'ee more 
formula 
(3) 
and the 
of Ro: 
he beam 
different 
hang Yi, 
(4) 
rad 
, which 
and the 
'en. The 
Model 
> AMTF 
rmed to 
which 
(7) 
vhich is 
ind the 
, which 
and the 
is the 
vhich is 
mwidth 
ratio of 
  
From the eq.6 and eq.7, the relationship graph of the 
dimensionless EIFOV N and the dimensionless scanning 
interval k (Fig.1) can be derived, which indicates the six 
relationship curve grahps of N and k under different 
dimensionless angular quantisations(assuming m, =0, m, =0.5, 
m,=1, my=15, my=2, m;=2.5). From Fig.l, we can see 
that N is minimum on the condition of k=0 , which is 
described as theoretical minimum E/FOV. Furthermore, we can 
also see that the function of N = f(k) is montone increasing 
function, which asymptotic line is the line of N =k. 
  
  
  
   
   
  
     
   
   
  
  
  
  
  
  
  
NI 
5 NF 
4 (Ms. JUS 
34 
Vg 
2 JWS" : MN nin 
Ne A (N,, FO bs m, = 1 
mt a 1 #15 
; Ju, BN in, N,) n Te 
| On Z0) Ü 
] m, = 2.5 
: n M) 
Q T T T T T T = T Y r 
0 1 2 3 4 s k 
  
Figure 1. The rationship curve graph of N and k 
3. THREE KINDS RELATIONSHIP AND SIMPLIFIED 
FORMULA 
In practice, the scanned point cloud is supposed to have the 
specific angular resolution which equals scanning interval or 
laser beamwidth. In addition, it is hoped to evaluate the scanner 
performance by its theoretical minimum EIFOV which is 
governed only by angle quantisation. From Fig.1, the condition 
of the N=k is k>1 (or k— +x ) can be shown, and the 
dimensionless EIFOV N is equal to the theoretical minimum 
value when & 20. Although Lichti(2006) have given that the 
condition of the N 21 is & 20.545 , and N,,, — 0.8594 on the 
condition of ignoring angular quantisation, which can be 
effected when the size of angular quantisation is appropriate as 
scanning interval. However, more biases could arise in 
computing results and dimensionless variable & could not be 
infinite in actual scanning parameter setting, the minimum & 
value should be deduced( k > 1 ). Here, we assume that k, and 
k, are the dimensionless scanning interval variables, the 
dimensionless EIFOV N is equal to N, when k=k, , the 
dimensionless EIFOV N is equal to 1 when k=k, , the 
theoretical minimum dimensionless EIFOV is denoted by N. 
min > 
  
and N, satisfy the equation be ky < 0.005 . 
Assuming the regulations of relative approximate error is 0.005, 
which is that the condition of N =k is 
  
x « 0.005 . As the 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B3, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
relationship of N and m is montone increasing function, 
which asymptote is N —k and satisfy that N>k and 
  
k € [0,oo) (Lichti, 2006). So the function of y = Nt is also 
= «0.005 when &»&, 
  
montone increasing function, and 
which is equivalent to N =k when k >k, . So we can obtain 
point cloud which angular resolution is closed to scanning 
interval by setting the scanning interval parameter more than 
k, » and can obtain point cloud which angular resolution is 
equal to laser beamwidth by setting the scanning interval 
parameter at k, . Furthermore, N,,, can be used to estimate 
relationships of minimum theoretical angular resolution from 
different scanners, and use the scanner having smaller Nm to 
accomplish the task of obtaining higher angular resolution point 
cloud. 
Analysed from above, it have great significance in deriving the 
relationship of k, and m, the relationship of &, and m , and 
the relationship of N_, and m . Moreover, we need to derive 
the simplified formula of calculating &, , X, and N,,, under 
knowing the value of the angular quantisation. The following is 
the three kinds of relationships and simplified formulas. 
3.1 The Relationship & Simplified Formula of 4, And 7 
Assuming N, up , shown as Fig. 1, az1. With the 
equation (6) and (7), we can gain the relationship of a , k, and 
  
  
  
  
  
  
m is 
sin(7-) 2J, GI Eo sin( ES ) 
-— (8) 
m T TM 7 
2a 2ak, 2ak, 
As sime) si, Em <1, with the equation (8), we can gain 
X X 
qu z 
sin(—) 2J.( ) sin ^ T 
2d yds __ 206 SER >— (9) 
E z^ 7. mi mm c 
2a 2ak, 2ak, 
sin(Z) 2J Gr Sin) 
Note: 4, = du EA With the equation (9), 
Zi m 
2 2k, 2k, 
we can gain 
sin(— 5 2J, GE = ui 
— 4 (10) 
T m c zm 
  
2a 2ak, 2ak,