multiresolution wavelet transform (Thuy and Tokunaga, 2004) and the multiscale
Hermite transform (Silván-Cárdenas and Wang, 2006).
The latter method is based on the so-called multiscale erosion operator defined
in the multiscale Hermite transform (MHT) domain. This method progressively
removes the contribution of above-ground objects to the transform coefficients, so
that when the inverse transform is applied, an approximation of the DTM results. This
chapter had the objectives of revising the theoretical basis of this method, discussing
the practical aspects of its operation, and presenting an extension of the original
method by revising the erosion operation and testing its performance with ISPRS data
sets. The chapter starts with the background section, where the scale-space representation is first discussed as the theoretical foundation of the MHT. This includes a
description of the MHT of discrete 2D signals. In this part the local spatial rotation in
the transform domain is also introduced. The extension of the single-scale case to
multiple scales is presented next by building upon previous definitions. The following
section presents the ground filtering method with emphasis on new features added to
the previous method. In this part, it is shown how the multiscale erosion operation,
which is the main mechanism to estimate the MHT coefficients of the bare terrain
from the MHT expansion of the DSM, can be generalized to higher order coefficients
through a scale-space shifting operation. This section also includes the parameter
selection and other processing steps needed by the ground filtering method. The last
part of the chapter presents some results from a few tests and relevant discussion.
14.2 BACKGROUND
14.2.1 Scale-Space Representations
The Gaussian scale-space theory was first developed by Iijima in the late 1950s
(Iijima, 1959), but it was not adopted by the computer vision and image processing
community until further developments by Witkin and Koenderink in the 1980s
(Witkin, 1984; Koenderink, 1984). For the latter, the theory served as a means to
generalize existing notions of Gaussian pyramids and as a well-founded way to
perform multiscale analysis.
The original axiomatics of scale-space theory postulated the scale space to (1) be
linear, (2) be shift invariant, (3) be scale and rotation invariant, (4) preserve
positivity, and (5) fullfil the semigroup property. The latter means that the scalespace representation is equipped with an associative binary operator, the convolution operator. These axioms led Iijima and others to conclude the Gaussian kernel
was the only possibility for the linear scale space, so that “Gaussian scale space”
was also a term to refer to “linear scale space.” However, a deeper look into the
matter led Felsberg and Sommer (2004) to discover another non-Gaussian linear
scale space based on a Poisson kernel. They concluded that for the Gaussian kernel
to be the unique kernel of the linear scale space a sixth axiom needed to be added to
the original list, one that requires the frequency response of the scale space kernel to
be continuosly differentiable at the origin.
BACKGROUND
269
Hermite transform (Silván-Cárdenas and Wang, 2006).
The latter method is based on the so-called multiscale erosion operator defined
in the multiscale Hermite transform (MHT) domain. This method progressively
removes the contribution of above-ground objects to the transform coefficients, so
that when the inverse transform is applied, an approximation of the DTM results. This
chapter had the objectives of revising the theoretical basis of this method, discussing
the practical aspects of its operation, and presenting an extension of the original
method by revising the erosion operation and testing its performance with ISPRS data
sets. The chapter starts with the background section, where the scale-space representation is first discussed as the theoretical foundation of the MHT. This includes a
description of the MHT of discrete 2D signals. In this part the local spatial rotation in
the transform domain is also introduced. The extension of the single-scale case to
multiple scales is presented next by building upon previous definitions. The following
section presents the ground filtering method with emphasis on new features added to
the previous method. In this part, it is shown how the multiscale erosion operation,
which is the main mechanism to estimate the MHT coefficients of the bare terrain
from the MHT expansion of the DSM, can be generalized to higher order coefficients
through a scale-space shifting operation. This section also includes the parameter
selection and other processing steps needed by the ground filtering method. The last
part of the chapter presents some results from a few tests and relevant discussion.
14.2 BACKGROUND
14.2.1 Scale-Space Representations
The Gaussian scale-space theory was first developed by Iijima in the late 1950s
(Iijima, 1959), but it was not adopted by the computer vision and image processing
community until further developments by Witkin and Koenderink in the 1980s
(Witkin, 1984; Koenderink, 1984). For the latter, the theory served as a means to
generalize existing notions of Gaussian pyramids and as a well-founded way to
perform multiscale analysis.
The original axiomatics of scale-space theory postulated the scale space to (1) be
linear, (2) be shift invariant, (3) be scale and rotation invariant, (4) preserve
positivity, and (5) fullfil the semigroup property. The latter means that the scalespace representation is equipped with an associative binary operator, the convolution operator. These axioms led Iijima and others to conclude the Gaussian kernel
was the only possibility for the linear scale space, so that “Gaussian scale space”
was also a term to refer to “linear scale space.” However, a deeper look into the
matter led Felsberg and Sommer (2004) to discover another non-Gaussian linear
scale space based on a Poisson kernel. They concluded that for the Gaussian kernel
to be the unique kernel of the linear scale space a sixth axiom needed to be added to
the original list, one that requires the frequency response of the scale space kernel to
be continuosly differentiable at the origin.
BACKGROUND
269