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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
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Figure 3. 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. If an operator would select the proper case,
it is possible to calculate the estimate parameters for the
expected Symmetric Relative Orientation.
5. 3-IMAGE PROCEDURE
In our approach, we choose to use three images to operate the
global procedure, in order to eliminate the human decision.
Actually to start the Absolute Orientation, we have to select
manually one among the founded four configurations.
Introducing the third image, we want to bypass the human
decision, turning it automatically.
The step of the Model Construction furnishes four admissible
solutions, as above said, and produce four distinct models
(called A,B,C,D). In case of three partially overlapped images,
this step can be repeated two times. Indeed the model 1-2 can
be formed by the images 1 and 2, and the model 1-3 can be
formed by the images 1 and 3.
5.1 Bridging the models
A 3D S-transformation allows to make the bridging of these
models, taking into accounts all the four models obtained
according to the admissible solutions founded in the Relative
Orientation. The procedure leads to sixteen different small
blocks, as it is shown below:
2A AB 2C 2D
1A | 1A-2A | 1A-2B | 1A-2C | 1A-2D
1B 1B-2A | 1B-2B | 1B-2C | 1B-2D
1C | 1C-2A | 1C-2B | 1C-2C | 1C-2D
ID | 1D-2A | 1D-2B | 1D-2C | 1D-2D
Figure 4. Models Bridging
The majority of these blocks are completely unlikely; indeed the
sigma naught of the 3D S-transformation adjustment is
enormous. This fact is reasonable because if and only if both
models (1-2 and 1-3) are congruent between themselves, the
bridging can be carried out successfully.
The set of congruent and incongruent combinations supplies
only two small blocks whose sigma naught is satisfying. The
two small blocks are originated from two different admissible
solutions in each four couple; (this means that) putting in a
square table all the sixteen solid structures, the two congruent
ones belong always to different rows and columns.
The analysis of the geometry of four admissible solutions
recognizes the high regularity of the presented values. As a
consequence, the two congruent small blocks present 3D
coordinates in two mirror reference frames.
6. OBJECT RECONSTRUCTION
6.1 Absolute Orientation Parameters
Starting from a roto-traslation in the space, a rational alternative
to classical Rotation Matrix is the Rodriguez Matrix.
R -(I-S) (I^ S) (9)
where I is the identity matrix of 3x3 dimensions, and S is an
emisymmetric matrix defined as follows:
0 e —b
S--c O0 a (10)
b -a 0
This emisymmetric matrix S permits to find the exact solution of
the absolute orientation problem, thanks to the solution of a
linear system, after a suitable substitution of variables.
After simple substitutions, we obtain a linear solution, showing
the direct proportion between the model coordinates
x = x(u°,v°,w°) and the object ones y z y( X,Y,Z):
yzRysü-Sy(-swx-(ü-Sywzü-Sk) (D
Reorganizing matrices and vectors, in a way which collects in a
unique vector the three unknown parameters, coming from the
above mentioned emisymmetric matrix, we obtain the following
final equation:
0 (e, eut.) —(Ÿ, -v°,)|a, X, -u^, (12)
ZZ, =) 0 et. -u?,) b. $y =v? =o
(T. y) X, zi) 0 l6; 2 -W?,
6.20 Exact Solution of the Absolute Orientation
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.
6.3 Absolute Orientation with 3-image procedure
If we have to manage two different small blocks in two mirror
reference frames, the qualitative comparison with the object
coordinates select automatically the congruent configuration. As
well known, a 3D S-transformation permits to compare model
and object coordinates, transforming the first coordinates in the
second ones. The 3D S-transformation can be done in a linear
way, in fact the whole procedure terminates with a unique
solution, which traces back all the path followed, enhancing the
