Since linear scale-space approaches, by acting isotropically in the image domain,
delocalize and blur image edges, nonlinear operators and nonlinear scale spaces have
been studied and applied in various image processing and computer vision applications. Following the pioneering work of Perona and Malik (1990), there has been a
flurry of activity in partial differential equation and anisotropic diffusion filtering
techniques (Weickert, 1998). For remote sensing applications, a number of anisotropic diffusion schemes have been proposed and applied to aerial and satellite data
sets (Lennon et al., 2002; Camps-Valls and Bruzzone, 2005; Karantzalos and
Argialas, 2006; Duarte-Carvajalino et al., 2007; Ouma et al., 2008; Plaza et al.,
2009), combined, in most cases, with pixel-based classification techniques. All their
scale-space formulations, though, were based on either diffusions during which the
average luminance value is preserved or geometrically driven ones formulated under a
variational framework. Although these formulations may reduce the problems of
isotropic filtering, they do not eliminate them completely: spurious extrema and
important intensity shifts may still appear (Meyer and Maragos, 2000; Karantzalos
et al., 2007; Tzotsos et al., 2011).
Therefore, another way to produce nonlinear scale-spaces is through mathematical
morphology and, in particular, with morphological levelings, which have been
introduced by Meyer (1998) and further studied by Matheron (1997) and Serra
(2000). Morphological levelings overcome the drawback of spurious extrema or
important intensity shifts and possess a number of desired properties for the
construction of elegant scale-space representations. Levelings, which are a general
class of self-dual morphological operators, do not displace contours through scales
and are characterized by a number of desirable properties for the construction of
nonlinear scale-space representations. They satisfy the following spatial and spectral
properties/axioms (Meyer and Maragos, 2000; Meyer, 2004; Karantzalos et al., 2007;
Tzotsos et al., 2011):
• Invariance by spatial translation
• Isotropy, invariance by rotation
• Invariance to a change of illumination
• Causality principle
• Maximum principle, excluding the extreme case where g is completely flat
In addition, levelings:
• Do not produce new extrema at larger scales
• Enlarge smooth zones
• Create new smooth zones
• Are particularly robust (strong morphological filters)
• Do not displace edges
Designing and formulating an optimal scale space framework are still active areas
of research. Recent efforts include studies on certain scale-space formulation (Nilufar
174
MULTISCALE SEGMENTATION AND CLASSIFICATION
delocalize and blur image edges, nonlinear operators and nonlinear scale spaces have
been studied and applied in various image processing and computer vision applications. Following the pioneering work of Perona and Malik (1990), there has been a
flurry of activity in partial differential equation and anisotropic diffusion filtering
techniques (Weickert, 1998). For remote sensing applications, a number of anisotropic diffusion schemes have been proposed and applied to aerial and satellite data
sets (Lennon et al., 2002; Camps-Valls and Bruzzone, 2005; Karantzalos and
Argialas, 2006; Duarte-Carvajalino et al., 2007; Ouma et al., 2008; Plaza et al.,
2009), combined, in most cases, with pixel-based classification techniques. All their
scale-space formulations, though, were based on either diffusions during which the
average luminance value is preserved or geometrically driven ones formulated under a
variational framework. Although these formulations may reduce the problems of
isotropic filtering, they do not eliminate them completely: spurious extrema and
important intensity shifts may still appear (Meyer and Maragos, 2000; Karantzalos
et al., 2007; Tzotsos et al., 2011).
Therefore, another way to produce nonlinear scale-spaces is through mathematical
morphology and, in particular, with morphological levelings, which have been
introduced by Meyer (1998) and further studied by Matheron (1997) and Serra
(2000). Morphological levelings overcome the drawback of spurious extrema or
important intensity shifts and possess a number of desired properties for the
construction of elegant scale-space representations. Levelings, which are a general
class of self-dual morphological operators, do not displace contours through scales
and are characterized by a number of desirable properties for the construction of
nonlinear scale-space representations. They satisfy the following spatial and spectral
properties/axioms (Meyer and Maragos, 2000; Meyer, 2004; Karantzalos et al., 2007;
Tzotsos et al., 2011):
• Invariance by spatial translation
• Isotropy, invariance by rotation
• Invariance to a change of illumination
• Causality principle
• Maximum principle, excluding the extreme case where g is completely flat
In addition, levelings:
• Do not produce new extrema at larger scales
• Enlarge smooth zones
• Create new smooth zones
• Are particularly robust (strong morphological filters)
• Do not displace edges
Designing and formulating an optimal scale space framework are still active areas
of research. Recent efforts include studies on certain scale-space formulation (Nilufar
174
MULTISCALE SEGMENTATION AND CLASSIFICATION