Hans-Gerd Maas
As a first test, the implemented technique was applied for matching 100 points which were manually chosen on buildings
in the overlap area of two crossing strips (Figure 3). The results, averaged over these 100 circular patches with a radius of
12m, are summarised in the following:
e Setting the acceptance criteria quite rigidly, 74% of the
matches were accepted.
e The average standard deviation of unit weight was 12.1
cm.
e The average standard deviation of the shift parameters
was 2.1cm/2.0cm/1.0cm in X/Y/Z. Related to the average
point density of 0.3 points per square meter, this corre-
sponds to a relative precision of the planimetric shift
parameters of ~'/g) point spacing.
e The maximum standard deviations were 3.0cm/4.0cm/
1.3cm in X/Y/Z.
Figure 3: Part of dataset Eelde with chosen patch centers
In particular, the estimated standard deviations of the horizontal shift parameters seem much too optimistic. This is a
result of the fact that the design matrix in LSM is derived from observations with stochastic properties; in combination
with the noise properties and the point density this causes the values in the covariance matrix to be too large. This
problem can only partly be compensated by filtering of the data, as filtering will often create non-existent gradients in the
vicinity of range data edges. Edge preserving smoothing would be a solution, but has not been implemented to irregu-
larly distributed data yet.
A more realistic figure on the precision of the determined shift parameters can often be obtained by comparing the
results of patches with the patch center point slightly shifted over a small range in X and Y. Averaged over the 74
successfully matched patches, this approach indicated a precision of 4.5cm/3.7cm/0.7cm for the shift parameters in
X/Y/Z. This figure may be too optimistic as well, as further outlined in the following chapter.
4. Refinement of the algorithm
A general requirement for the application of LSM techniques is the presence of a bandwidth-limited signal. This is not
fulfilled over flat areas and at height discontinuities such as building edges. While flat areas provide information for the
determination of the vertical shift parameter and tilted roof planes with gradients in at least two different non-opposite
directions provide information for the determination of the two planimetric shift parameters, discontinuities do not
contribute to the determinability of parameters and may lead to convergence problems or even biased solutions as
shown in chapter 2.3.
In the case of the example shown in chapter 3, this problem becomes obvious as inconsistencies between results for
neighbouring patches. Figure 5 (left) shows shift parameters for the successfully matched points of the test area shown
in Figure 3; a trend, which should be expected as a consequence of strip errors, is hardly recognisable in these results.
These inconsistencies are caused by local singularities, which emerge after the exclusion of irregular TIN-meshes in
regions of occlusions. A patch containing a building with a standard gable roof (Figure 4 - left), for example, will show a
singularity in the gable direction: the normal vectors on the ground and the two roof planes form a plane, and the shift
perpendicular to this plane (i.e. in gable direction) is not reliably determinable.
This problem does not become obvious from the covariance matrix obtained from LSM, or from an analysis of the local
variation of the patch center, as outlined in chapter 3. As a consequence of the noise in the laserscanning height data in
combination with the fact that the design matrix is generated from observations with stochastic properties, the estimated
standard deviations and correlation between parameters are too optimistic and often do not indicate these singularities.
In practice, even for patches extracted over perfectly flat terrain, the standard deviations of the planimetric shift parame-
ters were sometimes less than 10cm in a dataset with an average point spacing of 1.8m. The same applies for the standard
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 551
