International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part BS. Istanbul 2004 
  
lower uncertainty boundary, z' and z according to equation (3), 
evolve with time for an object approaching the sensors with 
constant speed. The track starts at time 7, with a value for the 
z-component of z, and reaches at the current time / the value z. 
For this diagram we waive the quantization of the position since 
for increasing difference in disparity this will be less important, 
but we have to keep the uncertainty range of «1 unit for the 
disparity. All possible trajectories with constant velocity which 
are consistent with the measured data are defined by the set of 
all straight lines which lie totally inside the boundaries z^ and z. 
The extreme cases, upper (v') and lower (v) boundary for the 
calculated velocity, just touch the position boundaries as shown 
in Figure 4. There is no reduction of the velocity uncertainty 
depending on the number of measured positions as known from 
regression due to the fact, that the position uncertainty is not a 
statistical error as discussed at the end of chapter 2. 
  
A 
distance 
  
  
  
v 
f, 
time 
  
  
  
Figure 4. Distance over time, position uncertainties (z^ and z), 
and limits for the fitted average velocity (v' and v) 
for an object approaching with constant speed. 
Obviously the difference between v* and v' decreases while the 
object approaches the sensors and therefore the position 
uncertainty at the current time ? decreases. As a measure for the 
uncertainty of the velocity we introduce the relative velocity 
error Av/v given by equation (4): 
  
  
  
Av yu 
(=, > z) =, J 
V ys 
2 2 2 
Z = Z 
Ele enl 
z z (4) 
z, max 0 max 
# 2 2 
max a = z bir Zg 
z max z 0 zZ max 
The minus sign in the first line of equation (4) is to get positive 
values for the relative velocity error since the velocities v' and 
v have negative values for approaching objects. The values z 
and zy in equation (4) have the same meaning as in Figure 4, 
except zo is not necessarily the start-value of the track, but only 
some earlier and therefore greater value than z. The difference 
between zg and z is the track-length taken into account for the 
velocity calculation, which can be the whole or only part of the 
track. Inspecting equation (4), we find that the best choice 
regarding a small value for the relative velocity error is to use 
for z the smallest possible distance, which is the current 
distance, but to vary zo up to a maximum value which is set by 
the finite track-length. We expect an improvement of the 
relative velocity error with increasing difference between z and 
zo. This is only true until zy reaches a certain distance z,, which 
is given by equation (5): 
; 1 
Z opt (z) = 7 3 7a ; (5) 
ZU zZ VE as LE 
max 
  
In fact, further increasing of zo worsens the relative velocity 
error. The reason for this is that the boundaries for the position 
uncertainty, z' and z, are of second order in z but the fitted 
average velocity is represented by a linear line. This means that 
the choice of zo = Zop: (if the track is long enough to allow this) 
leads to a minimal uncertainty for the object velocity achievable 
for the current distance z. 
In summary, we found two fundamental limitations for the 
velocity determination. First, for an object at a current distance 
z we need a minimal track-length which must be in conformity 
with a difference of 2 units of disparity and second, the relative 
velocity uncertainty is limited by a minimal value achieved by 
selecting the optimal track-length (z,-z) for velocity extraction. 
In Figure 5 these results are shown in a diagram. The lower 
solid line shows the minimal track-length, which is necessary 
for velocity extraction, depending on the current distance given 
by the normalized value z/z,4,. The track-length is given by the 
normalized value (zo-zYz,,;: The upper solid line shows the 
optimal track-length, which is necessary to reach the minimal 
velocity uncertainty. The dashed lines show the relative 
velocity uncertainty obtained by choosing a track-length 
between the two limits. 
Of course, the track-length taken into account for velocity 
extraction is limited by the length of the whole track. For 
example, this limit is shown in Figure 5 by the dotted line for an 
object detected first at z/Zmax = 0,27. As this object approaches 
the track-length increases. At the distance z/z,4, 7 0,17 the 
whole track-length is long enough to recognize that the object 
approaches. While the object further approaches we are free to 
select any track-lengths for velocity extraction between the 
minimal track-length and the maximal track-length which is 
either given by the whole track-length or the optimal 
track-length. For example we can choose the whole 
track-length, which increases, until it exceeds the optimal 
track-length, which then decreases. This yields the best velocity 
uncertainty achievable at every current distance of the object 
but at the expense of a rather long track-length taking into 
account for the averaged velocity, what means that the velocity 
is averaged also over a rather long time interval. Or we can 
choose a fixed velocity uncertainty and after reaching this 
  
    
   
   
   
   
   
   
   
   
    
   
     
    
     
  
  
  
   
     
  
  
    
    
    
     
      
   
  
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