9-11 Nov. 1999
between the two sur-
axis which may not be
's. Take the extreme
values contain no in-
rfaces are.
e comparison method
rences along surface
e data sets to a com-
m statement, the pro-
ibe the mathematical
,P2;--. ‚Dn} be a sur-
s p that are randomly
Qm) be a second sur-
ributed points q. Sup-
? describing the same
| Systems. In the ab-
| is the model system
ct space system. After
| — $5, except for dif-
the observed points p
arise from the discrete
or example, n - m.
different distribution
ted surface. Suppose
sets are known to be
| problem is now to es-
the two sets such that
as similar as possible
| is cast as an adjust-
et of points q is trans-
e differences between
dditionally, the orien-
$1 and S» can also be
nces assures the best
erences in surface nor-
ints q be transformed
first set by a 3-D simi-
(19)
fined by the shortest
1. Two scenarios are
Let us first approx-
q' by a plane, for ex-
points p confined to
tch). Then, the short-
ce patch, expressed in
ree directional cosines
1, is
) (20)
servation equation for
r = (sRq—t)-h-p-d (21)
The observation equations are not linear, hence, approx-
imations for the transformation parameters are neces-
sary. Habib and Schenk (1999) describe an elegant
method of obtaining transformation parameters and sur-
face differences.
If the points p of the surface patch cannot be satisfacto-
rily approximated by a plane, then a second order sur-
face can be used. Should this also fail, then surface
patch is not suitable (not smooth enough) for the pro-
posed procedure and no observation equation is formed
for this particular point q. If it can be sufficiently approx-
imated, then the situation depicted in Fig. 14 applies.
The distance from q to the surface is measured along
the surface normal.
point of second surface p q
/
/ surface normal
data points
!
|
|
residuals
Figure 14: Illustration of determining the shortest dis-
tance between point q and surface patch SP,.
4.3 Post-Processing
The raw 3-D laser point data sets represent physical sur-
faces in a discrete manner. Because they lack an explicit
description of surface properties, meaningful informa-
tion must be extracted. For most applications, the ex-
plicit knowledge of surface properties, such as discon-
tinuities (in elevations and surface normals), piecewise
smooth surface patches, and surface roughness, is es-
sential.
Fig. 15 depicts a post-processing schema of raw laser
data points. Depending on the application, some or all
of the steps are followed. In this paper we have stressed
the fact that there is no redundant information available
for computing the 3-D position of laser points. On the
other hand, the discrete representation of surfaces by
randomly distributed laser points is highly redundant.
For example, three points suffice to define a planar sur-
face patch, but most likely, hundreds of laser points are
available. Statistical blunder detection methods exploit
this redundancy. Apart from this traditional methods,
there are also reasoning-based approaches for checking
the data. Such methods would try to explain the data
given the hypothesis of a segmented surface.
Thinning is related to the redundancy of the points.
From a practical point of view, thinning is recommended
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
raw 3-D laser points other sensory input
blunder detection
thinning
I
ground/non-ground separation
gridding
/
segmentation
I
model-based object recognition
Figure 15: Schematic diagram of the most important
post-processing steps of raw laser data sets. By and
large, the application determines whether all or even ad-
ditional steps are necessary.
to reduce the size of the (huge) data sets. Since some
points carry more relevant surface information than oth-
ers, the cardinal question is what points can safely be
eliminated. Theoretically, thinning should be treated
as a resampling problem with the objective function to
minimize the loss of surface information. This, in turn,
requires knowledge about the surface topography—a
problem that segmentation tries to tackle.
Most every post-processing scheme includes the interpo-
lation of the quasi-randomly distributed laser points to a
regular grid (gridding), motivated by the fact that there
is a plethora of algorithms available to process gridded
data. It is almost equally popular to convert the inter-
polated elevations at the grid posts to grey levels. The
resulting range images are now in a suitable representa-
tion for image processing.
Segmentation is the next step in our processing schema.
The goal is to make surface properties explicit. Such
properties include surface discontinuities (e.g. abrupt
elevation changes, abrupt changes of surface nor-
mals), piecewise continuous surface patches, and sur-
face roughness. Surface properties are essential in ob-
ject recognition. As Csathó et al. (1999) point out, seg-
mentation is not yet a standard procedure in processing
laser data sets. Data thinning and blunder detection,
frequently performed in an ad-hoc manner and with pro-
prietary algorithms, is immediately followed by an at-
tempt to detect objects, for example, buildings. It is well
known in computer vision that such shortcuts are dan-
gerous. Success with one example does not guarantee
generalization of the method.
A more general approach to object recognition requires
