The main idea of the scale-space representation of a measured signal, such as
the DSM, is to embed the signal into a one-parameter family through Gaussian
smoothing, where the parameter controlling the width of the Gaussian kernel is
termed the scale. Furthermore, if scale is equated to time, the evolution of the scalespace family can also be described by the diffusion equation, a differential equation
that governs heat diffusion in a homogeneous medium. In this sense, the scalespace representation is the result of letting an initial heat distribution, given by
the original signal, to evolve over time, so that fine-scale features in the signal will
disappear monotonically with increasing time or scale. The diffusion equation
description was also convenient because the Gaussian kernel and all its derivatives
are solutions to this equation, which had the important implication that Gaussian
derivatives are natural operators of the linear scale space (Koenderink and Van
Doorn, 1992).
In light of the scale-space theory, decomposing an input signal in terms of
Gaussian derivatives at multiple scales is not only desirable but also convenient
because these basis functions exhibit a wide range of orientation and scale characteristics that make them more efficient to detect primitive structures, such as ridges,
edges, and lines. Moreover, psychophysical and biophysical evidence has showed
that early processing of the visual signal by the human visual system performs similar
operations. At this point, it should be noted that a DSM can be treated as an image
because many editing processes may require visualizing and interpreting the DSM as
an image. Furthermore, digital derivatives of the DSM have more natural meaning
than for images. For instance, the derivative along the steepest ascent direction corresponds to the slope around the derivation point. Also, level contour lines have a
close relation with the gradient field.
14.2.2 Multiscale Hermite Transform
Several decompositions based on Gaussian derivatives have been developed
(Martens, 1990; Reed and Bloom, 1996; Silván-Cárdenas and Escalante-Ramírez,
2006); however, most of them differ in the selection of scaling constants and
implementation. One such decomposition is the multiscale Hermite transform,
which is an overcomplete signal decomposition based on the difference of Gaussian
filters that are further decomposed in scale and rotated Gaussian derivatives (SilvánCárdenas and Escalante-Ramírez, 2006). Its development had an inspiration on
models of the receptive fields of ganglion cells in the human visual system. It can be
implemented in a pyramidal fashion, similar to a wavelet transform, with the
additional property of being steerable, that is, it can be computed in a rotated
coordinate system as a linear combination of the unrotated version.
The MHT is defined for one, two, and higher dimensions and for both continuous
and discrete domains. Of particular interest is the case for 2D discrete signals, which
is relevant for the ground filtering problem. For that particular case the theoretical
development has been revised in several studies, including for ground filtering
(Silván-Cárdenas and Wang, 2006) and for building detection (Silván-Cárdenas
and Wang, 2011) from Lidar measurements.
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MULTISCALE APPROACH FOR GROUND FILTERING FROM LIDAR
the DSM, is to embed the signal into a one-parameter family through Gaussian
smoothing, where the parameter controlling the width of the Gaussian kernel is
termed the scale. Furthermore, if scale is equated to time, the evolution of the scalespace family can also be described by the diffusion equation, a differential equation
that governs heat diffusion in a homogeneous medium. In this sense, the scalespace representation is the result of letting an initial heat distribution, given by
the original signal, to evolve over time, so that fine-scale features in the signal will
disappear monotonically with increasing time or scale. The diffusion equation
description was also convenient because the Gaussian kernel and all its derivatives
are solutions to this equation, which had the important implication that Gaussian
derivatives are natural operators of the linear scale space (Koenderink and Van
Doorn, 1992).
In light of the scale-space theory, decomposing an input signal in terms of
Gaussian derivatives at multiple scales is not only desirable but also convenient
because these basis functions exhibit a wide range of orientation and scale characteristics that make them more efficient to detect primitive structures, such as ridges,
edges, and lines. Moreover, psychophysical and biophysical evidence has showed
that early processing of the visual signal by the human visual system performs similar
operations. At this point, it should be noted that a DSM can be treated as an image
because many editing processes may require visualizing and interpreting the DSM as
an image. Furthermore, digital derivatives of the DSM have more natural meaning
than for images. For instance, the derivative along the steepest ascent direction corresponds to the slope around the derivation point. Also, level contour lines have a
close relation with the gradient field.
14.2.2 Multiscale Hermite Transform
Several decompositions based on Gaussian derivatives have been developed
(Martens, 1990; Reed and Bloom, 1996; Silván-Cárdenas and Escalante-Ramírez,
2006); however, most of them differ in the selection of scaling constants and
implementation. One such decomposition is the multiscale Hermite transform,
which is an overcomplete signal decomposition based on the difference of Gaussian
filters that are further decomposed in scale and rotated Gaussian derivatives (SilvánCárdenas and Escalante-Ramírez, 2006). Its development had an inspiration on
models of the receptive fields of ganglion cells in the human visual system. It can be
implemented in a pyramidal fashion, similar to a wavelet transform, with the
additional property of being steerable, that is, it can be computed in a rotated
coordinate system as a linear combination of the unrotated version.
The MHT is defined for one, two, and higher dimensions and for both continuous
and discrete domains. Of particular interest is the case for 2D discrete signals, which
is relevant for the ground filtering problem. For that particular case the theoretical
development has been revised in several studies, including for ground filtering
(Silván-Cárdenas and Wang, 2006) and for building detection (Silván-Cárdenas
and Wang, 2011) from Lidar measurements.
270
MULTISCALE APPROACH FOR GROUND FILTERING FROM LIDAR