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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B4, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
91
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Samples from boresight, pixels
Figure 9. Difference between the corrected (xc) and distorted
(xd) pixel location in the down track direction derived from
WAC to NAC co-registration. Notice that this method also
identified a slight twist in the mounting of the WAC (~0.094°).
3.2.2 Derivation of an Improved Camera Model: The
WAC camera model is composed of several interdependent
clements that impact the accuracy of the map projection (Table
1). In order to successfully derive a new camera model, each
parameter must be solved for simultaneously. A custom
MATLAB function was composed that calculated a root mean
squared (RMS) error of the difference between the corrected
pixel location defined by the NAC image and the corrected
pixel location calculated from the distorted pixel location in the
WAC image for a camera model with a given set of parameters.
An optimization function was then used to identify the set of
camera model parameters that minimized the overall RMS.
Camera Element Parameters
Camera Pointing a, B,y
Focal Length fl
Boresight location Xo Ve
Distortion Model k;, k;
Or
Xo, o, K2, ks, ka, pj,
D» s, and s»
Table 1. Camera model elements and corresponding parameters.
3.2.3 Distortion Modeling: To account for the pincushion
distortion present in the WAC optics, a radial distortion model
was empirically derived before launch. In this distortion model,
the radial distance each pixel is away from the optical axis, r,
was calculated and used to derive the coordinates of the
undistorted, or corrected, pixel:
Xe = xaf (1 +kr? +kor)
2 3
Ye =va/(1+kr + kyr
| (3)
|
where — x, y, coordinates of undistorted, or corrected, pixel
X4, va = coordinates of distorted pixel
k;, k> = radial distortion coefficients
r = distance the distorted pixel is from the optical axis
After the launch of LRO, small band to band offsets (< 2 pixels)
in map projected WAC color images were observed. In
addition, the accuracy of the pre-flight distortion model near the
edge of the CCD had residual displacements of 1 to 3 pixels in
some bands, which was most likely due to the twist in
orientation between the CCD and the flight direction (Figure 9).
This latter displacement was noticeable in monochrome images,
which span the entire 1024 pixel CCD array. To correct for
these small offsets, a variation of the Brown distortion model
was used [Brown, 1966; Brown, 1971]. This distortion model
accounts for not only the radial distortion, but also corrects
decentering in the optics and tilt of the CCD array using the
following set of equations:
Xe = Xd +xa[kar? +kır? thy?) pir? +257) +
, ! 2
2P2X4Y4 +817
2 3 4 2 2 (a)
Ve =Yd +ya[kar *kar kar MZ *2xj J+
2p1x151 +sy”
where x',y', -decentred coordinates (ie. x'47 x,- X)
kj, k», k — radial distortion coefficients
P1, P2 = decentring distortion coefficients
51, S2 = tilting distortion coefficients
4. CURRENT RESULTS AND FUTURE WORK
4.1 NAC Calibration Results
Prior to implementing the camera pointing corrections outlined
in section 3.1, absolute NAC pointing was good to within +833
uradians cross-track and +612 uradians down-track (42 m and
31 m, respectively, from a 50 km altitude), while the relative
offset between NAC-L and NAC-R images acquired
simultaneously has 70-280 pradians (7-28 pixels). After
applying the pointing corrections, the absolute pointing error
was +639 pradians cross-track and +635 uradians down-track
(33 m and 32 m from a 50km altitude). The relative offset
between the two cameras was reduced to +5 radians (0.5 pixels
or 25 cm from the 50 km orbit) thus providing a seamless
boundary between the two simultaneously acquired NACs.
Using the new camera pointing solution, the locations of surface
hardware from the Apollo and Soviet landers were calculated
(Table 2 and 3). In Table 3, the locations of the LM and the
central station of the Apollo Lunar Surface Experiments
Package (ALSEP) were identified. The Delta True column
contains data for these three sites that have a Laser Ranging
Retroreflector (LRRR). In these cases, the “true” LM and
ALSEP positions were determined by calculating the exact
camera pointing required to place the LRRR in the correct
location. Using that vector, the coordinates of the other objects
were derived. The variation of these “true” coordinates between
images was +1.5m.
Standard
Deviation, m
Object Lat Lon Lat | Lon
Luna 16 -0.51351 | 56.36377 | 18.6 | 18.4 11
Luna 17 38.23758 |324.99816 | 15.9 | 10.9 16
Lunokhod |! | 38.31500 | 324.99169 | 13.3 | 20.4 15
Luna 20 3.78665 | 56.62414 | 15.8 | 13.6 9
Luna 21 25.99963 | 30.40923 | 6.5 | 77.7
Lunokhod 2 | 25.83273 | 30.92246 | 22.4 | 17.4
Luna 23 12.66706 | 62.15113 | 13.4 | 10.0
Luna 24 12.71439 | 62.21285 | 13.7 | 119
#of
Images
Calculated Location
[conto
Table 2. Location of Soviet hardware derived from NACs.
483