the bai 
* n § fte Leica 
e ments were 
scanner. The 
1 cm distance 
0 r eveal the 
length, i 
guii 
1 5 m up to 
:rv ed for the 
■ r e used as 
rements. 
n from the 
wiling the 
¡wall were 
1 at the sled 
ith the laser 
was first 
led by the 
around the 
rhat would 
irget center 
¡ed for the 
:.g~1.5|im 
es between 
vere made 
was! 
the target distances. The reference distance (i.e. a sort of origin) 
for the obtained relative distances was thus the first 
measurements about 1 m from the scanner. 
Next, the relative distance differences of the tacheometer 
measurements were subtracted from the relative distance 
differences resulted from the laser scanner measurements. The 
comparison method was thus a relative one, not an absolute one. 
The standard deviation of the mean of four errors w (Figs. 6 and 
7, and Eq. 4) was also calculated. For the error calculations, we 
considered the tacheometer measurements as exact. 
4. FORMULAS AND ERROR FUNCTION BY FOURIER 
ANALYSIS 
Calculations of the error function were carried out by using 
Fourier analysis (Joeckel and Stober, 1989, Thibos, 2003, 
Herman, 2002). Formulas are valid for equal sampling intervals. 
Let v n =d nL - ¿/„r where d nL and d nT , n=0,...,N-\, are the relative 
distances measured by the laser scanner and tacheometer, 
respectively. The measurements were made at approximately 
equally spaced intervals d„=nS, n=0,...,/V-l, where <5=0.1 m for 
relative distances less than 4 m and <5=0.45 m for relative 
distances over 4 m. The discrete Fourier transform of the 
centered observations w„= v n - v , n=0,...,N-\ is given by 
5. RESULTS 
All measured relative distance errors are shown in Figure 5. 
The y-axis describes the relative distance error of the scanner 
v n =d„L - d n j, and the x-axis describes the relative distances 
where the reference distance is about 1 m away from the 
scanner. Relative distances of the x-axis are distances from the 
reference distance. 
12 ( mm ) 
8 * 
■ Measured relative errors 
N-1 N-1 
u k = rd„) 
ZTo NS 
(1) 
for k=0,...,N/2-\ (N is even). For the interval 0 m < d„ < 4 m 
and 4 m <d„< 29 m , the highest maxima of the absolute value 
of U k are located at wavelengths as explained in the following 
tables (Tables 2 and 3). 
4 .♦ Ì* * * 
* ♦ , ♦ ♦♦ ♦♦ 
\* ♦ 
Ot-CM COTI- 
Relative distance (m) 
Relative distance 0-4 m 
Corresponding 
wavelength 
00k/8 
20.4/m 
3.14/m 
1.57/m 
0.31 m 
2.0 m 
4.0 m 
Table 2. Angular frequencies and wavelengths of highest 
maxima at relative distances (0-4 m). 
Relative distance 4-29 m 
cOk/0 
Corresponding 
wavelength 
0.24/m 
25.6 m 
5.14/m 
1.22 m 
Table 3. Angular frequencies and wavelengths of highest 
maxima at relative distances (4-29 m). 
The Fourier series yields a correction function for the relative 
function is given by: 
Figure 5. Measured relative distance errors. 
Figure 6 illustrates measured relative errors between relative 
distances 0 and 4 m, meanwhile Figure 7 shows comparable 
relative errors for relative distances between 4 m and 29 m. The 
standard deviation w„ (see Eq. 4) of the mean of relative errors 
shown in graphics as error bars (Figs. 6 and 7) is calculated 
from the four scanner measurements and from the mean value 
of relative tacheometer measurements at each relative distance. 
Scanner vectors at each relative distance were calculated so that 
the 1 st measurement at the 1 m reference distance d() /fL was 
subtracted from the 1 st measurement of n th distance d„ iiL , 2 nd 
measurement of the reference distance do l was subtracted from 
the 2 nd measurement of n th distance d„ , and so on. 
2' 
Corresponding relative distances d„ T for the tacheometer were 
subtracted from these results. Average a n of four differences 
were counted and then corresponding w n . 
((k„L -d 0itL )-d nT )-a n f +...■+ ((k 4ii -d n<L )-d nT )~a„f 
v 
v„ = v +—£W cos K« + ^)= 
N U 
2 M 
v+ — Y.a k coa(a> k n)+b k sm(a> k n) 
Nim