fol. XXXVIII, Part 7B 
In: Wagner W., Szekely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
95 
i using the kNN algo- 
osest point pair enters 
distance dij between 
; smaller then the min- 
e time the 3D distance 
cm), (4) 
umber of laser points, 
cal points are consid- 
xpected that the mean 
iroximately zero. 
t density, compared to 
easons. First, from an 
be arbitrary close to- 
while the vehicle can 
ually more laser scan- 
e at the same time. It 
it drive-lines can have 
sition time is different 
y have changed. Also 
fferent quality. There- 
points are investigated 
e data set. 
rent scanners (scanner 
verlapping drive-lines 
;ntical points are ana- 
possibly identify sys- 
i geometric attributes, 
points is investigated. 
erpolate a DTM. The 
:rse Distance Weight- 
>han and Toth, 2008). 
topics, however they 
The main emphasis is 
i a number of individ- 
Huaxing, 2008). The 
terrain points (FD1), 
t (FD2), terrain point 
FR) and interpolation 
ted from the existing 
)rs (FD1, FD2, FD3) 
’he fourth influencing 
:d to the interpolation 
zero (Kraus et al., 2006). First a grid of 1 x 1 m size is laid over 
the terrain laser points. For grid cells, which include 4 or more 
terrain laser points, a tilted plane is modeled in a Least Square 
sense by a first order polynomial as given in Eq. 5: 
Z = do d - a\X + a 2 Y. (5) 
Here X, Y, Z are the coordinates of the terrain laser points (ob 
servations) that are included into the plane computation and 00, 
ai and a2 are the unknown plane coefficients. The graphical rep 
resentation of each term in Eq. 5 is shown in Fig. 3. To make the 
Z Y/ 
■z 'it 
/ (X.Y.Z Æ 
/ '7/ 
/fM,zyy 
/ V 
z=a 0 x 
Z=ajX X 
Z=a,Y 
Figure 3: A graphic representation of terms given in Eq. 5; after 
(Li et al., 2005) 
least squares computation more efficient, a new coordinate sys 
tem is used with the interpolation grid point (Xg,Yg) as the ori 
gin; therefore the method is called Moving Least Squares (MLS) 
adjustment (Karel and Kraus, 2006). The equation of a plane 
(Eq. 5) simplifies, so the plane coefficient ao becomes the eleva 
tion of the grid point itself, as written in Eq. 6: 
Zg = ao (6) 
The mathematical model of Moving Least Squares for linear sur 
face fitting is then given in matrix vector notation as in Eq. 7b: 
y TU A- X (7a) 
' Z\ 
’1 Ax -X G Yi - Yg 
z 2 
1 X 2 - Y 2 - Y g 
ao 
= 
a i 
(7b) 
a 2 
Zn_ 
1 X n — Xg Y n — Yg_ 
Where Xi, Y i: Zi for i = 1... n are the coordinates of the n 
original laser terrain points included in the plane computation. 
Then the unknowns in vector x and their variance-covariance ma 
trix Sài are computed in a least squares adjustment as written in 
Eq. 8 and Eq. 9 respectively: 
¡6 = (A r E- 1 A)“ 1 A T E- 1 y (8) 
E** = (A T E^A) 1 (9) 
Where Y yy is the variance matrix of observations. Here, the the 
oretical height precision of the laser points azi computed in Sec 
tion 2.2 is used. Besides, the vertical distances between the orig 
inal terrain points and the modeled plain, are computed. Those 
residuals e are applied to calculate the Root Mean Square Error 
(RMSE) as written in Eq. 10 for each plane: 
RMSE = 
(10) 
precision of the original laser points (FD2), their density (FD1) 
and distribution (FD3). The second term a\ represents the qual 
ity loss due to the representation of the terrain surface. In this 
research the RMSE is considered as a measure of the terrain sur 
face roughness (FR) with respect to the plane modeled by the cho 
sen random-to-grid MLS interpolation (FI). Therefore a e simply 
equals the RMSE as computed in Eq. 10. 
3 RESULTS AND DISCUSSION 
In this section the results of the quality evaluation of both LMMS 
laser point heights and a derived DTM are discussed for a LMMS 
data set representing a stretch of Dutch coast. Before these results 
are given, first this data set is described in more detail. 
3.1 Data description 
The LMMS data set was acquired on the Dutch coast near Egmond 
aan Zee using the StreetMapper system owned by provider Geo- 
maat (StreetMapper, 2010, Geomaat, 2010). The acquisition took 
place on November 27, 2008 at the time of low tide. Within 2 
hours a stretch of beach of 6 km long and 180 m wide was cov 
ered. The point cloud consists of about 56 million laser points. 
As experienced by Geomaat, the 3D laser point coordinates and 
the classification into terrain and non-terrain points can be done 
within 2 days. In this research a smaller representative test area 
of 213x101 m was chosen, which is covered by 8 drive-lines, 
see Fig. 4. The data set consists of 1 220 825 laser points. Each 
record of a laser point has 15 attributes, which are: 3D laser point 
position X, Y, Z, intensity I, class number C, scanning angle 
0, time of point acquisition T, drive-line number DL and scan 
ner number SC, range R, incidence angle a, footprint diameter 
Df p , range error due to scanning geometry 5R, measuring pre 
cision az,m and geometrical precision az,sr- The second data 
set used in this research composes of eight trajectories positions 
within the test area (black lines in Fig. 4). 
x 10 s 
5.1651 
5165 
5.1649 
^ 5.1648 
E 
ra 5.1647 
c 
'■§ 5.1646 
Z 5 1645 
5.1644 
5.1643 
5.1642 
1.0316 1.0318 1.032 1.0322 1.0324 1 0326 1.0328 1.033 1.0332 
Easting [m] x 10 5 
Figure 4: The digital photo of the test area [GoogleMaps]. The 
black dashed lines mark the trajectories driven downward i.e. 
from the north to the south and the solid lines mark the trajec 
tories driven in the opposite direction. 
3.2 Results of height differences of identical points 
)06), a grid point ele- 
lear interpolation (FI), 
es and co-variances of 
to estimate the quality 
ly speaking the preci- 
a standard deviation 
ors are assumed to be 
To finally predict the DTM quality uotm, a mathematical model 
after (Li et al., 2005) as written in Eq. 11 is used. 
o 2 dtm=°1 0 +o-l (11) 
Here the standard deviation of the constant plane coefficient o ao 
represents the quality of the original data and accounts for the 
By analyzing the attributes of the identical point pairs, i.e. the 
scanner and drive-line number, it is concluded that the major 
ity of identical point pairs belongs to the same scanner and the 
same drive-line. Fewer identical point are found in the scanner 
overlap and drive-line overlap (see Table 1). In Table 1 the re 
sults of height differences for the three cases are presented. The 
mean (avg) of height differences AZ is very close to zero for the