=Paper=
{{Paper
|id=None
|storemode=property
|title=An interpretation of the L-Fuzzy Concept Analysis as a tool for the Morphological Image and Signal Processing
|pdfUrl=https://ceur-ws.org/Vol-972/paper7.pdf
|volume=Vol-972
|dblpUrl=https://dblp.org/rec/conf/cla/AlcaldeBF12
}}
==An interpretation of the L-Fuzzy Concept Analysis as a tool for the Morphological Image and Signal Processing==
https://ceur-ws.org/Vol-972/paper7.pdf
An interpretation of the L-Fuzzy Concept
Analysis as a tool for the Morphological Image
and Signal Processing
Cristina Alcalde1 , Ana Burusco2 , and Ramón Fuentes-González2
1
Dpt. Matemática Aplicada. Escuela Univ. Politécnica
Univ. del Paı́s Vasco. Plaza de Europa, 1
20018 - San Sebastián (Spain)
c.alcalde@ehu.es
2
Dpt. Automática y Computación. Univ. Pública de Navarra
Campus de Arrosadı́a
31006 - Pamplona (Spain)
{burusco,rfuentes}@unavarra.es
Abstract. In this work we are going to set up a new relationship be-
tween the L-fuzzy Concept Analysis and the Fuzzy Mathematical Mor-
phology. Specifically we prove that the problem of finding fuzzy images or
signals that remain invariant under a fuzzy morphological opening or un-
der a fuzzy morphological closing, is equal to the problem of finding the
L-fuzzy concepts of some L-fuzzy context. Moreover, since the Formal
Concept Analysis and the Mathematical Morphology are the particular
cases of the fuzzy ones, the showed result has also an interpretation for
binary images or signals.
Keywords: L-fuzzy Concept Analysis, Fuzzy Mathematical Morphol-
ogy, Morphological Image Processing
✶ ■ t✁✂❞✄☎t✆✂
The L-fuzzy Concept Analysis and the Fuzzy Mathematical Morphology were
developed in different contexts but both use the lattice theory as algebraic frame-
work.
In the case of the L-fuzzy Concept Analysis, we define the L-fuzzy concepts
using a fuzzy implication and a composition operator associated with it. In the
Fuzzy Mathematical Morphology, a fuzzy implication is also used to define the
erosion but a t-norm also appears to introduce the dilation.
On the other hand, both theories have been used in knowledge extraction
processes in data bases [14–16].
Next, we will show a brief description of them.
c 2012 by the paper authors. CLA 2012, pp. 81–92. Copying permitted only for private
and academic purposes. Volume published and copyrighted by its editors.
Local Proceedings in ISBN 978–84–695–5252–0,
Universidad de Málaga (Dept. Matemática Aplicada), Spain.
82 Cristina Alcalde, Ana Burusco and Ramón Fuentes-González
✷ ❆ t✁✂✁✄✁ t☎
2.1 ▲-fuzzy Concept Analysis
The Formal Concept Analysis of R. Wille [28, 17] extracts information from a
binary table that represents a formal context (X, Y, R) with X and Y finite sets
of objects and attributes respectively and R ⊆ X × Y . The hidden information
is obtained by means of the formal concepts that are pairs (A, B) with A ⊆ X,
B ⊆ Y verifying A∗ = B and B ∗ = A, where ∗ is the derivation operator that
associates the attributes related to the elements of A to every object set A, and
the objects related to the attributes of B to every attribute set B. These formal
concepts can be interpreted as a group of objects A that shares the attributes
of B.
In previous works [11, 12] we have defined the L-fuzzy contexts (L, X, Y, R),
with L a complete lattice, X and Y sets of objects and attributes respectively
and R ∈ LX×Y a fuzzy relation between the objects and the attributes. This is
an extension of the Wille’s formal contexts to the fuzzy case when we want to
study the relationship between the objects and the attributes with values in a
complete lattice L, instead of binary values.
In our case, to work with these L-fuzzy contexts, we have defined the deriva-
tion operators 1 and 2 given by means of these expressions:
∀A ∈ LX , ∀B ∈ LY A1 (y) = inf {I(A(x), R(x, y))}
x∈X
B2 (x) = inf {I(B(y), R(x, y))}
y∈Y
with I a fuzzy implication operator defined in the lattice (L, ≤) and where A1
represents the attributes related to the objects of A in a fuzzy way, and B 2 , the
objects related to all the attributes of B.
In this work, we are going to use the following notation for these derivation
operators to stand out their dependence to relation R:
∀A ∈ LX , ∀B ∈ LY , we define DR : LX → LY , DRop : LY → LX
DR (A)(y) = A1 (y) = inf {I(A(x), R(x, y))}
x∈X
DRop (B)(x) = B2 (x) = inf {I(B(y), Rop (y, x))}
y∈Y
where we denote by Rop the opposite relation of R, that is, ∀(x, y) ∈ X × Y,
Rop (y, x) = R(x, y).
The information stored in the context is visualized by means of the L-fuzzy
concepts that are some pairs (A, A1 ) ∈ (LX , LY ) with A ∈ fix(ϕ), set of fixed
points of the operator ϕ, being defined from the derivation operators 1 and 2 as
ϕ(A) = (A1 )2 = A12 . These pairs, whose first and second components are said
to be the fuzzy extension and intension respectively, represent a set of objects
that share a set of attributes in a fuzzy way.
