MS EE RE 
  
rte EEE ro 
which convert to 
1/9 = 02 ( h1’2 + h2’2 + h1"2 + h2"2 ), (2.3.1) 
if the a priori variances ox? - oy? = 02 of the 
measured coordinates are equivalent. Adjustment and 
error computation correspond therefore to the rules 
of customary adjustment of weighted observation 
equations. In this way, also more than eight points 
can easily be used for image correlation without 
adjustment of the calculation of Z where the con- 
dition det(Z)=0 must be obeyed (Haggren and Niini 
1990). Thus Z can only deliver approximate values 
of relative orientation. 
The results of the adjustment will be the solutions 
da'Tz(dé'dX?), da''z(dQ"de"dK") 
and the matrix of dispersion 
Qui Qı2 
Sa = 02Q = ©? ; (2.3.2) 
Qu27 Q22 
containing instead of the estimate s? the known a 
priori variance c? and the submatrices  Q11 be- 
longing to P' and Q22 belonging to P". Q2 in- 
dicates the stochastic correlation between the 
images, which influences the reconstruction of the 
model but not the transformations into the normal 
case. Hence the dispersion of the rotation P'--»Py' 
will be 
0 0 0 0 0 0 
Sa’= 02 = 02Qa’ = 0 Oeo Ook 
0 Qi 0 Gek OkkK 
and of the rotation P"-->Py" 
One Ope Ook 
Sa’ = 0?Q22 = 02Qa" = | Ope 000 Ook 
OQK OK OKK 
3. NORMAL CASE 
3.1 Transformation 
By means of the calculated elements of relative 
orientation, the transformation (1.1.4) will yield 
image coordinates xw of the normal case. Using now 
eight points xu for a correlation of the trans- 
formed images, the result must be, because of 
R’=R"=E, the easily predictable matrix 
0 
ZN-CN-EBEZ-E 0. — 
1 
ooo 
Oo =O 
m 
u 
ooo 
O + O 
0 
Ü - 
1 
as a global check of the whole procedure. The 
detailed test may be performed by the inverse 
transformation x = tR'xu from the normal case to 
the real situation or analogous to (1.1.5) 
1. XN J.xn 
and y s -C 
K. XN K. XN 
(3.1.1) 
  
  
x = -C 
These formulas will be needed also for the inevi- 
table transformation of pixels from the normal to 
the original images in connexion with the inter- 
polations of grey levels by resampling. 
The search for homologous points (pixels) is to be 
executed now in P’ along the epipolar line h’.x’= 0 
with 
704 
o 0.0 X 0 
h’= Cux” = | O0 0-1 yd zc is 
0 1:0 {{-c y" 
i.e. cy? - cy" z D (3.1.2) 
or  y'zy"'zy and in P" along the epipolar line 
h".x"z 0 with 
0 00 lix' 0 
h"z OTx' = 0 O0 1 y*I z-lc |, 
0 +40 |=c y? 
1l.o. "— cy" 4 cy' =z 0, (3.1:3) 
hence y'zy"zy too. This implies that, of course, 
all homologous points are situated at identical 
parallel epipolar lines in the very same plane 
(Haggren and Niini 1990). 
3.2 Propagation of errors concerning transformation 
The influence of small variations onto (1.1.4) is 
implicitely given by 
dtXN + TdXN = TdAXN + Rdx 
or in scalar notation after regrouping 
e1.dx - xudt 
e2.dX - yudt 
es.dx * cdr. 
T( dxN + yNdK + cdé ) 
T( dyn —- xndK - cd) 
T( xnd® - ynd2 ) 
dt can be eliminated by the third equation and, 
considering Tt=-e3.x/c from the third component of 
(1.1.4), the differential relation 
  
dxn = Bada + Bxdx, (3.2.1) 
with 
1 XN YN -C2+xn2  -ynC de 
Bada-—— dé 
c | c?*yu? —XN YN XNC dK 
and 
1 XNis+ci1 XNj3+Cj1 dx 
Bxdx= dy 
es.x| ynis+ci2 YNj3+Cj2 
results, where Ba contains the well-known co- 
efficients of small rotations and Bx indicates the 
influence of small coordinate shifts in the origi- 
nal image. If these differential movements are 
stochastic quantities, the uncertainty of xw re- 
sults from the expectation Sn=E{dxndxnT} (Pelzer 
et al. 1985) because of E(dadx')zO (da and dx are 
independent!) as 
E((Bada)(Bada)') + E{(Bxdx)Bx dx)T } = 
BaE(dada' )Ba? + BxE{dxdxT}BxT = 
BaSaBaT + BxO2EBxT = 
02 (BaQaBaT + BxBxT). (3.2.2) 
SN 
Assuming that the original images are very close to 
the normal case, i.e. R * E , the second part of 
Ig) 
M 
nn -— 
(3.2.2) converts ause of e3.X--c and 
cC 
1 Cau 0 
Bxdx = — 
C 0 c 
LD, 
to BxBxT=E. In this case, the uncertainties of the 
coordinate measurement add directly to the un-