describes a transition from one scale to another, and upscaling or downscaling is an
essential protocol in the transition (Krönert et al., 2001).
Characterized by irregularity and scale independence, fractals are recognized as
a suitable method to capture the self-similarity property of the spatial structure of
interest (Zhao, 2001). Self-similarity represents invariance with respect to scale. In
geoscience, the property of self-similarity is often interpreted as scale independence
(Clarke, 1986). However, most environmental phenomena are not pure fractals at
all scales. Rather, they only exhibit a certain degree of self-similarity within limited
regions and over limited ranges of scale, which is measurable by using statistics
such as spatial auto-covariances. The underlying principle of fractals is to use strict
or statistical self-similarity to determine the fractal dimension (FD) of an object/
surface, which is often used as an indicator of the degree of irregularity or
complexity of objects. When fractals are applied to remote sensing, an image is
viewed as a complex “hilly terrain surface” whose elevations are represented by the
digital numbers. Consequently, FDs are readily computable and can be used to
denote how complicated the “image surfaces” are. Remote sensing studies assume
that spatial complexity directly results from spatial processes operating at various
levels, and higher FD occurs at the scale where more processes operate. With FDs,
the spatial processes that occurred at different scales are measurable and comparable. Compared to other geospatial algorithms in image analysis such as landscape
metrics, fractals offer a better benefit in that they can be directly applied to raw
images without the need for classification or land cover feature identification, in
addition to their sound mathematic bases. Therefore, it is not surprising to see a
growing number of researches utilize fractals in remote sensing image analysis (De
Jong and Burrough, 1995; Emerson et al., 1999, 2005; Lam, 1990; Lam and De
Cola, 1993; Myint, 2003; Qiu et al., 1999; Read and Lam, 2002; Weng, 2003).
Fractal-derived texture images have also been used as additional layers in image
classification (Myint, 2003).
Spatial resolution has been another focus in remote sensing studies. It is necessary
to estimate the capability of remote sensing data in landscape mapping since the
application of remote sensing may be limited by its spatial resolution (Aplin, 2006;
Buyantuyev and Wu, 2007; Ludwig et al., 2007). Imagery with finer resolution
contains greater amount of spatial information, which, in turn, enables the characterization of smaller features better. The proportion of mixed pixels is expected to
increase as spatial resolution becomes coarser (Aplin, 2006). Stefanov and Netzband
(2005) identified weak positive and negative correlations between the normalized
vegetation index (NDVI) and landscape structure at three different resolutions (250,
500, and 1000 m) when they examined the capability of the Moderate Resolution
Imaging Spectroradiometer (MODIS) NDVI data in the assessment of arid landscape
characteristics in Phoenix. Asner et al. (2003) examined the significance of subpixel
estimates of biophysical structure with the help of high-resolution remote sensing
imagery and found a strong correlation between the senescent and unmixed green
vegetation cover values in a deforested area. Agam et al. (2007) sharpened the coarseresolution thermal imagery to finer resolution imagery based on the analysis of the
relationship between vegetation index and land surface temperature. The results
4
CHARACTERIZING, MEASURING, ANALYZING, AND MODELING SCALE
essential protocol in the transition (Krönert et al., 2001).
Characterized by irregularity and scale independence, fractals are recognized as
a suitable method to capture the self-similarity property of the spatial structure of
interest (Zhao, 2001). Self-similarity represents invariance with respect to scale. In
geoscience, the property of self-similarity is often interpreted as scale independence
(Clarke, 1986). However, most environmental phenomena are not pure fractals at
all scales. Rather, they only exhibit a certain degree of self-similarity within limited
regions and over limited ranges of scale, which is measurable by using statistics
such as spatial auto-covariances. The underlying principle of fractals is to use strict
or statistical self-similarity to determine the fractal dimension (FD) of an object/
surface, which is often used as an indicator of the degree of irregularity or
complexity of objects. When fractals are applied to remote sensing, an image is
viewed as a complex “hilly terrain surface” whose elevations are represented by the
digital numbers. Consequently, FDs are readily computable and can be used to
denote how complicated the “image surfaces” are. Remote sensing studies assume
that spatial complexity directly results from spatial processes operating at various
levels, and higher FD occurs at the scale where more processes operate. With FDs,
the spatial processes that occurred at different scales are measurable and comparable. Compared to other geospatial algorithms in image analysis such as landscape
metrics, fractals offer a better benefit in that they can be directly applied to raw
images without the need for classification or land cover feature identification, in
addition to their sound mathematic bases. Therefore, it is not surprising to see a
growing number of researches utilize fractals in remote sensing image analysis (De
Jong and Burrough, 1995; Emerson et al., 1999, 2005; Lam, 1990; Lam and De
Cola, 1993; Myint, 2003; Qiu et al., 1999; Read and Lam, 2002; Weng, 2003).
Fractal-derived texture images have also been used as additional layers in image
classification (Myint, 2003).
Spatial resolution has been another focus in remote sensing studies. It is necessary
to estimate the capability of remote sensing data in landscape mapping since the
application of remote sensing may be limited by its spatial resolution (Aplin, 2006;
Buyantuyev and Wu, 2007; Ludwig et al., 2007). Imagery with finer resolution
contains greater amount of spatial information, which, in turn, enables the characterization of smaller features better. The proportion of mixed pixels is expected to
increase as spatial resolution becomes coarser (Aplin, 2006). Stefanov and Netzband
(2005) identified weak positive and negative correlations between the normalized
vegetation index (NDVI) and landscape structure at three different resolutions (250,
500, and 1000 m) when they examined the capability of the Moderate Resolution
Imaging Spectroradiometer (MODIS) NDVI data in the assessment of arid landscape
characteristics in Phoenix. Asner et al. (2003) examined the significance of subpixel
estimates of biophysical structure with the help of high-resolution remote sensing
imagery and found a strong correlation between the senescent and unmixed green
vegetation cover values in a deforested area. Agam et al. (2007) sharpened the coarseresolution thermal imagery to finer resolution imagery based on the analysis of the
relationship between vegetation index and land surface temperature. The results
4
CHARACTERIZING, MEASURING, ANALYZING, AND MODELING SCALE