Image Processing Reference

In-Depth Information

later. This theory was introduced by Zadeh and the first article on the subject dates

back to 1965 [ZAD 65]. See [DUB 80, KAU 75, ZIM 91] which contain most of the

theory.

8.2. Definitions of the fundamental concepts of fuzzy sets

8.2.1.
Fuzzy sets

S

be the universe or the space of reference. This is a classical (or crisp) set.

Its elements will be denoted by
x
,
y
, etc. In image processing,

Let

S

will typically be the

space in which the image is defined (

Z

n
or

R

n
, with
n
=2
,
3, etc.). The elements

of

are then the points of the image (pixels, voxels). The universe can also be a set

of values taken from characteristics of the image such as the scale of gray levels. The

elements are then values (gray levels). The set

S

can also be a set of primitives or

objects extracted from the images (segments, areas, objects, etc.) in a representation

on a higher level of image content.

S

A subset
X
of

S

is defined by its characteristic function
μ
X
, such that:

μ
X
(
x
)=
1

if
x

∈

X

[8.1]

0

if
x/

∈

X

The characteristic function
μ
X

is a binary function, specifying for each point of

S

whether it belongs to
X
.

Fuzzy set theory deals with gradual membership. A fuzzy subset of

S

is defined

in [0
,
1]
1
. For any
x
of

by its membership function
μ
of

,
μ
(
x
) is the value in [0
,
1]

that represents the degree to which
x
is a member of the fuzzy subset (often referred

to simply as “fuzzy set”).

S

S

Different notations are often used to refer to a fuzzy set. The set

{

(
x, μ
(
x
))
,

, completely defines the fuzzy set and is sometimes denoted by
S

x

∈

X

}

μ
(
x
)
/x
,or

in the discrete finite case
i
=1
μ
(
x
i
)
/x
i

where
N
indicates the number of elements

in

S

.

Since knowing the set of all of the pairs (
x, μ
(
x
)) is completely equivalent to

having the definition of the membership function
μ
, from now on we will simplify

the notations and use the functional notation
μ
(a function of

S

into [0
,
1]) to refer to

both the fuzzy set and its membership function. We denote by

F

the set of fuzzy sets

defined on

S

.

[0
,
1]

1. The interval

is the most commonly used, but any interval or any other set (a lattice,

typically) can be used.

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