"art B3. Istanbul 2004
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
| Experiment | GN I (px) | REI(px) | GNG (m) | REG (m)
A 1 0 0.001 0
B 1 ] 0.001 0.001
Table 1: Experiments description.
est fie iagonal mean error mean error
on A :
ygon 027 %
A
Table 2: Stability of V.
(B) introduces rounding errors, which are non Gaussian and are
much more realistic. For each experiment, on each geometry,
1000 trials have been made.
Table 1 shows the different experiments. The legend for Table 1
is:
e GN I : Gaussian Noise on all Images observations, points
and lines;
e RE I: Rounding Error on all Images observations, points
and lines
e GN G : Gaussian Noise on Ground Control Points,
75 materialized GCP e REG: Rounding Error on Ground Control Points.
0.1 mm. :
Stability of accuracy prevision First, we have verified that
the matrix V is stable, i.e., that it does not change when data
and solution change a little. Using all trials, we can compute
a mean value and a standard deviation for each element of V.
Experimental results are shown in Table 2. V diagonal mean error
is the mean, for all elements of the diagonal of V, of the standard
deviation divided by the mean value of this element. V mean
error is the mean, for all elements of V, of the standard deviation
divided by the mean value of this element.
These results show that V is very stable, and is not influenced
much by usual noise. It seems that some simulations with lines
are less stable than the one without lines. Anyway, for real cases,
the matrix V obtained after convergence can be used directly.
Accuracy previsions vs measured accuracy The goal of this
part is to verify that the observed errors and the computed ones
are coherent. Using all trials, it is possible to compute the vari-
ance and co-variance of all unknowns X;, and to compare it with
V. The results are shown in Table 3. Legend for Table 3 is :
Test field Mean SDE | Mean CE
Polygon A 0.016 % 0.025
Polygon B 0.037 % 0.025
Facade A 0.6 % 0.09
Facade B 0.5 % 0.17
Table 3: Accuracy simulation results.
SDE : Standard Deviation Error is :
VIP — Vy |
yobs
i
with à € [1, Nu] (28)
A stereo rig mounted on i
of a facade. 30 GCP are where V25° is the observed value of V;
tie lines have been manu-
CE : Correlation Error is Corr?! — Corr;,;| for all un-
knowns (4,j € [1, Nu])), where Corr?"
correlation between unknowns X; and X;
is the observed
Camera Position | Rotation | 3D Point
(mm) (mrad) (mm)
FE; mean 39 3.8 27
sd 3 0.7 2.8e-02
E» mean 32 25 2.6
sd 2 0.6 2.8e-02
Table 4: Camera rotation and position accuracy with 30 GCP
(line E1) and with tie lines (line E»).
Camera Position | Rotation | 3D Point
(mm) (mrad) (mm)
E; mean 84 6 74
sd 12 2 45
E» mean 41 1 18
sd 9 0.2 8
Table 5: Camera rotation and position accuracy with 6 GCP and
24 tie points (line E1) and with tie lines (line E>).
Results are very promising : observed and computed variance are
very similar. So the matrix H^! is a very good estimate of the
variance matrix V. The variance of any unknown can be predicted
with an error inferior to 196, even if noise is not perfectly gaus-
sian. Linear features seem to degrade the quality of the estimation
of correlation between parameters, especially when rounding er-
ror is applied (Facade B). Anyway, this method enables us to get
the precision of each parameter of the system.
5.3 Mobile Vehicle
This real case study corresponds to the facade test field. In the
first experiment (Table 4) all GCP have been used in the bundle
adjustment algorithm. The second experiment (Table 5) analyzes
the result of the bundle adjustment process with only 6 GCP (24
points are set as tie points in our process). For both experiments,
significance number a is set to 0.95. Statistics are made on the
length of the longest axis of each 3D ellipsoid.
The comparison of the angle between the longest ellipsoid axis
and the Oz axis stresses the fact that using vertical tie lines pro-
vides a longest ellipsoid axis closer to the Oz axis by 2-5 degrees.
In the second experiment, one should note that all 24 control
points, which are not used in the bundle, fall inside the estimated
covariance ellipsoid.
5.4 Discussion
Simulation and real experiments have stressed the importance of
lines in order to increase parameter estimation accuracy in a bun-
dle adjustment process. We have observed that the gain in ac-
curacy was depending on the orientation of the given lines. The
error propagation modeling has also been validated by our exper-
iments.
6. CONCLUSION AND FUTURE WORK
We have presented a technique to compute a bundle adjustment
with points and lines. We provide an error propagation algo-
rithm in order to assess the reliability of the estimated parameters
(viewing position, viewing rotation, 3D points and lines). Exper-
iments on simulated data and on real data are described. Many
extensions and further research on this topic can be done:
e Error propagation with non-Gaussian distributions. De-
pending on the algorithm used to select points and lines the
error distribution is non-Gaussian. Moreover, very often
we must face a bundle adjustment problem with small data
sets. In these situations, specific statistical tools have to be
used in order to estimate parameter accuracy (large devia-
tion, etc.)
