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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
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Z
uncertainty
volume
t NS
e
eat
position
ur
sensor | sensor 2
Figure 2. Position uncertainty volume with upper (z') and
lower limit (z) for the calculated position with
coordinate z.
For each object position in the two images the position
uncertainty is +): pixel. This leads to an uncertainty of +1 for
the disparity. For small deviations of the disparity the deviation
in the z-component of the calculated three-dimensional position
is given by equation (2):
Oz z^
E Ad Ad (2)
Ac mic.
od Zz
max
It shows that for a fixed uncertainty in disparity, i.e. Ad = +1,
the uncertainty of the z-component (Az) increases with
quadratic order in z. Equation (2) is a good approximation for
the position uncertainty at short ranges, but at long ranges the
uncertainty of +1 for the disparity is not small compared to the
absolute value of the disparity. There we use the exact values
for the z-component of the farthest point (z') and the
z-component of the nearest point (2°) of the uncertainty volume
given by equation (1) but decreasing and increasing the
measured disparity by one unit, respectively, as shown in
equation (3).
EHI sigo ]
C
d ST 1 l/z jee Vz.
(3)
= z max ]
edf 1 Va Mr
max
It is important to note that this uncertainty is not a statistical
error, but a systematic error. This implies that it can not be
decreased by subsequent independent measurements. To
explain this fact, we assume that we have found the same
disparity in several subsequent stereo-image pairs. With that the
probability for an object being near the average position is not
higher than being somewhere else in the uncertainty volume,
which means that the uncertainty volume does not decrease
compared to the uncertainty volume for a single stereo-image
pair. We have to remind this fact when we discuss the
uncertainty for the velocity estimation in the next chapter.
3. VELOCITY ACCURACY
In this chapter we want to discuss the fundamental limitations
for the accuracy of the object velocity based on the accuracy of
the object position given in chapter 2. Since for objects at long
ranges the position uncertainty in the z-component is always
much greater than the position uncertainty perpendicular to it,
we restrict our discussion to the z-component of the velocity
and use for that component the variable v.
For the extraction of the object velocity we need subsequent
position determinations for the object at different points of time.
When an object is moving in space, it successively hits the
same or other uncertainty volumes, which means that the
calculated position located in the middle of each uncertainty
volume jumps from time to time by rather large amounts.
Figure 3 shows an example of this situation. All three tracks in
Figure 3 lead to alternating values of the disparity of 3 and 4.
Even though the tracks differ in the ratio of the number of these
disparity values, for long ranges this ratio is affected by
disturbance of the atmosphere and not quite reliable. Since the
difference in disparity of £1 unit can be caused by approaching
objects as well as by departing objects, this difference yields no
information about the sign of v.
calculated
positions
Figure 3. Different object tracks together with their
corresponding uncertainty volumes. The calculated
quantized positions and disparity values are
indicated.
Only after the disparity reaches a difference of +2 or -2 units it
makes sense to calculate a velocity value. Of course, the
uncertainty of the velocity at this time is rather large, but it
improves while the absolute difference of the disparity
increases. Figure 4 shows how the position and the upper and
