the range error can be calculated using the following equation:
σ
σ
σ
σ
D
l
x
r
x
r
f
D
x
D
x
D
f
D
l
r
r
2
2
2
2
2
2
2
=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
θ
σ
θ
p
p
2
2
(5)
where:
∂
∂
= −
D
x
D
f B
l
l
2
∂
∂
=
( )
+
( )
+
−
( )
+
D
x
Dtan
f x tan
D f x tan
f B f x tan
r
p
r
r
p
l
l
p
l
r
r
p
θ
θ
θ
θ
2
( )
∂
∂
=
+
( )
−
+
( )
+
( )
D
f
D
f x tan
D x f tan
f B f x tan
r
r
r
p
l
l
p
l
r
r
p
θ
θ
θ
2
1
1
∂
∂
=
( )
+
+
( )
−
( )
+
D Dx tan
f x tan
D tan
f
p
r
p
r
r
p
p
l
θ
θ
θ
θ
2
2
2
f x x
f B f x tan
r
r l
l
r
r
p
+
(
)
+
( )
θ
.
In the above equation,
σ
D
is the standard error of range
D
;
σ
x
l
is
the coordinate measurement error on the left image, and
σ
x
r
is
the coordinate measurement error on the right image. Accord-
ing to theoretical analysis, pixel measurement error can reach
an accuracy of approximately one-third of a pixel. Since the
resolutions of images acquired by the surveillance camera and
PTZ
camera may be different, the following strategy is em-
ployed. The pixel measurement error on the images acquired
by the surveillance camera is estimated to be one-third of a
pixel, while the pixel measurement error on the images ac-
quired by the
PTZ
camera is estimated to be a multiple of one-
third pixel depending on the times of the focal length of the
PTZ
camera with respect to that of the surveillance camera. It
should be noted that in the proposed dual camera system the
pixel size of the surveillance camera and the
PTZ
camera dif-
fer.
σ
f
r
is the standard error of the focal length of the
PTZ
cam-
era, which can be determined from an antecedent zoom lens
calibration and modeling procedure.
σ
θ
p
is the standard error
of the pan angle of the
PTZ
camera, which can be estimated
from the pan control system of the camera.
From the above equation, the range error at a specific range
D
is dependent on the baseline length
B
,
PZT
camera focal
length
f
r
,
PZT
camera pan angle
σ
p
, and the image coordinates
x
l
and
x
r
. For a given set of
D
,
B
,
f
r
, and
θ
p
,
x
l
is dependent on
x
r
, so that it can be represented by
x
r
.
To investigate the measurement accuracy in different sce-
narios, the accuracy analysis was divided into the following
three steps. First, the relationship between the range error and
different baseline length
B
at different ranges was investigat-
ed. In this step, the focal length
f
r
of the
PZT
camera was set to
a constant length, and the pan angle
θ
p
of the
PZT
camera was
set to 0°. Second, the relationship between the range error and
PTZ
camera focal length at different ranges was investigated.
In this step, the baseline length
B
was set to a constant length,
and the pan angle
θ
p
of the
PZT
camera was again set to 0°.
Third, the relationship between the range error and the
PZT
camera pan angle
θ
p
at different ranges was investigated. In
this step, both the baseline length
B
and the focal length
f
r
of
the
PZT
camera were set to constants.
As the focal length of the
PTZ
camera changes during the
process of zooming, the range measurement accuracy at each
specific range may vary from place to place in the image, due
to the involvement of uncertainties (
σ
f
r
) in the calibration and
modeling procedure of the
PTZ
camera’s focal length, and be-
cause
σ
f
r
will be magnified by the location of the target point
on the image. Normally, the closer of the target point to the
image center (or principal point), the smaller the influence of
σ
f
r
on the range measurement accuracy will be. This can be
verified by the following error-propagation derivation when
representing all the items of
x
l
in Equation 5 using a function
of
x
l
(
θ
p
is set to 0 to simplify the derivation).
σ
σ
σ
σ
D
l
x
r
x
r
r
f
D
B f
D
B f
D x
B f
l
r
r
2
4
2 2
2
4
2 2
2
4 2
2 4
2
=
+
+
.
(6)
In the above equation, it can be seen that the first and sec-
ond components in the right part of the equation are constant
for a given set of
D
,
B
,
f
l
,
f
r
, and coordinate measurement er-
rors, whereas the third component is dependent on
x
r
. When
x
r
= 0 (i.e., when the target point is located at the image prin-
cipal point), the range error
σ
D
reaches a minimum. To find a
unified comparison analysis of the measurement accuracy, we
have focused on finding the attainable accuracy of the point
located at the image principal point (
x
r
= 0) in the following
discussion. This is also meaningful since in real applications
of the proposed dual camera system, such as target tracking,
the
PTZ
camera would always be pointing at the target and
would locate the target around the image center area.
Using the camera parameters listed in Table 1 as references
and Equation 5, three figures were generated to show the
range measurement error in the three previously mentioned
scenarios. Figure 3 shows the relationship between the range
error and baseline length
B
at ranges between 10 m and 60
m. In this case, the focal length
f
r
and pan angle
θ
p
of the
PZT
camera were set to 60 mm and 0°, respectively. The baseline
length varied from 0.5 m to 1 m. Figure 4 shows the relation-
ship between the range error and focal length
f
r
of the
PZT
camera at different ranges, where the baseline length was set
to 0.8 m and the pan angle
θ
p
of the
PZT
camera was set to 0°.
f
r
varied from 10 mm to 100 mm in this case. Figure 5 shows
the relationship between the range error and pan angle
θ
p
of
the
PZT
camera at different ranges, where the baseline length
was set to 0.75 m and the focal length
f
r
of the
PZT
camera
was set to 60 mm.
θ
p
varied from −45
° to
45
°
in this case
.
σ
θ
p
was set to 10", based on the
PTZ
camera control parameters. It
should be noted when considering the above parameters that
we were aiming for better than 1 percent measurement accu-
racy within a normal observation range (e.g., 60 m) in indoor
or outdoor environments.
The range measurement error decreased with increased
baseline length, as shown in Figure 3, and this trend was
more significant over long ranges, which was consistent with
traditional photogrammetric theory despite the disparity in
focal length between the two cameras. For a surveillance
camera with a focal length of 8 mm and a
PTZ
camera with a
focal length of 60 mm, a minimum baseline length of 0.75 m
was required to reach a measurement accuracy of 1 percent
within a range of 60 m (0.6 m range error). Increase in the
surveillance camera’s focal length would improve the mea-
surement accuracy; however, this would reduce the
FOV
of the
surveillance camera, which deviated from the original inten-
tion of the proposed dual camera system to monitor a wide
scene. While decreasing the surveillance camera’s focal length
would offer an even wider
FOV
, however it would be more dif-
ficult to reach a better than 1 percent measurement accuracy
within a normal observation range of 60 m through a feasible
stereo configuration. An 8 mm focal length for the surveil-
lance camera was selected as a sensible mid-point value, as
a trade-off between the monitoring
FOV
and the attainable
measurement accuracy based on the theoretical analysis and
actual experiments in this paper.
Changes to the focal length of the
PTZ
camera improved the
measurement accuracy, especially at long ranges, as shown in
222
March 2015
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
