Point number Residuals Residuals
linear solution | improved solution
1 0.0009 -0.0001
2 0.0032 -0.0034
3 0.0027 -0.0011
4 0.0042 0.0009
5 0.0070 0.0018
6 0.0020 -0.0030
7 0.0051 0.0009
8 0.0079 0.0026
9 0.0028 -0.0005
10 0.0027 -0.0014
11 0.0072 0.0024
12 0.0023 0.0006
std error: 0.006 std error: 0.002
Table 1 The bias in the linear solution
32 Criteria for Outlier Classification
Once the parameters are estimated, erroneous
observations should be classified as outliers by some
criterium. For the LS estimates using all observations,
statistical methods based on standardised residuals, v/0,,
are well established. Other estimation methods, based on
different minimising functions, uses other test statistics or
criteria.
3.2.1 Data Snooping: The method of data snooping
uses the standardised residuals, v/o,;, for outlier
detection, where the
0,; = 00 4 0
is computed from the LS estimated covariance matrix of
the residuals,
O,, = Qu - A(A'PA) A"
The matrix A(A'PA J^ A' is the estimated covariance matrix
of the observations, called the A’matrix in
photogrammetry and geodesy and the hat-matrix in
statistics. When og is not known a priori but estimated
from the observations the following test statistics is used
[Forstner, 1985]
cap =Y; s —Vi Pi
WET TET
Goi On Go; Jn
The estimated oy is calculated as
(Ev pv)-vi p. ir
r-1
0
The test statistics Iwil is compared to a critical value,
which depends on the significance level of the test. The
experiments in this study are tested on a level of 99%.
3.2.2 Least Median Squares, LMedS: The method of
LMedS [Rousseeouw, 1987] minimises the squared sum
of the medians of the residuals, minmed(? ) . The
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
estimate is found by a repeated search algorithm using a
subset or minimum configuration of the observations. The
method has a high theoretical breakdown point but the
search algorithm is, in practice, only useful for low
number of unknowns as in the case of relative orientation.
The way the estimated oo is calculated is partly based on
empirical investigations. An observation is accepted if the
test statistics w; = r/ O9, < 2.5, where
3, 7 14826(14-5/ (n — p) med 1}
When the number of unknowns grow, the number of
possible combinations of observations grow dramatically.
For a given maximum fraction of outliers, it is however
possible to estimate the number of combinations required
to reach a given certainty level. In the case of linear
relative orientation with eight unknowns and 18
observations, there are 43758 combinations but at a
maximum fraction of 40%, the number of combinations
needed to get an error-free sample at a certainty level of
95% is only 177.
3.2.3 Minimum Description Length, MDL: The
basic idea in MDL states that if the observed data are
dependent or non-random, Le., is possible to model, then
the expected description length of the modelled data will
be less than the description length, DL, of the un-
modelled data itself. Enough but no redundant
information should be provided for decoding and
restoring the data.
When using the MDL criterion as an estimator with robust
properties, the parametric model is fixed. The different
models which are compared are instead the different
combinations of data belonging/not belonging to the
parametric model. The data is modelled to the parametric
model in such a way that the MDL is found.
When the parametric model is fixed and not compared
with other models, several parts of the DL are constant,
like e.g. the description length of the parameters. The
remaining parts which have to be computed are:
D Lotal =
ial +2 DL for the n, outliers
€
An wR + DL for the nn model points
€
DL(deviations) DL for the gaussian noise
where
Ne the number of outliers
Im the number of model points
Ne + ng the total number of observations
? Here Ib is the logarithmic function to basis 2, i.e., Ib x — "log x.
The resulting unit for measuring information is called bits.
R
€
Random co
relative orie
algorithms.
computed f
removed ur
combinatior
1500 T
1250 T—
1000 t
750 +
MDL in bits
500 +
250 +°°
0 ka
332
fig. 1 Illus
calc
For the cor
generated.
orientation:
translations
was added
random gr
process, in
encountere
Data Se
I
II
III
IV
Table 2
Each data
sets of r
configurati
were of tw
at a rando
errors with
The noise
oo of 10H]
small form
equivalent
The relati
calculated
