Michael Cramer 
2 PRINCIPLES OF GEOREFERENCING 
2.1 Indirect method 
Up to now the indirect image orientation is the favoured approach for the orientation of traditional image based frame 
sensors (e.g. photogrammetric cameras). Within this approach the exterior orientation of each image is treated as 
unknown and estimated in a bundle adjustment process. This is the only way to determine the sensor position and 
orientation if no additional orientation sensors are used during the flight and only rough estimations of the exterior 
orientation of the imaging sensor, e.g. from flight mission planning, are available. Using the indirect method of image 
orientation, the six unknown orientation parameters are estimated from a number of ground control points and their 
corresponding image coordinates. For the evaluation of multiple images, this orientation determination is solved by 
aerial triangulation, where adjacent images are connected by measuring homologous points. Enforcing intersection 
constraints between multiple images, a reduced number of ground control points is sufficient to estimate the parameters 
of exterior orientation. Assuming the standard model of central perspective the georeferencing is based on the 
photogrammetric collinearity equation (e.g. Kraus (1994)). This equation defines the mathematical model for the 
physical process of image formation. The model relates the image coordinates to the object coordinate system, where 
the camera geometry itself is given by the parameters of interior orientation (principal point, focal length of the imaging 
sensor), and the camera station with respect to the global object coordinate frame is described by the time dependent 
parameters of exterior orientation. 
Usually, the interior orientation is measured via laboratory calibration and assumed to be a known quantity for the 
bundle adjustment process. Nevertheless, to refine the model and to obtain highest object point accuracy, additional 
self-calibration parameters are introduced into the bundle adjustment. Ebner (1976) and Grün (1978) for example 
proposed additional orthogonal polynomials to overcome remaining systematic errors caused by effects like non- 
flatness in the focal plane, non-modeled lens distortions or anomalies in refraction. In the approach given by Brown 
(1971) physical meaningful error terms are estimated using appropriate coefficients to describe the changes in the 
geometric values of interior orientation and additional parameters like scale, shear, radial and decentering distortions. 
The expanded bundle adjustment with self-calibration is a very effective method to compensate systematic errors and is 
widely spread for highest accuracy applications. Therefore, using AT for image orientation not only the exterior 
orientations for each image are estimated as unknown parameters but also the coordinates of new object points and - if 
necessary - additional parameters for camera self-calibration are determined simultaneously in a closed solution. 
Since the exterior orientations are estimated as free unknown parameters and can be strongly correlated with some of 
the used self-calibration terms, the estimated values might be different from the actual true physical position and 
orientation during exposure. To derive the influence of varying parameters in AT on the exterior orientations and the 
object point coordinates, different parameter are introduced in the bundle adjustment and the corresponding empirical 
object point accuracy is calculated in comparison to given reference values. For these investigations a photogrammetric 
image block (wide-angle, image scale 1:13000) consisting of three long strips (7 images each) with 60% forward/side 
overlap is analyzed. Using nine ground control points located at the border and in the centre of the block the coordinates 
of 122 well distributed and signalized object points are calculated and compared to given reference coordinates to 
determine the empirical accuracy. Overall, eight different bundle adjustments are computed using two different sets of 
interior orientation parameters and four different approaches for self-calibration. Details about the used parameters for 
the bundle adjustments are given in Table 1. For the first four runs the interior orientation from lab calibration is used. 
In Versions 5 - 8 the correct values are manually falsified by errors of about óc — 20jum for the camera focal length and 
dx'y = 10um, dy'y = -5um for the principal point coordinates. Concerning the influence of self-calibration, two 
adjustments (Version 1, 5) are done without any additional parameters, first. For the following versions the three 
geometric parameters of interior orientation are added (Version 2, 6). Since only the correction of focal length is 
significantly estimated, the principal point coordinates are eliminated for the final adjustment run. After this the full 
parameter set proposed by Brown (1971) (Version 3, 7) and finally the 12 parameter polynomial approach defined by 
Ebner (1976) is applied in Version 4 and 8. Similar to Version 2 and 6, only the significant parameters are used in the 
final adjustment. As it can be seen from the results given in the table, the estimated o a posteriori values are consistent 
and in the range of 4.2 - 4.7um. Although the approaches for self-calibration are quite different, there is no major effect 
on the empirical accuracy in object space. The additional errors introduced on the interior orientation parameters for 
Versions 5 - 8 are of almost no influence on the obtained object point accuracy. The horizontal accuracy from check 
point analysis is in the range of 4 - 7cm for all runs. For the vertical accuracy some differences are visible. Using the 
subset of Brown's parameters for self-calibration the remaining systematic effects in height are modeled and the 
accuracy could be improved. Since the different variations used for the bundle adjustment are of minor effects on the 
object points, their influence is strongly correlated on the estimated exterior orientation parameters. To evaluate these 
variations, the different estimated orientation parameters from AT are compared, where the results of Version 3 provide 
the reference orientations and object coordinates. The statistics of the obtained differences (RMS values) are listed in 
Table 2. The influence on the estimated orientations is clearly visible. Depending on the used self-calibration terms, the 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 199