Model (10) is the viewing angle model for each element in
the
CCD
line array. This model absorbs the influence of various
geometric distortion factors.
( )
( )
tan
tan +
0
0 1
2
2
3
Ψ
Ψ
x
x
i
i
i
i
i
i
i
ax ax s s ax s s ax s s
=
(
)
+ × −
(
)
+ × −
(
)
+ × −
(
)
3
0
0 1
2
2
3
tan
tan +
Ψ
Ψ
y
y
i
i
i
i
i
i
i
bx bx s s bx s s bx s s
=
(
)
+ × −
(
)
+ × −
(
)
+ × −
(
)
(
)
3
i
=1,2,3
(10)
In the empirical test of the
GFXJ
, the
CCD
line array is di-
vided into three segments. In model (10),
s
is the id order of
the
CCD
element, and
s
i
is the id order of the first
CCD
element
of segment
i
. The image space coordinates of
CCD
element
s
are (tan(
Ψ
x
), tan(
Ψ
y
), –1). Before flight, the laboratory calibra-
tion values are (tan(
Ψ
x
0
), tan(
Ψ
y
0
), –1), where
Ψ
x
0
and
Ψ
y
0
are
laboratory-calibrated viewing angles for
CCD
element
s
. The
comprehensive influences of various deformation factors are
ax
i
0
+
ax
i
1
×(
s
–
s
i
) +
ax
i
2
×(
s
–
s
i
)
2
+
ax
i
3
×(
s
–
s
i
)
3
and
bx
i
0
+
bx
i
1
×(
s
–
s
i
) +
bx
i
2
×(
s
–
s
i
)
2
+
bx
i
3
×(
s
–
s
i
)
3
. For the
GFXJ
camera, the ad-
ditional calibration parameters are independent and different
for each
CCD
line array. The meanin
Table 1.
For the
ADS40
camera, the viewin
elements are measured every 2–5 d
pending on the length of the
CCD
line and the focal length of
the camera. The viewing angles for other
CCD
elements in be-
tween these measurements are interpolated numerically. The
measurements are converted to
CAM
files for each
CCD
line,
and these files represent the
x
/
y
coordinates for the center of
each
CCD
element with respect to the focal plane coordinate
system (Fuchs 2010). However, for the
GFXJ
, this laboratory
calibration is not efficiently carried out prior to flight. Thus,
Ψ
x
0
takes the nominal viewing angle value, and
Ψ
y
0
is calcu-
lated according to the
CCD
element pixel size and each
CCD
element’s id order.
At the
CCD
segment boundary, the viewing angles of the
adjacent segments
i
and
i
+ 1 should satisfy the equivalent
constraint,
ax ax s s ax s s ax s s
ax ax s
i
i
i
i
i
i
i
i
i
0 1
2
2
3
3
0
+1
1
+1
+ × −
(
)
+ × −
(
)
+ × −
(
)
=
+
× −
(
)
+
× −
(
)
+
× −
(
)
+ × −
(
)
s
ax s s
ax s s
bx bx s s
i
i
i
i
i
i
i
i
+1
2
+1
+1
2
3
+1
+1
3
0 1
+ × −
(
)
+ × −
(
)
=
+
× −
(
)
+
×
bx s s bx s s
bx bx s s
bx
i
i
i
i
i
i
i
i
2
2
3
3
0
+1
1
+1
+1
2
+1
s s
bx s s
i
i
i
−
(
)
+
× −
(
)
(
)
+1
2
3
+1
+1
3
i
,
=1,2,3
(11)
where
ax
i
0
,
ax
i
1
,
ax
i
2
,
ax
i
3
,
bx
i
0
,
bx
i
1
,
bx
i
2
,
bx
i
3
are the additional
parameters (APs) for
CCD
segment
i
and
ax
0
i
+1
,
ax
1
i
+1
,
ax
2
i
+1
,
ax
3
i
+1
,
bx
0
i
+1
,
bx
1
i
+1
,
bx
2
i
+1
,
bx
3
i
+1
are the APs for
CCD
segment
i
+ 1.
