The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part B4. Beijing 2008 
Figure 3 Data Sets 
Ql(X[,Y/,Z') is determined as follows: 
X\ = {X iA +X,)l2 
Y!=Vi-i + y>)li (3) 
z;=s(x;,y;) 
where , Y { is the planar coordinate, S{X' h Y{) denotes the Z 
coordinate of the surface S at X t , Y t . Actually, Y[ = Y t _j = Y t . 
Q\ is much closer to P than Q t and Q t _j (See the blank circle). 
Specially, if the spatial Euclidean distance between Q t and 
is less than a prearrange threshold, Q\ is considered as to 
be the anticipated intersection point P . 
3 SURFACE MATCHING WITH NDCC 
The distance between the point s • R/7' +1 on the transformed 
second surface and its corresponding point P i on the first 
surface along the normal vector n can described as: 
Dist t = (P t -(s-RPj+t))w 0 (4) 
where n Q is the unit normal vector of h, • denotes the scalar 
product. 
Clearly Dist should be zero in ideal case when the match is 
reached. Then, an object function that minimizes the sum of 
squared Dist can pull the two surfaces close to each other. 
min ^ Wf • Distf (5) 
where w,, the weight of Dist,, 0 or 1, is used to deal with the 
question caused by the partial overlapping(Li, Xu et al. 2001). 
According to the principle of least square, the surface can then 
be matching with an iterative behavior. 
The terminated conditions of the iteration are: 
1) The difference of the estimated transformation parameters 
between two successive iterations is less than per-arranged 
threshold; 
2) Or reach the maximum iteration number. 
This algorithm is called least normal distance algorithm (LND). 
Using LND, the two surfaces are pulled close to each other 
along the surface normal vector. 
4 EXPERIMENT ANALYSES 
In order to provide a better understanding of the performance of 
NDCC, we implemented the proposed algorithm, and the 
iterative closet point (ICP) algorithm with the desired rotation 
angle and translation are respectively less than 0.1" and 0.01m 
between two successive iterations and the maximum iteration 
number is 70 for a comparative study based on simulated data 
set (Figure 3). It is a typical landform surface. It is grided data 
set containing 100x120 with an interval distance equal to 10m. 
The second surface is derived from the first surface by firstly 
applying the per-arranged transformation (rotation angle is 2° 
and translate is 50m) to it, and then adding zero-mean Gauss 
noise with a standard deviation equal to 0.2m. 
Both algorithms are directly applied to the data set without any 
pre-processing, feature extraction, and also without knowledge 
about the overlapping. Thus, the experimental results based on 
such data set are objective and they represent typical surface 
conditions. 
The performance indices of interest in this paper are 
convergence and computational efficiency. Then all 
experimental results are given below in turn 
4.1 Convergence 
To give an in-depth discuss of the convergence of our algorithm, 
the method for computing the convergence rate will be given 
briefly first. The distance E between two surfaces can be 
measured using the mean of the distance between all 
corresponding points: 
where «(> l) is the number of iteration, ||| is the Euclidean 
distance, p i and p'j are corresponding points locating on the 
first and second surface, the sub-script i, j are the number of 
point, N is the total number of the corresponding pairs, ii(o) is 
the surface distance before matching. Note that the 
corresponding points are not construct by the matching 
algorithm, but the known corresponding points of two surfaces 
to be matched. Therefore, E is an objective value and is 
independent of the matching algorithm. 
During the beginning phase, E is very big owing to the large 
error existing in the transformation parameters. With the 
increasing of the iterations, E reduces to a small positive 
number, not zero. 
The convergence indicator (Cl) can then be computed according