A quick description of the algorithm according to the pre-
ceding formulation is:
a. the matrices Ni N,, N; Ny, Ny, Ny, and uj, Uj, uy, are
formed for each photo as it is introduced into the
adjustment,
b. the matrices R;, R;, R; and r; T; are computed, and
c. the contributions (R,-R, R1 RT and r; -R;R 1 rj of
the i th photograph is added to the system of normal
equation.
Then the solution of the normals (in the case of minimum
constraints (15)) is given as
x=(R+HTH)_r+ETSz (17)
where S - (H ET - H ET)-! and E, E are arbitrary full rank
matrices with rank equal to that of the matrices H and H
respectively, so as to fulfil the relations À ET - 0 and
A ET-0.A typical choice of E, E is that of inner con-
straints (see Dermanis, 1991 for analytical representa-
tions).
The corrections X; as well as the values of the additional
parameters y; are computed for each photo separately
through the relations
X,- Ri (d-R|3) (18)
y2Nym,-NIx-NIE) . (19)
After the computation of the vectors x; and Yi for each
^ > "
photo, the vector v; of the estimates for the observation
errors can be evaluated as
V; b; - ÀÁ; x - A; X; Di yi (20)
and the quadratic form is,
^ T
Qi- Vj P; v; (21)
and for all photographs
Ms
P=, &i= Vi PiVi . (22)
ç
Il
-
A
The a-posteriori variance of unit weight is given by
^ m
vss UA (23)
e
=
where DF are the degrees of freedom, which can be
calculated as
m m m
oren-ee = Zn (nt Zt Zeilen (24)
i=1 il el
where n is the number of observations in all photos, r is
the sum of r; (total number of coordinates of control
points), r? (total number of e.o. parameters), rs (total
number of additional parameters) and k (number of con-
straints for the definition of the reference frame).
Consequently, the covariance matrices are computed ac-
cording to
^ ° A 9 A ° ° e e
C(x)= 62 À = 02 { (R+HTH)-HETS STE} (25)
fori=1,2 ..m
C(&,X) = Q;=—-02 { QR; - ETS STE}
CG)- 9 0,20 (Rr H7 RT Q RR; -ELSS E]
C$) = -62 Q5, 2-9 Q Ny, Ny!
C) - 02 Q5, - E RLRT QN, Ny Q6)
Ch) = € Q5, 2 (Ny! - NI NS, 0 N Ny]
and
Ch) = 0 Qs
where
Q Q; Qs,
Qu- P;- [à;x;n,]| 97 Q: 05, || 4 |. 0»
oT oT p?
05, 05, 05,
2.4 Inclusion of geometrical constraints into the
adjustment
Theoretically such geometrical constraints can be imposed
either on x or on X. Practically, however, constraints on
exterior orientation means that we know the values of
these parameters quite accurately, which only rarely (if
ever) is the case. On the other hand, geometrical con-
straints on ground coordinates of points (eg. parallelism,
perpendicularity, coplanarity, points on arc, etc.) is very
usual. Such constraints can then be used to improve our
solution (eg. Ethrog, 1984). Hence,
Gx=d (28)
Then the solution is given by
x©®=x-QGT(G Q 6") (Gx-d)
0929 [0-0c'(G0c cQ)
