14.3 PROBLEM FORMULATION
Variations in elevation can be due to natural terrain relief, height variation in
vegetation, and human-made features as well as to transitions between ground
and nonground surfaces. From the ground filtering problem point of view, the latter
are the most important elevation variations. These variations can occur at various
scales, that is, within a few meters or over long distances. Linear operators such as
those employed in the MDHT are effective tools for detecting elevation variations that
occur at different scales.
A terrain signal t x; y is assumed embedded in the DSM surface z x; y through
addition with the nonground feature height h x; y and a vertical noise e x; y from the
acquisition process, or
z x; y = t x; y + h x; y + e x; y
(14.3)
Furthermore, in the MDHT domain the surface model above can be written as
z n;m p; q = t n;m p; q + h n;m p; q
(14.4)
where the noise component is assumed negligible and the scale and angle indices
k; q have been omitted to simplify notation. Hence, the filtering problem is reduced
to a source separation problem with two components, namely the height of objects
above the ground and the terrain elevation. Evidently, even in the case of a negligible
noise component, the problem is still ill-conditioned as there are more unknown
variables than equations. Therefore, no unique solution exists, and different solutions
will differ in the way unknown variables are further constrained.
14.4 FILTERING METHOD
The ground filtering method described here is based on the MDHT expansion of the
DSM. The method was first introduced by Silván-Cárdenas and Wang (2006), so only
the major additions are detailed here. The major processing steps of the ground filtering
method are depicted in Figure 14.3. The process takes as main input the original point
cloud and produces as output a labeled set of points. Some intermediate products are the
eroded DSM and the ground/nonground mask. The original method performed a single
decomposition of the DSM. However, a new feature has been added, which includes the
possibility for repeated application of the multiscale erosion operator to an estimated
DTM. This part is indicated with dashed lines in Figure 14.3. The following sections
describe the processing steps indicated in the flowcharts.
14.4.1 Point-to-Raster Conversion
Point-to-raster conversion is a necessary step for processing in the raster mode. The
conversion is carried out in two steps. In the first step, a grid covering the same area of
274
MULTISCALE APPROACH FOR GROUND FILTERING FROM LIDAR
Variations in elevation can be due to natural terrain relief, height variation in
vegetation, and human-made features as well as to transitions between ground
and nonground surfaces. From the ground filtering problem point of view, the latter
are the most important elevation variations. These variations can occur at various
scales, that is, within a few meters or over long distances. Linear operators such as
those employed in the MDHT are effective tools for detecting elevation variations that
occur at different scales.
A terrain signal t x; y is assumed embedded in the DSM surface z x; y through
addition with the nonground feature height h x; y and a vertical noise e x; y from the
acquisition process, or
z x; y = t x; y + h x; y + e x; y
(14.3)
Furthermore, in the MDHT domain the surface model above can be written as
z n;m p; q = t n;m p; q + h n;m p; q
(14.4)
where the noise component is assumed negligible and the scale and angle indices
k; q have been omitted to simplify notation. Hence, the filtering problem is reduced
to a source separation problem with two components, namely the height of objects
above the ground and the terrain elevation. Evidently, even in the case of a negligible
noise component, the problem is still ill-conditioned as there are more unknown
variables than equations. Therefore, no unique solution exists, and different solutions
will differ in the way unknown variables are further constrained.
14.4 FILTERING METHOD
The ground filtering method described here is based on the MDHT expansion of the
DSM. The method was first introduced by Silván-Cárdenas and Wang (2006), so only
the major additions are detailed here. The major processing steps of the ground filtering
method are depicted in Figure 14.3. The process takes as main input the original point
cloud and produces as output a labeled set of points. Some intermediate products are the
eroded DSM and the ground/nonground mask. The original method performed a single
decomposition of the DSM. However, a new feature has been added, which includes the
possibility for repeated application of the multiscale erosion operator to an estimated
DTM. This part is indicated with dashed lines in Figure 14.3. The following sections
describe the processing steps indicated in the flowcharts.
14.4.1 Point-to-Raster Conversion
Point-to-raster conversion is a necessary step for processing in the raster mode. The
conversion is carried out in two steps. In the first step, a grid covering the same area of
274
MULTISCALE APPROACH FOR GROUND FILTERING FROM LIDAR