CMRT09: Object Extraction for 3D City Models, Road Databases and Traffic Monitoring - Concepts, Algorithms, and Evaluation 
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3.2 Slant range referencing 
To complete the back and forth projection process, we also need 
a computational way to reference a point, given in ground range 
coordinate, back to slant range coordinate. We simply use the 
second order polynomial linking a ground range position to its 
longitude - latitude coordinates to find back its geographical 
position. These geographical coordinates are then converted in 
Cartesian coordinates in the Earth center coordinate system and 
the range is derived computing the distance between the 
position of the sensor on its orbit and the Cartesian coordinate 
of the considered point. 
Special attention was drawn to this back and forth referencing 
process to ensure reliability and accuracy in accordance with 
VHR context. In practice, mathematically speaking, the 
referencing process can easily reach centimeter precision. 
4. SAR SCENE SIMULATION 
4.1 From aperture to simulated intensity 
As explained previously, from a DSM projected in ground 
range, we build a structure allowing to define either dihedral or 
surface backscattering. To each point on the ground range 
sampling grid, we associate what we have called an aperture, 
which is an evaluation of the incident energy intercepted by the 
dihedral or the considered surface element. Therefore, from a 
DSM, we build what might be called an aperture map. 
Each point of the ground range mesh is thus considered as a 
point scatterer to which is associated a point scatterer response 
backscattering an energy proportional to the incident one. 
The ground range mesh is then referenced back in slant range 
and the corresponding projection map is built. For each 
destination point in slant range, the projection map contains the 
coordinates of all intervening points in ground range coordinate. 
Intervening points are those that have to be taken into account 
to extrapolate the projected value at the considered slant range - 
azimuth position. After this step, we know the location of the 
centre of each intervening point scatterer response with respect 
to a given slant range -azimuth position. 
The pixel at that position receives from a given point scatterer 
response, an energy that is the integral of the impulse response, 
limited to the slant range pixel area. This integral will be the 
weight attributed to the contribution of the considered point 
scatterer response. The simulated energy is obtained summing 
all contributions of all intervening point scatterers responses for 
a given slant range pixel. 
In SAR, the impulse response or point scatterer response in 
slant range - azimuth is a sinc-like function generally 
approximated by a sine function (Bamler 1993). In terms of 
energy, we thus deal with a square sine function and our 
apertures map in slant range must then be considered as a mesh 
of square sine functions of different heights. 
In practice, computing the integral of bi-dimensional squared 
sine function on a given interval is highly complex. Therefore, 
we approximate our point scatterer response by a Gaussian 
having the same width at half maximum. The advantage is that 
calculating the integral of a bi-dimensional Gaussian on a given 
interval is straightforward (Fig. 6). One drawback is that strong 
side lobes issued from dihedral backscattering process are not 
modelled. 
♦ 
Figure 6: Integration of an approximated point scatterer 
response limited to a target slant range pixel 
Figure 7 shows the square root of the simulated image obtained 
in slant range starting from our seed DSM given in ground 
range and following the whole procedure described here-above. 
The real detected SAR image is shown on the right of the figure 
for qualitative comparison. For the sake of clarity and to 
improve contrast, the square roots of the simulated intensities 
are represented. 
If, from a macroscopic point of view, similar structures are 
roughly observable, the simulated image does show a level of 
details very far from the one of the detected SAR image. 
Reasons of having apparently so poor results may have three 
distinct origins: the seed DSM quality, the structure model used 
for estimating the local backscattered energy and the used 
parameters. 
Figure 7: Simulated SAR scene based on seed DSM structure 
When projecting the InSAR DSM onto ground range to build 
the seed DSM, available parameters are on-ground resolution 
cell dimension, semi-major and semi-minor axis of the ellipse 
used to find intervening points, weighting method and 
interpolation method. These parameters have great influence on 
the smoothing and the quality of the seed DSM. 
In the reverse process, when referencing the backscattering 
structure toward slant range, parameters are the azimuth and 
slant-range resolution to determine the point scatterer response 
width, semi-major and semi-minor axis of the ellipse used to 
find intervening points and the resolution cell dimension of the 
targeted simulated image. 
5. DSM ITERATIVE MODIFICATIONS 
At this stage, we have the tools required to link slant range and 
ground range geometries allowing a back and forth process. The 
DSM in it self is now in ground range geometry and allows 
generating a simulated SAR intensity image in slant range 
geometry to be compared to the really detected one.