about how to sufficiently measure its textural property in images at a single scale/
resolution or across multiple scales/resolutions is not yet available. Characterized by
irregularity and scale independence, fractal measurement can offer great potential in
describing the spatial complexity of landscape features in images and in providing
insights into the issue of scale and resolution in remote sensing.
Spatial objects in an image scene are often characterized by their dimensionality. In
classical or Euclidean geometry, integers are used to define this characteristic since
the shapes of spatial objects are assumed to be composed of straight lines, for instance,
one dimension for a road, two dimensions for a lake, and three dimensions for a
building. However, the shapes of many natural and man-made objects tend to be
irregular and rarely are composed of straight lines. To avoid the gross simplification
embedded in Euclidean geometry, fractal measurement makes use of the fractional
value reported by the fractal dimension (FD) to define the dimensionality at a more
detailed degree that is otherwise impossible with classic geometry. Typically, FD
values range from 1.0 for a simple curve to 2.0 for a tortuous curve that ultimately fills
two spaces and appears two dimensional. The values can also change from 2.0 for a
simple field to 3.0 for a field that is so complex that it finally fills three spaces and
looks three dimensional. In remote sensing, images can be viewed as a “hilly terrain
surface” whose elevations are represented by the digital numbers (DNs). So the FD
value varies from 2.0 for a perfectly smooth surface to 3.0 for a very rugged surface
that eventually fills a volume. Overall, the more spatially complex an object or image
surface, the higher its FD. FD values can therefore be used as an indicator of the
overall spatial pattern in terms of the degree of irregularity or complexity of an object
or image surface.
To calculate FD, the characteristic of self-similarity or scale independence,
meaning invariance with respect to scale, must be considered. For an ideal selfsimilar object, or a fractal, it is made up of copies of an infinite number of copies of
itself at reduced scale(s). Research has indicated that many natural and man-made
objects such as a mountain, a forest, drainage, agriculture fields, and an urban
landscape exhibit this self-similarity property (Mandelbrot, 1983). The underlying
principle of fractal measurement is to capture the self-similarity of an object/surface,
using statistics, to determine its FD. However, most spatial objects, including remote
sensing images, are not pure fractals at all scales. Rather, they only exhibit a certain
degree of self-similarity over limited ranges of scale, which can have important
implication for the understanding of the operational scale of those spatial objects.
With this in mind, the spatial complexity suggested by a FD value can be assumed to
result directly from the spatial processes of a given object operated at a certain scale.
Therefore, the higher FD may imply a scale where more processes operate or at a level
more approximating to the operational scale of a single process. The significance of
FD for studies concerning not only spatial analysis but also the scale resolution issue
mentioned above is thus apparent. Therefore, it is not surprising to see a growing
number of researches utilize fractal measurement in remote sensing analysis (De Jong
and Burrough, 1995; Emerson et al., 1999, 2002, 2005; Lam, 1990; Lam and De Cola,
1993; Myint, 2003, 2007; Qiu et al., 1999; Read and Lam, 2002; Weng, 2003). A
review of the literature indicates that FDs have been frequently employed to
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231
resolution or across multiple scales/resolutions is not yet available. Characterized by
irregularity and scale independence, fractal measurement can offer great potential in
describing the spatial complexity of landscape features in images and in providing
insights into the issue of scale and resolution in remote sensing.
Spatial objects in an image scene are often characterized by their dimensionality. In
classical or Euclidean geometry, integers are used to define this characteristic since
the shapes of spatial objects are assumed to be composed of straight lines, for instance,
one dimension for a road, two dimensions for a lake, and three dimensions for a
building. However, the shapes of many natural and man-made objects tend to be
irregular and rarely are composed of straight lines. To avoid the gross simplification
embedded in Euclidean geometry, fractal measurement makes use of the fractional
value reported by the fractal dimension (FD) to define the dimensionality at a more
detailed degree that is otherwise impossible with classic geometry. Typically, FD
values range from 1.0 for a simple curve to 2.0 for a tortuous curve that ultimately fills
two spaces and appears two dimensional. The values can also change from 2.0 for a
simple field to 3.0 for a field that is so complex that it finally fills three spaces and
looks three dimensional. In remote sensing, images can be viewed as a “hilly terrain
surface” whose elevations are represented by the digital numbers (DNs). So the FD
value varies from 2.0 for a perfectly smooth surface to 3.0 for a very rugged surface
that eventually fills a volume. Overall, the more spatially complex an object or image
surface, the higher its FD. FD values can therefore be used as an indicator of the
overall spatial pattern in terms of the degree of irregularity or complexity of an object
or image surface.
To calculate FD, the characteristic of self-similarity or scale independence,
meaning invariance with respect to scale, must be considered. For an ideal selfsimilar object, or a fractal, it is made up of copies of an infinite number of copies of
itself at reduced scale(s). Research has indicated that many natural and man-made
objects such as a mountain, a forest, drainage, agriculture fields, and an urban
landscape exhibit this self-similarity property (Mandelbrot, 1983). The underlying
principle of fractal measurement is to capture the self-similarity of an object/surface,
using statistics, to determine its FD. However, most spatial objects, including remote
sensing images, are not pure fractals at all scales. Rather, they only exhibit a certain
degree of self-similarity over limited ranges of scale, which can have important
implication for the understanding of the operational scale of those spatial objects.
With this in mind, the spatial complexity suggested by a FD value can be assumed to
result directly from the spatial processes of a given object operated at a certain scale.
Therefore, the higher FD may imply a scale where more processes operate or at a level
more approximating to the operational scale of a single process. The significance of
FD for studies concerning not only spatial analysis but also the scale resolution issue
mentioned above is thus apparent. Therefore, it is not surprising to see a growing
number of researches utilize fractal measurement in remote sensing analysis (De Jong
and Burrough, 1995; Emerson et al., 1999, 2002, 2005; Lam, 1990; Lam and De Cola,
1993; Myint, 2003, 2007; Qiu et al., 1999; Read and Lam, 2002; Weng, 2003). A
review of the literature indicates that FDs have been frequently employed to
INTRODUCTION
231