Internat 
C(d) (m?) 
  
  
90 1 f 1 { 3 
68% confidence intervals | | 
mean of products per bin | 
80 __ exponential model | 
| 
70 | 
| 
60 | 
50 | 
| 
| 
40 | 
| 
30 | 
| 
| 
20 - i 
10. | 
-10. ; . i j 
-50 0 50 100 150 200 250 300 350 400 450 
distance d (m) 
Figure 1: Covariogram Fitted with Exponential Model 
The standard error of prediction is given by (Heiskanen 
and Moritz, 1967, p. 267): 
a? (OQ) 2 Co - 2X5 20iCoi + S IER Nick (9) 
where the subscripts refer to horizontal distances between 
points (i.e.: ?k is the distance between I and K), O is the 
point interpolated 
and [ = Kids c KEN... u), 
are points of known properties used to find the standard 
error of the interpolated property of C 
The method is adapted for the matching programme. The 
property of the variables is the height. The number of 
points used to estimate the error is limited to the three 
vertices of the enclosing triangle, and can be found ana- 
lytically with, for a point O interpolated in an enclosing 
tr 
c?(O) 
iangle of vertices P, Q and ft: 
Co TE 2(A1 Co + AaCo+4 = AC) (10) 
+ AM A2Cp-q 3 A A3Cp—r = AgA3Cr—q) 
+ Co(A2 + À + 45) 
where the weights À; of the covariance factors are deter- 
mined using: 
Lo\Yr — Ye Le Yo — Yr + T7 (Yg — Yo \ 
Ne Tot = Ya) + #alYo — #r) + Tr — 99). (qq) 
ta (Ur — yg) + Tale — y) + Erg = Yo 
vy(yr ve yq) + T4 (Yp = Yr) ds Tr (Va = Un) 
Tyr FR Yq) FF Tap TES Ur) = ys =r Up) 
The covariance factors are estimated from the covariance 
function shown in Figure 
| for the distances shown in the 
subscripts. Co 18 the covariance for the nil distance, that 
is. the mean of the square of the products of the height of 
the points (the points 
the distance is nil). 
are multiplied with themselves when 
The exponential model (or Gaussian 
model) is given by (Mikhail, 1976, p. 405): 
Cd) e eer À (12) 
ional Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
The value of k — 0.0032728 in the exponential model is 
determined by a least squares method. Note that the model 
does not fit the experimental data accurately for the longest 
distances (see Figure 1): however, out of the 1719 triangles 
generated by the Delaunay triangulation of 5;, only 6 had 
sides larger than 240m. 
3. EXPERIMENT AND RESULTS 
3.1 Aim 
The aim of this experiment is to demonstrate the ability of 
the matching algorithm to register S» in the coordinate sys- 
tem of S,. Given the characteristics of the data (8 3.2), 5» 
was first matched to S,. The two sets were assumed then 
to be registered in the same coordinate system (8 3.3). The 
experiment (i.e.: the testing of the algorithm) was under- 
taken: So was transformed with known parameters, then 
matched with the algorithm. The ability of the algorithm 
to return S back to its registered spatial position was mea- 
sured by: 
|. comparing the parameters of the initial transformation 
and the parameters of the matching transformation, 
2. computing the mean of the absolute displacement of 
the coordinates of 5S». 
32 Data Characteristics 
The surface Ss is a data set of 27,748 points. The set was 
extracted from a laser project covering the Greater New- 
castle area. The accuracy reported for the Greater Newcas- 
tle project included a mean elevation difference of 0.1m 
based on direct observations of 12 test points. A com- 
parison to 12 derived test points (interpolated from sur- 
face model) produced a standard deviation of 0.25m. Only 
height accuracy was estimated. The position of the set was 
fixed by GPS/INS with two survey control points, and one 
reference point situated at the aerodrome of departure of 
the plane used for the survey. 
The reference set $4 of 884 points was sampled with a GPS 
system using a stop-and-go kinematic method. Its accu- 
racy varies from point to point but averages approximately 
30mm both horizontally and vertically. The matched sets 
are shown in Figure 2. 
3.3 Data Preparation 
The experiment tests the ability of the matching programm 
to register a large dense ALS set using a sparse GPS set. 
The GPS set is made up of six clusters of dense data. The 
Delaunay triangulation of 5; generates small triangles! 
the clusters, and large triangles which do not represent d^ 
curately the shape of the terrain between the clusters. Alter 
normalising the values of the data to minimise numeric 
errors (Pilgrim, 1991), S» is matched to Si, resulting ind 
small adjustment. An adjustment can be expected to occu! 
as the ALS set registration method is prone to planimetrt 
1174 
  
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