'anbul 2004
Fa ha
= 0. The
lue À; is a
lecomposi-
(12)
blem is ad-
estimation
inar homo-
ast squares
h =vec(H)
is observa-
L^ (3)
e observa-
1e solution
pproxima-
«plained in
of the pla-
e potential
been taken
ingent ad-
ch angle a
rizontal or
mal vector
jhy (6) de-
| distance c
imate zero
y weak. In
s has to be
cing these
the adjust-
cts can be
image are
parameters
be situated
ses (initial
(14)
the param-
e observa-
ail, 1976).
| and (11).
" H)! five
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
parameters have been introduced although just the fraction
H/Z is determinable. Depending on the actual calibration
phase (initial or update) either H or Z have to be fixed
by prior information. Because of the assumption of i. i. d.
observation groups the normal equation system for the ad-
justment model (14) can be built-up sequentially.
To make sure, that the necessary prior information has a
constant contribution to the solution, the relative weighting
between the observations and the prior information can be
controlled by a regularization factor A. An ad-hoc solution
is À = tr(N)/tr(P,,) (Press et al., 1992) with the traces
of the normal equation matrix N and the prior weights P,,,
for the observed’ parameters. Again, a conditioning of
the problem is advisable by a translation and scaling of the
image quantities and the principal distance respectively.
Kalman Filter. The sequential build-up of the normal
equation system offers the possibility of introducing a dis-
crete Kalman filter (Welch and Bishop, 2002) for the cali-
bration update phase. This is equivalent to a recursive pa-
rameter estimation process. To prevent a numerical over-
flow and the solution to bite, a memory length term k can
be introduced, which controls the amount of memory used
for the actual solution. With k = 0.9 for instance, 90 9o of
the past observations will be used at the present time. Af-
ter every evaluation step the normal equation matrix, the
right-hand-side vector, the sum of squared residuals and
the number of conditions have to be updated. The latter
becomes real-valued which is as yet practically irrelevant.
The parameter k may not affect the unknown object heights
H, as this parameter can vary from scene to scene.
4 EXPERIMENTAL RESULTS
4.1 Observations and Reference Calibration
Observations. For the evaluation of the approach an im-
age of a lecture room was recorded, showing a seating ar-
rangement of chairs of indentical heights (cf. fig. 3). The
camera used has an image format of 960 x 1280 picture el-
ements. The image measurement of the foot points of the
chair legs and the top points of the chair backs was done
by an operator.
Reference Calibration. For the evaluation of the ap-
proach a reference calibration has been carried out for the
intrinsic camera parameters as well as for the exterior ori-
entation.
After the recording of the image a calibration field has im-
mediately been captured on location. The intrinsic param-
eters are then taken from a bundle adjustment. Table 1
summarizes the results of the parameter estimation for the
intrinsic parameters.
For the determination of the exterior camera orientation the
image points representing the corners of the tables have
been measured. Together with the world coordinates of
the corresponding points 0.74 m above the ground plane
1071
ae
uw
Figure 3: shows the observed corresponding foot and head
points as well as the estimated horizon line, its point of
gravity and its hyperbolic error band (3o intervals).
and the interior orientation given in table 1 a spatial resec-
tion has been accomplished assuming a standard deviation
of 0.02 m for the object coordinates and 2 pel for the im-
ages coordinates. From the estimated matrix for the rota-
tion from the object to the camera coordinate system the
roll and pitch angle result from (1) and(2). The estimated
accuracies result from error propagation and are listed in
table 2. The estimated height of the camera above ground
has been verified with the help of a measuring tape.
parameter | estimation | estim. std. dev.
principal dist. ¢ 1328.86 pel 2.577 pel
scale factor mn 0.9962 3.37740 !
principal pt. Axg -1.35 pel 1.458 pel
principal pt. Ay -4.90 pel 1.389 pel
Table 1: summarizes the results from the intrinsic camera
calibration with a test field.
parameter estimation | estim. std. dev.
pitch angle a 31.2324 deg 0.4479 deg
roll angle ^ 0.4847 deg 0.5341 deg
camera position .X 3.0611 m 0.0961 m
camera position Y -2.2095 m 0.0397 m
camera height Z 2.5583 m 0.0830 m
Table 2: summarizes the results of the exterior reference
calibration.
4.2 Calibration Results
A height of H = 0.77 m have been determined for the
chairs in the scene. The results of the direct solution (12)
and of the constrained advancement with (13) are summa-
rized in table 3.
For the following calibrations prior information has to be
used in order to introduce metric information. For the