In addition, the first derivatives at the boundary should
remain equal due to the following smoothing constraints:
ax ax s s ax s s
ax
ax s s
a
i
i
i
i
i
i
i
i
1
2
3
2
1
+1
2
+1
+1
2
3
2
3
+ × −
(
)
+ × −
(
)
=
+
× −
(
)
+
x s s
bx bx s s bx s s
bx
bx
i
i
i
i
i
i
i
i
3
+1
+1
2
1
2
3
2
1
+1
2
3
2
× −
(
)
+ × −
(
)
+ × −
(
)
=
+
2
+1
+1
3
+1
+1
2
3
i
i
i
i
s s
bx s s
× −
(
)
+
× −
(
)
(
)
i
=1,2,3
(12)
Based on the piecewise self-calibration model (10)–(12), a
strict geometric imaging model is established for the airborne
line array
CCD
image according to the collinear constraint
equation as follows:
( )
tan
Ψ
x
S
j
S
j
S
j
S
j
S
j
a X X b Y Y c Z Z
a X X b Y Y
= −
−
(
)
+ −
(
)
+ −
(
)
−
(
)
+ −
(
)
+
1
1
1
3
3
c Z Z
a X X b Y Y c Z Z
a X X
S
j
y
S
j
S
j
S
j
S
j
3
2
2
2
3
−
(
)
( )
= −
−
(
)
+ −
(
)
+ −
(
)
−
(
)
+
tan
Ψ
b Y Y c Z Z
S
j
S
j
3
3
−
(
)
+ −
(
)
(13)
Refer to Table 1 for symbols and definitions.
For the airborne line array
CCD
image, each scanning
line has an individual perspective center and a different
set of
EO
elements. In
AT
, the number of
EO
elements to be
solved is far larger than the known number of observations.
Theoretically, the
EO
elements of each scanning line cannot be
solved. Therefore, a suitable mathematical model is needed to
simulate the position and attitude of the perspective center.
Currently, commonly used models are the DGR, PPM, and
Lagrange polynomials with variable orientation fixes (LIM)
models (Hinsken 2002; Kocaman 2008).
Empirical tests on the
GFXJ
show that the LIM model has
the highest precision and is the most stable (Ebner 1992;
Gruen 2003). Therefore, the LIM model is adopted in
AT
and
camera calibration for the
GFXJ
camera. The
EO
element model
for orientation fixes is as follows. The
EO
elements of scanning
line (
j
) are computed from their neighboring orientation fixes
(
k
) and (
k
+ 1) plus a correction term (
δ
X
j
,
δ
Y
j
,
δ
Z
j
,
δω
j
,
δφ
j
,
δκ
j
) as follows (Hofmann 1982; Müller 1991; Hinsken 2002):
X c X
c X X
Y c Y
c Y Y
S
j
j S
k
j
S
k
j
S
j
j S
k
j
S
k
j
=
+ −
−
=
+ −
−
+
+
(
)
(
)
1
1
1
1
δ
δ
j
S
k
j
j
k
j
j
j
k
c Z
Z
c
c
c
+ −
−
+ −
−
= + −
+
+
+
(
)
(
)
(
)
1
1
1
1
1
1
δ
ω δω
φ φ
φ
−
= + −
−
+
δφ
κ
κ
κ
δκ
j
j
j
k
j
k
j
c
c
(
)
,
1
1
(14)
where
X
k
s
,
Y
j
s
, …,
κ
j
are the
EO
elements for orientation fix (
k
);
X
s
k
+1
,
Y
s
k
+1
, …,
κ
k
+1
are the
EO
elements for orientation fix
(
k
+ 1); and the interpolation coefficients
c
t
t
t
t
j
k
j
k
k
=
-
-
+
+
1
1
are a
function of the time differences from the neighboring orienta-
tion fixes.
