A, 9-11 Nov. 1999
ing surfaces is concerned
lata. It is well known that
cipate in a least-squares
ice the solution. How ro-
| in this respect? Step 3
1es for parameter t; with
5. The values are entered
uppose now point q; is
wrong parameter values
accumulator array are in-
d from the peak. It fol-
pact on the solution—an
oach that can be applied
ep, involving the explicit
it remain unlabeled have
ct solution of a transfor-
s are obviously not part
tion; they can be labeled
ange detection. Here, we
ribution of blunders and
)jetween the two surfaces
concentrated.
on parameters
| through 7
hing
ction
istment of
rameters
sis
detection,...
f surface matching, blun-
e iterative determination
ers is accomplished by a
r space, described above
| is obtained by repeating
nown parameters. At the
cted and labeled accord-
simultaneous adjustment
rs, using the previous re-
er steps may follow, for
Jications such as change
International Archives of Photogrammetry and Hemote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
5 Experiments
In order to test the feasibility and performance of the
proposed surface matching method, we have performed
several experiments with synthetic and real data. This
section briefly summarizes the most pertinent results.
5.] Tests with Synthetic Data
Fig. 4 depicts the synthetic data set. Following the no-
tation used in the previous sections, data set S» con-
sists of the points q;,i = 1,2,...,30. Surface S; on
the other hand is given in form of five surface patches
SPy,...,SPs. The true correspondence of points q to
the surface patches is known in this simulation, as well
as the transformation parameters. The tests served the
purpose of recovering the parameters and the corre-
spondences. Moreover, the convergence rate was ex-
amined as a function of surface topography.
Figure 4: Synthetic data sets for simulation studies with
the proposed matching method. Data set 5, is given
by the five surface patches SP,... , SPs5 and data set 5»
is represented by 30 points. The figure also shows the
correct correspondence of points to surface patches.
The initial values of the parameters were set off by 3?
for the angles, 2 meters for the translation parameters,
and 4096 for the scale factor. All parameters were de-
termined correctly. Fig. 5 shows the accumulator array
for the scale factor. The number of non-zero elements in
the accumulator array corresponds to the number of cor-
respondences evaluated—in our example 30 x 5 = 150
(every point q with every surface patch SP). The distinct
peak with a value of 30 indicates that for all points one
correct correspondence was found.
Not all parameters exhibit the same behavior as a closer
examination of Fig. 4 reveals. Take the shift parameter
along the Y —axis, for example. It can only be deter-
mined from a correspondence to SP3; all other surface
normals have no Y — component. Thus, the accumulator
array has a peak value of six, referring to the correct
correspondence q;3,... ,q18 to SPs.
Finally, Fig. 6 shows the change of parameters as a func-
tion of number of iterations. As expected, the conver-
30
25 F
15 F
10 |=
1 1 1 i
0.90 0.95 1.00 1.05 1.10
Figure 5: One-dimensional accumulator array (his-
togram) for the scale parameter. The peak value of 30
indicates that all 30 points of data set S» contributed in
one correspondence to the correct scale factor.
gence rate depends on how separable a parameter is.
The translation parameters are linear hence fewer itera-
tions are required. By the same token, the angular ele-
ments need more iterations because of the highly non-
linear rotation matrix.
2.0
1
1
1
11.0
0.0
parameter X
1-1.0
eo ve of a
iterations iterations
Figure 6: Change of parameters as a function of num-
ber of iterations. After initial fluctuations, the parame-
ter stabilizes after a few iterations. The iterations are
terminated once the changes become marginal.
There is another factor that greatly influences the con-
vergence rate, however. Remember that we impose the
coplanarity condition to compute the transformation pa-
rameters. In essence, the distance d; in Eq. 2 is set to
zero. The distance is parallel to the normal of the surface
patches. To obtain a good solution for our transforma-
tion problem surface patches with normals oriented in
all directions are necessary. The topography of S, is im-
portant. As shown in Schenk (19993), the surface slopes
should point in different directions. The slope angle di-
rectly influences the goodness of the solution.
5.2 Experiments with Real Data
As reported by Csathó et al. (1998), ISPRS Technical
Commission Ill has acquired a multisensor/multispectral
data set over Ocean City, with several laser data sets pro-
vided by NASA Wallops, and aerial imagery flown by NGS
(National Geodetic Survey). The data provide an excel-
lent opportunity to test the proposed procedure on a real
world problem; how well does a laser surface agree with
