSON] described a camera with an affine chip model and developed a combined approach for taking camera and instrument error calibration into account [11]. [PERSON], in contrast, extended the collinearity equations using additional camera parameters to describe the pixel-to-angle relationship in IATS [61]. [PERSON] expressed central projection as affine transformation without theodolite errors for the use of different focus lens positions with fewer parameters [13]. The abovementioned methods were compared by [PERSON] [14]. All methods are based on a set of virtual control points for data acquisition during the calibration process. In reality, this set consists of only one \"real\" point, measured multiple times by rotating the telescope with a camera [8]. When appropriate calibration is provided, the obtained images and video frames from IATS are accurately georeferenced at any time. They are particularly suitable for structural and geo-monitoring, i.e., for objects and Earth surface areas which are very hard to approach. It is also possible to use additional vision technology for leveling applications by reading and analyzing digital leveling staff code patterns [62]. The major advantage of IATS is the fact that the methods and approaches of photogrammetry can be used in combination with the precision distance and angle measurements of TS. Together with the station coordinates and orientation of the instrument, each captured image is directly georeferenced and can be used for direction measurements with no need for object control points or further photogrammetric orientation processes. A universal description of the relationship between a homogeneous 3D coordinate (Equation (3)) and its image projection (Equation (4)) was given by [PERSON] [55]. The intrinsic camera parameter \(K\) and the extrinsic parameters (the rotation matrix \(R\) and the translation vector \(T\)) are thereby combined into a single, compact \(3\times 4\) projection matrix \(\mathrm{P}\) (Equation (5)). \[\widetilde{\mathbf{X}}=[X,Y,Z,1]^{\mathrm{T}}. \tag{3}\]  
\[\widetilde{\mathbf{x}}=[x,y,1]^{\mathrm{T}}. \tag{4}\]  
\[\widetilde{\mathbf{x}}=\mathbf{P}\widetilde{\mathbf{X}}=\mathbf{K}[\mathbf{R} ]\widetilde{\mathbf{X}}. \tag{5}\]  
The extrinsic parameters \(R\) and \(T\) shown in Figure 10 are composed of three different Euclidean transformations [63]:  
1. Transformation from the world or object coordinate system into the theodolite coordinate system (\(R_{w}\), \(T_{w}\));
2. Transformation (pure rotation) into the telescope coordinate system (\(R_{t}\));
3. Transformation into the camera coordinate system (\(R_{c}\), \(T_{c}\)).