an obvious new state (increase or decrease) and thus should
be regarded as thresholds to uncover the detailed variation of
fractal features due to the use of different window sizes. With
them, all window sizes can be classified into three groups in
terms of their contributions to fractal texture feature extrac-
tion as well as the
ML
classification:
a. Texture forming Group (windows smaller or equal to 27 –
31). This group includes the smallest windows used in this
study and is characterized by the lowest average correla-
tion with three spectral bands, the highest average mutual
information, the medium average coefficient of variance,
the highest average skewness, and kurtosis statistics, the
lowest average local variance, the lowest average
FD
values,
the lowest average Moran’s I, and the highest overall
JM
distances that are mostly the same as (for water) or greater
than (for non-water LULCs) those from the spectral clas-
sification. Clearly, as windows sized up from 13 to 31, the
effect of varying window sizes on fractal-based texture
extraction and classification is best demonstrated because
the results of all evaluation measures fluctuate the most
when compared to the following two groups (b and c). This
indicates the windows involved in this group tend to be
too small to capture stable fractal textures, which is not
achieved until approaching the windows of 27 – 31 with a
Landsat
ETM+
multispectral image. As it turns out, the 27
window reports the highest overall
JM
distance between
all
LULC
classes as well as one of the highest Khats, and
thus one of the greatest improvements for the medium
density residential class after comparing to the spectral
classification. Besides, the 31 window proves to be even
more significant in the texture analysis of the Landsat data.
When all windows are considered, it always marks the
starting point of the greatest change (increase or decrease)
that ends at the 33 window as observed from all mea-
surement criteria. This window reports the best textural
classification that is with the highest overall accuracy and
Khat and is statistically significantly different from those
derived from most of other windows and the spectral clas-
sification. It also reports the highest Khat for cropland and
pasture as well as for low density residential and thus the
greatest enhancement for the two classes as compared to
the spectral classification. In Marion County, the median
farm size was about 93,000 m
2
(2002 Census of Agricul-
ture) and the typical lot size for low density residential
was roughly between 670 m
2
and 4,000 m
2
(Wilson
et al.
,
2003) at around the time when the image was captured.
A 31 window size (930 × 930 m
2
) could account about 9
times of typical farm size and about 215 to 1300 times of
representative low density residential subdivisions in the
study area. It may thus imply that this window could offer
the scale that might best observe the fractal characteristics,
namely, the operational scale, of the two
LULC
types de-
picted in the image. In this way, the relationship between
the operational scale and the measurement scale (the pixel
size) for different
LULC
types recoded in an
ETM+
image can
be possibly investigated. In general, the robustness of the
31 window was confirmed by all fourteen criteria as the
best moving window to generate the most useful texture
features and to best support
ML
classification when using
only the multispectral bands from the Landsat
ETM+
image.
This should be used as a guideline for future studies if a
similar approach is adopted.
b. Texture Stabilizing Group I (windows 33 – 47 or 49): This
group contains the medium level window sizes and is
characterized with the medium average correlation with
three spectral bands, the medium average MIs, the lowest
average coefficient of variance, the lowest average skew-
ness coefficient and the medium average kurtosis statistic,
the highest average local variance, the medium (for
variogram and triangular prism) or highest (for isarithmic)
average
FD
values, the medium average Moran’s I, and the
lowest overall
JM
distances. In general, the window effect
is less visible within this group since their results given by
most evaluation measures are quite similar to each other. It
thus suggests that once a stable fractal texture is extracted,
this texture seems to be so distinct and invariant to scale
change determined by window sizes.
c. Texture Stabilizing Group II (windows 49 or 51 – 59): This
group has larger window sizes than the first two groups
and is characterized with the highest average correlation
with three spectral bands, the lowest MIs, the highest aver-
age coefficient of variance, the medium average skewness
coefficient and the lowest average kurtosis statistic, the me-
dium average local variance, the highest (for variogram and
triangular prism) or the medium (for isarithmic) average
FD
values, the highest average Moran’s I, and comparably high
overall
JM
distances as that for the Texture Forming Group.
In general, this group also portrays little within-group-vari-
ation yet its average values are either similar to or rather
higher/lower than those reported in the Texture Stabilizing
Group I. It thus implies that another robust and reliable
fractal feature that is insensitive to scale change become
possible when windows of 49/51 – 59 were used. As a
result, it may suggest the extracted features behave very
much like a fractal at the two levels of spatial scales speci-
fied by the window sizes used to define these two groups.
In another word, the Landsat
ETM+
image used in the study
seems to be statistically self-similar over the limited range
of 33 – 47/49 and again 49/51 – 59 windows. This informa-
tion could help to understand the long asking question of
“are remotely sensed images fractal” that is critical in the
application of fractal analysis in the field of remote sensing
(Sun
et al.
, 2006).
The window size classification discussed above does not
consider the 61 window, the largest size applied in the cur-
rent study. This is because the results reported at this window
always are rather different from others. Overall, most of the
evaluation criteria report very low, if not the lowest, results
at this window, presenting a sharp decrease between the
windows of 59 and 61. The only exception includes those
based on the kurtosis and variogram fractal analyses that
tend to concern more about the stochastic (but not repeating)
nature of texture, as their outcomes increase instead. Interest-
ingly, the trends between the 59 and 61 windows denoted by
all evaluation criteria in terms of direction and magnitude
echo those observed from either the 31 – 33 windows or the
47 – 49 windows. Since only one data point is reported with
the 61 window, it is not clear whether this window might
mark the starting point of another cycle that represents a new
state of the texture feature extraction. More studies are hence
needed to shed more light on the issue.
The Window Effect on Individual LULC Classes
by the Maximum Likelihood Classification
A total of sixteen texture features by the triangular prism
fractal algorithm were applied to perform the
ML
textural clas-
sifications. Based on the
JM
distances, it is found the window
effect on both forest and grass’ separability remain rather
limited. All other classes’ separability tend to be impacted
by the window sizes at a certain degree, particularly at the
27 – 31, 39 – 41, and 59 – 61 windows. This is especially true
for cropland and pasture and water, followed by commercial
and industrial and three residential groups. However, the
statistics of accuracy assessment seem to tell a different story
in that, the Khats for water and high density residential, along
with grass, show little variation across all windows instead.
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November 2018
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