Minimizing the objective function of Equation 15 is equiv-
alent to minimizing the sum of squared image coordinate re-
siduals of the collinearity equations (i.e., the classical bundle
adjustment error function) (Fusiello and Crosilla, 2015).
Weight Matrix of the 3D Coordinate Observations of Tie Points
Forward intersection aims to produce a metric 3D reconstruc-
tion of a scene as it is being observed by the binocular cam-
eras. In general, the baseline of the binocular cameras is short,
and the intersection angle is smaller when the features are
farther from cameras. In Equation 16, we should analyze the
reconstruction precision of features at different distances and
design a weight matrix of the tie points. In that case, the tie
points play an important role in the new rover pose estima-
tion method.
Suppose that
F
L
is the reference frame and that the im-
age coordinates of the left camera and right camera are
(
x
L
,
y
L
),(
x
R
,
y
R
), respectively; the 3D coordinates of the tie
points (
T
1
,
T
2
, …,
T
n
) denoted by (
X, Y, Z
) in
F
L
can then be
expressed as
X
=
N
1
X
L
,
(21a)
Y N Y N Y B
L
R y
=
+
+
1
2
1
2
(
) , and
(21b)
Z
=
N
1
Z
L
,
(21c)
where
N
B Z B X
X Z X Z
N
B Z B X
X Z X Z
x R z R
L R R L
x L z L
L R R L
1
2
=
−
(
)
−
(
)
=
−
(
)
−
(
)
,
,
denotes the projection coefficient of the stereo vision system,
B
y
is equivalent to
X
Y
Z
x
y
f
X
Y
Z
x
L
L
L
L
L
L
L
R
R
R
R
R
=
−
=
R
R
,
y
f
R
R
−
,
R
L
is
the identity matrix,
R
R
equals
R
containing
φ
0
,
ω
0
,
κ
0
, and
f
is
the focal length of the binocular cameras. In (Di and Li, 2007),
the authors propose that the accuracies denoted by the stan-
dard errors (
σ
X
,
σ
Y
,
σ
Z
) of the coordinates could be calculated
with measurement error of parallax (correlation/matching
error) and the measurement errors of the image coordinates
in the horizontal and vertical directions. When considering
the errors of the camera-external parameters
Q
xx
, it is crucial
to calculate
σ
X
,
σ
Y
,
σ
Z
based on Equation 21, and the specific
expression follows as:
σ
σ
σ
φ
X
L
L
L
b
L
L
L
L
N
x
b
x b
f
x b f
f
x b y
z
=
−
+
− +
+
−
1
2
2
2 2 2
2
0
(
)
(
)
L
L
L k
f
y
+
2
2
2 2
0
0
σ
σ
ω
, (22a)
σ
σ
Y
L
L
L
b
L
L
L
L
N
y
b
x b
f
x b x b
f
f
z
=
−
+
−
−
+
1
2
2
2
2
2
2
(
)(
)
+
−
+
+ −
−
(
)
2
2
2
2
0
0
2
2
2
2
σ
σ
φ
ω
(
)
x b y
f
f b
y
y
b x b
y
L
L
L
L
L
L
L
L
2
2
1
2
2
0
4
σ
σ
k
b
N
y
+
, (22b)
σ
σ
σ
φ
Z
L
L
L
b
L
L
L
L
N
f
b
x b
f
x b f
f
x b y
z
=
−
+
− +
+
−
1
2
2
2 2 2
2
0
(
)
(
)
L
L
L k
f
y
+
2
2
2 2
0
0
σ
σ
ω
. (22c)
where
b
=
x
L
–
x
R
≈
B
/
N
1
.
Taking the images (1024 pixels × 1024 pixels) of the CE-3
lunar rover’s stereo vision system as an example, we set the
moving distance of the rover at approximately 6 m, which is
efficient for navigation, and the intersection angle between
the optical axis and the ground plane is approximately 30°.
When the accuracies of
σ
X
and
σ
Y
are simulated by using
Equation 22a, and 22b, it is worth noting Figure (3a and 3b),
which shows that the image coordinates containing maximal
standard errors are selected at the edge of the overlapping
area of the binocular camera
FOV
, so
σ
X
will increase in the
-axis direction (horizontal) and
σ
Y
will increase in the -axis
direction (vertical). Owing to the image coordinates with 2D
characteristics, we obtain the accuracy of
σ
Z
by using
the equation
σ
Z
i
n
m i
Z Z
n
=
−
(
)
−
=
∑
1
2
1
instead of Equation 22c.
Z
m
is the actual measured value of coordinate
Z
based on the
experimental data, and the even distribution of all measured
3D feature points (
n
= 92) concentrates upon the opposite
direction of the optical axis.
In Figure 3a, all image points lie on the horizontal line (the
values of
x
range from −385 to 512, and
y
is 0), and the paral-
lax is 127 pixels, while the distance of all feature points is
approximately 4.5 m. In Figure 3b, the distance of the feature
points (the values of
y
range from −512 to 512, and
x
is 0)
theoretically ranges from 0.8 m to 6.2 m. As shown in Figure
3, the accuracy of coordinate
X
is equal to coordinate
Y
and
is obviously greater than coordinate
Z
. Thus, the accuracy
of coordinate
Z
is sensitive to the distance from the feature
points to the cameras. In addition, the numerical curve of
σ
X
and
σ
Y
is approximated as a straight line, which indicates
that the accuracy is steadily decreasing slowly. In contrast,
the numerical curve of
σ
Z
follows an exponentially increasing
law. In a word, the weight matrix
W
n
of the coordinate (
X,Y
)
observations of the tie points
j
(
j
= 1, …,
n
) can be set as 1.0.
Correspondingly, matrix
W
n
of the coordinate observations
should be indicated as:
W Z m
W
Z
m Z m
W
j
j
j
j
j
j
=
≤
= − × −
(
)
< ≤
1 0
2 0
1 0 0 3
2 0 2 0
4 0
. ,
.
;
.
.
. , .
.
,
= × −
(
)
>
0 4
3 0
4 0
2
.
.
,
.
.
Z
Z m
j
j
.
(23)
It is important to note that the general form of matrix
W
n
comes from experience and is not the only one. Therefore,
Equation 23 indicates that the weight of coordinate
Z
is sig-
nificantly lower than coordinates (
X,Y
).
In the above analysis, the point is that we set the error of
the parallax as 1/3 pixel and the errors (
σ
X
,
σ
Y
) of the im-
age coordinates as 0.12 pixels and 0.18 pixels by using the
self-calibration bundle adjustment algorithm. In addition, the
standard errors of the camera-external parameters in Equation
22 come from a previous section.
Experimental Results
The experimental results are presented in two sub-sections.
First, the epipolar geometry of the binocular cameras is evalu-
ated by using the image sequences and ground truth data from
CE-3 lunar rover. Second, the proposed localization method
is compared with the
BA
method based on the ordinary
LS
(Alexander,
et al
., 2006; Di,
et al
., 2008), parallaxBA (Zhao,
et al
., 2015), and
LM
algorithms (Li,
et al
., 2016) in terms of
accuracy, efficiency and convergence.
The lunar rover works in a ground test field that is similar
to the lunar surface environment. The 3D coordinates of all
control points in control reference frame
F
C
are measured by
two electronic stations whose angle measurement accuracy
is 20 and whose coordinate measurement accuracy is 0.12
″
mm. The feature points on the rover and twelve of the control
points in the reference frame
F
IGPS
are then measured using
610
October 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
