garding the
earch of the
m in all its
een recently
hich supplies
y repeating a
The Relative
s homologue
orresponding
ric Relative
vhich are the
cond image,
b, is already
factor À. In
to define 9i,
arameters of
| ©, is Missed
tation, as the
Ive previous
rameters. It is
tting. Let us
implies often
surement. If
of Relative
f all possible
sed group (in
tations in the
of Relative
searched, but
had to use a
ever with the
e Symmetric
ric ones, and
b. — cos Q9, cos k, 9, — arcsin b,
b, =COS Q9, sink, b, am
; k, = arctan —
b. = sing, i
RT (@,0:k, 0.0, )= R; (@,0,k, JR, (m, e)
(11)
Ri (mo. ) = RT (m.p. k. b.b, JR! (mo )
The convergence of linearization of trigonometric functions is
acceptable as far as values lower or near II/4. Therefore we
decided to explore all the admissible values for rotation angles
with a step of II/4, as shown below:
Q1; K, OF Pa Kk,
I1/2 = =
11/4 e e
0 e e e o e
[1/4 ® 6 e o e
11/2 = e e = e
3II/4 e e e
II e e e
511/4 e e e
31/2 e e e
711/4 e e e
Table 3. Exhaustive Research for Symmetric Relative
Orientation parameters
where & k,=0 if p,=+I1/2 and/or k,=0 if p,=+I1/2
As known, if the ¢ angle is around +Il/2, we can not
individuate the & rotation, which is fixed equal to zero. Indeed
in the polar zones (we assumed their range in a circle of one
degree), the two angles are identical or quasi identical, and this
fact produced singularity or ill-conditioning.
The exhaustive research explored 5x8x8x5x8-12800
possible configurations. For each case, a linear system was
solved, using the values of this configuration (case), as
preliminary values of the parameters of the Symmetric Relative
Orientation.
Examples were carried out in all the middle points of the
possible configuration. Considering the 5 parameters of the
Symmetric Relative Orientation, the angles «;, 6», K; are defined
in a complete rotation (8 configurations), whilst Qi, @, are
defined in a half rotation (5 configurations), which led to the
above mentioned 12800 cases.
Each linear system.solution gave us the estimate parameters for
the Symmetric Relative Orientation. The convergence to
admissible values is when o, is small enough. Considering only
the distinct solutions, we found four analytical acceptable
configurations.
Î
|
v
A
Figure 4. The 4 final possible configurations
These configurations are really different, so it is not so difficult
to have information about the initial position of the images, in
every specific case. Selecting the chosen case, it is possible to
calculate the estimate parameters for the expected Symmetric
Relative Orientation.
6. OBJECT RECONSTRUCTION
In our procedure for the Absolute Orientation, the object
reconstruction does not need preliminary parameters, because
we can reach the exact solution, by solving the linear system,
mentioned in an above paragraph. We tested this procedure,
considering 208 possible configurations. These cases come
from an object rotation following the global attitude angles (Q,
©, K), with a step of II/4. Exam was performed analyzing the
rotation in the space of a cube with 27 control points, regularly
distributed.
7. NUMERIC EXPERIMENTS
To verify precision, accuracy and reliability of these
techniques, a program in FORTRAN 95 language (compiled
and assembled with Lahey-Fujitsu FORTRAN 95 version 5.6)
was written, implemented and tested. It runs on a Pentium 3 PC,
with 933 MHz — 262 Mb / RAM — 30 GB / Hard Disk. The
exhaustive research for the Symmetric Relative Orientation
works in 4 - 5 seconds, while all others procedures are
immediate. In all the examples, we introduced random errors,
with standard deviation of 20 pm, as usual in photogrammetry.
