DLT is defined by [McGlone at al 1989]:
IX + L,Y + L:Z + L,
XL Ly] (6)
L5X + L;Y + L,Z + Lg
SNELL IU (7)
where L, ,...,L4 are the transform parameters.
The DLT transform is non-linear but by making the
assumption that corrections to the observed image
points are small and negligible, a linear equation
system results. This linear equation system is used to
determine the unknown parameters. A more
rigourous method is to treat the equations as non-
linear and compute the parameters iteratively. Point
coordinates in object space are calculated by treating
the equations as non-linear.
2. METHOD
The basic idea is to simulate an ideal test where con-
trol points, check points and cameras are simulated,
and to use a unit camera, i. e a camera with a unit
vnl à ; : 2
principle distance c and a unit reference variance oy.
Control points have been defined in two different con-
figurations with 6 and 13 points. Check points are
defined in an 11 by 11 by 11 grid. The coordinates of
both the control and check points are defined to be
without any error. Cameras have been defined in two
configurations with two and four cameras, see table 1.
Comparator readings of the image points have been
simulated using the collinearity equation without any
distortion component.
test number of | control
cameras points
1 2 6
2 4 6
3 2 13
4 4 13
Table 1: The two methods have been compared using four different
tests with two or four cameras and 6 or 13 control points
The comparison of DLT and Bundle adjustment has
been done using linear and iterative solutions of DLT-
parameters. The collinearity equations have been used
in two ways: interior and exterior orientation para-
meters unknown, and interior orientation parameters
known and exterior orientation parameters unknown.
In total, four cases have been tested and compared, see
table 2.
case | method solution of parameters and number of
parameters parameters
A |DLT linear L1,L11 (11)
B |DLT iterative L1^L11 (11)
C |Bundle | iterative Xo, Yo, Z9, À, ®, K,
Xp; Yp. € (9)
D |Bundle |iterative Xo, Yo, Zo, Q, ®, K (6)
Table 2: The four tested methods and their parameters.
21 Control points, check points and cameras
The control point grids have been configured accor-
ding to figure 1.
14 Anne Ru
e. n 7 | 2 '
er n f 1 A i
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ec--------- €-----€----«
X a) b)
Fig1 Control points used for the calculation of parameters.
Configuration a) has 6 points and b) has 13 points.
The checkpoints have been arranged into an equal
spaced grid of size 11 by 11 by 11, see figure 2. Side
length of the control point grids and the check point
grid have been selected to one.
Fig2 Check points arranged in an 11 by 11 by 11 grid.
The simulated cameras have been set to a height one
unit above the control or check point grids, pointing to
the centre of the grids, see figure 3.
oa) 10
Wd
uni 1.0
1.0
Fig3 Configuration of the simulated cameras. Test 1 and 3 uses
cameras a and b while test 2 and 4 uses all four cameras. The came-
ras are pointing to the middle of each grid configuration.
36
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