2.2 Space Intersection and Back-projection 
Stereo pairs can be formed by CE-1 and CE-2 images of differ- 
ent looking angles. Based on the rigorous sensor model, the 3D 
coordinate of a ground point in LBF can be calculated by space 
intersection from the image coordinates of the conjugate points 
in stereo images, and the image coordinate can be calculated 
from 3D coordinate by back-projection. Ideally, using the 3D 
coordinate from space intersection, the back-projected image 
positions should be the same as the measured image points 
which are used in space intersection. However, due to the orbit 
uncertainty and interior orientation uncertainty, the back- 
projected image points are different from the measured point. 
The differences are called back-projection residuals. Some 
regular patterns usually appcar in these residuals, which are ex- 
tremely useful for analyzing and finding out the error sources 
and eliminating the inconsistencies. 
3 SENSOR MODEL REFINEMENTS 
In order to reduce the inconsistencies (back-projection residuals) 
of stereo images and improve mapping precision, we propose 
two methods to refine the rigorous sensor model: 1) refining 
EOPs by correcting the attitude angle bias, 2) refining the inte- 
rior orientation model by calibration of the relative position of 
the two linear CCD arrays. 
3.1 Refinement of EOPs 
The internal structure of CE-1 stereo camera is very stable be- 
cause it is implemented in one area array sensor. Thus, the 
back-projection residuals of CE-1 images are mainly caused by 
errors of EOPs. For satellite images with high altitude, the posi- 
tion parameters and the attitude parameters have strong correla- 
tion. So correcting attitude angle bias can also compensate er- 
rors caused by position parameters. 
For each image, Equation (3) can be rewritten into Equation (4), 
and further simplified as Equation (5). 
  
  
  
X-X x 
x =A-R-| y (4) 
Z-Z ~7 
s 
PAS 
u, le | 
ues y -/[ 
jul =A y «cry 6 
us[x-X, Y-Y,Z-Z[] 
  
zJX-XY40G-Yy)(2-22Zy 
  
  
u, 
where R is the rotation matrix from image space to LBF. Then 
we get the partial derivatives 0R/0@,0R/0@,0R/ 0k in the 
form of 3 X3 matrix. 
The observation equation for attitude angle bias correction can 
be represented as Equation (6) (Yuan and Yu, 2008). 
v=A-x-L (6) 
OR. , GR. » BR 2 
where 4=[— 4, — uu — u] 
Sp ‘Be Ox 
L=-R-u +u, 
x -[do' do' dk T 
u =" (i=1,2) 
lu; | 
Given a set of initial values of attitude angle biases 
asdp=0,dw=0and dk =0, rotation matrix R can be calcu- 
lated using Euler angles 9-- do , o do , and x - d« . In each 
iteration, the correction values are added to the bias values as 
shown in Equation (7). 
do € do* dp 
do «€- dodo 
dk «— dk - dk 
Iteration stops when the correction values are less than a pre- 
defined threshold. And finally the attitude angle biases can be 
figured out. 
3.2 Refinement on Interior Orientation 
Because the two linear CCD arrays are separately assembled on 
the focal plane of the CE-2 CCD camera, their relative position 
may be changed in orbit, which can cause significant errors of 
the sensor model. As a result, back-projection residuals appear 
to have a regular pattern in image space. These residuals can be 
used to refine the interior orientation by calibration of the rela- 
tive position of the two linear CCD arrays. 
Assuming the position of forward-looking CCD array and the 
principal point are fixed, the relationship between the actual 
and the theoretical positions of backward-looking CCD array 
can be modeled by scale and translation along the array direc- 
tion on the focal plane. So the interior orientation formula for 
backward-looking images in Equation (2) can be rewritten into 
Equation (8). 
y 7 y, — (col — 5,): pixsize : (1 scale) + offset (8) 
where scale and offset are the scale and translation parameters 
for the calibration of the backward-looking CCD array. The 
problem is how to determine the value of scale and offset. Ac- 
cording to the residuals distribution, we can fit a least-squares 
line (LSL) to describe the error trend so that to estimate the two 
parameters. Let col; and r are the column position and residual 
of point k, a line function can be represented as Equation (9). 
By minimizing the sum of the squares of the r,, the two pa- 
rameters a and b can be estimated. 
r,=a-(col, —s,)+b (9) 
The estimated coefficients a and b of the LSL for both forward- 
and backward-looking images are summed up as the scale and 
offset values for the backward-looking CCD array, since we as- 
sume the position of forward-looking CCD array and the prin- 
cipal point are fixed. 
292 
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