AN ANALYTICAL TREATMENT OF THE PROBLEM OF TRIANGULATION BY STEREOPHOTOGRAMMETRY

Abstract. An analytical solution for the basic problem of triangulation by stereophotogrammetry is derived. The most general case is defined as the problem of determining simultaneously the orientations of the two cameras, whereby all eighteen unknown parameters of the orientation are considered with no limitations on the camera orientations. The process of triangulation is treated as the computation of spatial coordintes as functions of the elements of orientation and the corresponding plate measurements. The least squares solution derived is based on rigorous mathematical expressions which connect the plate measurements with the unknown parameters. In contrast to the conventional approach, the separation of the orientation problem into the two phases of relative and absolute orientation is avoided. This analytical solution can be based on a few basic theorems of solid analytical geometry. Because the observation method — monocular or stereoscopic — does not influence the formulas expressing the rigorous geometry, it is possible to make use of absolute control points which are not common to the area covered by the two photographs under consideration. Thus more favorable geometry is introduced into the problem of the double-point intersection in space. The least squares solution derived is suitable for any number and any combination of absolute, partially absolute and relative control points. In addition, any one of the elements of orientation — including the base line components — may be enforced in the solution. By applying the concept of pseudo-residuals and by introducing cross-weights, it is possible to treat the least squares solution like a problem involving independent indirect measurements. Furthermore, it is shown that the normal equation system can be formed step by step. This method has merit when electronic computes are used since the number of points carried in the solution has only a slight effect on the amount of memory space needed. The introduction of rotational auxiliaries which are essentially direction cosines and the combination of these with the plate coordinates as linear auxiliaries render the coefficients of the observational equations as partial differential quotients in terms attractive for an analytical treatment. The process of triangulation is treated as a part of the process of orientation as well as an independent computational procedure. A special chapter deals with the determination of the mean errors of the observations, of the elements of orientation, and of the triangulation results. Finally, the application of the proposed analytical method to the problem of control extension is discussed in principle. It is shown that the method used on the universal plotters for a strip triangulation procedure is an approximate solution only, because it is based on incomplete conditional equations. The rigorous geometry for the problem of extension is interpreted as the condition that rays originating from three consecutive camera stations have to intersect for at least one point located in the area common to the three photographs under consideration. The corresponding conditional equations are derived and the coefficients, of the corresponding observational equations are given. It is shown that it is now possible to include in the extension models which are formed by the combination of photographs taken at every other camera position. The thus extended base line provides for a favorable base-height ratio otherwise obtainable only by convergent photography. As stated before, the final normal equation system can be formed step by step. Attention is called to the fact that the matrix of the unknown parameters is filled in the neighbourhood of the diagonal only, thus making it possible to use an iterative subroutine. The method presented for a strip triangulation is useful in a least squares treatment of a block triangulation also.