The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
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So, the dimension of the 3D grid is both based on the full extent 
of the image and the elevation range of the terrain. The grid 
contains several elevation layers uniformly distributed, and the 
points on one layer have the same elevation value. The coarsest 
subdivision both for 2D grid definition and for layers spacing is 
dependent on the need to point out a sufficient points number 
for the RPCs estimation; on the other hand, the finest 
subdivision depends on the incompressible error of the 
geometric reconstruction used to generate the RPCs, so that a 
very fine discretization is unuseful and an upper discretization 
limit also exists. 
The RPCs least squares estimation (Tao et Hu, 2000) is based 
on the linearization of the generic RPFs equations, which can 
be written as (1): 
!n +b l^n I n +- + b 17^n I n +b 18 b n I n ~ a 0 ~ a l<Pn --- a l8^n -a 19 b n = ° (1) 
Jn +d i^nJn +...+d 17 X,nJ n +d 18 hnJ n -c 0 -CjiPn -...-c 18 X.n -c 19 hn =0 
where a,, b„ c 15 dj are the RPCs (78 coefficients for third order 
polynomials), I n , J n and cp n , ^ ,h n are the normalized coordinates 
obtained thought the equation (2) with scale and offset factors 
computed according to the equations (3): 
calculated so that the columns of the matrix B1 eiKmxr 
in AP=[B t B 2 ] are “sufficiently independent”) 
• B1 is the matrix used to estimate the RPCs 
Moreover, the statistical significance of each estimable 
coefficient is checked by a Student t-test so to avoid over- 
parametrization; in case of not statistically significant 
coefficient, it is removed and the estimation process is repeated 
until all coefficients are significant. In most of the cases the 
‘degrees of freedom’ are high (more than 100), thus there could 
be considered infinite, converting the t-Student distribution in a 
normal standard distribution. The confidence interval chosen is 
95%, so the value of the Student-t distribution was taken fixed 
to 1.96 (Millard, 2001). 
Finally, the generated RPCs are used for the image orientation; 
in the SISAR software an algorithm is implemented for the 
RPCs application that allows also for a possible refinement 
process based on shift or affine transformation. 
For each investigated image the RPCs (SISAR RPC) are been 
extracted using the known sensor model, implemented in 
SISAR software, with a specific number of GCPs and with a 3D 
grid (9x9x9), both sufficient conditions to have an accuracy 
assessment. The number of SISAR RPC are about 1/3 with 
respect to the standard number used in third order polynomial 
(78 RPC). The accuracy is, in worse case, close to 1.5 pixel 
when just 5 points are used (Tab. 4). 
T n = T - T ° ffset where T 
^scale 
^offset = min(w k ) 
w scale = max(w k )-min(w k ) 
^offset = J offset = ^ 
Igeale = n°Column -1 
Jscale = n°ro w — 1 
where 
<pA,h,I,J 
w = (p,A./h 
IMAGE 
n° SISAR 
RPC 
RMSE CPIpixj 
AFT 
FORE 
AFT 
FORE 
I 
J 
I 
J 
Rome 
22 
24 
0.93 
0.61 
1.26 
1.04 
Castelgandolfo 
24 
23 
1.04 
0.71 
0.97 
0.68 
Warsaw 
21 
26 
0.81 
0.59 
0.95 
0.69 
Mausanne 
22 
23 
1.35 
1.08 
1.43 
1.07 
where k is the number of available ground control points (GCP) 
and n° column/row are the overall columns/rows of the image; 
the normalization range is (0, 1). 
Deeper investigations underlined that many RPC coefficients 
are correlated; Tichonov regularization is usually used. On the 
contrary, in this work the Singular Value Decomposition (SVD) 
and QR decomposition are employed(Giannone, 2006). 
For a system of linear equations (Ax=b), with AeiPmxn (m>n), 
a SVD-based subset selection procedure, due to Golub, Klema 
and Stewart (Golub et al., 1993; Strang et al., 1997), proceeds 
as follows: 
• the SVD is computed and used both to calculate the 
approximate values of RPC to normalize the design 
matrix A and to determine the actual rank r of its; the 
threshold used to evaluate r is based on the allowed 
ratio between the minimum and maximum singular 
values; reference values are 1O* 4 ^-1 O' 5 (Press et al., 
1992) 
• an independent subset of r columns of A is selected by 
the QR decomposition with column pivoting QR=AP; 
in a system of linear equations (Ax=b), if A has a rank 
r, the QR decomposition produces the factorization 
AP=QR where R is diagonal matrix, Q is orthogonal 
and P is a permutation (the permutation matrix P is 
Tab 4. SISAR RPC number and RMSE on CP for investigated 
images 
3. IMAGE MATCHING 
The image orientation is only the precondition for the geometric 
correct use of the image information. One important issue of the 
stereo satellite Cartosat-1 is the generation of height models. 
With the spectral range from 0.50 up to 0.85pm wavelength 
large parts of the near infrared are included, giving optimal 
conditions also over forest areas. 
An automatic image matching has been made with the 
Hannover program DPCOR. It is imbedded in the measurement 
program DPLX allowing a fast check of the matched points. 
DPLX is using a least squares matching, having no accuracy 
limitations for inclined areas like the image correlation. The 
least squares image matching includes an affine transformation 
of the sub-matrix of one image to the sub-matrix of the other 
image. In addition a constant shift and linear changes of the 
grey values with both coordinates are included, leading to 9 
unknowns. The precise matching by least squares has a 
disadvantage of a low convergence radius - the corresponding 
image positions must be known on a higher level. In DPCOR 
this is solved by region growing. Starting from at least one 
corresponding point, the neighboured points are matched. Such 
a seed point may be a control point, which has to be measured