have the 
(20) 
parameters 
(21) 
unknown 
' equations 
equations. 
arameters, 
nine para- 
=0 (22) 
-0 (23) 
he general 
s identical 
and B is 
d by: 
(24) 
Solution of parameters: The parameters for each 
camera are calculated separately for each of the four 
cases A, B, C, and D. The parameters in the linear DLT 
equation are calculated by [Mikhail 1976]: 
A=N 1 (25) 
where 
N = B'WB (26) 
t - B'Wf (27) 
W = Q! = I (28) 
The image measurements are defined to be indepen- 
dent and with unit weight, thus Q = I in equation (28). 
The cofactor matrix of the residuals used in the esti- 
mation of internal reliability is : 
2 1 
Q,-Q-BN!B' -I-BN' B (29) 
The non-linear DLT parameters and the Bundle 
adjustment parameters are calculated by: 
N = É W.Jg (30) 
t = Je We f (31) 
W, = (Ja QM) = (Jah)! (32) 
The cofactor matrix of the residuals o is: 
Q^, - JAW, (I- Jg N! Jg Wo)JA (33) 
The cofactor matrix of the computed parameters for 
both the direct and the iterative methods is: 
= N" (34) 
The cofactor matrix o. is important as it is used in the 
following step of check point calculation together with 
the calculated parameters. 
3.2 Adjustment of object point coordinates 
Object points from the DLT: Object coordinates are 
calculated for case A and B in table 2 by using non- 
linear functions similar to equations (10) and (11), in 
the following denoted F'4 and F’4. The equation is dif- 
ferent in that all eleven parameters are treated as 
observations together with the comparator readings, 
i. e. L4,...,L4, have residuals added to them in the same 
way as the comparator readings x and y. The linearized 
DLT equation is the same as in equations (15) and (16), 
39 
with partial derivatives for the object coordinates X, Y, 
and Z added. The unknown object coordinates are 
calculated using the general least square adjustment in 
equation (17). The check points are calculated separa- 
tely but with data from all cameras. The A and B 
matrix has the same structure as in equation (18) and 
(20) but the jacobian J,, for camera i is: 
  
  
  
  
  
  
GF, OF; GF, OF; OF; 
8Ly 8Ly ^ OL 8x y, 
Jai = / , , , 7 (35) 
BP, BF, — SP, BF, OF, 
8Ly; Oly ~~ Ly Ox y, 
and the jacobian J. for camera i is: 
SF, BF, BP. 
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Object points from the collinearity equation: 
Analogous to DLT, equations (12) and (13) are used 
with the modifications that all parameters calculated 
in the previous step have residuals added to them in 
the same way as the measured image points. The 
modified functions are denoted F's and F';. Also the 
linearization is analogue, i.e. partial derivatives for 
the object coordinates X, Y, and Z are added to equa- 
tions (22) and (23). The unknown object coordinates 
are calculated in the same way as with DLT but the 
jacobian of the observation coefficients J ,. is: 
all, Ey Ne FT 
  
  
àQ 30 7 Sy, & X by 
Jai = 7 , , , , , (37) 
F, BF, o BF, BF OF, oF 
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The jacobian Jy, is the same as in equation (36) except 
that functions F's and F’, are used instead of F'4 and 
F',. 
Solution of cofactor matrix: The basis for estima- 
. a. . . c 
tion of precision is the cofactor matrix Q,, of the com- 
puted check points. In case A, B, C, and D the cofactor 
matrix is given by: 
c 
Q, - N! - (W,Js)! (38) 
We = (Ja Qua) (39) 
Matrix Qj is the a priori cofactor matrix of the observa- 
tions, i.e. the calculated parameters from the first step 
and the measured image coordinates of the check