CLOSED-FORM SPACE RESECTION USING PHOTO SCALE VARIATION 
by 
Riadh A. H. Munjy 
Mushtaq Hussain 
Calgis, 
1477 E. 
Fresno, 
Shaw Ave., 
Inc. 
Suite 110 
CA 93710 
USA 
Commission V, Working Group 2 
KEY WORDS: Algorithms, Close Range, 
Convergent, 
Orientation, Transformation. 
ABSTRACT 
Space resection is 
photograph are determined. However, 
the process by which the 
because the observation equations 
approximations of parameters are necessary at the beginning of the adjustment. 
spatial location and orientation of a 
are non-linear, 
A closed 
solution means that there is no need for any initial values or estimates of the spatial 
position and the orientation of the photograph. 
A general solution for closed space 
resection is proposed and described. The solution is based on the analysis of the scale 
variations of the image distances between the control points. The proposed approach has 
been tested in close range photogrammetry project with highly oblique and convergent 
photography. 
INTRODUCTION 
The problem of space resection involves the 
determination of the spatial position and 
the orientation: of. a. camera | exposure 
station. In photogrammetric practice, the 
solution of this problem is most commonly 
arrived at through a least squares 
adjustment based on the collinearity 
condition. Since this. is. a nonlinear 
model, approximations of the exterior 
orientation parameters are necessary in 
order to initiate an iterative adjustment 
solution. To obviate the need for such 
initial estimates for the parameters, 
especially in the case of convergent or 
oblique photography which is often used in 
the case of close range photogrammetric 
applications, various efforts have been 
made in the past by photogrammetrists and 
mathematicians to develop a closed-form 
space resection solution. In. addition, 
there has been considerable interest in 
such a solution in the field of robot 
vision since the early days of its 
application to the three-dimensional 
analysis (Sobel 1974). 
Closed-form space resection can be achieved 
by using the Direct Linear Transformation 
(DLT) as was proposed by Abdel Aziz and 
Karara (1971). But a minimum of six object 
control points are needed. Other closed- 
form solutions introduced special 
additional constraints such as the 
assumption that the object plane is nearly 
parallel to the image plane (Rampal, 1979). 
Fischler and Bolles (1981) and Zeng and 
Wang (1992) both proposed a somewhat 
similar closed-form space resection 
solution: that leads to a fourth degree 
polynomial in a variable that does not 
represent the spatial camera position nor 
its orientation. This leads to complexity 
of the solution resulting in multiple roots 
390 
for the which are 
imaginary. 
polynomial, two of 
Our approach for a closed-form resection 
solution is based on the well known fact 
that the scale in a perspective photograph 
is variable across the image plane. One 
explanation for this variation is the fact 
that during the imaging process in a frame 
camera, the third dimension (z image 
coordinate) in the image space is being 
forced to remain equal to the focal length 
of the camera at each image point. 
Conversely, a constant scale in the image 
space can be enforced, if the camera focal 
length is allowed to vary across the photo. 
If the scale between a set of observed 
image points, such as for some object space 
control points, is forced to remain 
unchanged, it will cause a move in the 
coordinates in the image space. This will 
result in a new set of image coordinates 
and a different focal length  (z image 
coordinate) at each control point. The new 
set of coordinates may be viewed as a 
representation of a scaled and rotated 
three dimensional model of the object 
control. points... Using. a. closed | three 
dimensional transformation between the 
ground coordinates and the newly formed 
three dimensional coordinates will result 
in. the camera Spatial position and 
orientation parameters. 
The collinearity equation which describes 
the geometrical relationship between the 
object point, camera spatial position and 
its exterior and interior orientation, and 
the image coordinates of the point, can be 
derived from a three dimensional conformal 
transformation. With this derivation the 
constant scale in the three dimensional 
transformation is eliminated and constant 
focal length is imposed to reflect the 
camera geometry. The resulting image is a 
three dimensional model with constant 7 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996 
  
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