The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3b. Beijing 2008 
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Figurel. Geometry of the 3D building with the position of 
the sun and the satellite 
2. STEREO VS MONO IMAGE FOR HEIGHT 
EXTRACTION 
In a typical imaging system, the 3D real world object after 
passing through the camera/sensor is project onto a 2D image. 
The mathematical model that links the 3D object coordinates to 
the 2D image coordinates is known as the camera model. 
(l,s) = CameraModel(<f>,A,h) 
Where ((f), A,,h) is the 3D object coordinates and (/, s) is the 
2D image coordinates. A popular replacement camera model 
that is used by IKONOS and other satellite systems is the 
Rational Polynomial Coefficients (RPC): 
(l,s) = RPC(</>,A,h) 
Whether rigorous Camera Model or RPC, one can see that to 
invert the process, i.e. to compute 3D object from 2D image 
coordinates with one image, one has three unknowns 
((f), A,h) and two observations (/, s) which cannot be solve 
uniquely. The usual solution is to use another image that covers 
the same object, i.e. use of stereo imagery. 
If we know the height, e.g. from known DEM, the latitude and 
longitude of the object which is on the ground (in our case, the 
base of the building) can be determined. 
With the simple geometry as seen in Figure 1, we can deduce 
that the height of a building (relative height coordinates) can be 
computed without knowledge of the DEM, provided the 
building is vertical, i.e. same latitude and longitude coordinates. 
The DEM is required to determine the latitude and longitude 
coordinates of the base (which is same as the latitude and 
longitude of the top of the vertically standing object). 
The last fact is very important as it allow us to measure the 
building location and height with any coarse DEM and to 
improve the accuracy of the location and height (which is 
unchanged) when more accurate DEM is available without the 
Figure2. 2D satellite image of a building with the base and 
shadow clearly located, p is the top, b is its base and s is its 
shadow. 
need to remeasure all the buildings. A function of the software 
for this task has been implemented in the package as well. 
3. RPC SENSOR MODEL REFINEMENT 
For high accuracy determination of object coordinates from 
image coordinates or vice versa, there may be a need to 
improve the camera model with ground control points (GCPs). 
In traditional photogrammetry, the GCPs are used to refine 
physical orientation parameters such as the rotation angles 
and/or translation shift. For RPC, the rotation and translation 
parameters are all absorbed into the cubic polynomials. Luckily 
there are other indirect techniques for refinement of the RPC 
camera model. 
One way is to finetune the normalized sample and line 
parameters of the RPC in image coordinates (Fraser et al., 2003, 
Grodecki et al., 2003) with an affine transformation: 
s' = <ar n + a, • s + gn • / 
(1) 
1' - b 0 + by • s + b 2 ■l 
where 
s' and /' are sample and line coordinates of the GCP 
calculated from RPC; 
s and / are the sample and line coordinates of GCP 
observed in the satellite image; 
a 0 , a,, a 2 , b 0 , by, b 2 are the adjustable coefficients. 
This is akin to early days of aerotriangulation strip adjustment 
with polynomials. 
We have developed a method to refine RPC in space domain by 
finetuning the object space latitude, longitude and height 
parameters of the RPC with the following set of equations: