parameter 
decreased 
e when 2 
yarameter 
ition and 
s is not a 
ght) 
images 
image). 
on 
ht 
images 
tion). 
model in 
lelled. In 
opted to 
nstead of 
ynomials 
accuracy 
istics of 
rt arcs of 
of orbits 
iS not as 
pairs, the 
used but 
nce. The 
han the 
oriented 
riding 14 
observation equations, for the 12 parameter solution. The 
number of control points may be decreased if a poorer 
orientation is accepted. An along track pair of images, 
taken in the same orbit, can theoretically be oriented with 
only 4 ground control points per image, for the 13 
parameter solution, giving a solution with some 
redundancy. A reduction in the number of ground control 
used and of the orientation parameters settings, results in 
a non-convergence of the algorithm or unacceptable 
results. 
2.3 Conjugate points 
Conjugate points are control points identified in more 
than one image, whose ground coordinates are assumed 
unknown. Each ray pair gives one coplanarity equation. 
The ground coordinates (Xa, YA, Za) of a conjugate 
point A are unknown and the method consists in 
orienting a pair of images so that a conjugate point forms 
two ray pairs coming from each image that will intersect 
in space. These have thus to lay in the same plane, which 
is described by the so-called co-planarity equation. 
In figure 3, Aj and Aj denote de vectors originating at 
exposures centers of each image, i and j, passes through 
the images at points aj and aj and ends at point A. The 
vector B extends from one projection center to the other. 
For the two rays vectors A; and Aj to intersect in space 
the following condition must apply: 
(A XA,).B «0 [1] 
Still from figure 3, the vectors Aj and Aj can be 
expressed as: 
A, - (X, - X)i « (Y, -Y2).j * (Z, - Zi). 
A, - (X, - X/)i (Y, -Y]).;j « (Z, - ZD). 
Let 
Uu, = rad tt V PE 
y Xp y 
then [2] can be written in the form 
Ai 7 À,.(uj.l vj. j  w;.K) 
A;= A, (u,i+v,.j+ w,.k) 
and 
BzQX -xymWy 258: 5 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
The coplanarity condition of equation [1] in then satisfied 
if the following determinant is nil, as in [6]. 
The lack of coplanarity results in a residual Fj, which can 
be expressed as the observation equation [7]. 
UT) 1 510 
U; V; w, |=0 [6] 
(X7 — X. (vw; vw) 
(Y/ -Yj.(w..u; -w;.u) * [7] 
7 zy Suv zr 
À 7 0j 0$ Ys Zi 
9j 05 X 4) 
f f 
  
e 
  
(XV,yj) a, 
(xj.y 
> | 
  
  
Fig 3- Coplanarity vectors. 
This information is expected to improve the relative 
orientation of the model. It does not influence the 
absolute orientation of the model, and it can be used in 
two different ways. 
First, it can be used in conjunction with ground control. 
The ground control can be reduced and the number of 
observation equations required for the model orientation 
may be completed using conjugate points. In this case, 
only the relative orientation of the model is expected to 
be improved. 
Second, conjugate points can be used alone and a relative 
orientation of the model should be possible. However, 
the program is rather different from classical methods of 
orienting aerial photography, and this was impossible to 
achieve with the algorithm described.