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.