where
X
is the vector of the changes in the unknown
EO
ele-
ments;
X
g
is the vector of the changes in the
GCPs
;
X
s
is the
vector of changes in the additional calibration parameters
ax
i
0
,
ax
i
1
,
ax
i
2
,
ax
i
3
,
bx
i
0
,
bx
i
1
,
bx
i
2
,
bx
i
3
; X
r
is the vector of changes
in the unknown
GNSS
lever arms (
Δ
u
,
Δ
v
,
Δ
w
); X
i
is the vec-
tor of changes in the unknown
IMU
boresight misalignments
(
Δ
e
x
,
Δ
e
y
,
Δ
e
z
);
V
x
is the residual vector of image point coordi-
nates;
V
s
is the residual vector of additional calibration param-
eters;
V
1
and
V
2
are the residual vectors of the equivalent and
first-order constraints;
V
g
is the residual vector of the ground
coordinates;
V
r
is the residual vector of the
GNSS
lever arms;
V
i
is the residual vector of the
IMU
boresight misalignment; (
A
,
B
,
C
,
A
1
,
A
2
,
E
s
,
E
g
, R
a
, R
i
) are the corresponding design matrices;
L
g
,
L
s
,
L
1
,
L
2
,
L
r
, and
L
i
are the corresponding discrepancy vec-
tors; and
P
x
,
P
s
,
P
1
,
P
2
,
P
g
,
P
r
, and
P
i
are weight matrices.
The integral calibration model (17) considers the
EO
ele-
ments, the
GNSS
lever arms, and
IMU
boresight misalignment,
and the deformation factors of the forward/nadir/backward
CCD
line arrays, fully describing the
of the
GFXJ
camera. However, althou
tion model (17) is comprehensive, t
obtained by different experimental
are still some unavoidable correlati
integrated calibration process of
EO
elements,
GNSS
lever arms,
and
IMU
boresight misalignment, and additional calibration
parameters, resulting in random assignment of positioning
errors and instability of calibration values. Therefore, we pro-
pose that the
AT
of
EO
elements, the calibration for
GNSS
lever
arms and
IMU
boresight misalignment and the additional self-
calibrating parameters should be performed independently
and iteratively.
An iterative two-step scheme is put forward for the
AT
and calibration process. The whole process is divided
into the inner calibration process and the outer calibration
process. The outer calibration process comprises the inner
calibration process and the
GNSS
lever arms and
IMU
boresight
misalignment calibration. The inner calibration process deals
with the
AT
and the camera lens and
CCD
line calibration.
When the inner process is completed, the
CAM
files are
generated, and the scheme turns to the outer calibration
process. The flowchart of the iterative two-step scheme is
illustrated in Figure 6.
The details of the iterative two-step calibration scheme are
as follows:
1. First, the rigorous imaging model (15) is set up for the
GFXJ
image, and the
GNSS
/
IMU
observations are converted into
the UTM map projection coordinate system (either the lo-
cal tangent plane coordinate system or geocentric Carte-
sian coordinate system can also be used).
2. Then, the GPU-accelerated automatic image matching
algorithm is implemented on the
GFXJ
image to extract a
large number of tie points for
AT
.
3. Next, using
GCPs
and tie points, an
AT
block is con-
structed, and the LIM model is used during
AT
. The inner
calibration process starts. The whole inner calibration
process contains the following steps.
a.
AT
is performed based on the first and fifth equations in
model (17) with
GNSS
/
IMU
observations as initial values
for
EO
elements. After
AT
,
EO
elements for each orienta-
tion fix and unknown
EO
correction vector (
Δ
X
s
,
Δ
Y
s
,
Δ
Z
s
,
Δ
ω
,
Δ
φ
,
Δ
κ
) are ascertained.
b. Keeping
EO
elements for each orientation fix constant,
APs calibration is performed according to the second,
third, and fourth equations in model (17) for
ax
i
0
,
ax
i
1
,
ax
i
2
,
bx
i
0
,
bx
i
1
,
bx
i
2
.
c. Based on the calibration values of APs
ax
i
0
,
ax
i
1
,
ax
i
2
,
bx
i
0
,
bx
i
1
,
bx
i
2
, image pixel coordinates (tan(
Ψ
x
), tan(
Ψ
y
))
of each element on the forward/nadir/backward
CCD
line arrays are calculated by the piecewise calibra-
tion model (10) and written into
CAM
files for the
GFXJ
camera.
Using
CAM
files, the image pixel coordinates (tan(
Ψ
x
),
tan(
Ψ
y
)) can be updated. Using updated image pixel coor-
dinates, the
AT
process is carried out again according to step
a). The inner calibration process is carried out from step a) to
step c) until the
EO
elements (
Δ
X
s
,
Δ
Y
s
,
Δ
Z
s
,
Δ
ω
,
Δ
φ
,
Δ
κ
)
and
additional calibration parameters
ax
i
0
,
ax
i
1
,
ax
i
2
,
ax
i
3
,
bx
i
0
,
bx
i
1
,
bx
i
2
,
bx
i
3
tend to be stable and the changes between two itera-
tions are less than the threshold.
4. After the inner calibration process is finished, the scheme
turns to the outer calibration process.
Figure 6. Flowchart of the iterative two-step calibration
scheme for the
GFXJ
.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
September 2019
649
