BI
is and outliers using
gy (iii), eight points
nts, MDL
BI
errors
s and outliers using
x points algorithm,
N
n parameters in the
ich the geometry of
dependencies and
meters in a very
d indeed also small
1s or outliers can
re not claimed to be
; and comments are
rs and making the
estimate have been
jety over the years,
on that they look at
type of LS estimate
g some criteria are
r given new weights
o more points are
ns and test statistics
rarup et al 1980] to
like data snooping
and the Balanced L1-norm suggested by [Kampmann &
Wolf, 1989]. The major drawback with these methods is
that they will fail if the first estimate is too far away from
the true solution. In the relative orientation problem a few
number of outliers may be enough to make the solution
degenerate completely. In the tests carried out in this
study it can be seen very clearly that one, or possibly two,
errors can be found. When the number exceeds two or
three points, or the fraction of outliers goes beyond 10-
1596 these methods are very likely to fail. In these cases
other strategies must be applied.
Algorithms based on repeated calculations using a small
sample of data, sometimes referred to as RANSAC or
bootstrap methods, are in many cases able to find a
solution close to the optimal one and to identify large
fractions of outliers. Since only a small sample of data is
used for the solution, expectancy values and standard
errors are not possible to calculate as in a LS estimate.
Depending on the purpose of the calculation, the
estimates can of course be improved by a LS adjustment
on the remaining data after removal of the outliers. If the
sample is chosen by random, as in all cases in these tests,
it is very likely that the selected solution contains
observations close to each other. Residuals of
observations far from these tend to be high since the
model coordinates are extrapolated. Due to this, errors of
type I are more common than for methods using all
available data. If enough precautions are taken to ensure
that observations are not removed by mistake and one has
an awareness of the limitations of the solution, these
methods are well suited for limited tasks like the relative
orientation.
The third strategy, to include the errors in the model and
calculate a cost function, shows a very nice behaviour
both for few numbers of outliers as well as for many.
Some additional information must be provided in order to
compute the DL's that is not needed for the other
methods. This information defines the range or bounds of
the observations and its resolution. The calculations of the
DL's are not very complicated and could be considered as
an alternative in some implementations.
For autonomous systems with arbitrary orientations,
estimates based on linear algorithms using repeated
calculations on small samples of data seem to be a fruitful
way of getting robust results. For standard aerial images,
these methods can be used as well, but standard methods,
like data snooping and iterative five points algorithms,
are more stable as long as the number of outliers are low.
The answer to the question put forward with this study,
whether to remove or add outliers, is, not very surprising,
thus depending of the application and the expected types
of errors and error fractions that might occur.
7. ACKNOWLEDGEMENTS
The author is grateful for the help and support given by
Johan Philip in supplying algorithms and for
implementations of the six- and eight-point solutions.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
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