International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B3, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
airborne system over relatively flat horizontal terrain and cannot
be applied for terrestrial laser scans.
The shortcomings of the aforementioned approaches mandate
the development of alternative methods for the local point
density estimation. In this paper, new approaches are proposed
for the estimation of local point density indices. These
approaches aim to derive unique/accurate point density values
for individual LiDAR points while considering the 3D
relationships among them and physical properties of the
surfaces they belong to. Furthermore, the implication of
considering the varying point density in different LiDAR data
processing activities will be highlighted in this paper.
The paper starts with the introduction of the proposed
approaches for local point density estimation. In the following
section, the impact of considering the local point density indices
on some of LiDAR data processing activities is pointed out and
discussed. In the next section, the performance of the proposed
methods for local point density estimation and the impact of
considering these local indices on LiDAR data processing
results are evaluated through experimental results using
airborne and terrestrial LiDAR datasets. Finally, concluding
remarks and recommendations for future research work will be
presented.
2. METHODOLOGY
This section introduces alternative methodologies for the
estimation of local point density indices. These methods try to
overcome the shortcomings of previous approaches by
considering the 3D relationships among the points and the
physical properties of the surfaces enclosing the individual
points.
In the following subsections, the detailed explanation of these
approaches is presented and their advantages and disadvantages
are pointed out.
2.1 Approximate Method
In this method, the local point density index for each point is
computed while only considering the distribution of the points
within its spherical neighbourhood. This neighbourhood is
defined to include n-neighbours of the point in question
( Figure 1), where n is a pre-specified number of neighbouring
points.
Figure 1. 3D neighbourhood of the point in question (POI)
For a given LiDAR point, the local point density (LPD) is
estimated as follows:
part (1)
Ar
n
Where zr, is the area of the circle centred at the point in
question with a radius (r,) that is equivalent to the distance from
this point to its n"-nearest neighbour.
This approach provides a unique estimate of the local point
density for all the points in a LIDAR dataset in a fast and simple
manner. Therefore, this approach can be efficiently utilized for
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in-flight quality assessment of the collected LiDAR point cloud.
However, it suffers from the following shortcomings:
- lt does not consider the physical properties of the
surface which the point in question and its neighbouring
points belong to.
— ltassumes a uniform distribution of the points in the 3D
space defined by the point in question and its
neighbouring points.
2.2 Eigen-value Analysis of the Dispersion of Neighbouring
Points
In order to resolve the first drawback of previous method, this
approach estimates the local point density only when the point
in question and its neighbouring points define a planar region.
In this method, a spherical neighbourhood of the point in
question is initially defined. This neighbourhood encloses the n-
nearest neighbours of the point in question, where n is the
number of points needed for reliable plane definition while
considering the possibility of having some outliers. The
planarity of the established 3D neighbourhood is investigated
using the eigen-value analysis of the dispersion matrix of the
(n1) points within the spherical neighbourhood of the point in
question relative to their centroid (C) (Figure 2).
Figure 2: Dispersion of the points within the established 3D
neighbourhood relative to their centroid
This dispersion matrix (Ci) is computed as follows (Pauly et al.,
2002):
1 n+1 > — = 1
c eR iz (5, EX. F nod = Veentroid y
n+1 (2)
where TI = IX, y. Z2 I
> 1 n+l_,
, and T. ntroid = n zl E 1;
The eigen-value decomposition of the dispersion matrix (Cj)
results in three eigen values (Ai, A5, A34). For planar
neighbourhoods, one of the eigen values will be quite small
when compared to the other two eigen values. This eigen value
corresponds to the eigen vector which is perpendicular to the
plane passing through those points. Once the planarity of the
established neighbourhood is confirmed, the local point density
is estimated in the same way as the approximate approach. This
approach overcomes the first drawback of the approximate
method by considering the planarity of the surface enclosing the
point in question. However, it still assumes a uniform
distribution of the points within the established 3D
neighbourhood. This approach also has the following
shortcomings:
— It does not check whether the point in question belongs
to the planar surface passing through the neighbouring
points or not.
— When the majority of the points in the established 3D
neighbourhood are coplanar, non-coplanar points are
considered in the estimation of the local point density.
To resolve these problems, the planarity of the established 3D
neighbourhood is checked using the eigen-value analysis of the
dispersion matrix of the neighbouring points within the
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