According to the geometric rotation relationship, these
three rotation matrices satisfy the following mathematical
relationship (Hinsken 2002):
R
b
m
(
α
,
β
,
γ
) =
R
c
m
(
ω
,
φ
,
κ
) ·
R
b
c
(
e
x
,
e
y
,
e
z
)
(4)
The
OPK
angle system is adopted in this paper. Accord-
ing to the angle system definition, the following observation
equation can be set up for angles (
α
,
β
,
γ
):
α
β
γ
=
′
′
(
)
=
′
( )
=
′
′
(
)
-arctan /
arcsin
-arctan /
23 33
13
12 11
r r
r
r r
(5)
In Equation 5, the left-hand side is the
IMU
observations for
(
α
,
β
,
γ
). The right-hand side is the calculated values for (
α
,
β
,
γ
)from model (4). If the
IMU
boresight misalignments (
e
x
,
e
y
,
e
z
)
are zero or have exact values, Equation 5 holds. Without
accurate calibration for (
e
x
,
e
y
,
e
z
), Equation 5 does not hold.
Equation 5 is the theoretical basis for the calibration of the
IMU
boresight misalignment.
If a flight lasts a long time, the drift parameters should also
be considered in the
GNSS
/
IMU
angle observation model (6).
α
β
ω
ω
φ
=
′
′
(
)
(
)
=
′
( )
(
-arctan / +
arcsin +
23 33
13
r r a t - t b
r a t - t
+
+
0
0
)
=
′
′
(
)
(
)
b
r r a t - t b
φ
κ
κ
γ
-arctan / +
12 11
+
0
(6)
By linearizing the
IMU
angle observation Equation 6, the
following
IMU
boresight misalignment calibration model (7)
can be set up:
v
v
v
B
B
e
e
e
x
y
z
α
β
γ
ω
φ
κ
=
1
2
+
∆
∆
∆
∆
∆
∆
+
+
0 0 0
0 0 0
0 0 0
∆
∆
∆
∆
∆
∆
X
Y
Z
a
a
a
S
S
S
ω
φ
κ
+ −
(
)
t t
b
b
b
L
0
∆
∆
∆
ω
φ
κ
+
,
(7)
where
B
1
is the coefficient matrix for unknowns (
Δ
ω
,
Δ
φ
,
Δ
κ
),
and
B
2
is the coefficient matrix for unknowns (
Δ
e
x
,
Δ
e
y
,
Δ
e
z
).
The other symbols are listed in Table 1. Because the
IMU
angle
observation model (6) does not involve the
EO
linear elements,
the coefficient matrix is a zero matrix.
L
is a constant matrix
whose values are the differences between the
IMU
angle obser-
vations and the calculated values from Equation 5.
In the
IMU
boresight misalignment calibration process, the
EO
elements are assumed to be constants, and the specific
calibration model for the
GFXJ
camera is proposed as follows:
v
v
v
B
e
e
e
a
a
a
x
y
z
α
β
γ
ω
φ
κ
=
+
2
∆
∆
∆
∆
∆
∆
+ −
(
)
t t
b
b
b
L
0
∆
∆
∆
ω
φ
κ
+
(8)
If the
IMU
observations do not drift over time or the route is
short, the last two terms on the right side in model (8) may be
ignored. Mostly, the last two terms are ignored.
Piecewise Self-Calibration Model
Based on the CCD Viewing Angle
Before flight, the sensor’s technical parameters (including the
principal point, focal length, and optical distortion) are cali-
brated in the laboratory. However, in practical flight, imaging
environment variations and device deformation factors will
inevitably induce changes in the sensor imaging parameters,
causing systematic errors in image positioning. (Tempelmann
2000; Hinsken 2002; Kocaman 2006, 2008; Casella 2008a,
2008b; Fuchs 2010).
The optical distortion error of the
GFXJ
camera lens mainly
includes radial distortion, decentering distortion, and plane
distortion. Radial distortion is deviation along the radial
direction caused by the camera lens; decentering distortion is
mainly caused by the inconsistency between the lens opti-
cal center and the geometric center; and plane distortion
includes image plane unflatness and in-plane distortion. The
CCD
deformation errors mainly include changes in pixel size
and translation and rotation of the
CCD
line array in the focal
plane. The changes in pixel size mainly affect the imaging
scale. The overall translation of the
CCD
line array will cause
the offset of the principal point. In addition,
CCD
rotation,
bending, and scaling will also have an effect on the offset of
the image point coordinates. For the
ADS40
camera, the Brown
model is usually used for camera lens and
CCD
distortion
calibration, and different physical geometric calibration mod-
els have also been proposed for other line scanning cameras
(Fraser 1997; Wang 2012a, 2012b, 2012c).
All of the above physical geometric models were estab-
lished based on the characteristics of various distortions
existing during camera imaging. An advantage of these
models is clear illustration and definition of each distortion
source. Their shortcomings are that too many parameters
will lead to overparameterization and strong intercorrelation
between parameters (Kocaman 2008; Jama 2011). As a result,
it is difficult for camera calibration to obtain stable and reli-
able calibration values, and previous experimental studies on
the
GFXJ
camera confirmed this finding. Therefore, we use an
empirical mathematical model to comprehensively describe
the effects of various geometric distortions regardless of the
specific physical meaning of each distortion source.
Considering the strong correlation between the camera’s
focus and other distortion parameters (such as the
CCD
scaling
factor) and adopting the
CAM
file definition of the
ADS40
camera (Schuster 2000), we put forward the normalized
viewing angle model for the
GFXJ
. For image point p, its image
space coordinates (
x
′
,
y
′
,
z
′
) are expressed as
′ = =
( )
′ = =
( )
′ = −
x
x
f
y
y
f
x
y
tan
tan
1
Ψ
Ψ
z
,
(9)
where (
x
,
y
) are the image plane coordinates and (
x
′
,
y
′
,
z
′
) are
the normalized image space coordinates. The image point
p
(
x
′
,
y
′
,
z
′
) and the ground point
P
(
X, Y, Z
) satisfy the collinear
equation model.
Ψ
x
and
Ψ
y
are the viewing angles for the cor-
responding
CCD
element, where
Ψ
x
is the viewing angle along
the flight track (flight direction) and
Ψ
y
is the viewing angle in
the direction vertical to the flight track (
CCD
direction).
For satellite-borne
CCD
line array sensors, geometric dis-
tortions such as
CCD
translation, rotation and scaling, focus
variation, and radial distortion can be described by low-order
geometric distortion curves due to a small field of view (FOV)
and long focus. High geometric distortion can be ignored.
However, the
GFXJ
has longer
CCD
line arrays and broader cov-
erage, which involves more complicated distortion factors. To
accurately describe the geometric deformation of the
GFXJ
, this
paper proposes a piecewise self-calibration model (10)–(12)
based on the
CCD
viewing angle.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
September 2019
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