NETWORK OPTIMIZATION IN INDUSTRIAL PHOTOGRAMMETRY 
A, Chibunichev 
Photogrammetry Departmant of Moscow Institute of Geodesy, Aerial Surveying and 
Cartografy (MIIGAIK). Gorochovsky By-str., 4 ,103064, Moscow, Russia. 
Commission V 
Abstract: 
The desiner formulates an initial network geometry, the precision of the image coordinate measurements and 
precision of the object coordinates. A solution is obtained for the optimal geometry and the number of camera 
stations. An ideal matrix of normal equations (for required precision) is used as a target function in the 
optimization process. The optimization is carried aut 
by a least squares method wich minimizes the 
discrepancies between the real normalequations matrix of a given initial network, and the corresponding ideal 
matrix. The constrains for values of the exterior orientation elements for camera stations are introduced into 
the optimization process by means of substitution of variables. 
KEY WORDS: analitic, industrial, photogrammetry, design, network, optimization, criterion, matrix. 
INTRODUCTION 
Actualy, photogrammeiry is widely applyed, for 
investigation of various industrialobjects. Oneofthe 
main problems in this case is photogrammetry net: 
work design, which must provide required precision 
ofabject point coordinates and a minimum number of 
camera stations. The design is commonly solved by 
means of an interactive method through the process 
of netwokr simulation and bundle block adjustment 
(C.S.Fraser, 1982, 1984, 1987). 
This paper deals with the algorithm of the analytical 
solution of the optimization problem, which permits 
fo solve the network design automaticaly. The main 
idea is to provide the required precision of the object 
points coordinates by shifting the camera stations. A 
criterion (ideal) matrix is used as a target function. 
This matrix is formed the way an ideal covariance 
estimated matrix of the object point coordinates is. 
K.R.Koch (1982) was the first to uss a criterion 
matrix as a target function for geodetic network 
design. The similar approch for photogrammetric 
networks was used by S. Zinndorf (1989), A. 
Chibunichev (1990) and D.Pritsch, P. Grosilla (1990). 
Mathematícal formulation of the 
photogrammetric network optimization 
Optimization process is based on solving the fol- 
lowing target function: 
K; - Kj - 0, (1) 
where K; isa real covariance matrix of the estimated 
unknown (i) object point coordinates, Ki is the 
corresponding ideal (criterion) matrix. K;is obteind 
from phototriangulation using the results of network 
simulation. 
Ni-N = 0, (2) 
where N; and N; are correspondingly real and 
criterion matrices of normal equations. 
Eq. (2) can be solved with respect to photographs 
orientation elements which will correspond to the 
required precision of the object points. The diagonal 
elements of ihe criterion matrix of function (2) may 
be computed in the follo wing expressions: 
a 2 
E edi ceat ui safe 
PL) M A (5 
where m is standard error of unit weight, which 
caracterizes the image point coordinates precision; m, 
; Wy, mz is the required precision of the object point 
coordinates. Nondiagonal elements of matrix N;are 
egual to zero. In more detail this problem is des- 
cribed in A.Chibunichev (1990). 
To solve eq. (2) it is to be expressed in a Taylor 
series restricted to linear terms: 
Ba+Le=V (4) 
with 
, bin aks; © 
B-l ... ^ a = | 8%; |* L=W.--N. 
bg. bmn ! ; ; t 
a En 
j eft .....s); ie] ,..., ky; s is the number of photographs 
in a block; k is the number of object points; n is the 
number of unknown parameters À . m - 6k is the 
number of equations; b= ix are partial derivatives 
of eq.(2) with respect to unknown elements of 
photographs orientation OG Ys su 
By Hy Haz; 
N; = By; Hy, Hyg; 
Bc Ry R2: 
Since N; is a symmetrical matrix, any object point 
give 6 independent equations with 6q unknown. 
Here q is the number of photos on which the point i 
appears. 
Takign into account, that 
N = A" A, (5) 
the partial derivatives of N with respect to the 
elements ef evterinr orientation of nhotos ran he 
M PANO 
ox 735 3X IX (6) 
The analitical expressions of the coefficients of the 
matrix A are very simple for the collinearity equ- 
ations. Therefore analitical expressions of eq.(6) 
are simple as well. 
The optimization problem is solved by means of the 
least-squares method ( V'V - min ). 
The parameters of photographs to be determined may 
be restricted io certain intervals (because of the 
camera format, the inability to move the camera to a 
certain position and so on) can be expressed by the 
inequalities