X is taken
ferent ac-
the distri-
. (12)
as well as
finity and
ve obtain
=X X y =
Y
d
e matrix
;, p) for
ctor y! -
tical lines.
OU;
j ox!
we here
nd. Y =
"the 3D
1erefore,
ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002
4.3.0 Observed Horizontal Lines We have the vector
yl = aj, X.) as uncertain observations. We, therefore,
obtain from (8)
Oe;
Ai(y;) 7^ Aj(l;, Xo.) — = = (1; X).
and
; Oe;
B;(y;, P) = B;(L, Xoo7, P) = red = EX GP) ;
Jj
4.3.3 Observed Points We have the vector y, =
(xp! , X]) as uncertain measurements. Therefore, we ob-
tain from (9)
de *
Ar (Un) = Ak(xi, Xx) = m =8, oX].
The other can be obtained by
der > =
Br XP) = 30: = (Spx.ı SP). 5
This holds for arbitrary point correspondences.
4.4 Optimization function
Using the assumed covaraince matrices for the observed
entities and the derived Jacobians, we can derive the co-
variance matrices Xe ¢;, Xe, e; > and ¥., ., of the constraints’
residuals e;, e; and ey, as given in (3).
We now have all components at our disposal necessary to
estimate P. Thus, we need to solve iteratively
190) (80V Al:
DA (Vi ) Xe Ai(y;)
+
~\V a)
2
c
~(v eU) ~(v ~(v
50) (82) A Jp =a
k
where p contains the unknown parameters of P. The itera-
tion can be initialized by results from a direct estimation of
p as in (11), which is equivalent to setting Xe,e, = M eje;
— X, e, — |. The fitted observations y;, y;, and y,. are
initialized by the actual observations in the image and the
drawing.
5 EXPERIMENTS
In this section we describe the experiments conducted to
evaluate the characteristics of the presented method. We
first show the correctness of error prediction by testing the
method on synthetic data. Then, the practical usability
of self-diagnosis tools is demonstrated by showing results
of application to images and a drawing of a high voltage
switch gear.
5.1 Synthetic Data
In order to guarantee realistic test conditions we use geo-
metric dimensions which come very close to those given in
the real data experiments in the following section. We use
a drawing of 400 x 400 pixels, the camera is placed off
the drawing at approximately (252, —222, 108)[pixel] and
has a viewing angle of approximately 50 degrees. We gen-
erate uniform distributed points on and above the ground,
vertical lines, and horizontal lines. We may then project all
features into the camera, using a known realistic projection
matrix, to have an ideal set of observations.
As a first indication for correctness of the estimated covari-
ance of P, 355, we compare the estimated back-projection
error of a 3D point with its true covariance under well de-
fined noise conditions. Therefore, we estimate the pro-
jection matrix from randomly perturbed observations, and
back-project a perturbed 3D point using the estimated pro-
jection. Using a perturbation of 04 = 0.5[pixel] for obser-
vations from the drawing, and c; = 1.2[pixel] for image
features, we repeat the estimation of P n = 5000 times
using 10 vertical lines, 10 horizontal lines, and 10 points.
The resulting Gaussian distributed point cloud is depicted
in Fig. 3. The experiment shows that the predicted 90%
confidence region from (5) and the empirically obtained
one are very close. Using the same setup, we now consider
the Mahalanobis distance between the estimated projection
parameters and the true ones p:
= >. EATEN x =
IP — DI, = 6 — BD) Xos(P — P)
If the model is correct, the Mahalanobis distance is X7;
distributed, since the projection matrix has 11 degrees of
freedom. The histogram in Fig. 3 shows the distribution of
the Mahalanobis distance based on 1000 experiments. The
comparison with the analytically plotted x3, distribution
shows good conformity.
5.2 Real Data
A large field of application for the presented pose esti-
mation method is as-built reconstruction in industrial en-
vironments. Therefore, we chose images of a high volt-
age switch gear and the associated top view drawing to
demonstrate the feasibility of the algorithm. The drawing
has a dimension of 1220x820 pixel. Its scale [pixel]:[m]
is 1:0.012, i. e. 1 pixel is equivalent to 12mm in the real
world. The images have a size of 1530x 1018 pixel. We
select feature points and lines in the image and in the draw-
ing with a standard deviation of 1 pixel and 0.5 pixel, re-
spectively. We assume that the selection process itself on
the drawing does not have any uncertainty, since drawings
are in general vectorized documents. However, drawings
reflect the as-built situation only up to a certain accuracy.
In our case, the drawing depicts the real world installation
up to approx. + 6mm, which results in the mentioned stan-
dard deviation.
As observations in the image we select 6 vertical lines, 5
horizontal lines, 3 points on the ground and 2 above. Once
