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The decomposition of the rotational matrix 
into the rotational elements can be 
performed by investigating corresponding 
trigono metric “function ‘values of that 
element as presented by Shih (1990). 
RESULTS 
To test this closed-form space resection 
algorithm, a program was written in C, and 
was implemented in a close range 
photogrammetry application software. Two 
image data sets were acquired using a 35 mm 
non-metric camera, with focal lengths of 
11.000 mm and 51.142 mm. Each set used a 
different control field that contained a 
number of points. Table 1 and 2 shows the 
data for these sets and the results of the 
resection solution obtained by using the 
proposed newly developed algorithm and that 
obtained from the iterative collinearity 
solution. Using the new algorithm only 
three control points were used. This 
resulted in two possible resection 
solutions. The collinearity solution was 
then obtained by using the algorithm 
solution as an initial estimate . The 
collinearity solution that result in the 
lowest image residuals is adopted as the 
final solution. 
CONCLUSIONS 
The proposed new mathematical model has 
been tested and implemented in a newly 
developed software for close range 
photogrammetry applications. Most of he 
users of this software are not formally 
trained photogrammetrists, and consequently 
a closed-form space resection solution is a 
functional software requirement. 
The minimum number of object control points 
required for the proposed solution are 
three. In general this will lead to two 
possible solutions. But in this new 
approach, the correct solution is achieved 
by using the proper focal length sign, 
eliminating the more tedious need for 
testing and searching for the correct 
Spatial position and orientation elements. 
Since the proposed approach is based on the 
Scale variations of the image distances 
between the control points, it can be 
modified to work with machine coordinates 
instead of photo coordinates. This will 
Provide an alternative approach to the use 
of DLT for processing non-metric imagery. 
393 
This approach also can be modified to 
process imagery taken with non-conventional 
with cameras, such as panoramic cameras and 
fish eye lenses. 
REFERENCES 
Abdel "Aziz, 'Y.T ' and" H.M. Kärara, 1971. 
Direct Linear Transformation into Object 
Space Coordinates in Close Range 
Photogrammetry, Proceedings, Sym. on Close 
Range Photogrammetry, pp 1-18. 
Dehn, E., 1960, Algebraic Equations, Dover 
New York. 
Fischler, M.A. ‘and ‘\R.C.: Bolles, 1881, 
Random Sample Consensus: A Paradigm for 
Fitting with Applications to Image Analysis 
and Automated Cartography, Communications 
of ACM, Vol. 24, No. 6, pp 381-395, 
Rampal, K.K., 1979. A Closed Solution for 
Space Resection, Photogrammetric 
Engineering & Remote Sensing, PP 1255-1261. 
Shih, T.Y., 1990, The Duality and Critical 
Condition in the Formulation and 
Decomposition of a Rotation Matrix, 
Photogrammetric Engineering and Remote 
Sensing, Vol. 56, No.8, pp. 1473-1179. 
Sobel, I., 1974, On Calibrating computer 
controlled cameras for perceiving 3-D 
scenes. Artificial Intelligence Journal, 
5:184-198. 
Zeng, 2. and X. Wang, 1992, A General 
Solution of Closed Form Space Resection, 
Photogrammetric Engineering and Remote 
Sensing, PP 327-338. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996