topography. This suggests to determine the systematic
errors based on known deviations of the computed sur-
face. Take Eq. 16, for example. This simple equation
permits to compute the effective angular error ó', such
as a mounting bias error. Another example is the de-
termination of a positional error of a profiling system.
Eq. 11 can be used to determine the effective positional
error 5'. However, this would require that we determine
elevation differences Az between the true and the com-
puted surface—a trivial problem at first sight. A closer
examination reveals a fundamental problem: how do we
compare two surfaces that are represented by irregularly
distributed points? Section 4.2 addresses this general
problem.
Other challenges of how to calibrate ALR systems loom
ahead. Some systematic errors are correlated. For ex-
ample, one can obtain the same elevation differences
or surface deformations either with a positional error or
angular error. Imagine the mounting bias error is in the
flight direction. Now, the effect of the angular error is
very similar to a positional error—the two error sources
cannot be separated. The calibration of cameras is also
confronted with the same problem. Here, the parame-
ter dependency is solved by choosing a proper calibra-
tion surface and decoupling those parameters that are
closely correlated. Choosing a suitable topography of
the calibration surface is an important issue in calibrat-
ing ALR systems. There is no consensus as to what type
of surfaces should be used to determine systematic er-
rors and calibration procedures remain ad hoc. Part of
the problem is related to the sheer impossibility of iden-
tifying the laser footprint on a known surface. Another
subtlety is the mathematical model that relates the laser
surface to the true surface. As demonstrated in Sec-
tion 3, most systematic errors cause surface deforma-
tions. Consequently, a simple similarity transformation
would not properly describe the relationship. 2
Filinet al. (1999) propose a new calibration scheme that
addresses some of the issues raised here and offers so-
lutions. Itis a laudable attempt to make laser calibration
more transparent and, at the same time, better suitable
for quality assessment.
4.2 Comparison of Surfaces
Comparing surfaces is a fundamental, frequently occur-
ring problem when generating DTMs. Calibrating ALR
systems is a good example. Here, a known surface is
compared with the laser point surface and based on the
differences, calibration parameters are determined. An
interesting application is change detection where sur-
faces, determined at different times, are compared in
order to identify surface changes that may serve as a
basis for volumetric calculations. Merging two or more
data sets that describe the same physical surface is yet
another standard task. All these cases have in common
that the discrete points that describe the same surface
are spatially differently distributed, may have different
sampling densities and accuracies.
The standard solution is to interpolate the data sets to
a common grid followed by comparing the elevations
at the grid posts. Although this popular approach is
straightforward it is not without problems. For one, the
elevations at the grid posts are affected by the interpo-
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
lation. Moreover, the differences between the two sur-
faces are expressed along the z — axis which may not be
very meaningful for tilted surfaces. Take the extreme
example of a vertical surface; Az values contain no in-
formation about how close the surfaces are.
Schenk (1999) describes a surface comparison method
that is based on computing differences along surface
normals, without interpolating the data sets to a com-
mon grid. We present the problem statement, the pro-
posed solution and briefly describe the mathematical
model.
Problem Statement Let S; = {p1,P2,... ,Pn} be a sur-
face described by n discrete points p that are randomly
distributed. Let S» = {q1,d2,-.. ,Gm!} be a second sur-
face described by m randomly distributed points q. Sup-
pose that the two sets, in fact, are describing the same
surface but in different reference systems. In the ab-
solute orientation problem, set S» is the model system
and set Sy is referenced in an object space system. After
proper transformation we have S, — S», except for dif-
ferences due to random errors of the observed points p
and q. Yet another difference may arise from the discrete
representation of the surfaces, for example, n - m.
Even in cases where n = n, the different distribution
may cause a differently interpolated surface. Suppose
further that no points in the two sets are known to be
identical (same surface point). The problem is now to es-
tablish a transformation between the two sets such that
the two surfaces S, and S, become as similar as possible
in terms of closeness and shape.
Solution The problem described is cast as an adjust-
ment problem where the second set of points q is trans-
formed to the first set such that the differences between
the two surfaces are minimized. Additionally, the orien-
tation of surface normals between S, and S» can also be
minimized. Minimizing the distances assures the best
positional fit while minimizing differences in surface nor-
mals assures the best shape fit.
Mathematical Model Let the points q be transformed
into the coordinate system of the first set by a 3-D simi-
larity transformation
'2 sRq-t (19)
The observation equations are defined by the shortest
distance from q' to the surface S1. Two scenarios are
feasible for expressing surface S;. Let us first approx-
imate S; in the neighborhood of q’ by a plane, for ex-
ample by fitting a plane through points p confined to
a small spatial extent (surface patch). Then, the short-
est distance d from q' to the surface patch, expressed in
Hessian normal form, using the three directional cosines
and the distance p from the origin, is
d-q-h-p (20)
with h = [cos œ, cos B, cos y ]"
The following expression is the observation equation for
point q
Internai
The observation
imations for th
sary. Habib ai
method of obtai
face differences
If the points p c
rily approximat
face can be us
patch is not su
posed procedur
for this particul:
imated, then tl
The distance fr
the surface nori
data poir
Figure 14: Illus
tance between
4.3 Post-Proc
The raw 3-D las
faces in a discre
description of :
tion must be e:
plicit knowledg
tinuities (in ele
smooth surface
sential.
Fig. 15 depicts
data points. De
of the steps are
the fact that the
for computing
other hand, th
randomly distr
For example, tt
face patch, but
available. Stati
this redundanc
there are also r
the data. Such
given the hypo
Thinning is re
From a practica
