The simulated cameras are defined to be independent
of each other, i.e. the parameters for each camera can
be calculated independently.
3.1 Adjustment of the parameters
DLT Parameters: Parameters of the linear DLT, case
A equation (8) and (9), are calculated using adjustment
of indirect observations of the form [Mikhail 1976]:
v+BA =f (14)
where v is the vector of residuals of the measure-
ments, B is the matrix of parameter coefficients, A is
the vector of parameter estimates, and f is the vector of
observations.
In order to calculate the parameters of the iterative
DLT, case B, equations (10) and (11) are linearised by a
Taylor series using only the first order terms. The
linearized DLT equations look like:
LT
+ $= +...
aea i,
MO E M (15)
Ple TX dx + —"dy =
ôL11 H Ox x dy y
0 OF, OF,
F tie, +...
cn
Sm dL Sr d Sm dy —0.. (16
+ == += dx + =
nn d 5 y (16)
The parameters are calculated using general least squ-
are adjustment. Expressing the coefficients of the
measured image points by matrix A, and the para-
meter corrections by vector A results in:
Av + BA =f (17)
Matrix A and B are based on equations (15) and (16). A
have the structure:
Ref (18)
Tax
where sub-matrix J,, is the jacobian for image
measurement of control point i:
SE, GE.
öx dy
0e
à
öx Sy
(19)
38
The number of control points is k. Matrix B have the
structure:
Jp;
J
B=| 12 (20)
Um
where the sub-matrix J. is the jacobian of parameters
for control point i:
BF, BF, BF,
SL, So 7 84
Ji = (21)
BF, SF, BF,
dL; OL, - dL;
Collinearity equation parameters: The unknown
parameters are calculated by linearization of equations
(12) and (13) in the same way as for the DLT equations.
Case C uses nine and case D six unknown parameters,
see table 2. The resulting equations for nine para-
meters looks:
Renee Dw
+ —dQ + —d® + —dK +
> Ag 8% ôK
ôF ôF ôF ôF
— dX, * — dY, * — dZ, + —dx, +
dX. dY, oZ. x, P
Sr d si S ac eve ee z0 (22)
oy, = Ox dy 7
0 OF, OF, OF
— qo. — — qK
Fo + 35d 307 ak +
OF, SF, SF, oF,
— dy —Ó = d
ES av d taz i s
Se n, poii on e qf ovium
oy, tk EC dy y e»
The system of equations is solved using the general
case least square in equation (17) where A is identical
to the jacobian in equation (18) and (19) and B is
identical to equation (20) but J, is substituted by:
Fs 8E Fs ôFs
60 606 ^ 6 oc
Joi = d (24)
Fg OF, OF, OF,
80 86 " By, à
Soli
camera
cases A
equatic
The in
dent a:
The cc
matior
Qi
The t
adjust
The co
Qr
The c«
both t
À
The cc
follow
the ca
3.2
Objec
calcul
linear
the fo
ferent
obser
i e. L4
way a
DLT €