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 :