In: Wagner W„ Szekely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
„MlWi 1 
372 
where e ] , e 2 and e 3 indicate three independent and orthogonal 
eigenvectors; A,, Aj and A, are eigenvalues with respect to the 
eigenvectors e ] , e 2 and e 3 . The eigenvalues are real and 
A, > A 2 > Aj if T p is a positive-semidefmite tensor. Most of 
LiDAR systems provide only three-dimensional Cartesian 
coordinates of points, and implied vector information cannot be 
directly obtained. The tensor voting algorithm presented here 
can be applied for deriving the vector information. The kernel 
of the tensor voting is the tensor communication among points. 
Each point receives vector information from its neighbouring 
points and stores the vector information by the tensor addition 
rule. The total tensor can be expressed as follows: 
T =E>' T ' (2) 
w(s) = exp(-5 2 / k 2 ) (3 ) 
directional difference of the normal vector of that point are less 
than the corresponding thresholds. Then, the point with the 
second largest cl-value in LiDAR data, excluding all extracted 
points in the region associated with the first seed point, is 
adopted as the second seed point for growing the next region. 
This region-growing procedure proceeds until no more seed 
points are available. Figure 1 illustrates segmentation result 
after region growing. 
2.4 Ridge lines and boundary points 
The method for extracting ridge lines used here is based on the 
intersection of two adjacent roof faces segmented from LiDAR 
data, as recommended by Maas and Vosselman (1999). The 
accuracy of the ridge line is about 0.4° in spatial angle. 
According to the rule that the triangles on the outer boundary of 
a triangular irregular network (TIN) mesh have only one or two 
neighboring triangles(Pu and Vosselman 2007), a TIN structure 
is adopted to extract boundary points. 
where w is a Gaussian decay function, s is the distance between 
the receiving site and the voting site, and k is a scale factor that 
controls the decay rate of the strength. In our experiments, the 
scale factor k is equal to 1.2 times the search radius, and the 
search radius region includes at least 20 points. 
2.2 Tensor decomposition 
After the tensor voting procedure is completed, the geometric 
feature information, such as planar, linear and point features, 
can now be detected according to the capture rules of geometric 
features mentioned in Medioni et al.(2000). However, the 
eigenvalues A, , A^ and Aj are generally smaller in the border 
region of an object than in the central region of the same object, 
because the points in the border region collect fewer votes than 
the points in the central region do. To reduce the effect of the 
number of votes, the planar feature indicators A, - A 2 may be 
normalized as 
(V^) 
1 \ 
(4) 
The normalized value of planar strength are introduced for the 
planar feature extraction and the region growing in this study, 
since it is the sensitive indicator for planar features. 
2.3 Region growing with principle feature 
The region-growing method is adopted to collect the points with 
similar planar features. The region-growing method used here is 
based on the homogeneity of the principal features. The 
principal features of interest are the planar feature strength and 
the corresponding normal vectors in this study. In region 
growing, only the points with the planar feature strength Ci 
greater than a threshold can be adopted as seed points. Since the 
strength of planar features in building areas is often greater than 
0.96 in our experiments, the threshold is recommended to be 
0.96 or larger. First, the point that has the largest c r value is 
chosen as the seed point for the planar feature extraction. A 
point is merged into the region if both the Ci-value and the 
Figure 1. LiDAR points and TIN structure 
3. FUSION OF LIDAR AND MAP 
3.1 Registration 
The first step for fusing LiDAR data and topographic map 
information is to transform these two datasets to a common 
coordinate system. The discrepancies between boundary points 
and building outlines are depicted in Figure 2. To determine the 
transformation parameters, we use robust least squares 
matching with the objective function which consists of the sum 
of squares of the distances from boundary points to building 
outlines on a local xy-plane. 
search direction 
Figure 2. Boundary points and ridge lines 
In order to register the boundary LiDAR points of buildings to 
the corresponding outline segments, a 2D similarity trans 
formation is adopted as a mathematical tool in this study: 
x' 
_ 
w 
u 
X 
+ 
r 
y_ 
-u 
w 
_y. 
s