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P'PS = /z-tan or, (5) 
D = POP' + P'PS = (h/tana 2 ) + {h• tana x ) (io) 
D = POC P' + P'PS = {h-tan a x ) + (h/ tana,)(6) 
h _ D tan^) ^sin (2or,) 
tan 2 («!) + l 2 
(7) 
3.4 Height estimation from combined SAR/optical data 
As it will be shown in this paragraph, estimates for the heights 
of elevated objects can also be deduced from combined optical 
and SAR data. Simply, the distance between the displaced 
bridge in the optical image and the displaced bridge in the SAR 
image has to be measured. 
OPT SAR 
Figure 8: Height estimation of an elevated object by means of 
combined optical and SAR imagery 
In Figure 8 an elevated object P with height h is imaged by both 
an optical sensor (OPT) and a SAR sensor (SAR). The two 
sensors acquire the object with two different off nadir angles 0] 
and 0). The corresponding depression angles are cii and a2. 
Object P is projected to two different locations in the images 
due to the different viewing geometries of the sensors. On the 
one hand, the optical sensor measures angles and hence P is 
imaged to point PO in the optical image. On the other hand, the 
SAR sensor measures distances from a side-looking perspective. 
Therefore, P is projected to point PS in the SAR image 
(layover). Object P’s displacement can be expressed as (8) in 
the optical image and as (9) in the SAR image (assuming a flat 
wave front). The measured distance D between the displaced 
object P in both images is the sum of the two separate 
displacements in the optical and in the SAR image. Adding the 
two separate shifts leads to equation (10). Finally, the elevation 
of object P can be estimated with (11) (Inglada & Giros,2004). 
(8) 
(9) 
h = D 
tan (a 2 ) 
tan(a,)-tan(« 2 ) + l 
(ii) 
These equations have shown that only three parameters have to 
be known in order to estimate the height of an elevated object 
from combined optical and SAR imagery: the incidence angle 
of each sensor and the distance between the displaced object in 
the optical image and in the SAR image. In our case we assume 
that all terrain in the imagery has already been accounted for by 
ortho-rectifying the images with a DEM since we are only 
interested in the bridge’s height. None-the-less, equation (11) 
also allows for the estimation of three-dimensional surface 
models in case no DEM is available. In such a case the images 
would have to be registered using a local transformation. A 
disparity map could be drawn and the terrain heights could be 
deduced from the displayed shifts. Again, for such kind of 
application a locally flat earth has to be assumed between the 
projections PO and PS of object P. 
4. SUMMARY AND FUTURE PERSPECTIVES 
A comparison of bridges over water and bridges over land has 
shown that the appearance of bridges in SAR imagery strongly 
depends on their environment. Due to multi-path signal 
propagation at bridges over water, a height value can be 
determined directly from a single image. Such multiple bounces 
do usually not occur at bridges over land. First concepts of this 
long term project now have shown that a three-dimensional 
modelling of elevated man-made objects is possible from a 
single SAR image and from combined optical and SAR data. 
In a next step the equations shown in this paper will be used in 
order to obtain estimates for the railroad bridge near Zellingen 
(Figure 4, 5, and 6). A field campaign will be done in order to 
measure precisely the bridge height at each pillar. Hence, 
estimates obtained from the imagery can be compared to the 
real heights and thus be evaluated. 
So far all displacement measurements are done manually. 
Further steps of this project will be the integration of automatic 
feature extraction and object based image registration 
techniques. The bridges will be modelled on object level. 
ACKOWLEDGEMENTS 
The authors thank Dr. Essen, Dr. Schimpf and Mr. Brehm of 
FGAN-FHR for providing the SAR test images of the 
MEMPHIS sensor and Intermap Company for providing the 
InSAR data set. 
REFERENCES 
Cadario, E., Schulz, K., Gross, H., Hammer, H., Thiele, A., 
Thoennessen, U., Weydahl, D.J., Soergel, U.,2008. Feature 
extraction and change detection for bridges over water in 
airborne and spacebome SAR image data. In: Proceedings of