9-11 Nov. 1999
natical model is not linear,
order to be solved in a
ing a Taylor series to the
quation for each point of
nz ; dz dei ] 6G
| 3$ x dy
AX AY]
fferences between the two
ons, while gradients are
istructing a small surface
urface, using a planar or
| avoiding the need to
regular grid. The decision
sed on an analysis of the
duals are obtained by
er order surface is sought.
is clear that in some
meters can be accurately
ong them. The number of
epends mainly on surface
two horizontal planes for
may be determined. The
is based on an analysis of
he surface spectrum. A
r another paper.
algorithm is suitable for
s. Areas with steep slopes
ons as laser measurements
errors.
ng the matching algorithm
a synthetic data set, where
| surfaces is known, allows
n flaws. Using real data
n to actually determine the
aser surface and a surface
nents consists of a surface
hown in Figure 1. A large
nts on this surface were
face. These points are
network. To simulate the
| noise was added to the
maller set of 30 randomly
he same manner, which
e points were shifted and
erefore it was possible to
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
check the quality of the parameters resulted from the surface
matching algorithm. The surface gradients were extracted from
the target surface by calculating local bilinear surface parameters.
A stable solution for all selected sets of parameters was obtained
within 3-4 iterations of the adjustment procedure. The
determined values of the parameters were similar to those used
for simulating the transformation of the source surface. The
elevation differences between points on the transformed source
surface and the target surface were smaller than the noise that was
added initially to the target surface.
In another test with the synthetic data, the influence of large
errors in some points of the source surface was assessed when no
efforts were made to recognize or eliminate the outliers. It was
found that outliers in one or two points on the source surface do
not affect the ability of the algorithm to determine the correct
parameters.
Figure 1: Synthetic test data.
3.2 Experiments with real data
Further testing was undertaken using airborne laser data and
aerial imagery covering an urban site in Ocean City, Maryland.
These data contain points acquired by the Airborne Topographic
Mapper (ATM) laser system and panchromatic aerial
photography from NGS. The ATM is a conical scanning laser
altimeter developed by NASA for precise measurement of surface
elevation changes in polar ice sheets, ocean beaches and drainage
systems. The instrument combines high pulse rate with a scanning
capability.
Figure 2 shows part of an aerial image that was taken as a test
area. For this testing, planar surfaces of sloped roofs were
measured manually on a digital photogrammetric workstation.
Several combinations of these segments were used as the target
surface. The results of this testing are shown here using two of
these sets. The first set contains one row of buildings and the
second set contains two parallel rows of buildings.
Each planar segment was defined as a mathematical surface,
Which was determined using a least-squares adjustment. For each
segment, the appropriate points from the laser data set were found
and elevation differences between the laser points and the roof
segment were used as observations. The mathematical
formulation enables the calculations of both these elevation
differences and the gradients required for performing the
matching procedure
Figure 3 shows the laser points overlaid on the planar roof
segments. A histogram of the height differences between the roof
segments and the laser points was created, as shown in Figure 4.
The relatively large average difference in elevation between the
two data sets can be clearly observed. The histogram also shows
the existence of outliers. These outliers are actually points on the
ground, which have been mistakenly included in the areas being
matched. The outliers were eliminated using a median filter with
respect to the expected laser precision. The threshold is marked
on the histogram by a pair of vertical lines.
The algorithm described in Section 2 was applied to this data set.
Due to the geometric characteristics of the surface (the set of roof
planes), only three shifts can be determined accurately. The laser
points were transformed using the determined parameters, and are
shown in Figure 5. The respective histogram of the elevation
differences is shown in Figure 6. It can be seen that the
systematic shift of the elevations has been eliminated.
The results for the second set, which covers a larger area, are
shown in Figures 7-10. It can be seen that the systematic error is
similar to the smaller set, i.e., it is consistent over the area.
Figures 9 and 10 that the transformation was recovered in this
case as well.
Estimation of the accuracy of the parameters showed that their
standard deviation is better than 10 cm both in horizontal and
vertical directions. An elevation difference of approximately
110 cm in elevation (consistent with the visual inspection of the
histogram) and 40 cm in flight direction have been found. As
mentioned earlier, no attempt has been made at this stage to
analyze the source of these errors.
Figure 2: Image coverage of the test area.