By substituting model (14) into the imaging model (13), we
obtain a more detailed geometric imaging model (15).
Z
c
1
1
Y
Y
−
+
1
1 )
)
+
+
Y
c
+
+
+ −
−
−
j
j
X b
1
1
X b
−
+
−
+
a X
a X
a X
( )
tan
Ψ
x
j S
k
j
S
k
j
j GPS
k
j
a X c X
c X X b Y c Y
c
= −
−
+ −
−
+ −
+ −
+
1
1
1
(
(
(
)
))
(
(
(
δ
)
(
(
(
)
))
(
(
c Z c Z
c Z
Z
c X
GPS
k
GPS
j
j S
k
j
S
k
j
j S
k
−
+ −
−
−
1
1
1
3
1
δ
+ −
−
+ −
+ −
−
+
(
)
))
(
(
(
)
))
1
3
1
3
c X
Y c
Y
Y c
j
S
k
j
j GPS
k
j
GPS
k
GPS
j
δ
(
(
(
)
))
(
(
(
)
Z c
Z
Z
c X
c X
j S
k
j
S
k
j
y
j S
k
j
S
k
−
+ −
−
( )
= −
−
+
1
1
1
2
δ
tan
Ψ
+
+
−
+ −
+ −
−
+ −
+
1
2
1
2
1
δ
X b Y c Y
c Y
Y c Z c Z
j
j GPS
k
j
GPS
k
GPS
j
j S
k
))
(
(
(
)
))
(
(
(
)
))
(
(
(
)
))
(
(
1
1
1
3
1
3
−
−
+
+
c Z
Z
c X
c X
Y c Y
j
S
k
j
j S
k
j
S
k
G
δ
δ
PS
k
j
GPS
k
GPS
j
j S
k
j
S
k
j
c Y
Y c Z c Z
c Z
Z
+ −
−
+ −
+ −
−
+
+
(
)
))
(
(
(
)
))
1
3
1
δ
(15)
By linearizing model (15), the following equation is ob-
tained:
v c
v c a dX a dY a dZ a d a d a d
c
x j
S
k
S
k
S
k
k
k
k
j
=
⋅
+
+
+
+ +
+ −
(
)
(
11
12
13
14
15
16
1
ω ϕ
κ
) (
⋅
+
+
+
+
+
+
+
+
+
+
a dX a dY a dZ a d
a d
a d
S
k
S
k
S
k
k
k
k
11
1
12
1
13
1
14
1
15
1
16
ω
ϕ
κ
+
− − − −
=
⋅
+
+
+
1
11
12
13
21
22
23
2
)
(
a dX a dY a dZ l
a dX a dY a dZ a
x
y j
S
k
S
k
S
k
4
25
26
21
1
22
1
23
1
1
d a d a d
c a dX a dY a dZ
k
k
k
j
S
k
S
k
S
k
ω
ϕ
κ
+
+
+ −
⋅
+
+
+
+
+
)
(
) (
+
+
+
− − − −
+
+
+
a d
a d
a d
a dX a dY a dZ l
k
k
k
y
24
1
25
1
26
1
21
22
23
ω
ϕ
κ
)
,
(16)
where
l
x
and
l
y
are constant terms and
dX, dY, dZ
compose the
ground coordinates vector.
Iterative Two-Step Calibration Scheme for the GFXJ Camera
Combining models (3), (6), (13), (16), the integral calibration
models for the
GFXJ
camera are as follows:
V AX BX CX
L P
V
E X
L P
V
A X
L P
V
A X
L P
V
E
x
g
s
x
x
s
s s
s
s
s
s
g
=
+
+
-
=
-
=
-
=
-
=
1
1
1
1
2
2
2
2
g g
g
g
X
L P
V
L P
V
L P
-
-
-
r
r
r
r
i
R X
R X
a
i i
i
i
,
(17)
648
September 2019
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
