OUTLIER DETECTION IN RELATIVE ORIENTATION -
REMOVING OR ADDING OBSERVATIONS
Peter Axelsson
Department of Geodesy and Photogrammetry, Royal Institute of Technology
100 44 Stockholm, Sweden
e-mail pax Q geomatics.kth.se
KEY WORDS: Image Orientation, Robust Estimation, Error Identification, Reliability
ABSTRACT:
Outlier detection in relative orientation has been studied using three different strategies, (7), removing bad
observations, outliers, using data snooping, (ii), adding good observations using Least Median of Squares estimates
and (iii) finding an optimum between good and bad observations in a cost function using minimum description
length criteria, MDL. To be able to compare the strategies, relative orientation algorithms based on linear or closed
formulae were used. Two algorithms for calculating relative orientation without any provisional values were applied.
For a LS solution a linear solution based on eight unknowns was used. For a direct solution with minimal point
configurations the same algorithm was used but estimated without any redundancy, i.e., with eight observations. In
addition to this, two other algorithms were applied to some of the data in order to get the results verified and
compared. In total, four algorithms tested on a large number of simulated data configurations. The results show that
large fractions of outliers can be detected for the strategies (ii) and (iii) even with arbitrary image orientations, while
strategy (i) shows very good stability for normal aerial geometry with few outliers.
1. INTRODUCTION
Procedures that do not need provisional values for the
calculation of relative orientation parameters of a stereo
pair of images are of great need in general applications
where approximate locations and attitudes of the cameras
are unknown. When automating point selection and point
identification, these procedures must also be more robust
against large fractions of outliers than in the case of
manual measurements.
The traditional way of handling outliers in
photogrammetry is by investigating diagonal elements in
the least squares, LS, estimate of the covariance matrix of
the observations and their residuals. The observations are
then kept or removed depending on some statistical test,
e.g., data snooping [Baarda,1967]. In such a procedure,
all observations are part of the initial LS estimate and
outliers are removed one at a time. A different strategy is
to start with a minimum configuration of observations,
adding points as long as they fulfil some criterion. The
solution of the minimum point configuration is often
repeated with different random sets of points and the
"best" solution chosen, e.g., least median squares, LMedS
[Rousseeuw,1987] or RANSAC [Fichler and Bolles,
1981]. Both strategies regard outliers as observations not
belonging to the model, ie. not having the same
statistical properties as good observations. A third
strategy is to extend the mathematical model to include
also outliers and by a cost function locate a minimum
where the optimal number of outliers is found, Such a
cost function has been formulated within the minimum
description length, MDL, principle [Rissanen, 1983] and
used as an estimator with robust properties [Axelsson,
1992].
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
2. AIMOF THE INVESTIGATION
A comparison of the three strategies was made regarding
the robustness against outliers when calculating the
relative orientation of a stereo pair of images. The three
strategies are:
i Removing bad observations, outliers, using LS
estimates and data snooping
ii Adding good observations using LMedS
iii Finding an optimum between good and bad
observations in a cost function using MDL
To be able to compare the strategies, only relative
orientation algorithms based on linear or closed formulae
were considered. For the LS solution a linear solution
based on eight unknowns were used [Tsai and Huang,
1984], [Philip, 1989]. For the direct solution with
minimal point configurations the same algorithm as for
the LS solution was used, but estimated without any
redundancy, i.e., with eight observations.
In addition to this, two other algorithms were applied to
some of the data in order to get the results verified and
compared. In total, four algorithms were used in the
investigation:
1. Linear overdetermined LS estimate [Philip, 1991]
Linear LS estimate with minimal point
configuration.
3. Closed six-point formula [Hoffman-Wellenhof,
1979].
4. Iterative LS solution with five unknowns
3.1 Esti
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