Tutorial: Dealing with rotation matrices and translation vectors in image-based applications: A common reference system for cameras

This tutorial follows and completes a previous work concerning how to solve the factors of ambiguity intrinsic in the definition of rotation matrices, to describe the relative orientation of one or more cameras acquiring images from different points of view. The two works, jointly, aim to provide an error-avoiding methodology to describe the whole relative acquisition geometry (pose). This is a crucial task for those aerospace applications based on the use of imaging devices, like flying objects detection and tracking, automatic guidance, tridimensional reconstruction, images georegistration, and change detection. To describe the pose, not only the orientation but also the relative shift should be taken into account. This can be conveniently modeled through a translation vector. Using it together with a rotation matrix it is possible to achieve the goal, but it is also prone to ambiguity. In this contribution, the main factors that result in such ambiguity are addressed. Through a detailed analysis, it is shown how to solve them, in order to manage the interaction between translation vectors and rotation matrices properly. This avoids the errors that frequently occur in practical applications, whenever it is required to find the transformation that makes the reference system of a camera coincident with the reference system of another, or to switch from the expression of the coordinates of a point of the scene in the reference system of a camera to its expression in the reference system of another. One of the reasons why errors are likely to be made in describing the pose is the lack of a commonly adopted choice for the camera reference system. To obviate this lack, the manuscript also presents a recommended definition for it. For the sake of completeness, the definition includes a second reference system for identifying points on the camera sensor plane, in order to permit describing also the projective transformation operated by the camera during the image formation process.