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

k