Figure 4. At relatively short ranges (e.g., 10 m and 20 m) an
increase in the
PTZ
camera’s focal length from 10 mm to 100
mm only slightly improved the measurement accuracy. At
longer ranges, increasing the
PTZ
camera’s focal length well
improved the measurement accuracy when the
PTZ
camera’s
focal length itself was relatively short (e.g., 10 to 40 mm).
But the improvements was not significant when its focal
length was already relatively long (e.g., 60 mm). This could
be explained that the focal length of the surveillance camera
(e.g., 8 mm) influenced the measurement accuracy in this
case, and when the
PTZ
camera’s focal length was close to the
surveillance camera’s focal length increasing of the former
would improve the measurement accuracy largely. But, when
the
PTZ
camera’s focal length was far longer than the surveil-
lance camera’s focal length the latter dominated the measure-
ment accuracy and further increasing of the former would not
improve the accuracy notably.
From Figure 5, it can be seen that the pan angle of the
PTZ
camera affected the range measurement accuracy, and that this
trend was more significant over long ranges. For object points
along a specific range, the point located in the range corre-
sponding to the middle of the baseline generated the maxi-
mum convergent angle. The convergent angles of points to the
left or right of that point were smaller. Therefore, when the
PTZ
camera pointed to the point corresponding to the middle
of the baseline, this maximum convergent angle caused the
best measurement accuracy. Measurement errors increased
when the
PTZ
camera changed its pan angle to point to other
places along the specific range. The pan angles with the best
measurement accuracies were dependent on the baseline
length and on the range. For the baseline length of 0.75 m
used in Figure 5, the pan angles with the best measurement
accuracies were 2.15
°,
1
.
08°, 0.72°, 0.54°, 0.43°, and 0.36° for
ranges of 10 m, 20 m, 30 m, 40 m, 50 m, and 60 m, respective-
ly. These values are marked with arrows in Figure 5.
In the above theoretical analysis, estimates of the uncer-
tainties in the error propagation were made, based on general
considerations. A more comprehensive analysis based on
Monte Carlo simulation that incorporated additional factors is
described below.
Monte Carlo Simulation
Monte Carlo simulation is an efficient way to investigate the
numerical properties of a complex mathematical model with
respect to artificial noise in the input data (Luhmann, 2009).
Noise is added based on statistical distributions and typi-
cal noise levels, so that the resulting output data varies in a
realistic way.
A simulation program was created for this study. To reduce
the number of possible combinations, only the most critical
impact parameters were investigated. For each numerical
simulation, 10,000 variations were computed. Each simula-
tion was based on normally distributed input noise with an
internal threshold of 2 sigma, and image point measurement
noise with an internal threshold of
⅓
pixel. The detailed steps
of the Monte-Carlo simulation were as follows:
1. The exterior orientation parameters (
EOP
s) were gener-
ated. The origin (0, 0, 0) of the local reference frame
was located at the perspective center of the surveil-
lance camera, and the coordinates of the
PTZ
camera’s
perspective center were (
B
, 0, 0). The three rotation
angles were set to (
θ
T
,
θ
P
, 0).
2. The 3
D
coordinates of targets in object space were
generated, at different distances from the center of the
baseline. In this study, we choose 10 m, 20 m, 30 m, 40
m, 50 m, and 60 m as the target ranges.
3. The image coordinates of the target points on the
surveillance and
PTZ
camera images were calculated
using the
EOP
s (generated in Step 1) and the
IOP
s, which
were based on the geometric model of the dual camera
system (Equations 1 and 2).
4. 10,000 Gaussian random noise data sets were gener-
ated, with means of 0 and standard deviations of the
estimated pixel measurement errors, and added to
image measurement coordinates from the two camera
images. A further 10,000 Gaussian random noise data
sets with 0 means and 1 percent
f
r
standard deviations
were generated and added to the focal length
f
r
of the
PTZ
camera. Finally, 10,000 Gaussian random noise
data sets with 0 means and tan
-1
(1 pixel/
f
r
) standard
deviations were generated and added to the tilt angle
θ
T
and pan angle
θ
P
of the
PTZ
camera.
5. The 3
D
coordinates of the target points were calculated
10,000 times by space intersection, using the above
parameters and image coordinates with noise based on
Equations 1 and 2.
6. The calculated coordinates and the true 3
D
coordinates
of the targets were compared and analyzed statistically.
The above simulation was performed for each combination
of baseline length
B
, focal length
f
r
of the
PTZ
camera, and pan
angle
θ
P
of the
PTZ
camera at different target ranges. Figures 6,
7, and 8 were generated to show the range measurement error
with respect to these three parameters.
From the figures referenced above, it can be seen that the
simulation results are generally consistent with the results of
the theoretical analysis. The results from the former are rela-
tively inferior to the results from the latter as can be noticed
from the more fluctuant curves in Figures 6, 7, and 8 com-
pared with those in Figures 3, 4, and 5. The general consis-
tency between the results from these two methods verifies the
correctness of the range calculation equation (Equation 4) and
the corresponding error propagation derivations.
From the theoretical analysis and Monte Carlo simulation
results, the following summary can be derived:
1. The baseline length was the main factor influencing the
range measurement accuracy. This influence was more
significant for long ranges than for short ranges.
2. The focal length of the
PTZ
camera also influenced the
range measurement accuracy. But the influence was
related to the surveillance camera’s focal length. When
the
PTZ
camera’s focal length was far longer than the
surveillance camera’s focal length further increasing of
the former would not improve the accuracy notably.
3. The pan angle of the
PTZ
camera influenced the range
measurement accuracy. This influence was more sig-
nificant at long ranges than at short ranges. The highest
accuracy was achieved when the
PTZ
camera was
pointed towards a target located along the middle line
of the two cameras.
Actual Experimental Analysis
To verify the previous theoretical analysis and Monte Carlo
simulation of the attainable measurement accuracy of the
proposed dual camera system, two sets of experiments were
conducted, one in an indoor facility with a relatively short
range and the other in an outdoor environment with a rela-
tively long range.
The indoor facility, shown in Plate 2a and 2b, contained 68
cross-shaped targets, mounted on two perpendicular walls.
To determine the orientation parameters of the
PTZ
camera at
different focal lengths, 29 targets were densely mounted near
the center of the study area, which ensured that at least eight
targets would be present in the image acquired by the
PTZ
camera at a long focal length. The average distance between
the camera system and the targets was approximately 10 m.
224
March 2015
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
