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Title
Modern trends of education in photogrammetry & remote sensing

192
most
: 1 cal
I i zed
irger
963).
i ob-
¡1 low
ated
a
0
0
0
2 us 1
he
ts
n-
asic
d to
rows
and
only
be
the
five
tion
; Pr
oor-
o be
ints
the
jmns
Un
ions
r me
Fig. 5: Relative orientation of projective
coordinates uk r and uk 3 of the epipoles, the main
elements of the following relative orientation.
2.3 Relative orientation
The relative orientation is to be performed in a
model space Pm 3 , where the object may be recon
structed by means of the inhomogeneous relations
y = yoR + pRx R = yos + oSx 3 ,
which are analogous to the reconstruction in an
orthonormal 1 zed space of analytical photogrammetry
and wherein yoR and yos are the corresponding
centers of projection. Introducing four-dimensional
homogeneous basic matrices Tr and Ts, formally the
equivalent expressions
y = trTr*x r = TsTs*x 3 (2.3.1)
or else
irxr = ÏRy , Tsxs = Tsy (2.3.2)
arise. They include the Important relations
TRyoR=0,Tsyos=0, '(2.3.3)
because, evidently the transformation of yo must
yield the corresponding internal origin x=0. More
over, the projections
TRyx = tk r xk r , Tsy« = TK 3 XK S
(2.3.4)
of any preassigned point yx of the epipolar axis
produce the epipoles of Pr and Ps, which are known
already from correlation.
All these equations show clearly that relative
orientation corresponds here with the computational
determination of two projective matrices T{tik}.
Their setup starts from the selection of four of
the homologous points, which must be well-distri
buted in space 1n order to define a unit detra-
hedron ei M of Pm. Three of them are also points of
bundles
the unit triangles of Pr and Ps (fig.5). In this
way the spaces Pm 3 and Pr 2 or P3 2 ( the following
will be derived without r or 3 ) have corresponding
base vectors
t i bn i = bi
( see section 1.3 and fig. 5 ), but since the bi
are linearly dependent, only the reciprocal vectors
bn 1 exist and therefore only the (regular) matrix
product
1/to
1/ti
1/T2
1/T3
can be obtained. According to (2.3.2) and con
sidering (1.3.2), the relation u=M _, um between the
affine coordinates u of the image and the spatial
affine coordinates um t = (1,um 1 ,um 2 ,um 3 ) of the model
will be required. The inversion yields


1
TO T1-T0 T2-T0 T2-T0
1
u 1
=
0 Ti 0 0
UM 1
u 2
0 0 T2 0
UM 2
0
0 0 0 T3
UM 3
or, using T of (2.3.2) but adapted for the use of
affine coordinates,
TM
BmB* -
r
u = Tum .
The fifth point now 1s the preassigned point ukm on
the epipolar axis, because this 1s the only ray
where assumptions do not disturb the intersection
of the bundles. Insertion of uk and ukm into
(2.3.5) furnishes the values of the components of
T:
1 “UK 1 -UK 2
UK 1
UK 2
10 - ,
l 1 “
1 *-Z -
1-UKM 1 -UKM 2 -UKM 3
UKM 1
Uk m :