information within a surface (and a laser surface in particular)
and the intensity values in imagery allows the concepts from
image matching methods to be used in developing a suitable
algorithm for surface matching (Kilian ef al., 1996).
The mathematical formulation is derived by dividing the
transformation into two steps, i.e., horizontal and vertical. The
latter also includes leveling parameters. A similar approach is
taken in the absolute orientation procedure in traditional
photogrammetry (Kraus, 1993). Let a target surface be
represented by 7 irregular distributed points with coordinates
(x,y,z), and a source surface be represented by ; irregular
distributed points with coordinates (x",y",z"). These two
surfaces represent the same real surface, but they may have been
captured by different methods, which might introduce some
systematic errors between the two data sets. The problem is to
determine the transformation parameters required for
transforming the source surface into the coordinate system of the
target surface.
The horizontal transformation between the two surfaces may be
described by two horizontal shifts, a rotation parameter and a
scale factor. [Each point from the target surface may be
transformed to the source surface by these parameters. The
coordinates in the source surface are calculated by
x" cos& sink y x") (AX
y" —sink cosK | y'J ( AY
where AX,AY are the horizontal shift parameters, xis the
horizontal rotation and m is the scale factor.
Once a horizontal transformation has been performed, the
elevation shift AZ and leveling slopes a,b are introduced, to
relate the two surfaces by
z(xLy")eexrby rz" "y "HAZ. (2)
The differences between the two surfaces in the case described
here are assumed relatively small. In particular, small leveling
angles are assumed. Based on this assumption, Equation 2 is
rewritten as
z(x^y?)sdgx doy zx" v"rdz, (3)
where dp,d@ are the small leveling angles.
The planar coordinates (x",y") from Equation 1 may be
substituted into the right hand side of Equation 3 and in this way,
the relationship between the height differences and the planar
orientation parameters is established. Assuming further that the
horizontal rotation is also small, and that the scale factor is close
to 1, provides the following expression:
z(x^y)2dgx'* doy dZ +
C ENS :
+=} s : + ).
y’| |-dkK dm|| y’| | AY
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
As can be clearly observed, this mathematical model is not linear,
and therefore must be linearized in order to be solved in a
standard least-squares method. Applying a Taylor series to the
equation above, a linear observation equation for each point of
the source surface is given by
: ] dant dz 02 497, da dz
y dx. gy ox 3$ dx dy
[dp do AZ dk dm AX Nad
It can be observed that the elevation differences between the two
surfaces are considered as observations, while gradients are
required for forming the design matrix.
The gradients are calculated by reconstructing a small surface
patch around a point in the target surface, using a planar or
bilinear surface generation approach, avoiding the need to
interpolate neither of the surfaces to a regular grid. The decision
concerning which surface to use is based on an analysis of the
surface residuals. If large residuals are obtained by
reconstructing the small surface, a higher order surface is sought.
Using the described approach, it is clear that in some
circumstances not all the seven parameters can be accurately
determined, due to high correlation among them. The number of
parameters that can be determined depends mainly on surface
geometry. In the case of matching two horizontal planes for
instance, only a difference in height may be determined. The
decision about which parameters to set is based on an analysis of
the variance-covariance matrix and the surface spectrum. A
discussion about such analysis is left for another paper.
It should be noted that the proposed algorithm is suitable for
surfaces with relatively moderate slopes. Areas with steep slopes
should be eliminated from the calculations as laser measurements
in these areas may be affected by large errors.
3 EXPERIMENTAL RESULTS
Experiments were conducted by applying the matching algorithm
to both synthetic and real data. Using a synthetic data set, where
the 3D transformation between the two surfaces is known, allows
the elimination of any implementation flaws. Using real data
indicates the capability of the algorithm to actually determine the
transformation parameters between a laser surface and a surface
generated by photogrammetric means.
3.1 Experiments with synthetic data
The synthetic data used for the experiments consists of a surface
with a known parametric function, as shown in Figure 1. A large
number of randomly distributed points on this surface were
calculated to represent a target surface. These points are
represented in Figure 1 as a triangular network. To simulate the
affect of errors on the data, random noise was added to the
elevations of these points. Another smaller set of 30 randomly
distributed points was created in the same manner, which
constitutes the source surface. These points were shifted and
rotated by selected parameters, and therefore it was possible to
International
check the quality of th
matching algorithm. TI
the target surface by calc
A stable solution for all
within 3-4 iterations
determined values of th
for simulating the tran
elevation differences be
surface and the target su
added initially to the tar;
In another test with th
errors in some points of
efforts were made to re
found that outliers in or
not affect the ability of
parameters.
Figur:
3.2 Experiments with 1
Further testing was ur
aerial imagery covering
These data contain poin
Mapper (ATM) lase
photography from NGS
altimeter developed by |
clevation changes in po
systems. The instrument
capability.
Figure 2 shows part of
area. For this testing
measured manually on
Several combinations o
surface. The results of
these sets. The first se
second set contains two
Each planar segment
which was determined 1
segment, the appropriat:
