A line-photogrammetric mathematical model for the reconstruction of polyhedral objects

Man-made objects often have a polyhedral shape. For polyhedral objects it is advantageous to use a line-photogrammetric approach, i.e. lines are observed in the images instead of points. A novel line-photogrammetric mathematical model is presented. This model is built from condition equations with image line observations and object parameters in the form of the coordinates of object points and the parameters of object planes. The use of plane parameters significantly simplifies the formulation of geometric constraints. Object line parameters are not included in the model. The duality of the point and plane representation in space is exploited and leads to linear equations for the computation of approximate values. Constraints on the parameters are used to eliminate the rank deficiency and to enforce geometric object constraints. The exterior orientation of the images is assumed to be approximately known. The rotation matrix is parametrised by a unit quaternion. The main advantages of the presented mathematical model are the use of image lines as observations and the way in which it facilitates the incorporation of all types of geometric object constraints. Furthermore, the model is free of singularities through a combination of over-parametrisation and constraints. The least squares adjustment allows a rigorous assessment of the precision of the computed parameters and allows for statistical testing to detect possible errors in the observations and the constraints. Examples demonstrate the advantages of the proposed mathematical model and show the effects of the introduction of geometric constraints.