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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3b. Beijing 2008 
pixel displacement. In principle, this way is not measuring the 
main resolution. But to make a procedure tractable, resolution is 
measured without considering the effects of other parameters. 
The geometric error, a positional error in correspondents, is 
calculated by making a grid over the whole image area for both 
images. This grid is transformed by the real transformation 
parameter values and by the estimated transformation parameter 
values for the reference and the candidate image respectively. 
The maximum, minimum and the number of displacements 
larger than one pixel are also recorded. Our final decision is 
based on having no displacement larger than one pixel. 
For each percentage of the amount of the disturbance, 
parameters are estimated and geometric and parameter errors 
are calculated. To find the boundaries, the amount of errors is 
increased until the estimated parameters result in a wrong 
transformation. 
3.4 Failure Mechanism 
One obvious reason why the DE method may result in a wrong 
transformation is when the obtained transformation has a lower 
energy value, given the simulated errors, then the energy value 
that corresponds to the real transformation. I.e. in this case the 
minimum of the energy function is no longer corresponding to 
the real solution. 
Another case occurs when the estimated parameter values are 
wrong although their energy is higher than the energy of the 
real parameter values. This case corresponds to the failure of 
the optimizer in finding the global minima with our settings 
even if the minimum is shifted. If the DE setting parameters are 
not sensitive enough, if may be necessary to increase the 
number of generations, NG, in combination with using a smaller 
multiplication factor, F, and a small cross over probability, pc, 
to find the global minimum. In our parameter space, the special 
projective parameters v, and v 2 (i.e. the fifth and sixth 
parameters) are less sensitive than the other transformation 
parameter of Equation 1 in changing energy value especially by 
increasing amount of disturbances. This sensitivity is reduced 
by increasing the amount of disturbances. Changing the 
optimizer settings in these cases is likely to succeed, of course 
at the cost of increasing the computational effort. The added 
value seems not high which results either in very little 
increasing acceptance boundary or very little increasing the 
rejection boundary. We consider this case also as a failure. 
The amount of the disturbances is increased until the global 
minimum no longer corresponds to the real parameter values. 
Then the amount of errors is reduced and the optimizer is run a 
few times till the result of all runs are correct. Otherwise the 
errors are reduced and run again. 
The above-mentioned procedure is done for both simulation 
types to find the minimum amount of disturbances cause failure. 
The method therefore can handle disturbances lower than this 
amount. 
4. RESULTS 
Our image sequences are recorded from a non-stable platform, 
in this case a helicopter hovering above a highway. These 
image sequences are used to collect statistics concerning the 
behavior of drivers of all vehicles on a highway stretch in busy 
(nearly congested) traffic during an elongated period of time. 
Typically, we record highway stretches with a length of 
300~500m during one hour or more. We use a b/w camera with 
1392 x 1040 pixels which gives a ground resolution of approx. 
25-40cm, at a frame rate of 15 fps. The transformation 
parameters (S, 0, t t , t 2 , v /( v 2 ) used for the simulation in this 
paper are: 
[0.9942 -0.7184 6.3931 8.1876 1.1395e-5 -2.4079e-5] 
The number of generation, NG, population size, q, 
multiplication factor, F, and cross over probability, pc, are 
respectively 50, 16, 0.6, and 0.55. All the calculations are done 
in a second fine image scale of an image pyramid and the 
results are scaled up. The range of the parameters for the 
maximum 10 pixel movement is: 
[1-0.0912 -0.8232 -10 -10 -2e-5 -3.7e-5] 
[1+0.0912 0.8232 10 10 2e-5 3.7e-5] 
respectively for lower and higher band. The resolution of the 
parameters for one pixel movement is: 
[1+0.0091 0.0823 1 1 2e-6 3.7e-6] 
Figure 1 and Figure 2 demonstrate the maximum amount of 
allowed moving objects and illumination variations respectively 
for two different data sets. The result represents the fact that the 
second data set (highway crossing) can handle both a large 
amount of moving objects and a larger amount of illumination 
variations. The boundaries of the acceptance of the method are 
represented in Figure 3 and Figure 4 as the amount before the 
star on the x-axis for the moving objects and illumination 
variations respectively for two the different data sets. The 
amount after the star indicates the rejection boundaries either 
because of the failure of the optimizer within our settings, the 
amount between the star and the rectangle, or because of the 
real failure, the amount after the rectangle. The star is an 
example of the optimizer failure. The y-axis demonstrates the 
absolute parameter error divided by the resolution. This error is 
visualized for each parameter. 
Figure 1: Moving objects before failure in data set 1 (up) and 
data set 2 (down). The left figures are the reference images and 
the right ones are the candidate images. The difference between 
the images is the transformation of the whole image and object 
motion.