International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
gramme to form the normal equations of the least squares.
The others either did not find correspondence or were elim-
inated as outliers. Experience shows that better matching
results are obtained with the points of S, sampled in the
periphery of Ss, but this characteristic was not shown in
this experiment. 55 is now assumed to be registered in 51
coordinate system. The two sets are now ready for the ex-
periment.
3.4 Measure of the Accuracy of the Matching
After the adjustment, the programme is tested by trans-
forming S5 with known parameters. This transformation is
referred to as the initial transformation. Sa is then matched
to S,. The process is undertaken with different values for
the parameters of the initial transformation. A measure
of the accuracy of the matching (i.e.: the ability of the
programme to return 5» to its original spatial position) is
demonstrated by comparing the parameters of the match-
ing transformation which undo the initial transformation
(i.e: the inverse operation).
The matching accuracy is not easily assessed by interpret-
ing the simultaneous effect of the differences between the
three rotations, the three translations and the scaling of
both initial and matching transformations. An additional
measure of the accuracy is therefore given by the mean
and standard deviation of the mismatches of the points of
Ss. To obtain this mismatch, the coordinates of Sa in its
original position are compared to the coordinates of 5» af-
ter the initial and matching transformation. The absolute
mismatch between each point is obtained with the distance
formula. The mean 77 of the mismatches of the 27,749
points of 55 is computed with the standard deviation o,,.
3.5 Results
Four trials were undertaken for the experiment. The re-
sults of the trials are shown in Tables 1, 2, 3 and 4. The
Tables show the parameters of the initial transformations
and the parameters of the matching transformation. The
absolute values of the differences between the parameters
are shown in the last column. The last three rows of the
Tables indicate the number of iterations undertaken by the
matching programme, the mean of the mismatches 777 and
the standard deviation 0,
From the four trials, it can be observed that the largest ro-
tational error was K = 0.0255° in trial 2, while the largest
translation error was Tx = 0.0798m which occurred in
trial 4. The largest mismatch occurs in trial 4 where 5»
was shifted by an initial transformation including a trans-
lation of 2m.
4. CONCLUSIONS
This article reports on a weighted least squares matching
algorithm developed at the University of Newcastle, Aus-
tralia. The algorithm measures the separation between the
surfaces along the normal distances between the points of
1176
Table 1: Matching Results
peu [initial | results | | A |
QJ 0 -0.0019 | 0.0019
D 0 -0.0013 | 0.0013
K 0 -0.0005 | 0.0005
Tx -| 0.9629 | 0.0371
Ty -| 0.9897 | 0.0103
Tz -10 9.9875 | 0.0125
s 1 1.0002 | 0.0002
[ Iterations 10
m 0.0479
| of ty 0.0178
(w, ¢ and & in decimal degrees)
(Tx Ty, Tz, M and o,n in metres)
Table 2: Matching Results (cont.)
fon | initial | results FTA| 1
w 0 -0.0029 | 0.0029
Q 0 -0.0019 | 0.0019
K 0 0.0255 | 0.0255
Tx 1 «1.0122 | 0.0122
Ty 1 -1.0618 | 0.0618
Tz 10 -10.0452 | 0.0452
MARE | 1.0004 | 0.0004 |
Iterations 5 al
m 0.0980
| Om 0.0426
(w, $ and & in decimal degrees)
(Tx Ty, Tz, M and om in metres)
one surface to the plane patches of the other. The weight-
ing technique of the least squares involves the production
of a spatial covariogram. The weights are computed as the
inverse of the interpolation errors estimated from an expo
nential function fitted on the experimental covariogram.
The experiment presented in this article demonstrates the
ability of the programme to register an ALS data set of
27,749 points in the coordinate system of a data set of
884 points obtained with a stop-and-go GPS method. The
reference set is sparsely sampled, and contains 6 clusters
densely sampled. The weighting technique of the algo-
rithm efficiently allocates large weights to points interpo-
lated close to the vertices of the patches generated by the
Delaunay triangulation, where the interpolation error is min-
imal. Likewise, small weights are allocated to the nor-
mal equations of the least squares which involve points far
away from vertices where large interpolation errors occur.
Four trials involving different sets of parameters showed
that the programme is able to register the ALS surface with
mean errors approximating 100mm in the worst of the four
cases. The weighted least squares matching program cal
be used to orient data sets which may include buildings and
protruding features. The reference surface is sampled be-
tween the protrusions (i.e.: in the streets and parks) using
a precise and dense GPS method. The sampling is ideally |
patchy - the size of the patches is as important as the den- |
sity of the patches to the final accuracy of the matching
The programme is thus a useful tool for data reconstruc |
tion.
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