The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part Bl. Beijing 2008
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thought as initial attitude angles. The more precise camera
parameters are obtained by iterative process based on the least
square technique.
Different from traditional coordinate systems in
photogrammetry, the attitude angles in the celestial sphere
coordinate system are shown in figure 1 .cc 0 , S 0 and k denotes
the attitude angles (Wang, 1979). When compared to the
distance of the star to the Earth, the distance of star sensor to
the Earth can be omitted, so the geocentric position is always
considered be the projection centre (Xu, 1998), thus each image
has only three attitude angles as exterior orientation elements.
Figure 1. The attitude definition in the celestial sphere
coordinate system
Rigorous collinearity equation in the celestial sphere coordinate
system is shown as below:
y x +Ay~ f Q ' cosacos ^ + ^\ sinacos£+c,
,° <f cosacos8+b 3 sinacos<J+c 3 sin<!> ^
. r cu cosacosS+L sirmcos^+c, sin^)
y-y 0 +Ay = -f— 2 -
a, cos a cos o+b 3 sin a cos 8+c 3 sin 8
Where Cl = the right ascension of the star,
/3 = the declination of the star,
(a i , c i ) (i=i,2,3) = Nine elements in the attitude rotation
matrix.
Ax , Ay = the distortions of the CCD.
Ax^x-a^XV* 2 +kf 4 +k i r 6 ),
A T=(y-.yoXV 2 +V 4 +V 6 )
r = V(x-x 0 ) 2 +(T-T 0 ) 2
K 1 ,k 2 ,k 3 = the second-order and fourth-order and sixth-
order coefficients of r in the radial distortion
For single star point, if the image point is viewed as observed
value (v), and the principal point offset (x 0 , yo) and focal length
(/), and the attitude angle are considered as unknowns, the error
equation can be founded as follows:
Acp
dx dx dx dx dx dx
A co
dtp dco dK df 3y 0 <dy 0
A k
x-(x)
y,_
dy dy dy dy dy dy
¥
y-(y)
dtp dco dK df dx 0 dy 0
Ay 0
.Avo.
(2)
The differential coefficient of equation (2) can be derived. 2n
error equations can be given if there are n points. The optimal
unknowns are obtained by iterative process based on the least
square technique. The process is stopped when the correction of
unknown parameters are less than the predefined thresholds.
2.2 Experiment procedure
The experiment procedures are shown as figure 2. The real star
catalogue is adopted in this experiment to provide the right
ascension and declination of the guide stars. Suppose the
camera parameters before calibration and the real camera
parameters are known, the star image coordinates can be
simulated based on the imaging principle when the real attitude
angles are set. Initial attitude angles are computed with these
simulated star images coordinates and the camera parameters
before calibration, which are considered as initial attitude
angles. The more accurate camera parameters and attitude
angles can be obtained based on space resection with initial
camera parameters and attitude angles. The difference value of
the calibrated camera parameters and the real camera
parameters can be thought as the standard to evaluate the
calibration accuracy.
During the simulated process, the random position error within
0.1-0.5 pixel is added into image points individually. At present,
the extraction accuracy of the star image points has achieved
0.1 pixel (Quine et al., 2007), so the simulation is reliable.
Figure 2. Experiments procedure of on-orbit calibration