The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
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to the surface distance E , and the mean of all C/s is called the 
average convergence indicator (AC1). Both of them are of 
objective because they are based on the objective variable E , 
and are adopted in this paper to discuss the convergence rate in 
detail. 
Cl(n)= E{n)/E(n-\) 
ACI = 
1 
n — 1 
m-n 
Ÿ^Cl(m) 
(7) 
Using Cl and ACI, the convergence rate can be discussed in 
detailed and make comparison of different algorithms. The 
matching algorithm converges when Cl less than 1. The less Cl 
is, the faster it converges. ACI describes the convergence rate of 
the entire matching process. 
ICP need about 2 times iterations more than LND in average 
(Figure 4). Such experimental result illustrated that LND 
converge faster than ICP in general. The variety of the surface 
distance E and convergence indicator (Cl) are similar in three 
data sets. The experimental result on the data sets A (Figure 3) 
is given in Figure 5, and the variety of corresponding six 
transformation parameters is given in Figure 6. 
Figure 4 The iterations required by ICP and our method 
160 
120 
v 
^80 
I \ "k 
40 
V 
* ICP “ LND 
“iii 
30 40 
0 10 20 30 40 
3 
400 
Figure 6 The six transformation parameters against iterations 
Rx, Ry, Rz (in degree), and Tx, Ty, Tz (in meter) 
From the variety of E and Cl (Figure 5), LND always converges, 
and the surface distance reduces monotonously. At the 
beginning and ending phase, Cl of LND is close to 1. In these 
cases, LND converges slowly. In the middle phase, Cl of LND 
is rather small. It illustrate that LND converges very faster. The 
ICP gives another different behavior. Some iterations are not 
convergence, but divergence, although it convergence to correct 
matching at last. The surface distance is not always reduces 
during the iterations. Similar conclusion is also illustrated in 
Figure 6. As to ICP, the variety of six parameters is much more 
complex than that of LND. 
The ACI of LND and ICP is respectively 0.31 and 0.66. This 
result illustrate that LND converges faster than ICP. It is 
consistent with the result in Figure 4.