x
d
= (
sample
–
BORESIGHT_SAMPLE
) ·
PIXEL_PITCH
(3)
r
=
x
d
(4)
x
=
x
d
/ (1 +
k
1
·
r
2
)
(5)
where
PIXEL_PITCH
is the pixel size of the image,
BORE-
SIGHT_SAMPLE
is the principal point coordinate,
x
d
is the
distorted position (the measured position),
r
is the distance
between the optical center and image point,
k
1
is the distor-
tion coefficient, and
x
is the corrected focal plane position
in millimeters (mm). The
IO
parameters of the left and right
cameras (
NAC
-L and
NAC
-R) can be found in the
SPICE
kernels
of the
LRO
mission. The
NAC
cameras are line scanners (single-
line
CCD
);
y
d
is unmeasured and probably unimportant, and
thus
y
is assumed to be zero according to the
LROC
Instrument
Kernel file (NAIF 2014; Liu
et al.
2017).
The
RFM
is represented as the ratio of the polynomials
shown in Equation 6 as follows:
r
P X Y Z
P X Y Z
c
P X Y Z
P X Y Z
n
n n n
n n n
n
n n n
n n n
=
(
)
(
)
=
(
)
(
)
1
2
3
4
, ,
, ,
, ,
, ,
(6)
where (
r
n
,
c
n
) and (
X
n
,
Y
n
,
Z
n
) are the normalized image coor-
dinates and ground coordinates, respectively. The third-order
polynomial
P
i
(
i
= 1, 2, 3, and 4) has a general form as follows:
P X Y Z a a X a Y a Z a X Y a X Z a Y Z a X
i
n n n
n
n
n
n n
n n
n n
n
( , ,
)
= + + + +
+
+
+
+
1 2
3
4
5
6
7
8
2
a Y a Z a X Y Z a X a X Y a X Z a X Y
n
n
n n n
n
n n
n n
n n
9
2
10
2
11
12
3
13
2
14
2
15
2
+
+
+
+
+
+
+
a Y
a Y Z a X Z a Y Z a Z
n
n n
n n
n n
n
16
3
17
2
18
2
19
2
20
3
+
+
+
+
)
(7)
where
a
1
,
a
2
… to
a
20
are the coefficients of the polynomial
function
P
i
, named rational polynomial coefficients (
RPCs
).
The
RFM
of the
LROC NAC
imagery was established by least-
squares fitting with a large number of virtual control points
generated by
RSM
of the image (Di
et al.
2018).
points in a certain interval were created first in
as the virtual control points in image space, aft
elevation in the object space was divided into
ers and the planar ground coordinates of the virtual control
points in every layer were calculated using the
RSM
. Finally,
the
RPCs
were derived using these virtual control points via
least-squares fitting (Liu
et al.
2017).
Subarea Planar Block Adjustment
The accuracy of the constructed
RSM
of
LROC NAC
imagery
depends on the accuracy of the orbit and attitude of the
LRO
.
Consequently, the fitted
RFM
also contains errors at the same
level as that of the
RSM
. Benefiting from lunar gravity field data
of the Gravity Recovery and Interior Laboratory mission, the
LRO
orbit determination obtained an accuracy of ~20 m. The
accuracy was further improved to ~14 m after incorporating
crossovers of
LOLA
data (Mazarico
et al.
2012). The errors of
the
RSMs
and
RFMs
of the images cause positional deviations of
adjacent rectified images that should be reduced to a subpixel
level to better support engineering and science applications.
Photogrammetric block adjustment is an effective means
to improve the geopositioning accuracy of a geometric model
(Gwinner
et al.
2010; Wu and Liu 2017). Traditionally, three-
dimensional ground coordinates of the tie points are solved
using stereo block adjustment. However, if the stereo conver-
gence angle is very small (e.g., <10
°
), the normal equations of
the block adjustment will be ill-conditioned, and as a result,
the calculated ground height will be abnormal. This is wide-
spread in our experiments because of the lack of coverage of
stereo
LROC NAC
pairs.
The traditional
DOM
registration is also a widely used meth-
od to remove the geometric deviations between images with
low convergence angles. This is usually accomplished with
the help of high-precision control data. However, the lack
of high-precision control data of the lunar surface limits the
registration precision. It is hard to make the
NAC DOMs
geomet-
ric seamless by using the traditional registration method with
control points from presently available
DOM
mosaics or
DEMs
.
To resolve the problem, a
DEM
-aided planar block ad-
justment was developed to refine the
RFMs
of the
LROC NAC
images. In order to ortho-rectify the
LROC NAC
images and
support collaborative analysis of produced
DOM
mosaic and
existing
DEM
in various science applications, the geometric
models of the images should be corresponded to the reference
DEM
. Thus, in each subarea block adjustment, a few control
points were manually selected using
SLDEM2015
as the refer-
ence. The control points are mostly centers of small craters
and are evenly distributed in the research area. The extracted
feature points in every
NAC
image are matched automatically
to obtain tie points. After that, the distribution of tie points is
checked carefully to ensure that every overlapping area has
evenly distributed points. If such a condition is not satisfied,
manually selected tie points will be used. Most of the
NAC
im-
ages (750 out of 765 images used) were taken in the afternoon
and have similar illumination conditions, and they can be
automatically matched for tie point selection. A very small
number of images (15) were taken in the morning and have se-
vere illumination differences with neighboring images taken
in the afternoon, and manual selection is necessary to obtain
tie points in these images. Image matching under different
illumination condition is still a challenging issue. Recently,
Wu
et al.
(2018) have done some related work in automatic
matching of planetary images using illumination-invariant
feature points. The issue is worth of further study to make the
adjustment process more efficient.
The error equations of the block adjustment are shown in
Equation 8. Compared to stereo block adjustment, planar block
adjustment is a method that calculates only the tie point ground
hile the elevation coordinates can be inter-
In the block adjustment, the affine transfor-
age space (Equation 8) is used to compen-
errors rather than recalculating the
RPCs
:
F e e sample e line x
F f f sample f line y
x
y
= + ⋅
+ ⋅
−
= + ⋅
+ ⋅
−
0 1
2
0 1
2
(8)
where the image coordinates acquired by back-projecting the
ground coordinates with the
RFM
(as shown in Equation 6) are
represented by
sample
and
line
,
while
the measured image
coordinates are shown as
x
and
y
;
e
0
,
e
1
,
e
2
are the affine trans-
formation parameters in the sample direction, and
f
0
,
f
1
,
f
2
are
the affine transformation parameters in the line direction.
Because the error equations of planar block adjustment are
nonlinear relative to the ground coordinates, a Taylor series
expansion is used to linearize the error equations as shown in
Equation 9. The unknowns to be solved include the tie point
ground planar coordinates (
lat
,
lon
) and the affine transforma-
tion model parameters as shown in Equation 9. The elevation
coordinates are interpolated from a
DEM
iteratively:
v
F
e
e
F
e
e
F
e
e
F
lat
lat
F
lon
x
x
x
x
x
x
=
∂
∂
⋅
+
∂
∂
⋅
+
∂
∂
⋅
+
∂
∂
⋅
+
∂
∂
⋅
0
0
1
1
2
2
∆
∆
∆
∆
∆
lon l
v
F
f
f
F
f
f
F
f
f
F
lat
lat
F
x
y
y
y
y
y
−
=
∂
∂
⋅
+
∂
∂
⋅
+
∂
∂
⋅
+
∂
∂
⋅
+
∂
0
0
1
1
2
2
∆
∆
∆
∆
y
y
lon
lon l
∂
⋅
− ∆
(9)
484
July 2019
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
