280 
x'=x+ax= 
MP,LM) 
p 2 (p,l,h) 
+ AX 
Y=Y+AY= 
MP,LM) 
M p ,LM) 
+ AZ 
(3) 
The adjustable functions and AT are typically 
polynomials of the image coordinates X and Y. Such methods 
are known to be particularly applicable to a system with a 
narrow field of view imaging a relatively flat area (Ian Dowman, 
2000). However, refinements to the RPC camera model are 
required that will be valid for systems with a large field of view 
and moderately hilly terrain (Jack Grodecki, 2003). In order to 
further improve the accuracy of the sensor model, more 
correction items can be added in image space. 
T Z 
Figure2. Adjustment of object space coordinate system 
Different from traditional approach of translating, rotating and 
scaling imaging parameters that describe the sensor system to 
remove the biases in the measured imaging parameters, the 
object space coordinate system may be mathematically rotated, 
translated and scaled in an attempt to remove the biases in the 
biased. The adjustment of the object space coordinate system 
may be implemented as an adjustment of object space 
coordinates before the application of the given RFM by using 
nine physical parameters. Therefore, refining the RFM then 
only requires adjustments to nine physical parameters and 
processing of significantly fewer GCPs. The RFM with the set 
of coefficients is then applied to the set of new object space 
coordinates to result in a pair of image space coordinates. One 
refined RFM defined as the follow equations can be adopted: 
Since the sensors of Beijing-1 small satellite are a linearity 
pushbroom CCD camera, the focal plane is one line of detection, 
in contrast to frame camera, wherein the focal plane is a two 
dimensional image plane. If the flight direction is perfectly 
oriented in the north-south direction, a more correct formulation 
for refinements to RFM of a pushbroom camera is detailed 
below. 
p 
>0' 
1+5, 
0 
0 " 
"hi 
"hi 
~P 
>' 
L' 
= 
4 
+ 
0 
1 + ^2 
0 
"hi 
"ha 
L 
+ 
L 
H' 
A. 
0 
0 
1+S3 
."hi 
"hi 
"hi. 
H 
H 
However, it is very unlikely that the flight direction is oriented 
perfectly in the north-south direction. This lack of perfect north- 
south orientation may be accounted for by skewing the latitude 
and longitude coordinates by an orientation angle on the 
skewing the latitude and longitude coordinates and then reverse 
skewing the latitude and longitude coordinates by the same 
orientation angle[27SUN]. 
p 
^0 
cos 9 
sin# 
0' 
1 + 5, 
0 
0 
m u 
m n 
«7,3 
0 
P t 
L 
= 
A, 
+ 
-sin# 
COS# 
0 
0 
1 + $2 
0 
A«,, 
m n 
"hi 
L, 
+ 
0 
H 
H 0 
0 
0 
1 
0 
0 
l + s, 
m >: 
m„ 
H 
0 
where 
# = arctan 
ÔP 
av 
,8P 
p t ' 
cos# 
sin# 
0' 
P' 
L, 
= 
-sin# 
cos# 
0 
L 
H s 
0 
0 
1 
H 
In practice, most of the object space coordinates can be in a 
linear measurement for easting or northing. So equation (4) and 
related equations can be simplified to minimize the use 
engineered parameters and thus provide a more generalized 
solution. With small rotational angles including pitch angle 60 , 
roll angle ^ , and yaw angle K and scaling factors 
equation can be rewritten as the follow equations. 
P 
M 
cos y 
sin^ 
0" 
'•*1 
-K 
CO 
'0' 
M 
L 
= 
4 
+ 
-sin?' 
cosy 
0 
K 
S 1 
-<p 
4 
+ 
0 
H' 
A. 
0 
0 
1 
-CO (p 
s i - 
H 
0 
L = 
H 
where ^ °’ °’ °‘ are translating factors of physical parameters; 
(sl,s2,s3) are scaling factors of physical parameters; Both of 
them are acted for adjusting latitude, longitude and height, 
respectively ( m wi m \2, m M, m n, mm i\t m n," 1 33) arg 
rotating factors that are organized in a rotational matrix 
determined from physical parameters which are called 
rotational angles including pitch angle ^ ; ro n angle ^ , and 
yaw angle ^ . The rotating factors are determined from the 
rotational angles in more than one relation. One standard 
relation expresses as follows (Sun Jiabin, 2003): 
m,, m, 2 m, 3 
i"n "hi “a = 
m,, my, 
which may be expanded and generalized as: 
P =a 0 +q-P+a 2 -L+a } -H+P 
• L — /?Q+^’.P+ô,'Z/+7y//+/. 
H —c 0 +c l -P+c 2 -L+c i -H+H 
From which 12 adjustable coefficients the equations may be 
identified. However, not all of the 12 coefficients are 
independent. The 12 adjustable coefficients are determined 
based on the nine physical parameters including three 
translation factors, three rotation angles and three scaling 
factors through equations (4) and (5) before simplification and 
equations (8) after simplification. 
To improve the geometric accuracy of refinement to the RFM, a 
given set of GCPs may be added to the refined method. So, 
optimizing the 12 adjustable coefficients for a given sensor can 
begin with initializing the physical parameters and calculating 
the initial values of the adjustable coefficients. This kind of 
orthorectification based on RFM can diminish the influence of 
Po 
l + s. 
0 
0 
"’ll 
m n 
m,. 
~P~ 
4 
+ 
0 
l + s 2 
0 
m u 
m 22 
m 2 . 
L 
A. 
0 
0 
1 + s, 
_m 3 , 
m 12 
H 
COS COS AC — COS 40SÜ1AC+ sin ç>sin COCOS AC sin ty sin AC+cos <psin ¿yCOS K 
COS ty sin AC COS ^7COS AC+sin^siniysin AC —SUI Ç/COS AC+COS ^JSÛl ÛASÜl AC 
-sin ¿y sin <pCOS ¿y COS (PCOS ¿y