Using Homogeneous Coordinates to Solve The Problems of Determining The 
   
Orientation Parameters of Non-Metric Cameras and The Reconstruction of Space 
Models 
Mohammed El-Shafei Abdel-Latif! and Ahmed M. Elsonbaty? 
lFaculty of Engineering, Assiut-EGYPT 
?[nstitute of Geometry, TU Vienna, AUSTRIA 
Commission III, Working Group 2 
KEY WORDS: Engineering, Orientation, Reconstruction, Close-Range, Non-metric, Metric. 
ABSTRACT: 
In this paper a new mathematical model is developed for the determination of the orientation parameters of 
a non-metric camera using at least five control points, among which four points are coplanar, while the fifth 
point should lie outside that plane. 
In case two images of the same space model, taken at different locations by the same camera, are given, the 
space model can be directly reconstructed without calculating the parameters. 
The methods used here are direct and simple. They need no linearization of equations nor complicated tech- 
niques. 
1. INTRODUCTION 
Orientation parameters of a camera play an impor- 
tant role in photogrammetric measurements. The ac- 
curacy of these orientation parameters are very im- 
portant in estimating the accuracy of the photogram- 
metric measurements. Therefore the main objective 
of this research work is to determine the orientation 
parameters of a non-metric camera which allows the 
use of the non-metric camera in photogrammetric 
work and enable many engineers and scientists in nu- 
merous fields to make full use of the technical and 
economical advantages of photogrammetry. 
In this paper a direct method to determine the ori- 
entation parameters of a camera is developed. This 
method can be used to calibrate a non-metric camera. 
Here lens distortion and film deformation are neglect- 
ed because they have no mathematical relationship to 
the central projective geometry. 
For the determination of the orientation parameters, 
at least five control points in general position are 
needed. Here we consider the case when four of them 
lie in one plane; the fifth not. 
The work is divided into two main parts: 
1. In the first part the mathematical model is de- 
veloped using homogeneous coordinates in the 
1 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
plane of the four coplanar points. This leads to 
a simple direct solution to the problem. 
2. The second part deals with the reconstruction of 
a space model subject to the above conditions 
using two images without calculating the orien- 
tation parameters. 
2. THE MODEL 
As shown in fig.(1), the first model consists of known 
control points A;, A9, A3 and A4 lying in one plane 
called the object plane o. 'The fifth point As lies out- 
side a. 
For simplicity, the space rectangular coordinate sys- 
tem O; X,Y,Z is chosen such that the X- and Y — 
—axes lie in o. 
The control points have the known coordinates: 
Al X; Y, 271) i= 12 55 
with Z5 zz 0 and Z; = 0 for? z 1,2,3,4. 
Further, an image of the above model, taken upon the 
plane 7, is also given. The coordinates of the image 
points 
AUT yn 1.2. D