Geoscience Reference
In-Depth Information
Figure 13.2
Illustration of calculating the cover-type-conversion degrees of membership.
assigned to 1. On the other hand, if
, then it can be stated that the mapped pixel
is wrongly classified, and the degree of membership of
x
is different from
y
x
to become
y
would be 1. The above
statements can be summarized as follows:
M
(
x
Æ
x
) = 1
if x =
y
(13.3)
a
M
(
x
Æ
y
) = 1 and
M
(
x
Æ
x
) = 0
if x
π
y
a
a
Using a 3
¥
3 window, if there was a match between
x
and
y
, then it is reasonable to state that
the cover type of the more dominant pixels (
3 window was probably most representative.
However, if the mapped pixels were wrongly classified (e.g., no match between
x)
in the 3
¥
x
and
y
), then the
more dominant cover type
x
is, the higher the possibility that the mapped pixel with cover type
x
will have cover type
y
. Within that context, the cover-type-conversion degrees of membership
regarding
x
and
y
at the mapped pixel were computed as follows:
M
(
x
Æ
x
) =
n
/
9
if x =
y
(13.4)
b
x
M
(
x
Æ
y
) =
n
/
9 and
M
(
x
Æ
x
) = 1 - (
n
/
9)
if x
π
y
b
x
b
x
where
n
is the number of pixels in the 3
¥
3 window with cover type
x
. The ultimate degrees of
x
membership of cover types
at the mapped pixel were computed as the weighted-sum average of
those from the one-to-one and 3
¥
3-window-based comparisons as follows:
M
(
x
Æ
y
) =
w
• M
(
x
Æ
y
) +
w
• M
(
x
Æ
y
)
(13.5)
a
a
b
b
where
w
and
w
were weights for
M
and
M
, respectively, with
w
+
w
= 1 (note that
x
and
y
in
a
b
a
b
a
b
Equation 13.5 can be different or the same). In this study, we applied equal weights (i.e.,
w
=
w
a
b
= 0.5) for the two one-to-one and 3
3-window-based comparisons. Figure 13.2 demonstrates
how degrees of fuzzy membership of a mapped pixel were computed.
¥
13.2.4
Fuzzy Membership Rules
Here we integrate degrees of membership at individual locations derived from the previous step
into a set of fuzzy rules. Theoretically, a fuzzy rule generally consists of a set of fuzzy set(s) as
argument(s)
A,
and an outcome
B
also in the form of a fuzzy set such that:
k
If
(A
and
A
and … and
A
) then
B
(13.6)
1
2
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