Geoscience Reference
In-Depth Information
2.2 Evaluation of the enhanced PT generalization
To evaluate a point selection two different methods can be used, both
value for 2D- point data sets. The second method we extended to evaluate
3D- point data selection:
2.2.1 Comparison of the density values
The first evaluation method is a comparison of the density- values of the
kept points before and after the generalization. Furthermore a visual com-
parison of the density map with all points with the density maps of the kept
points can be done. We used Scotts formula for density calculation (for-
mula 6, Scott 1992). For the search ratio
h
(or Kernel Density bandwidth)
the mean distance of all original points was used.
Formula (6): general Kernel density function
)
x
−
x
1
N
()
()
∑
=
u
=
i
f
x
=
K
u
with
h
h
N
⋅
h
i
1
whereby we used for K the Gaussian-Kernel
:
1
1
⎛
−
⎞
2
K
=
⋅
exp
u
⎝
⎠
G
2
2
π
with
K
G
= standard Gaussian function
h
= smoothing parameter (bandwidth)
x
1
, x
2
,..., x
N
= points, placed within the Kernel Radius h
2.2.2 Comparison using the “Relative Local Densitiy”
Relative local density is used to express the density of an area over the
whole study region (Yan and Weibel 2008). The area can be a Voronoi
polygon (
Fig. 8
) and the relative local density of each point can be calcu-
lated as follows.
Formula (7):
Relative local density
r
i
:
R
r
=
i
i
n
∑
=
R
k
k
1
