Bas, Hüseyin Gazi 
A MATHEMATICAL MODEL FOR CORRECTING THE PHOTOGRAPHIC COORDINATES 
DUE TO ORIENTATION ERRORS (AN APPLICATION IN CLOSE RANGE PHOTOGRAMMETRY) 
Hüseyin Gazi BAS 
University of Sakarya 
Engineering Faculty 
Department of Civil Engineering 
Sakarya- TÜRKIYE 
Working Group V 
KEY WORDS: Photographic Coordinates, Orientation Errors, Close Range Photogrammetry 
ABSTRACT 
Photographic coordinates x and y are fundamental data in analytical photogrammetry. At the 
same time these coordinates are functions of interior and exterior orientation elements, and 
contain some errors. Therefore some corrections and reductions are applied to the image 
coordinates measured in comparators before they are used in mathematical models. Some 
corrections and reductions related to the physical error sources inherent in photogrammetric 
system are applied to the comparator coordinates, but any correction related to the exterior 
orientation elements is not applied to these coordinates. In this study, a mathematical model has 
been proposed to correct the photographic coordinates due to the small errors of exterior 
orientation elements of camera. The data obtained a close-range test field have been used in the 
offered mathematical model and the results have been criticized. 
1 INTRODUCTION 
There are two different techniques in restituting a photograph or photographic pair in 
photogrammetry. The first one is analog method which produces plain, profils or contour maps. 
The standart outputs of analog restitution method are in grafical form. Th second one is analytical 
or numerical method which depends on the analytical reconstruction of the bundles of rays and of 
the stereomodels. In analytical method the main data used in mathematical models are image 
coordinates obtained from comparators or digital plotters. These coordinates which called as “raw 
data” at the beginning, are functions of interior and exterior orientation elements. Therefore, every 
errors in these orientation elements will affect the image coordinates, and these errors will 
indirectly reflect to the object space coordinates. In this case, it is apperent that some corrections 
related with small errors in interior and exterior orientation elements must be applied to the image 
coordinates which will be used in mathematical models. Image coordinates must be especially 
corrected due to the small orientation errors for some photogrammetric applications which all of 
orientation elements or some of them are a priori known and are assumed errorless (Bas, 1985; 
Veress and Sun, 1978; Brandenberger and Erez, 1972); or for photogrammetric applications in 
which photos are taken from the same exposure station and are reconstructed with the same 
interior and exterior orientation parameters (Dauphin and Torlegard, 1977; Altan 1993). 
The aim of this study is to offer a mathematical model and its application in close-range 
photogrammetry to correct the image coordinates due to small orientation elements after 
transforming the comparator coordinates to the photographic coordinate system. 
2 REFINEMENT OF MEASURED IMAGE COORDINATES 
2.1 Corrections For Systematic Errors 
  
38 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
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