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' Figure 3. 
OPTIMIZING POINT NETWORKS FOR CLOSE-RANGE PHOTOGRAMMETRY: 
FIRST TEST RESULTS 
E.Tung **, J. Friedrich", F. Karsli*, E. Ayhan" 
* KTU, Department of Geodesy and Photogrammetry, 61080 Trabzon, Turkey — (etunc, jurgen, fkarsli, eayhan)@ktu.edu.tr 
Commission V, WG V/ 5 
KEYWORDS: Optimization, point networks, close-range photogrammetry, law of error propagation 
ABSTRACT: 
The goal of this paper is to introduce a simple method for optimizing point networks for close-range photogrammetry. The classical 
3D formulas are used to derive least square error estimates for a priori unknown field point coordinates X, Y, Z according to the law 
of error propagation. These estimates are dependent on assumed errors for all input parameters, such as 3D camera position and 
orientation errors as well as uncertainties of a camera's inner orientation. So far, the derived 3D formulas have been tested in the 
following ways. Firstly, the analytically gained derivatives were compared with numerically computed ones to ensure their 
correctness and secondly, the 3D formulas were checked with the corresponding 2D formulas. Results are positive and it is planned 
to implement the derived formulas in an interactive graphical user interface. 
1. Introduction 
Close-range photogrammetry includes many different 
applications like those for the documentation and restoration of 
historical buildings. For such a project, a simple method has 
been sought to optimize point networks under unchangeable 
restrictions like the minimum and maximum camera distances 
to objects. Especially the required number of pictures and stereo 
models should be minimized as long as 3D coordinate errors 
still remain above a given limit. 
Among the many different approaches to this problem, it was 
assumed that the classical 3D photogrammetric formulas in 
combination with the law of error propagation (Koch 1988) lead 
to a simple and effective method to fulfill the given 
requirements described before. 
2. Mathematical Model 
The starting point for the mathematical model considered here 
are the classical 3D photogrammetric formulas, which can be 
written as follows when neglecting the rotation matrix (Kraus 
1993): 
x-X»  Y-X.7-Z 
bx. : b'y mi b'z 
X-X, Y-Y, Z-Z 
bx b"y - bz 
where 
  
L-for Ist camera 
(I 
2 - for 2nd camera 
  
b'(x. y, Z) = R' *(x', y',—c") ; 
b'(x,y.2) = R" *(x'', y'',—c'') ; 
with 
X, Y, Z = 3D coordinates of object points, 
Xo.Yo, Zo = 3D coordinates of a camera’s focal point, 
R', R"' = Rotation matrices from a camera system to a 
local coordinate system, 
c — Focal length of a camera, and 
x', y' - 2D image coordinates of object points. 
The solution of equations (1) for the object coordinates X, Y 
and Z gives: 
b'y 
b'x 
  
*u Yum 
  
b'y by 
bx b'x 
ny 
Ao Y, 
b'"x = < 
  
  
  
  
  
  
  
  
x eX A *X, 
b'y b''y 
Y= (3) 
b'x hx 
b'y b"y 
Pre M eX, 
Z- b'z hz : 
b'x GE 
bz b'z 
The partial derivatives of equations (3) for the given parameters 
  
  
  
  
is given by: 
OX, Ax TB; 0X; A,-B,’ 
X. c] ex 10 3) 
OY, Ax Bg 0Y, Ay,-By 
ey 1 ay 1 
0%, A -B "0X, A -B 
gx A, XxX -B, (4) 
0Y, A,-B,'0Y, A,-B, 
—519—