the sake of quantifying the degree of independence of all im-
age bands that were selected to run image classification, these
image layers were also used to perform dissimilarity analysis
by the mutual information (MI) given in the following formula
(Amankwah, 2015):
MI
=
(
)
(
)
( )
( )
∑ ∑
, * log
*
,
*
1
E
Hist X Y
E Hist X Y
Hist X Hist Y
X Y
xy
xy
x
y
(1)
where
E
is the number of entries,
Hist
x
(
X
) and
Hist
y
(
Y
) are the
histograms and
Hist
xy
(
X,Y
) as joint histogram of image band X
and Y.
MI values start from zero, suggesting two image bands
are independent, and increase with no limit, implying the
uncertainty about one image band is decreased by the given
knowledge of another image layer.
Texture Bands Evaluation with Variation, Skewness, and Kurtosis Statistics
Many simple statistical measures have been employed to
assist in texture band assessment including the coefficient of
variation (
CV
), absolute skewness (
SK
), and absolute differ-
ence of kurtosis from kurtosis of Gaussian distribution (
KU
)
(Nyoungui
et al.
, 2002; Shaban and Dikshit, 2001). Since the
supervised
ML
classifier prefers to integrate features that have
lower coefficient of variance and small changes from skew-
ness (with value 0 for normal distribution) and kurtosis (with
value 3 for normal distribution) (Shaban and Dikshit, 2001),
the average values of the three statistics were computed for all
fractal-based texture bands in this research. For any texture
band i, the formulas of these statistics are given below (Sha-
ban and Dikshit, 2001):
CV
i
=
=
∑
cv
n
ij
j
n
1
(2)
SK
i
=
=
∑
| |
sk
n
ij
j
n
1
(3)
KU
|
|
i
=
−
=
∑
ku
n
ij
i
n
3
1
(4)
where
n
is the number of classes and
cv
ij
,
sk
ij
and
ku
ij
are coef-
ficients of variation, skewness, and kurtosis of class
j
for band
i
.
Texture Band Evaluation with Local Variation
By averaging standard deviations from all moving windows
at a selected size (typically 3 × 3) when they are applied to an
image, local variation (
LV
) is often used to assess the appro-
priate spatial resolution or scale of image data through the
examination of the local variation/resolution-cell size graph
(Cao and Lam, 1997; Maillard, 2003; Woodcock and Strahler,
1987). This spatial index has not been applied to evaluate
the quality of texture bands (such as those based on fractals)
before and thus was employed in the current study.
Texture Band Evaluation with Fractal Analysis
and Spatial Autocorrelation Analysis
To further assess whether
FD
values can be employed as a di-
agnostic indicator for spatial content of all kinds of images, all
created texture bands were used to perform fractal analysis.
The fractal nature of an image is displayed in a variety of
aspects such as size, shape, area, distance, correlation, and
power spectra. Since different algorithms are developed to
estimate
FDs
from various angles, it is very likely that their
results are not the same even for the same object. In order
to provide cross-validation of
FD
results, this research con-
sidered two other popular fractal algorithms in
ICAMS
, the
isarithm and the variogram approaches, plus the triangular
prism method that was already used to generate all texture
bands. The isarithm or walking divider method uses different
length segments (“step sizes”), as determined by image reso-
lution, necessary to traverse a curve vertically or horizontally
across that image to measure its
FD
. So the more irregular a
line, the greater increase in length as the step size decreases.
The theory of the variogram, however, differs greatly from the
isarithm and triangular prism methods since it does not apply
any “real” measurement unit. Rather, it fundamentally relies
on the use of the statistical relationship between the average
differences of pixel values and their related distance, which
is essentially described by the fractional Brownian motion
model (Zhao, 2001) that will not be discussed here. Based
on the previous research (Liang
et al.
, 2013), it was decided
that for the isarithm approach, the input parameters were set
as interval = 4, number of steps = 8, and orientation = both;
for the triangular prism method, the parameter was fixed at
number of steps = 5; and for the variogram method, its input
parameters were fixed to be 20 distance groups and sampling
every 10
th
pixel from both rows and columns of a regular grid.
Finally, the spatial autocorrelation index of Moran’s Is were
also calculated for all texture bands.
Supervised Image Classification
Upon the completion, multiple fractal features derived by
various window sizes that met more than one evaluation
criteria, which should indicate their potential as the input
of the
ML
classification, were selected. They were then used
to perform the
textural
classification along with the three
spectral bands (G, R, and NIR). To better understand whether
such a textural classification shows promise in
ML
classifica-
tion, another
spectral
classication was also carried out to use
only the three selected spectral bands. For both the textural
and spectral classification, the supervised logic was adopted
consistently. Table 1 lists eight
LULC
categories considered in
the investigation. The classification system used was modi-
fied from the USGS Anderson Level 1
LULC
classification
system (Anderson
et al.
, 1976), with the urban or built-up
land class subdivided into commerical and industrial as well
as high, medium, and low density residentials groups. The
selection of training for various classes was made using an
aerial photograph at a 1:12 000 scale. While the airphoto was
taken three years after the Landsat image, it is the only high
resolution raster reference data that is the cloest to the image’s
collection time. Therefore, the 2000 Zoning
GIS
layer (Wilson
et al.
, 2003) that details the land use of the study area was
also incorporated as additional reference data. The numbers
of training areas varied from nine to twenty-two for different
LULC
classes (Table 1). When choosing training areas to best
represent different
LULC
groups, a seperability analysis using
the attribute vector was also considered for both classifica-
tions. In particular, the Jeffries-Matusita (
JM
) distance was
calculated from the selected training examples using the fol-
lowing formula (Richards, 1986):
JM i,j
( )
= −
−
2 1(
)
e
B
ij
where
B 125 M M
M M 5 ln
ij
i
j
T
i
j
=
(
)
+
(
)
+
+
∑∑
−
0
2
0
0 5
1
.
.
. |
i
j
i
j
|
| || |
∑∑
∑∑
i
j
−
−
(5)
where
JM
(i, j) is
JM
Distance between class i and j, M
i
and M
j
are
the class mean vectors, and
∑
i
,
∑
j
are the class covariance matrices.
698
November 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
