The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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1 = (T t W 2 T)-'T t W 2 G 
3. REGULARIZATION METHOD 
3.1 Regularization Model 
I m =0,W m =W(I (0) ) = E 
C> = V.) + (T'K-J + A’sr'T’XflVo 
^,=^(/ w ).v w =G-r/ (S) 
(10) 
The denominators 5 ( . and Z) (i = 1,...«) change quickly in 
quantity when the GPS input distributed unevenly in calculation, 
which makes matrix T ill-condition in the equation, and thus the 
matrix T T W 2 T singularity. This case often happens when the 
rank of RPC polynomial is higher (eg, more than 2), which may 
result in not convergent during iteration. 
A unit matrix E could be added by the regularization method to 
improve the condition number of matrix T T W 2 T . 
T T W 7 T is a symmetric nonnegative definite matrix, and the 
eigenvalue of matrix T T W 2 T + h 2 E is in the range 
so the condition number of which 
it will reduce with the increase of h 2 . A regularization rule 
method is employed in this paper to improve the condition 
number of matrix to obtain stable numerical solution. The 
normal equation is calculated through iteration and can be 
ended when the condition be met (Neumaier,1998) . 
(T t W 2 T + h 2 E)I - T t W 2 G = 0 (9) 
Where: 
E is the unit matrix, h is the regularization parameter, s is the 
number of iteration. 
When the pixel measurement error is known, the covariance 
matrix P of parameter I could be calculated from the following 
equation: 
P = (T t W 2 T + h 2 E)-' T t W 2 R c W 2 T(T t W 2 T + h 2 E) r 
(ii) 
There are many ways to obtain the regularization parameter h. 
Different h will get different result, of which the optimal value 
is attained by trial method, here L curve method (Neumaier, 
1998) is employed. To get the optimal value of h, different 
value was tested in formula (9). The third power RPC was 
employed in the test, 50 GCP and their corresponding image 
points were selected for calculation of RPC, while 29 CkP were 
used for accuracy assessment. The result was shown as figure 1. 
Based on the test, less than 10 iteration times may get good 
convergence under most conditions when h was in the range of 
0.009 and 0.1. We also found that as long as h was in the range, 
the accuracy was not sensitive to specific h, that is to say the 
results were within 0.01 pixel (see table 1). So h=0.05 was used 
in the following text. 
Figure 1 Calculation for h—L Curve Figure