A Solution for Space Resection in Closed Form

Conventionally, space resection has been solved by iterative means, although for many years photogrammetrists have attempted to find a solution in a closed form. Recent work with the projective transformation approach as utilized in the Direct Linear Transformation (DLT) formulation has led to a closed solution for a plane object. Tests of this solution are reported together with a critical review of other closed solutions to the space resection procedure. INTRODUCTION What is Space Resection? According to Moffitt & Mikhail [1980]: The term space resection is the name given to the process in which the spatial position and orientation of photograph is determined based on photogrammetric measurements of the images of ground control points appearing on the photograph. Thus, space resection in photogrammetry is an analogy to the space resection in surveying [Masry, 1979]. In essence, the space resection mak~s use of image coordinates and heavily weighted or fixed object space coordinates to determine the positional and rotational elements of a photograph, or of a camera. Following this basic definition, space resection of a single photo can be extended to include the interior orientation parameters, or it can be reduced to include positional elements only. Therefore, the space resection may have 3 parameters (Xc' Y c, Zc), 6 parameters (Xc' Y c, Zc, 00, <p, lC), or more. What is a Closed Solution? The collinearity equation model provides the most conventional solution. Six parameters could be solved for rigorously, when using this model. An extension could be made easily to include interior orientation and other parameters. However, this approach requires linearization, therefore, the convergence relies on the closeness of the initial approximation to the 'true' values. Church, based on the image pyramid model, developed the well-known Church method [American Society of Photo gramme try, 1980] some 50 years ago. This model is an equivalent model to the collinearity equation model with a reduced parameter set. Here, only the positional elements are included. However, this model is also non-linear. A lot of research effort has been devoted to methods that avoid the requirement for initial values. Such approaches are termed closed solution. Rampal [1979] formulated an approach based on the image pyramid model and utilized the distance relations, which provide a closed form. However, one condition is assumed: the U object plane" is near parallel to the image plane.