A 0-0 fe
T -[e, &e-el]jo 13, 0 [e]. (1)
0-9 À, | €:
where e;, e; and e; indicate three independent and orthogonal
eigenvectors; Aj, A, and XA; are eigenvalues with respect to the
eigenvectors e;, e, and e;. The eigenvalues are real and
M>A,>; if T is a positive-semidefinite tensor.
The tensor voting method is used for deriving the implied
vector information in LiDAR point clouds. The kernel of the
tensor voting is the tensor communication among points. Each
point receives vector information from its surrounding points
and stores the vector information by the tensor addition rule.
The resultant tensor can be expressed as follows:
T= Y wr Q)
i=l
where w is a Gaussian decay function depending on the
Euclidean distance between the receiving site and the voting
site.
After the tensor communication is completed, the geometric
feature information, such as planar, linear and corner features,
can now be captured according to the rules of geometric
features mentioned in Medioni et al.(2000). However, the points
in the border region receive fewer votes than the points in the
central region do, so that eigenvalues Aj, A,, and A; are
generally smaller in the border region of an object than in the
central region of the same object. To reduce the effect of the
number of points, the planar feature indicators A;-A, may be
normalized as
cushy)
x (3)
The normalized value of planar strength is introduced for the
planar feature extraction and the region growing in this study,
since it is the sensitive indicator for planar features (You and
Lin 2011b).
The region-growing method is adopted for the segmentation of
the points with similar planar features. The region-growing
method used here is based on the homogeneity of the principal
features. The principal features are the normalized planar
feature strength c and the corresponding normal vectors in this
study. In region-growing, only the points with a normalized
planar feature strength c greater than a threshold can be adopted
as seed points. The threshold is recommended to be 0.96 or
larger in our experiments.
First, the point that has the largest c-value is chosen as the seed
point for the planar feature extraction. A point is merged into
the region if both the c-value and the directional difference of
the normal vector of that point are less than the corresponding
thresholds. Then, the point with the second largest c-value in
the LiDAR data, excluding all extracted points in the segment
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
associated with the first seed point, is adopted as the second
seed point for growing the next segment. This region-growing
procedure proceeds until no more seed points are available.
Figure 1 illustrates segmentation result after region growing.
In this study, ridge lines are inferred by the intersection of two
adjacent roof faces segmented from LiDAR data, as
recommended by Maas and Vosselman (1999). 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.
(b)
Figure 1. (a)LiDAR points and (b)the segmentation result.
3. DATA REGISTRATION AND TENSOR ANALYSIS
OF RESIDUALS
3.1 Data registration
In this study, LiDAR data and topographic maps are integrated
for building model reconstruction. Hence, data registration is
intended to transform these two datasets into a common
coordinate system. The discrepancies between boundary points
and building outlines before data registration are shown in
Figure 2. To determine the transformation parameters, the
robust least squares (RLS) matching of boundary points and
building outlines on a local xy-plane are used (You and Lin
2011a).
3 T
dala.
boundary points
building outline rs
Figure 2. Boundary points and ridge lines.
