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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