Figure 3: Illustration of a systematic positional and an-
gular error. The angular error causes the footprint in A
to be shifted by a to point B. The positional error q trans-
lates B to C which is the reconstructed surface point,
expressed by point vector p'.
p =p+q+R,RR;r (6)
3.1 Reconstruction Errors Caused by Positional Er-
rors
Profiler and Horizontal Surfaces Fig. 4(a) depicts a
laser profiling system during data acquisition over flat
ground and a vertical object, say a building. The re-
construction shown in Fig. 4(b) illustrates the effect of
a positional error. À timing error, At, causes a shift of
sS=A U (7)
with v is the velocity of the airplane. Assuming a con-
stant timing error, it appears that its effect is simply a
shift of the reconstructed points. This is not quite true
as the following analysis reveals.
Fig. 5 illustrates the typical situation where an area is
covered by adjacent flight lines, flown in opposite direc-
tions. We notice that the shift s in both flight lines is also
in opposite direction, causing a distortion of the recon-
structed object. Consequently, the reconstructed object
space is not simply a translation of the true surface but
includes angular distortions, even in the simple case of
horizontal surfaces flown by a laser profiling system.
Finally, Fig. 6 shows the relationship between the posi-
tional error q and the flight direction. With « the azimuth
of the flight trajectory, we have
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
AR ANY N° NY NN NY Mo NW
S (a)
(b)
Figure 4: The data acquisition of a laser profiling system
is shown in (a). The reconstruction in (b) shows a shift
s due to a positional error, caused, for example, by a
timing error between GPS clock and laser pulse emission.
Figure 5: The data acquisition of a laser profiling sys-
tem shows the typical flight pattern necessary to cover
an extended area. The reconstructed rectangular object
is distorted, because the shift s is in opposite direction
between two adjacent flight lines.
sige sin «
q = [= = lal fo s (8)
The magnitude in this equation is identical to the shift s
of Eq. 7. Eq. 8 demonstrates clearly how different direc-
tions of the flight lines affect the reconstruction. Only
under the unlikely condition of « being constant is the
reconstruction a simple shift. All other data acquisition
scenarios will cause angular distortions.
Figure 6: The
tional error q
line,.here ex[
Profiler and
the effect of
Let us NOW (
Fig. 7. As b
the reconstri
elevation err«
st
recon
1e —
Figure 7: Il
sloped surfa
slope angle )
Here, s’ is th
gradient of t
trajectory an
Fig. 8). Then
two azimuth
An analysis c
elevation err
firms our ini
