With the equation (10), we can gain 
1 2 
door (11) 
a m 
IN, - k| ; 
As — =a-1<0.005, we can gain 
1 
==> 0.9831 (12) 
a 
With the equation (11) and (12), we can gain 
1.9702 
AR FT (13) 
T 
With keeping two digit of decimals, indicated from the 
monotonicity of N and m ,derived from above deduction, the 
-k 
condition which makes Mk < 0.005 tenable is 4, > E , 
cT 
1 
  
  
i.e. 
i3 24 (5 sinit CT 
sin) C) sin) 1g 
> (14) 
Eo OR CE 
2 2h 72k 
1 
The above equation is very complicated. we need to get its 
simplified form for convenient calculation. Here, the least 
squares curve fitting method of 1000 uniform sampling points 
(m,k,) obtained by the equation (14) is used to derive the 
simplified formulas of &, and m. As up to now, the highest 
precision in point cloud data processing of terrestrial laser 
scanning is 0.01 folds laser beamwidth(Zhang Yi, 2008), and 
the maximum angular quantisation of different scanner is 2.08 
folds laser beamwidth(GIM, 2010). Therefore, we define that 
m € [0,2.5] and fitting precision is 0.005. Then, we can get the 
relationship graph of 4, and m (Fig.2) and the fitting formulas 
of k is 
k =a +b-(m—g) +h (15) 
where a = 30.8136 
b, =41.03034 
g, = 0.0008 
h, = 0.006 
From the equation (15), we can see that the relationship graph 
of k, and m is hyperbola, and the fitting errors of the equation 
(15) and m is Fig.3. From the plot of Fig.2, we can see that 
fitting errors is less than 0.005 when m > 0.01. so we can think 
that the equation (14) is approximately equaivalent with the 
equation (15). 
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 
  
  
  
  
  
Kk, 
16 
14 - 
12 
10 -] 
8 
6 4 
0 08 d 26 UM 
Figure 2. The rationship curve graph of k, and m 
0.0006 — 
0.0004 — 
0.0002 — 
0.0000 — 
-0.0002 + 
  
  
-0.0004 1 1 + T T T T T 
0.0 0.5 1.0 15 2.0 2.5m 
  
  
  
Figure 3. The rationship curve graph of fitting errors & 
(equation (15)) and m 
3.2 The Relationship & Simplified Formula of k, And m 
With the equation (6), (7) and N=1 , we can gain the 
relationship of k, and m that is 
  
  
  
ann sint) 
2 aa do 
Eu tinc) 
2 2 2 
where 0X k, < 0.545 
0€ m € 0.545 
The above equation is still very complicated. Its simplified form 
can be derived through the same method as above. we can get 
the relationship graph of k, and m ( Fig.4) and the fitting 
formulas of &, and m that is 
k, 2 a, -b,-(m+g,) + j,-m—h, (17) 
where a, = 0.35426 
b, =0.99264 
gj -0.0521 
J, =0.085793 
h, =0.047672 
    
  
  
   
   
     
    
   
    
    
   
     
     
    
  
    
    
     
     
    
     
   
      
  
   
Fro! 
ellij 
Fig. 
less 
app 
  
3.3 
Wi 
rel: 
Th 
as 
wh 
Fr 
hy