nbul 2004
rent defor-
originally
are mostly
| curvature
city of the
tions.
very sharp
larger than
ibration.
20
lations.
) indicates
imum, but
| expect in
the data.
X axis
Y axis
Z axis |
30 40
ions.
rtical axes
But in that
he sharper
purposes,
horizontal
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
0.34 rep quum 1 T IEEE
0.32
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Proportion of outliers
e o o
N nN no
N A o
:
e
N
0.18
0.16} re +
0.14! i eese i mi il. E pn ceci vn - -— i
-10 -8 26 -4 r2 0 2 4 6 8 10
Angle of rotation (degrees)
Figure 10: Evaluation function for rotations.
Curvature: the last deformation which have been investigated
in this preliminary study is a curvature of the laser points along
one of the planimetric direction, as illustrated with the figure 11.
This deformation is controlled with a parameter ó, which is re-
lated to the curvature radius R via the formula:
not ó*
=
For § = 0, the curvature radius is equal to the infinity, i.e. no
distorsion is applied to the laser points. The parameter § can
take positive and negative values. The two sharp minima on the
figure 12 indicate that this kind of deformation could also be es-
timated with this registration approach.
Figure 11: Parametrization of the curvature deformation: ó is the
vertical distortion at | meter from the curvature axis.
4 AUTOMATIC REGISTRATION
The convexity of the quality criterion demonstrated in the previ-
ous section led us to use a simple approach for the automation
of the registration. The Nelder-Mead simplex method has the ad-
vantage to be very fast to implement and does not require the
calculation of the gradient of the evaluation function (Nelder and
Mead, 1965).
Presently, the automatic registration procedure is limitated to the
search of the best planimetric translation, but it can easily be ex-
tanted to more complex deformations. The data used for this ex-
perimentation are presented on the figures 13 and 14. The scene is
in the suburb of Brussels and covers a surface of 270 x 340 m?.
The aerial image is in colour and has a resolution of 8 cm on
the ground. The laser points have been acquired with a oscil-
lating mirror system with a density of 0.32 pt/m?. Since both
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Parameter of the curvature distortion (mm)
Figure 12: Evaluation function for curvature around X and Y
axes.
data were already calibrated, we used the original position as the
ground truth.
To test the robustness of the approach, the simplex algorithm was
run from 8 initial positions in different directions at a distance
larger than 10 meters from the real position. The final positions
where close to the ground truth (see table 2), but we noticed a
relative dispersion of these local minima.
original simplex iterated simplex
(cm) X Y X Y
mean 13 0 6 7
std 18 15 2 7
[un max} |-[-3, 52} } [+35 1211 [3 91 | [-L 15]
Table 2: Statistics for 8 different starting positions with the orig-
inal and the iterated simplex methods.
In order to improve the stability of the registration process, two
solutions may be proposed at the expense of a higher cost in com-
putation time:
e since the simplex shrinks when approaching the minimum
of the evaluation function, it may be too small for the last
iterations: a solution is to re-run the algorithm from the last
position with a relatively large simplex,
e close starting positions may result in different local min-
ima: a solution is to run the simplex algorithm from different
starting positions and with different initial simplex size.
The table 2 gives the mean position, the standard deviation and
the interval of the 8 final positions obtained with the original sim-
plex procedure, and with the twice iterated procedure. We can no-
tice that with the second method all the final position are grouped
within a disk of radius 10 cm.
5 CONCLUSION
A 3D model of an urban scene can be reconstructed with airborne
laser data and a single aerial image using robust parameter esti-
mation techniques. We proposed to use the quality of the 3D
reconstruction to estimate the relative position of both data, and
further to automate their relative registration. A possible appli-
cation is the automatisation of the calibration process for the 3D
points acquired with an airborne laser scanning system.
