are neglected. In that study, such an erosion operator was derived from an explicit
model of a 1D surface in the continuous domain and then extrapolated for the 2D
discrete case. In contrast, here we show that the erosion operator is an approximation
of the local shifting operator expressed in Equation (14.6). Hence, this represents a
generalization of the erosion operator that allows shifting the higher order coefficients
as well, instead of simply setting them to zeros as in the previous work. Furthermore,
in the original formulation, the terrain elevation underneath nonground features was
assumed flat because the erosion operator was essentially a local spatial shifting. This
assumption is relaxed here.
Since the local spatial shifting is essentially a zero-order interpolation, a natural
generalization can be carried out through a truncated Taylor expansion with higher
order terms. In particular, here we use an expansion up to first order, which could
account for sloppy terrain underneath nonground features. This implementation uses a
Taylor expansion up to first-order terms which, after taking advantage of the relation
between the transform coefficients and the surface derivatives, can be expressed as in
Equation (14.8). Hence the erosion operator is implemented by computing the right
term of Equation (14.8) using the local spatial shifting of Equation (14.6):
z
q;N
n;m p; q ≈ z
q;N
n;m
p −
M
2
; q
−
M
4
ffiffiffiffiffiffiffiffiffiffi ffi
n + 3
p
z
q;N
n + 1;m p −
M
2
; q
(14.8)
Furthermore, one of the problems in representing local support signals with the
MDHT occurs along boundary cells. As with any neighborhood operators, the local
derivative operators used in MDHT are influenced by neighboring surface values.
Hence neighbours outside the support or coverage of the surface need to be
extrapolated. The way these cell values are created is called the boundary condition.
The boundary condition used previously was the symmetric extension, which consists
in reflecting the surface values horizontally around boundary cells. In this study an
alternative was adopted, which consists in reflecting both horizontally (x–y plane) and
vertically (z coordinate) the surface values around the boundary cells. This yields a
boundary condition called the antisymmetric reflection, which naturally extrapolates
preserving the surface slope beyond boundary cells.
14.4.3 Parameter Selection
There are many parameters that need to be properly defined for the filtering method to
work. First, the number of pyramid layers in the MDHT expansion must be selected
according to the largest above-ground feature that is to be removed from the input
surface. Since the erosion operator of the pyramid layer k erodes the features of size
2
k , then we can compute the minimum number of layers that will warrant the effective
erosion of all features up to a given size. Equation (14.9) is used to compute the
number of pyramid layers in terms of the width W of largest above-ground features
and the cell size d of the input surface. The factor of 2 in the formula accounts for the
fact that erosion is performed from two opposite sides. Hence the argument of the
logarithm in the formula is the largest semiwidth in number of cells. The maximum
FILTERING METHOD
277
model of a 1D surface in the continuous domain and then extrapolated for the 2D
discrete case. In contrast, here we show that the erosion operator is an approximation
of the local shifting operator expressed in Equation (14.6). Hence, this represents a
generalization of the erosion operator that allows shifting the higher order coefficients
as well, instead of simply setting them to zeros as in the previous work. Furthermore,
in the original formulation, the terrain elevation underneath nonground features was
assumed flat because the erosion operator was essentially a local spatial shifting. This
assumption is relaxed here.
Since the local spatial shifting is essentially a zero-order interpolation, a natural
generalization can be carried out through a truncated Taylor expansion with higher
order terms. In particular, here we use an expansion up to first order, which could
account for sloppy terrain underneath nonground features. This implementation uses a
Taylor expansion up to first-order terms which, after taking advantage of the relation
between the transform coefficients and the surface derivatives, can be expressed as in
Equation (14.8). Hence the erosion operator is implemented by computing the right
term of Equation (14.8) using the local spatial shifting of Equation (14.6):
z
q;N
n;m p; q ≈ z
q;N
n;m
p −
M
2
; q
−
M
4
ffiffiffiffiffiffiffiffiffiffi ffi
n + 3
p
z
q;N
n + 1;m p −
M
2
; q
(14.8)
Furthermore, one of the problems in representing local support signals with the
MDHT occurs along boundary cells. As with any neighborhood operators, the local
derivative operators used in MDHT are influenced by neighboring surface values.
Hence neighbours outside the support or coverage of the surface need to be
extrapolated. The way these cell values are created is called the boundary condition.
The boundary condition used previously was the symmetric extension, which consists
in reflecting the surface values horizontally around boundary cells. In this study an
alternative was adopted, which consists in reflecting both horizontally (x–y plane) and
vertically (z coordinate) the surface values around the boundary cells. This yields a
boundary condition called the antisymmetric reflection, which naturally extrapolates
preserving the surface slope beyond boundary cells.
14.4.3 Parameter Selection
There are many parameters that need to be properly defined for the filtering method to
work. First, the number of pyramid layers in the MDHT expansion must be selected
according to the largest above-ground feature that is to be removed from the input
surface. Since the erosion operator of the pyramid layer k erodes the features of size
2
k , then we can compute the minimum number of layers that will warrant the effective
erosion of all features up to a given size. Equation (14.9) is used to compute the
number of pyramid layers in terms of the width W of largest above-ground features
and the cell size d of the input surface. The factor of 2 in the formula accounts for the
fact that erosion is performed from two opposite sides. Hence the argument of the
logarithm in the formula is the largest semiwidth in number of cells. The maximum
FILTERING METHOD
277