Visual Error
(E
inter
)
proposed by Chen and Wang (2004), limits
the difference of the average spectral values between objects
by setting a threshold value. At the same time, it calculates
the composite heterogeneity of the segmented object and all
the adjacent objects by setting weights according to the length
of the coincidence boundaries of the segmented object and
the adjacent objects.
Goodness based on the local spectral difference in boundary
Goodness based on local spectral difference in boundary
mainly contains Edge Gradient Measure
(EG)
goodness and
Max Border Contrast
(MBC)
goodness (Zéboudj, 1988). The
larger the spectral gradient, or the spectral difference of the
pixels, on the coincidence boundary of the segmented object
and its adjacent objects, the stronger the heterogeneity be-
tween the objects. However, the index is less reliable with a
low degree of homogeneity of each object.
Goodness Based on Spectral Variance
Except for using spectral difference as goodness, it is also
possible to evaluate inter-region heterogeneity using spectral
variance. Nowadays, goodness based on spectral variance are
the most frequently applied hetrogeneity goodness indexes
in remote sensing field at the moment. Ming
et al
. (2006),
according to Variance Contrast Measure
(VC)
goodness, took
the ratio of the spectrum variance’s difference between two
adjacent objects, and the spectrum variance of the combined
adjacent objects, as the goodness to evaluate inter-region
heterogeneity.
VC
f f
f f
=
−
+
1 2
1 2
(33)
where
f
1
and
f
2
are the average gray-values of two adjacent
regions. The larger the
VCM
goodness value, the stronger the
inter-region heterogeneity.
An optional measure, Variance Contrast Across Region
Measure
(
VCAR
)
(Ming
et al
., 2006), can also be used as the
measure for inter-region heterogeneity. If the two adjacent
regions are quite different, the mean of gray value variance of
the two regions would be much less than that of one region
into which two regions are merged. For the whole image, the
sum of all contrast values of every pair of adjacent regions is
regarded as
VCAR
of the whole image. The larger the variance
contrast, the greater the difference of the regions.
VCAR v
v v
= −
+
1 2
2
(34)
where
v
1
and
v
2
are respectively gray value variances of the
adjacent two regions, and represents the gray value variance
of the merged region.
Currently, the most frequently used goodness index to
measure heterogeneity in remote sensing field is
Moran’s
I
. This goodness evaluates inter-region heterogeneity be-
tween objects, in line with the degree of spatial autocorrela-
tion between the segmented objects. The concept of spatial
autocorrelation, proposed by Fotheringham
et al
. (2000),
aims to quantitatively measure the similarity and degree of
dependency of the objects and its adjacent object within the
same region. Scholars found that spatial autocorrelation can
measure heterogeneity between objects with high efficiency.
Therefore,
Moran’s I
, as a common spatial autocorrelation
index, is widely used as a global spatial inter-region heteroge-
neity goodness index, while often being jointly applied with
the global local variance index (Espindola
et al
., 2006; Kim
et
al
., 2009).
The computational formula of
Moran’s I
goodness is as follows:
MI
n
w y y y y
y y
w
i
n
j
n
ij
i
j
i
n
i
i j
ij
=
−
−
−
=
=
=
≠
∑ ∑
∑ ∑ ∑
1
1
1
2
(
)(
)
(
)
(35)
where
n
represents the number of the segmented objects in
the whole image, and
w
ij
represents the index determining
spatial adjacency. If the segmented objects
S
i
and
S
j
involved
in the algorithm are adjacent,
w
ij
equals 1. If they are non-
adjacent,
w
ij
equals 0;
y
– represents the average spectral value
of the whole image, while
y
i
represents the average spectral
value of the segmented object
S
i
. The smaller the value of
Moran’s I, the stronger the heterogeneity between objects.
However, it is impossible to evaluate the inter-region
homogeneity of a single segmented object and its adjacent ob-
jects due to the fact that
Moran’s I
is a global goodness. John-
son and Xie (2011) used local
Moran’s I
goodness to calculate
local inter-region heterogeneity. But, local
Moran’s I
uses the
average spectral value of the whole image, resulting in an un-
reliable evaluation result. Zhang et al. (2015a) proposed
Geary
goodness to measure local inter-region homogeneity, trying
to improve
Moran’s I
by considering spectral information
pixels in coincident boundaries of a segmented object and its
adjacent objects.
The formula for
Geary
goodness is as follows:
LGI
w y y
i
j
j i
k
ij
i
j
=
−
= ≠
∑
1,
(
)
(36)
where
w
ij
and
y
i
have the same definition as in
Moran’s I
good-
ness. The larger the
Geary
value, the stronger the heterogene-
ity between objects.
Composite Goodness Indexes
In most situations, homogeneous goodness and heterogeneous
goodness only focus on local objects. In order to apply this
local goodness to an entire image, it is important to create a
composite of multiple local goodness into global goodness.
For some goodness indexes like ,
E
inter
, and
E
intra
, etc., the com-
positing process only requires a direct addition of all the local
goodness. Besides, most goodness based on the value of intra-
region spectral variance can be composited by area weighting
summation, such as
NU
,
WV
, and
LV
. Moreover,
MWC
goodness
which is based on spectral difference can also be composited
through the area weighting method. As for the heterogeneous
goodness based on boundaries, their local compositing re-
quires the weighting of coincident boundaries’ lengths.
In addition to the process of compositing a goodness index
from local to global scales, the compositing process of good-
ness can also transform intra-region homogeneous and inter-
region heterogeneous goodness, or other indexes, into a com-
prehensive goodness, which evaluates the segmented quality
in a holistic way. Espindola
et al
. (2006) standardized
LV
and
Moran’s I
and added them together, obtained a comprehensive
segmentation measure. He
et al
. (2009) improved Espindola
et al
. (2006)’s method, introduced the weight indices of
homogeneity and heterogeneity.
Kim et al
. (2009) jointly used
LV
goodness and
Moran’s I
to set the optimal segmentation
parameters for remote sensing images. Cánovas-García and
Alonso-Sarría (2015) normalized weighted
LV
goodness and
weighted
Geary
goodness and combined them in to a com-
posite goodness. It is worth mentioning that the compositing
of different types of goodness is not confined to the indexes
of intra-region homogeneity and inter-region heterogeneity;
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October 2018
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