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