586 
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi XXXVII. Part B5. Beijing 2008 
(c) Matching result from normalized reflectance image 
Figure 1: Comparison of the matching result between SPi and 
SP 2 from histogram equalization and normalization using SIFT 
method. 
In our method, we include the discrete geometric properties of 
the key points to exclude false matches from the registration. In 
particular, the Gaussian and mean curvature values are 
computed based on the method of Cohen-Steiner and Jean- 
Marie (Cohen-Steiner and Jean-Marie, 2003) because of its 
time-efficiency, accuracy, and generality. There are some open 
problems to address for computing geometric curvatures in 
scans: One is ensuring that the point is on an object and not part 
of the background. Another is that the objects in different 
depths have a different appearance in the range image. Objects 
that are closer to the scanner are described by more points and 
have a more dominant appearance compared to objects that are 
further away. Therefore, the scanner placement significantly 
influences the object representation in the data. This 
dependency leads in turn to a bias in the number of extracted 
feature-points, favoring closer objects, and causing same 
objects to be described by different curvature values. In 
addition, discrete curvature is second-order derivatives of the 
surface which is sensitive to noise and small perturbations. 
(a) Range ball (b) Object’s boundary(c) Gaussian curvature 
map 
Figure 2: Curvature estimation within the bounding ball. 
However, in the case of laser scanning point clouds, the scale, 
viewpoint and the relationship of neighbor points are known. 
Based on the above information, we use the Euclidean distance 
between neighbor points to distinguish the different objects in 
scenes avoiding erroneous calculation of curvatures. We 
propose an approach which estimates the curvature of a point 
and not only covers neighborhoods of variable size but also 
takes into account the topology of the surface in that 
neighborhood. As shown in Figure 2(a), our approach is based 
on a bounding ball whose center is at each point of the matches, 
whose radius represents the scale at which the shape is analyzed, 
and whose boundary intersects the object’s boundary (as shown 
in Figure 2(b)). In our method, we set the radius as 0.5m. We 
calculate the discrete curvature of a center point and also taking 
into account the Gaussian-weighted curvatures of its 
neighboring points within the radius. By doing that, most 
influence of the scanner placement and noise can be reduced. 
As far as correct matches are concerned, the curvature values of 
the pair points should be close. Ideally, the difference of pair 
points’ curvature values should be zero. However, for the 
reason that noise and errors do exist in scans, we can consider 
the pair points as correct matches when its curvature difference 
is relatively close to zero. Finding the standard deviation of 
curvature difference between two entire point clouds is 
unrealistic because they are not one-to-one correspondence 
between each other. In most cases, the standard deviation is 
estimated by examining a sample taken from the data set. The 
most common measure used is the Sample Standard Deviation 
(SSD). In our method, we set the threshold as 3 (T, where <T is 
the SSD of the curvature difference of the candidate matches 
and can be calculated as follows: 
P H >~ H 2^ < 6 > 
where K x and H u K 2 and H 2 are the Gaussian and mean 
curvatures of matches in SPi and SP 2 , respectively and n is the 
number of candidate matches. Due to the lack of texture 
information in the reflection image, we find the matched points 
on planar surface, such as wall and ground, usually turn out to 
be false matches. Therefore, we take the points as false matches 
if their curvature values are less than a certain threshold. In our 
method, we calculate the Sample Standard Deviation of the 
Mean (SSDM) of Gaussian and mean curvatures as the 
threshold: 
r K 2 =JlFlt^-*> 2 
To sum up, we regard the candidate matched points as correct 
matches if they meet following outlines, otherwise they should 
be excluded as false matches. 
1. Kt> r AND Hi> r H ; 
K 1 Hi 
2. K 2 > r K2 AND H 2 > r„ 2 ; 
3. < 3<T k ; 
4. \H X ~H 2 \ < 3<r H 
Therefore, in our method, we can safely exclude the false 
matches according to the deviation of geometric curvature of 
the matches. The full scene of Gaussian and mean curvature 
map of SP t are shown in Figure 3, where red color is for