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