(24)
TiS
1trol
otal
con-
] ac-
Q5)
i id
=
p
(26)
- Qn
the
yosed
ts on
es of
ly (if
con-
lism,
very
2 Our
(28)
(29)
Next the vector x is replaced by x“) in the relations (18)
and (19), and analogously the matrix Q by Q© in the rela-
tions (26).
The insertion of the constraints can follow a sequential
form (Rossikopoulos and Fotiou, 1990) in which case the
insertion of the k th constraint
grx=d (30)
where gT is the k th row of the matrix G, is as follows: The
quantities € and q*(€) is computed
6-gTX&-D— d- 5 (g, X7) - d (31)
r
OO = gr QC Pg= 5) gr gs (0), (32)
where x(-? is the r element of the vector x(-, g., g, are
the elements r and s respectively of the vector g and (Q&-
D), is the element of the r th row and the s th column of
the matrix QV, This matrices x&-1 and Q& 1 are the
solution of the normal equations with the previous k-1
constraints. When k is the first constraint to be introduced
then the matrices are the x and Q.
Next the coordinate X(9 in the vector X(? is computed by
BD DO ke 33)
and each elements (Q®); of the matrix Q® by
. fe 1 = fi Ai.
(Q0, - (0-5. — 26 2 (gr (QED), Ys QE Dy) |
(34)
The range of all the above summations refer only to those
coordinates of the control points which are included in the
constraint.
2.5 About combined adjustment
In case we have surveying measurements (angles, distan-
ces) we can use them into a combined surveying-pho-
togrammetric adjustment. Generally the inclusion of surv-
eying measurements into the bundle adjustment is straight-
forward and one has only to consider the updating of the
matrices N and ü of eq. (9).
A word of caution is however important here concerning
the relationship between the weights of measurements of
photo coordinates and those of surveying measurements.
This is a typical variance component estimation problem
well documented in the literature (eg. Kubik, 1967,
Fórstner, 1979, Schaffrin, 1983, Rao and Kleffe, 1988,
Dermanis and Rossikopoulos, 1991, Dermanis and Fotiou,
1992),
3. STATISTICAL TESTING SCENARIOS
During the adjustment of the measurements a number of
statistical testing is applied. These tests are applied both
for the evaluation of the imposed geometrical constraints
and the additional parameters. It is well known that in the
first case non-compatible constraints can lead to
divergence of the solution, while in the second high
correlation between additional parameters or between
those and e.o. parameters can lead to ill-conditioned
systems of normals.
These tests of course are besides the usual blunder dete-
ction module (Dermanis, 1990) which is included in SNAP.
3.1 Test of constraints
Testing of the compatibility of the imposed geometrical
constraints can be done both globally and one-by-one as
follows:
Global testing of constraints. This test is based on the
equation
S od seen.
_(Gx 4) (GQG) (Gx d p (35)
F q o2 ^ qDF
where q is the number of constraints, and the quantities Q
and 02 are coming from the solution without constraints.
In case this test fails (meaning that at least one constraint
is incompatible) one should perform a test for each
imposed constraint sequentially. Besides eq. 35, alternative
formulas can be used (see eg. Dermanis, 1986).
Sequential testing of constraints. This test follows the
general data snooping strategy. That is the testing of the
k th constraint gT x = d is based on equation
A
e2 Qo
"Eq hw em
F
where the quantities € 2 gT x&-D— d, qX(€) 2 gT Q€-D g,
0? and DF have been computed from the solution with the
previous k-1 constraints.
In order for the two tests (eqs. 35, 36) to be equivalent the
respective significance levels a and ao should be chosen
appropriately, according to Baarda's reliability theory
(Baarda, 1967).
3.2 Test of additional parameters
Let y, is the group of additional parameters of the i th
photograph, that we are currently testing. These parame-
ters are non-significant if:
AT Ql A
Yi Q5 Yı a
"ue Sf (37)
where Q55; is the submatrix of Q5.
If a single additional parameter y; of i th photograph is to
be tested, then the statistic used is:
12
Ez ua dT sS Fer
0? qX(yj :
or
(38)
