In: Wagner W„ Szgkely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
This is schematically shown in Figure 1. Object B is displaced 
differently depending on the incidence angles 6\ and 0 2 . By 
measuring the displacements p 2 and p 2 the relative height h of 
the object can be calculated with Eq. 1. 
h- (i) 
cot 0 2 ± cot 6 X 
The difficult and time consuming part of the StereoSAR method 
is the search of homologous points in both images. In Stereo 
SAR this is rather difficult because of the speckle noise and the 
large differences in geometry and radiometry between the two 
SAR images acquired under different incidence angles. 
Our search of homologous points is based on pyramid layers of 
the images. The search starts at the highest pyramid level, 
subsequently refining the search using lower pyramid levels. 
There are many different search criterions suggested for SAR 
(Tupin and Nicolas 2002), but in our experiments we achieved 
the best results using the two-dimensional normalized correla 
tion (see also Fayard et al. 2007) 
r = - L - L (2) 
(«-l)5 r 5 m 
where r is the reference image and m is the match image, r 
and m are the mean values of the reference and match image 
values inside the correlation window, s r and s m are the standard 
deviation of the values inside the correlation window and n is 
the number of pixels in the correlation window. In higher 
pyramid levels a smaller correlation window is chosen, but in 
lower pyramid levels the correlation window gets bigger. 
One problem of the pyramid level based approach is that small 
errors on high pyramid levels can propagate and cause large 
errors and outliers in the final result. In previous experiments 
we found outliers with height errors of more than 250 m in a 
rather flat terrain of our test area (Balz et al. 2009) caused by 
this. 
The SRTM dataset can be used to improve the accuracy of the 
process by providing reliable initial values for the iterative 
search process. Furthermore, the SRTM data can also be used to 
filter large outliers. 
3. USING THE RPC MODEL FOR STEREO 
RADARGRAMMETRY 
A sensor model is established to relate the image coordinates 
and corresponding object coordinates. Rigorous physical sensor 
models accurately represent this relationship, but using them is 
very time-consuming when positioning each pixel by a rigorous 
sensor model. Moreover, the sensor parameters are needed in 
rigorous sensor model, which may violate the confidential rules 
of commercial companies. Although the rigorous sensor models 
are more accurate, the development of generic sensor models 
with high fitting accuracy, real-time processing ability and sen 
sor-independent features is very useful. 
The rational polynomial coefficients (RPC) model is a typical 
generic sensor model which describes the relationship between 
image space and object space by ratios of polynomial functions. 
It has been successfully applied in geo-coding of high- 
resolution optical imagery, such as IKONOS, QuickBird, 
GeoEye-1, etc. (Volpe, 2004; Fraser & Hanley, 2005), and has 
become a standard component for processing optical data. 
Our investigation indicates that the RPC model for TerraSAR-X 
data has an excellent fitting accuracy (Wei et al. 2010). The 
RPC model uses two main forms for the forward (Eq. 3-4) and 
two main forms for the inverse transformation (Eq. 5-6). 
.. P t (XJ,Z) 
r — (3) 
p 2 (X,Y,Z) u 
p 3 (X,Y,Z) 
p 4 (X,Y,Z) { 
V pVr.c.Z) 
A = (5) 
p2(r,c,Z) 
p3(r ,c,Z) 
p4(r,c,Z) (6) 
r and c are the normalized row and column indices in image 
space, X, Y and Z are the normalized 3D object coordinates. The 
purpose of normalization is to improve the numerically stability 
of the equations. The image coordinates and object coordinates 
are both normalized to values between -1 and 1 using 
v — y c — c 
f. _ °rg offset _ org offset 
'scale 
''scale 
Y — Y Y —Y 7—7 
yr org offset y 1 or 8 1 offset ^ ^ org " offset 
(7) 
X. 
scale 
1 scale 
J scale 
where the subscript org means original coordinates, and offset 
and scale are parameters for normalization. 
In RPC-based geo-coding, the RPC parameters need to be 
solved in advance. A virtual regular object grid covering the full 
extent of the image with several elevation layers is established. 
Each grid point’s corresponding image row and column indices 
can be calculated using the rigorous physical sensor model and 
the satellite ephemeris data. After the image coordinates and ob 
ject coordinates are obtained, the RPC parameters can be esti 
mated using a least-squares solution. 
The accuracy of RPC-based geo-location is affected by the 
computation accuracy of RPC parameters directly. This is the 
difficult point, because the equations are usually very ill- 
conditioned. There are two main categories for solving ill-posed 
equations: biased methods, like ridge trace, GCV, or L-curve, 
and unbiased methods, like the iteration method by correcting 
characteristic value (IMCCV). The ridge trace method is quite 
time-consuming with a low accuracy. The GCV method is 
sometimes not converging, which is a major drawback. The L- 
curve method is fast with a high accuracy. But the method is a 
little less accurate than the IMCCV, while the IMCCV takes a 
lot of time for the iterative improvement and relies heavily on 
the initial data. Therefore, we use the L-curve results as the ini 
tial data for the IMCCV. This approach can get very accurate 
results fast. 
Given a over-determined linear system 
AX-L (8) 
and using the least square solution, the results of A! - will be 
X = (A T PA)-'A T PL (9)