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linear methods to solve for relative orientation' based on the
essential matrix (Longuet-Higgins, 1981) and on the
fundamental matrix (Faugeras et al, 1992), algorithms
integrating self camera calibration (Hartley, 1992), applications
to motion vision (Weng ef al., 1993) and estimation techniques
different for classical L.S. have been introduced. A review on
this subject can be found in Hartley and Zisserman (2000).
The impact of these solution to photogrammetry was twofold:
e solution to the problem of approximations, which are
found by applying linear method and then refined by
standard photogrammetric equations; ih this case
derivation of geometric from algebraic parameters is
needed (see Hattory & Myint, 1995; Pan, 1999).
e preliminary gross outlier rejection, integrating high break-
down robust techniques (Torr & Murray, 1997).
Several matching techniques were developed to fin homologous
points and features. While in photogrammetry orientation
equations are prevalently point-based, machine vision
techniques also exploit other kinds of constraints, such as lines,
surfaces and angles.
Nevertheless object reconstruction from a pair of images suffers
from low redundancy, being the control on the extracting of
homologous point by matching techniques limited to epipolar
constraints. To overcome this drawback, a formulation of the
relative orientation of a triplet of images has been established
through the so called frifocal tensor (also referred to as trilinear
tensor), introducing a higher redundancy (Spetsakis &
Aloimonos, 1990; Hartley, 1994)". A review can be found in
Ressl (2000). Application of trifocal tensor to solve for
approximations in standard orientation procedures (relative
orientation; AT) is very useful, because it allows to deal
effectively with large fraction of blunders, such those resulting
from automatic extraction of tie points by matching techniques.
However as in case of relative orientation, derivation of
geometric parameters must be computed.
1.3 Three-image orientation through exhaustive research
In this work, we tried to solve the non-linear problem of three-
image orientation based on the classical background of
photogrammetry. Solution such as those based on the trifocal
tensor could be very useful for stand alone problems such as
those of machine vision. When approximate values of geometric
orientation parameters of a standard photogrammetric block
have to be found, a solution giving this parameterization (or a
similar one, e.g. requiring only a 3D transform) would be better.
In Mussio & Pozzoli (2003a,b) a solution of relative orientation
problem based on exhaustive research of the preliminary values
of parameters has been proposed.
Exploring the 3D space with a step of II/4 is possible to find all
the preliminary values of the unknown orientation parameters.
This idea want to avoid the linearization of the orientation
functions supplying the lack of information about the position
and the attitude of an image.
In reality the paper of Thompson (1959), coming from
photogrammetry, already proposed a linear method for
relative orientation which was similar to that of Longuet-
; Higgins.
- Also in this case (see note 1), formulation of the dependency
among three images was already published — in
photogrammetry by Rinner & Burkhardt (1972).
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
2. FROM IMAGES TO OBJECT VIA MODEL
The main function of photogrammetry is the transformation of
data from the image space to the object space. We can make this
transformation in a direct way, with collinearity equations, or in
two steps, with the formation of a model and, only in a second
time, reconstructing the original object (Kraus, 1993). First of
all, we have to take into consideration that:
- animage is not a map;
- at least two images are needed for reconstructing an
object.
A relation of roto-traslation with scale variation constitutes the
link between the coordinates of the point Q (x,y,z) in an image,
and the coordinates of the corresponding point P (X,Y,Z) in the
object space. Both reference systems are traditionally Cartesian
reference systems, but the same is true, with minor changes,
using a different reference system, suitable linked to the
previous ones. Let us show the above mentioned relation:
xt X X9
M mAh; y — f (1)
2 £o
where x^, y^. c = image coordinates and principal distance
X90. Yo, Zg 7 coordinates of projection center
X, Y, Z 7 object coordinates
À = scale factor, variable point by point
2
Figure 1. Reference Photogrammetric Systems
3. PROJECTION TRANSFORMATION
The photogrammetric technique is based on a transformation of
a perspective (or a couple of perspectives) in a quoted
orthogonal projection. In this transformation, we have non-
linear equations and, before starting the plotting, we need
information about the preliminary value of unknown
parameters. Our main aim is to find expressions working with
parameters easy to be obtained. We choose a ‘wo steps
procedure to orient two images in the 3D space. This procedure
does not use the classical collinearity equations (/2 parameters:
X, Yy Zi X» Yo, Z; -coordinates of the two projection
centers, and 6, Qi, K |, 0», Q2, K» -attituded angles of the two
sensors), but separates the model formation (Relative
Orientation) from the object reconstruction (Absolute
Orientation). In this procedure, we define the problem of
