field angle. In addition, some parameters are less significant in imagery geometric accuracy. If using physical measurement
model as the internal calibration model to calculate each parameter, the calculation equation would be seriously ill-
conditioned, and thus the reliability and the accuracy of the calibration could not be ensured. Therefore, although the
camera physical measurement model is rigorous in theory, it is not suitable for on-orbit internal calibration.
To solve the problem, a 2-dimensional detector directional angle model is adopted as the internal calibration model as it
shows in Figure 1. By calibrating the tangent of directional angle
,
x y
for each detector in the reference coordinate system determined by external calibration, the LOS of each
detector in the inertial coordinate system can be determined accurately.
O
X
Y
Z
Image
V
x
y
S l
Figure 1. Directional angle model of detector Polynomial model can be used to model the tangent of
directional angles of detectors. As the internal distortion is low- order because of its narrow field of view, we use an individual
three-order polynomial which has high orthogonality and low correlation as the internal calibration model.
Image cam
, , 1 tan
, , tan , , 1
T T
x y
V x y
s l s l
f f
3 where
2 2
2 2
3 3
1 2
3 4
5 6
7 8
9 2
2 2
2 3
3 1
2 3
4 5
6 7
8 9
tan , ,
tan
x y
s l a a s a l a s l a s a l a s l a s l a s a l
s l b b s b l b s l b s b l b s l b s l b s b l
,sl
is the detector’s image plane coordinate we define the original point is the centre of focal plane.
9
, , a
a
and
9
, , b
b
are internal calibration parameters
I
X
. Then, an on-orbit geometric calibration model for GF-4 can
be constructed as:
2000 2000
tan ,
tan ,
1
x g
body cam ins
J y
ins J
wgs g
body g
body
s l X
X s l
R R R
Y Y
Z Z
4
The external calibration parameter
, ,
cam E
ins
X R
roll pitch yaw
is used to compensate the camera installation angle and determine
the attitude of the camera coordinate system for internal calibration.
The internal
calibration parameter
9 9
, , , , ,
I
X a
a b b
is used to describe and
compensate camera internal distortion. A stepwise calibration is performed, external parameters
estimated, and then internal parameters estimated in a generalized camera frame determined by external parameters.
As some internal errors are included in the external calibration results, the reference coordinate system could not well represent
the real camera coordinate system. However, this does not affect the calculation of internal calibration parameters because
of the high correlation between external and internal calibration parameters on account of the narrow field angle. In addition, the
proposed flexible internal calibration model could well compensate the residual errors that caused by external
calibration, which would lower the precision requirement of external calibration. Once the internal parameters are
determined on-orbit accurately, there would be no need to update them frequently, because they are relatively more stable
than external parameters.
3.2 Estimation of the Camera Parameters
By matching the satellite image with the reference orthophoto and the corresponding DEM, we can automatically obtain GCPs.
It is necessary to use a certain number and evenly distributed GCPs to ensure the quality of the parameters estimation. To
guarantee the number and distribution of the matched GCPs, satellite image with no cloud and water cover should be selected,
and mountainous area will be better choice to achieve more texture information for auto-matching because of the relative
lower resolution. The coordinate of each control point
, ,
g g
g i
X Y Z
is in the WGS84 geocentric euclidean coordinate system, and the corresponding coordinate of image
point is
,
i
s l
in the image plane coordinate system.
N
is the number of GCPs.
According to Eq. 4 we can set:
2000 2000
1 1
1 2
2 2
3 3
3
, ,
g body
ins J
J wgs
g body
g bo
x y
dy cam
in z
s
X X
R R
Y Y
Z Z
A B C R
roll pitch yaw A B C
A B C U
U U
5
Then Eq. 4 can be transformed to Eq. 6 for external calibration.
1 1
1 3
3 3
2 2
2 3
3 3
tan ,
tan ,
x y
z x
x y
z x
y z
y x
y z
AU BU
C U F
s l A U
B U C U
A U B U
C U G
s l A U
B U C U
6
To determine external calibration parameters, we assume initial internal calibration parameters are “true”. We initialize
the external and internal calibration parameters
E
X
and
I
X
with on-ground calibration initial
E
X and
I
X . We define k the times of iteration.
Linearize Eq. 6 to get Eq. 7 as:
, ,
E k
i k i k
E
R A X
7
in which
This contribution has been peer-reviewed. doi:10.5194isprsarchives-XLI-B1-389-2016
391
, ,
, ,
k E
I E
i k k
E I
i k
F X X R
G X X
k E
k
pitch X
roll yaw
, ,
, ,
, ,
, ,
, i k
i k i k
i k E
i k i k
i k i k
i k E
F F
F F
X pitch roll yaw
A G
G G
G X
pitch roll yaw
where
k E
X
is the correction of the external calibration parameters obtained in
th k iteration.
, E
i k
R
is the residual error vector of
th i GCPs calculated by the current
,
k E
I
X X
in th
k iteration.
k E
X is calculated in least-square method:
1 k
T E
T E
E E
k k k
k k k
X A P A
A P R
8 where
1 T
k i
N k
A A
A A
1 T
E E
E E
k i
N k
R R
R R
1
, , ,
E E
E E
k i
N k
P diag p
p p
, E
i k
P
represents the weight of the observation value of th
i GCP in
th k iteration in external calibration.
Then
k E
X can be updated as
1 k
k k
E E
E
X X
X
9
We repeat the estimation iteratively until
1
-
k k
E E
X X
where
is a small positive. Eq. 4 can be transformed to Eq. 10 for internal calibration.
1 1
1 3
3 3
2 2
2 3
3 3
tan ,
tan ,
x y
z x
x y
z x
y z
y x
y z
AU BU
C U f
s l A U
B U C U
A U B U
C U g
s l A U
B U C U
10
After external calibration, we believe the modified
E
X
is true, and leave internal calibration parameters to be calibrated.
Insert the modified
E
X
into the Eq. 10 and obtain Eq. 11.
I i
i I
R B X
11
where
9 9
9
tan , tan
tan tan
tan tan
tan tan
tan tan ,
x x
x x
x I
i y
y y
y y
i I
i
d s l
d d
d d
dX da
da db
db B
d d
d d
d s l
da da
db db
dX
9 9
, , , , ,
T I
X a
a b b
E I
i E
i
f X R
g X
I
X
is the estimated internal calibration parameters.
I i
R is the vector of
th i GCP in the camera frame calculated by
current
E
X
. Then we can obtain the modified internal parameters
I
X
in least-square method:
1 T
I T
I I
I
X B P B
B P R
12
where
1 T
i N
B B
B B
1 T
I I
I I
i N
R R
R R
1
, , ,
I I
I I
i N
P diag p
p p
I i
P represents the weight of the observation value of th i
GCP in external calibration.
4. EXPERIMENT AND ANALYSIS 4.1 Experiment Data
