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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