ame origin
ned by lin-
> restricted
its can be
(man and
i)
Y)
nd a least
'ivial null-
| is added
de, 1997]
has nega-
t A. This
has over-
nd Kanade,
factoriza-
ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision", Graz, 2002
Figure 6: Section of the Eremitage se-
quence showing the tracked features
HS
Figure 7: Section of the Eremitage se-
quence showing the tracked features. It
is the frame following the frame shown in
Figure 6. Note the change in feature loca-
tions.
To solve this problem we propose to parameterize, Q as:
220 0
Q(e, À) = R(e) | 042 0 | R(e)” (11)
00)
where R(e) is a rotation matrix with the three Euler angles
denoted by e. The term aZ Qa, — bf Qb, = 1 is replaced
by det(A) — 1, such that the over all scale of A is much
more robust and less sensitive to the noise in a particular
frame.
Hence the estimation of Q is a nonlinear optimization prob-
lem in six variables, with a guaranteed symmetric positive
definite Q. Our experience shows that this approach to the
problem is well behaved with a quasi-Newton optimiza-
tion method.
5 EXPERIMENTAL RESULTS
We illustrate the capabilities of the proposed algorithm via
three sets of experiments. The first demonstrate the capa-
bility of using different error functions. This is followed by
a more systematic test of the tolerance for different kinds
of possible errors. Finally we show an example of why
the proposed method for Eucledian reconstruction is to be
preferred.
5.1 Different Error Functions
To demonstrate the capability of using different error func-
tions, we used an image sequence of the Eremitage castle,
see Figure 4. The 2D features were extracted via the Har-
ris corner-detector [Harris and Stephens, 1988], where-
upon the epipolar geometry was used for regularization via
RANSAC/MLESAC [Torr, 2000] followed by a non-linear
optimization [Hartley and Zisserman, 2000].
Figure 8: Section of the Eremitage sequence showing the
tracked features and residuals from the roof using the trun-
cated quadratic with the proposed method. The residuals
are denoted by the dark lines.
This enforcement of the epipolar geometry enhanced the
quality of the data, but it did not yield perfectly tracked
data. There are two main reasons for this.
First, the trees around the castle yield a myriad of pottential
matches since the branches look pretty much alike. The
restriction of correspondances to the epipolar line is not
sufficient to amend the situation as is shown in Figures 6
and 7.
Second, when filming a castle, one moves approximately
in a plane — both feet on the ground. This plane is paral-
lel to many of the repeating structures in the image, e.g.
windows are usually located at the same horizontal level.
Hence the epipolar lines are approximately located ’along”
these repeating structures and errors here cannot be cor-
rected by enforcing the epipolar geometry. In general the
sequence containes plenty of missing features, mismatched
featuress and noise.
The truncated quadratic [Black and Rangarajan, 1996], the
Hubers M-estimator [Huber, 1981] and the 2-norm were
tested as error functions. The reason the proposed method
was used with the 2-norm and not the original Christy-
Horaud method is, that there were missing features in the
data-set. These missing features are incompatible with the
Christy-Horaud approach, but the approach presented here
deal with them by modeling them as located in the middle
of the image with a weight 109 times smaller than the 'nor-
mal' data. It is noted that this approach for dealing with
missing features can not be expressed in the framework of
[Irani and Anandan, 2000].
In order to evaluate the performance the residuals between
the 2D features and the corresponding reprojected 3D fea-
tures were calculated. The desired effect is that the resid-
uals of the 'good' 2D features should be small, hereby in-
dicating that they were not ’disturbed’ by the erroneous
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