VxXy4 73k, dXy4 tbka i dYiacteraa dZga = Jin 
Vy =dg+1,2 dX Tb» dYyatera2 dZga - Je? 
(2.3) 
Where 
dpi 7 7 (ka ma; tfo *muyqia 
bea 7 7 (p mao fia *mi) qi 
Cp 7 7 (X mas fa miaYygka 
apa 2 7 7 (yi mai Hi *m qa 
bia» 7 7 (ea *ma2 tfo, *moyq a 
Crest. 7 7 (yj m3 fi *mayqia 
Jena Xu S fea Pina Aqu 
Jj 2 7 Yka * fea * Vka /dka 
In Equation (2.3), the a, b. c coefficients and J are 
evaluated at the approximations. In matrix form, 
Equation(2.3) is given as 
Vin = Aya dP, - Jua (2.4) 
Where 
Via (VXpa, Vyga)-. 
dPi-(dX,4, dY,.4, dZ. ) 
Ue. Qaa Ja 2). 
and 
de hi JP el €t Li) 
Aga | 
Gk + 1,2 bi. L7. Ch + 1 2/ 
With the statistical model given by Equation (2.1). 
we obtain the least squares estimation by using the 
Kalman filter(Kalman, RE. 1960) 
Piri =P FAP; 
7 P -Dy Air (Dur + Arr Dic Akt)” (Ain P-L) 
Dank = 
Dig = Dig, Aka (Dirt Arr Dix Aer)” Aca Dy. 
(2 
Un 
) 
Where Li (Xi. ya) . Dr; is the covariance matrix 
of observation xy; and yy; Dia is the covariance 
matrix of 3D object coordinates estimated by all k+1 
images; (Ac; Pi - Lim ) is the difference between the 
observations (image coordinates) of the (k+1)th 
image and the projected image coordinates from the 
3D coordinates Py to the (k+1)th image plane. 
By Equation (2.5), the updated 3D coordinates and 
their covariance matrix are calculated based on the 
new observation in the (k+1)th image and the 
previous 3D coordinates. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
3. THE OPTIMIZATION CRITERION FOR 
PRECISION AND RELIABILITY 
3.1 Precision Criterion 
According to Equation (2.5), it is obvious that 
À Du = Dix - Dirt = 
De A Du, t Au Dee) Ars Die 3.1) 
is a positive definite matrix only if the matrices Dy. 
and D,, are positive definite. This ensures that every 
observation Equation (2.3) will improve the 
precision of 3D coordinate estimates. The efficiency 
dependents mainly on both the Dy; and 3D object 
coordinate covariance Dy projected on (k+1)th 
image plane (Ai Dy A^ ). 
Generally, we consider the relative gain matrix 
Di. ^ Du D V 
D^ Aa (Dp Avy Dia Aa) 28a DR (3.2) 
and define 
TD A Du pa) = : 
Tri | - Di; (Dij * Aca Di Axa)! ] = maximum 
(3.3) 
as the precision criterion of optimization for (k+1)th 
image. 
The total precision criterion 1s 
Tr (Dirt ) = Minimum (3.4) 
Let u, be eigenvector of the matrix Di, with its 
eigenvalues given as A, (17 1, 2, 3) 
and suppose 
If we select A‘, ^ (ui,u;) in Equation (3.3), the 
precision gain becomes greatest. That means when 
the direction of the image observation is 
perpendicular to the 2D error ellipse plane defined 
by the eigenvectors u; and u, this particular image 
observation makes the greatest contribution to the 
enhancement of the accuracy of the 3D object 
coordinates. 
In the special case, when there are only two images 
intersecting the 3D object point, we have 
      
   
   
  
   
   
   
   
  
   
    
    
  
  
  
   
   
   
  
   
   
  
   
    
  
    
   
  
  
   
  
  
  
  
  
  
    
   
   
  
   
  
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