h algorithm using a
ie observations. The
down point but the
ıly useful for low
relative orientation.
d is partly based on
on is accepted if the
y med 7
ow, the number of
; grow dramatically.
tliers, it is however
mbinations required
the case of linear
nknowns and 18
binations but at a
er of combinations
a certainty level of
h, MDL: The
> observed data are
sible to model, then
> modelled data will
h, DL, of the un-
t no redundant
for decoding and
stimator with robust
fixed. The different
stead the different
t belonging to the
ed to the parametric
ound.
and not compared
e DL are constant,
1e parameters. The
iputed are:
outliers
model points
ussian noise
outliers
model points
er of observations
sis 2, Le., Ib x = log x.
ion is called bits.
1 1996
R the range of the data
(approx. the image area)
£ the resolution of the observations
Random combinations of points were selected and the
relative orientation computed with the eight and six points
algorithms. For each combination, residuals were
computed for all points. Points with high residuals were
removed until the DL had reached its minimum. The
combination giving the lowest DL was selected.
1500 T
ovd ETT
S 000 Tn ul gestu Totam auobe n NL Sigma
5 ns — — — Model
= 7504 Outliers
S Total MDL
$00 T —JÀ€— Minimum
250 +
0 2 A ru EM
1 2 3 4| 56 7 8 9 10 1] 12 13 14 15 16 17 18
Number of outliers
fig.1 Illustration of the different parts of the MDL
calculations
4. EXPERIMENT
For the comparison of the strategies, four data sets were
generated. Two sets having random translations and
orientations and two sets having typical aerial image
translations with 60% overlap, table 2. A Gaussian noise
was added to the initial simulated measurements and a
random gross error was added during the estimation
process, in one point at a time, until a false solution was
encountered, i.e., to the breakdown point.
Data Set | Type of Noise level | Type of
translation errors
I Random 0.5 %o Large
II Random 2.5 %o Large
II Aerial 0.05 %o Large
IV Aerial 0.05 %o Small
Table 2 Description of the four data sets
Each data set consisted of 100 point configurations, ie.,
sets of relative orientation points, and each point
configuration consisted of 18 point pairs. The gross errors
were of two kinds, large and small. The large errors were
at a random position within the image area and the small
errors within the neighbourhood of the observation.
The noise level of set I was approximately equivalent to a
co of 10pum and for set II of approximately 50um for a
small format camera. The noise level of set III and IV was
equivalent to a Gp of 5 um for a 23x23 cm aerial image.
The relative orientations of the point configurations were
calculated with the four different algorithms. The forth
algorithm, the 5-point iterative LS algorithm, were only
applied to set III and IV since the approximate
translations and rotations were assumed to be small and
known à priori.
5. RESULTS
The results from the calculations are presented in figures,
showing the number of successful relative orientations for
the different data sets. A relative orientation was
classified as successful if the correct number of outliers
was discovered and did not depend on how close the
estimated parameters were to a true value. Both errors of
type I, removing correct observations, and type II,
omitting to remove erroneous observations, are indicated
as failures in the histograms.
5.1 Strategy l: Removing bad observations
The linear LS algorithm was used together with data
snooping to remove erroneous observations (fig 2).
Linear LS solution, Data Snooping
EI
[p
EJ
DIV
Number of data sets
0 1 2 3 4 5 6 7 8
Number of introduced errors
fig. 2 Removing bad observations, strategy (i), LS linear
estimate, algoritm 1
As a comparison, an iterative 5 point LS algorithm was
applied to the data set of aerial configurations, set III and
IV (fig 3).The iterative LS solution could not be used on
set I and II since arbitrary orientations could not be
handled and approximate values were un-known.
Iterative LS solution, Data Snooping
Not applied
M Not applied
OI
DIV
Number of data sets
0 1 2 3 4 5 6 7 8
Number of introduced errors
fig. 3 Removing bad observations, strategy (i), LS
iterative estimate, algoritm 4
45
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
