The geometric constraints (12b) can be treated as stochastic variables with associated 
weights,instead as strict conditions. The results of the stochastic case with very large weights are 
similar to the results of the conditions. The latter have the disadvantage of being less flexible and 
they require more storage and CPU time, decomposition requires partial pivoting and the 
dimensions of the matrices and vectors of the normal equations increase by 2n, where n the 
number of images. Treating equations (12b) as observation equations results in 
-e,=Bx+t ; Pi (12c) 
with ee -N(0,0%c1) ; c>0 
The least squares solution for the system (12a), (12c) is given by 
» (APA B'Bjc) ^! (A pi Bl'tc) (13) 
x» 
The structure of the matrices B and N = (ATPA « BIB / c ) of equations (12c) and (13) is shown in 
Figure 1. 
AZ | ASHIFTS 4 SHAP. PARAM. AZ 4 SHIFTS 4 SHAP. PARAM. 
| f TL 
. . es 
I = identity matrix 
0 = zero elements 
A - non-zero elements 
a = elements modified by the geometric 
( stochastic ) constraints 
  
   
   
  
N 
  
  
  
  
  
  
  
Figure 1. a) Structure of matrix B, b) Normal matrix N for Pi - diagonal 
The residuals and the variance factor are obtained as 
v -Ax-I (14a) 
v=Bx+t (14b) 
T. T 
do im D v dip .  r.... redundancy (15) 
r 
By using a Z-approximation in the inverse form of the collinearity equations, approximate 
x,y-image coordinates are obtained. For all four shifts the approximate values are zero. For the 
two scales and two sheerings, values of one and zero respectively are assumed.