/ Nov. 1999
ig point and whose
irection of the laser
urface is expressed
th 7 the measured
ser coordinate sys-
stem for transform-
changes with every
orm it into the laser
z. Its origin is also
ntation depends on
, in nutating mirror
ear with the rotation
starting position of
canning system, the
1e scan plane with z
system to the refer-
e rotation matrix R;
ad. R; is determined
‘a profiler, R; is the
am, the rotation ma-
; two angles are nec-
g mirror system. Let
lual laser beam with
| interior orientation
ser system to the ob-
s required. Thus, we
of the laser footprint
(1)
ositional and Re the
| exterior orientation
nents are usually ob-
urements. We omit
relationship between
nd the laser system
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-17 Nov. 1999
for it does not offer new insight into the comparison
laser/photogrammetry reconstruction.
Eq. 1 expresses the basic reconstruction principle for
laser systems. Note that there is no redundant infor-
mation for p; if the range or the orientation is wrong
we will find out—if at all—only later by analyzing local
properties of the reconstructed surface.
We now turn to the reconstruction of surfaces by pho-
togrammetry. Fig. 2 illustrates the concept. To compare
it as closely as possible to the case just discussed, let us
begin with a local image coordinate system (x’, y’, z’)
that has its origin at the perspective center. Its z' —axis is
collinear to the light ray from the surface point P through
the perspective center. In this system, point P on the
surface is simply obtained by
r = Ad (2)
with d = [0,0,d]" the point vector of image P in the
local image coordinate system; À is a scale factor. Since
À is unknown, it is not possible to determine P from a
single image. The standard photogrammetric procedure
is to perform the reconstruction from multiple images.
Figure 2: Relationship of local image coordinate systems
and object space reference system for reconstructing
surfaces from images.
As discussed in the case of the laser system, the local co-
ordinate system changes from point to point. Let us in-
troduce a reference system, known as image or photo co-
ordinate system. The origin is the same, but the z—axis
is collinear with the camera axis. The transformation
from the local ray system to the reference system is ob-
tained by the rotation matrix R; which is defined by the
spatial direction of the image vectorr = [x, y, — f]! . In-
troducing the exterior orientation (c, R,), we find for the
reconstruction of P
p = c + AR R;d (3)
This equation is identical to Eq. 1, except for the un-
known scale factor A. The reconstruction from pho-
togrammetry requires two or more images. This is
clearly a disadvantage compared to ALR, because work-
ing with multiple images requires that object point P
is identified on all images involved. This task—known
as image matching—is not trivial if performed automat-
ically. On the other hand, the reconstruction is redun-
dant; each additional image increases the redundancy
by two.
To complete the comparison of determining surface
points by ALR or photogrammetry, let us briefly discuss
how the exterior orientation is obtained. In photogram-
metry, two possibilities exist. In the case of direct orien-
tation, the relevant parameters are derived from GPS/INS
observations, just as in ALR systems. The traditional
approach, however, is to compute the orientation pa-
rameters from known features in object space, such as
control points. There are two important differences be-
tween these approaches. If the exterior orientation is
computed from control points then the reconstruction
of new points in the object space becomes essentially an
interpolation problem. In the case of direct orientation
(no control points in object space), the reconstruction re-
sembles extrapolation. Why is this important? Extrapo-
lation has a much worse error propagation than interpo-
lation. Another subtle difference is related to the interior
orientation, here symbolically expressed by R;. Schenk
(1999) points out that errors in the interior orientation
are partially compensated in the indirect orientation, but
fully affect the reconstruction in the direct orientation.
3 Analysis of Systematic Errors
As discussed in the previous section, the computation
of 3-D positions from range measurement and GPS/INS
observations is not redundant. If one of the variables in
Eq. 1 is wrong we will not find out, except perhaps later
when analyzing the data. Thus, it is important to con-
sider the effect of systematic errors of a ALR system on
the reconstructed surface—the purpose of this section.
As illustrated in Fig. 3, we assume two systematic errors;
a positional error q and an attitude error, expressed by
the two angles w and @ which determine the rotation
matrix R4. Although this is a grossly simplified view it
captures the notion of reconstruction errors. A likely
source for a systematic positional error is rooted in the
problem of accurately synchronizing the GPS clock with
laser pulse generator. How significant is this error? Sup-
pose an average velocity of the airplane of 100 m/sec
and a timing error of 5 msec. Then, the resulting posi-
tional error would be half a meter—clearly something to
worry about for low altitude, high precision laser altime-
try projects. A typical example of a systematic angular
error is the mounting bias.
The airplane and the laser footprint in point A of Fig. 3 il-
lustrate the data acquisition. The reconstruction of point
A is affected by the angular error a and the positional er-
ror q. Thus, the total reconstruction error is
= q+a=q+Ryr (5)
With Eq. 1 we obtain for the reconstructed point C
