NON-LINEAR PROBLEMS OF ANALYTICAL PHOTOGRAMMETRY
Alice Pozzoli?, Luigi Mussio"
* TU of Milan, DIIAR, P.zza Leonardo da Vinci 32 - 20133 Milan, Italy — alice.pozzoli@polimi.it
? TU of Milan, DIIAR, P.zza Leonardo da Vinci 32 - 20133 Milan, Italy — luigi.mussio@polimi.it
KEY WORDS: Analytical Photogrammetry, Relative Orientation, Absolute Orientation, Non-Linear Problems, Education
ABSTRACT:
Photogrammetry equations are easily derived from analytical geometry, but as it is well known to solve the linear problems,
preliminary values of the unknown parameters are strictly required. It must be pointed out that linear problems are obtained after a
linearization of the observation equations. This requirement is particularly complex when we operate in two steps for the model
formation, i.e. in the Relative Orientation procedure, using the coplanarity equation, and for the object reconstruction, i.e. in the
Absolute Orientation, using a 3D S-transformation. On the other hand, the acquired additional information derived from the two step
procedure allows to solve the one step procedure for the External Orientation, using the collinearity equations. As far as in the two
step procedure, it requires an exhaustive research in the space of all admissible parameters of the Relative Orientation (taking into
account the Symmetric case only), and then it needs a suitable substitution of parameters of the Absolute Orientation (we adopt the
coefficients of the Rodriguez rational matrix), so that the same problem changes from a non-linear problem to a linear one.
1. INTRODUCTION
The mathematical foundation of photogrammetry is directly
derived from the relations in the 3D space of the analytical
geometry. This means that collinearity, coplanarity and 3D S-
transformation, as it is well known, are geometric problems and
they flow easily in photogrammetry, solving the problems of
External Orientation, Relative Orientation and Absolute
Orientation.
These problems are non-linear problems and they are usually
solved, after a linearization of the above mentioned equations.
In this way, least squares adjustments can run automatically and
more refined treatments (e.g. robust procedures, reweighted
least squares, generalized least squares) can be performed, step
by step, always solving linear systems. It is obvious that all the
methods can start only if the preliminary values of the unknown
parameters are a priori known.
The photogrammetric praxis suggests to proceed to the model
formation, solving the problem of Relative Orientation, and
then go to the object reconstruction, solving the problem of
Absolute Orientation. Take note that the problem of External
Orientation, in one step, is traditionally solved successively,
using additional information acquired by the two step
procedure.
Anyway this two step procedure requires, as already said, the
knowledge of the preliminary values of its unknown
parameters. In some cases, this fact is true, because the project
for data acquisition define these parameters with a sufficient
accuracy, or auxiliary measurements are available at the time
of data acquisition (e.g. surveying measurements, or
ephemeredes). In other cases, these parameters remain
completely unknown (e.g. non conventional data acquisition,
like non professional images, images coming from unknown
and old sources, equipped vehicles, robots, and many other
applications in close range photogrammetry, for architectural
and archaeological surveying as well as for industrial
applications).
210
In this work we tried to solve some non-linear problems of
analytic photogrammetry. These problems are also widely
flowed into the digital photogrammetry. The model formation
needs an exhaustive research of the preliminary values (of
parameters) for the Relative Orientation, while the object
reconstruction can be achieved through a substitution of
parameters able to transform the problem of the Absolute
Orientation in a simple linear problem. In the model formation,
we used the coplanarity condition and, even if we could not find
the correct solution, we search for (only for the Relative
Symmetric Orientation) all possible preliminary values and,
among these values, we managed to select only 4 possible
configurations. In the object reconstruction, using the rational
Rodriguez matrix, we managed to solve a linear problem
finding an exact solution.
2. FIELDS OF APPLICATION
There are various innovative methodologies and procedures for
acquisition, processing and plotting of data. In fact, taking into
account several data base sources, we can work with optical
images, remote sensored or not, special images coming from
digital sensors, like laser scanning and SAR interpherometry,
geodetic and geospatial sequences, digital maps, and many
other qualitative and/or quantitative georeferenced data, time
depending or not. Notice that intermediate and final products
can be easily managed with GIS or LIS, very widely diffused
nowadays.
Geodetic and geospatial techniques provide to georeference
data, continuously acquired, giving information about
topographic slope and bending, speed and acceleration
considering a deformation field, and the strain-stress tensor.
These observations are high quality, precise, accurate and
reliable but they need a constant quality control. In
photogrammetry, we focus our attention on block and joint
adjustment, using least squares and robust procedures.
Optic
[KORN
broac
plani
multi
perm;
passii
under
and r
contr
objec
adjus
The
sterec
interp
in thi
with
planir
traditi
DTM
auton
appro
planir
The p
surfac
produ
or wi
of vec
retrie
perfec
archiv
whole
inforn
The n
data f
this tr
or in
secon:
have t
A rela
links |
image
Z), in
Cartes
chang:
previo
*
Thi
mir
Sol
G.
Arc
par
