International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part BS. Istanbul 2004 
  
number of parameters in photogrammetric adjustment 
and strengthens it. 
    
Bh 
NX 
  
Figure 3: First and second steps of GPSSIT supported 
terrestrial photogrammetry 
3. MATHEMATICAL MODEL 
Bundle adjustment is the most appropriate adjustment 
method used in digital photogrammetery since it depends 
on analytical principals and is a numerical technique. 
Colinearity condition, which depends on a line aligning 
the points on object, perspective center of camera and on 
the image of the object, is the algorithm that forms the 
mathematical model of GPSSIT supported terrestrial 
photogrammetry. Respecting this condition, all the lines 
passing through a central point and touch to two parallel 
planes at the each ends of the line are divided in two parts 
by the camera perspective center and the rate of these two 
parts on the same line one to another will be the same and 
constant for the other lines as well. Point of 
photogrammetric view, the central point described here is 
the camera perspective center and the plane close to the 
point is called as image plane and it is a real plane. The 
important description here, which does not match the 
theory above, is that the other plane is not a plane in 
practice. It is often an actual 3D space. For a metrical 
evaluation, image and object mediums both must be 
referred to one apiece metric system. These two reference 
systems are called image and object reference systems in 
photogrammetry. In GPS supported terrestrial 
photogrammetry, image coordinate system is the same as 
image coordinate system in the current digital 
photogrammetry, in spite of that the object coordinate 
system is designated with respect to GPS coordinate 
system that uses WGS84 datum. The coordinates 
obtained in GPS coordinate system can be transformed 
into a desired local coordinate system if it is required. 
This requirement comes out only if there is an offset 
between the point determined by GPS antenna and the 
point that the camera perspective center is referred. In the 
other case, the mathematical model to be formed 
photogrammetrically can be described directly in GPS 
coordinate system without the need for such a 
transformation. In this case and even considering that the 
coordinate systems are not parallel one another, the 
equation, which expresses the relationship between image 
coordinate system and GPS coordinate system that the 
object is described in, can be given as below; 
m d — 
X aps. X) 7*9 X aps. j 
Y zm AR, y d Yo == Yops- ( 1 ) 
trs! E f Zps- j 
where, 
[Xars-i Yaps-i Zasil : Is GPS coordinates of r'th object 
point, : 
[Xarsj Yars-j Zaps] : is GPS coordinates of perspective 
center of j'th image, 
xj and y; : 1s image coordinates of i'th object 
point in j’th image, 
X, and y, : is image coordinates of image 
center, 
i : is scale factor of i'th point in the 
j th image, 
R; : is rotation matrix of j'th image 
with respect to GPS coordinate system and 
f : 1s focal length of the camera used. 
After moving the term of [Xaps-i Y Gps. Zi in 
equation 1 to the left side of the equation and then 
multiplying the both sides of the arranged equation by 
] 
—R , the new form of the equation can be written as 
pe ir 
below when it is divided the first and second rows of the 
equation by the third one. 
A 4 d 5 7 
zt n ri Xess -Xops;) * Tix; CYobs; 7 Yrs ) +r, +i (Zaps = Zins) > (2) 
  
   
  
Xi = 3 ; T = = t 0 
à Duos; = Xorsid * aaa Yorsi ^ Yursi) t Nai (Zaps z Zorsi? 
+ -f Ly; (Kips = Xorsi) * og Yon EY, sj) t Po (Zarss ~Zorsi) € ( 3 ) 
n hu (Xopsi -Xcrsj )* Fo (Yop 4 Yos; ) + hes -Zops;) T? 
, ; 2 T 
Cis Piper; are elements rotation matrix of R; 
between image coordinate system and WGS84 coordinate 
systems. Here it is worth to mention that the coordinates 
of camera perspective center (XGps-j, Yaps-j Zaps-) in the 
above equation are not any longer unknown and they 
enter to the equation above as constant values. Since they 
are determined by GPS antenna when GPSSIT supported 
terrestrial photogrammetry is used in the field. In this 
case, 7 unknown parameters in normal case reduce to 4, if 
the used camera is a camera whose image coordinates of 
its image center and its focal length (x,, y,, f) are known. 
If the used camera is not a metric camera and image 
coordinates of its image center and its focal length (Xo, Yo 
    
  
   
   
   
       
  
  
  
  
  
     
    
    
  
  
  
   
    
   
   
   
    
     
   
  
  
   
    
  
    
  
    
        
  
    
   
    
   
   
   
   
    
   
    
   
   
     
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