The set L = {(A, A1 )/A ∈ fix(ϕ)} with the order relation ≤ defined as:
L-Fuzzy Conc. An. for the Morph. Image and Signal Proc. 83
∀(A, A1 ), (C, C1 ) ∈ L, (A, A1 ) ≤ (C, C1 ) if A ≤ C( or A1 ≥ C1 )
is a complete lattice that is said to be the L-fuzzy concept lattice [11, 12].
On the other hand, given A ∈ LX , (or B ∈ LY ) we can obtain the associated
L-fuzzy concept. In the case of using a residuated implication, as we do in this
work, the associated L-fuzzy concept is (A12 , A1 ) (or (B2 , B21 )).
Other important results about this theory are in [1, 10, 25, 13, 24, 5].
A very interesting particular case of L-fuzzy contexts appears trying to ana-
lyze situations where the objects and the attribute sets are coincident [2, 3], that
is, L-fuzzy contexts (L, X, X, R) with R ∈ LX×X , (this relation can be reflexive,
symmetrical . . . ). In these situations, the L-fuzzy concepts are pairs (A, B) such
that A, B ∈ LX .
These are the L-fuzzy contexts that we are going to use to obtain the main
results of this work. Specifically, we are going to take a complete chain (L, ≤)
as the valuation set, and L-fuzzy contexts as (L, Rn , Rn , R) or (L, Zn , Zn , R).
In the first case, the L-fuzzy concepts (A, B) are interpreted as signal or image
pairs related by means of R. In the second case, A and B are digital versions of
these signals or images.
2.2 Mathematical Morphology
The Mathematical Morphology is a theory concerned with the processing and
analysis of images or signals using filters and operators that modify them. The
fundamentals of this theory (initiated by G. Matheron [22, 23] and J. Serra [26]),
are in the set theory, the integral geometry and the lattice algebra. Actually this
methodology is used in general contexts related to activities as the information
extraction in digital images, the noise elimination or the pattern recognition.
Mathematical Morphology in binary images and grey levels images In
this theory images A from X = Rn or X = Zn (digital images or signals when
n=1) are analyzed.
The morphological filters are defined as operators F : ℘(X) → ℘(X) that
transform, symplify, clean or extract relevant information from these images
A ⊆ X, information that is encapsulated by the filtered image F (A) ⊆ X.
These morphological filters are obtained by means of two basic operators,
the dilation δS and the erosion εS , that are defined in the case of binary images
with the sum and difference of Minkowski [26], respectively.
[ \
δS (A) = A ⊕ S = As εS (A) = A ⊖ S̆ = As
s∈S s∈S̆
where A is an image that is treated with another S ⊆ X, that is said to be struc-
turing element, or with its opposite S̆ = {−x/x ∈ S} and where As represents
a translation of A: As = {a + s/a ∈ A}.
84 Cristina Alcalde, Ana Burusco and Ramón Fuentes-González
The structuring image S represents the effect that we want to produce over
the initial image A.
These operators are not independent since they are dual transformations with
respect to the complementation [27], that is, if Ac represents the complementary
set of A, then:
εS (A) = (δS (Ac ))c , ∀A, S ∈ ℘(X)
We can compose these operators dilation and erosion associated with the
structuring element S and obtain the basic filters morphological opening γ S :
℘(X) → ℘(X) and morphological closing φS : ℘(X) → ℘(X) defined by:
γS = δ S ◦ εS φS = εS ◦ δS
The opening γS and the closing φS over these binary images verifies the
two conditions that characterize the morphological filters: They are isotone and
idempotent operators, and moreover it is verified, ∀A, S ∈ ℘(X):
a) γS (A) ⊆ A ⊆ φS (A)
b) γS (A) = (φS (Ac ))c
These operators will characterize some special images (the S-open and the
S-closed ones) that will play an important role in this work.
This theory is generalized introducing some tools to treat images with grey
levels [26]. The images and the structuring elements are now maps defined in
X = Rn and with values in R = R ∪ {−∞, +∞} or defined in X = Zn and with
values in finite chains as, for instance, {0, 1, . . . , 255}.
The previous definitions can be immersed in a more general framework that
considers each image as a point x ∈ L of a partially ordered structure (L, ≤)
(complete lattice), and the filters as operators F : L → L with properties related
to the order in these lattices [26, 19].
Now, the erosions ε : L → L are operators that preserve the infimum
ε(inf M ) = inf ε(M ), ∀M ⊆ L and the dilations δ : L → L, the supremum:
δ(sup M ) = sup δ(M ), ∀M ⊆ L. The opening γ : L −→ L and the closing
φ : L −→ L are isotone and idempotent operators verifying γ(A) ≤ A ≤ φ(A) .
Fuzzy Mathematical Morphology In this new framework and associated
with lattices, a new fuzzy morphological image processing has been developed [6,
7, 4, 8, 9, 21, 20] using L-fuzzy sets A and S (with X = R2 or X = Z2 ) as images
and structuring elements.
In this interpretation, the filters are operators FS : LX → LX , where L is
the chain [0, 1] or a finite chain Ln = {0 = α1 , α2 , ..., αk−1 , 1} with 0 < α1 <
... < αk−1 < 1.
In all these cases, fuzzy morphological dilations δS : LX → LX and fuzzy
morphological erosions εS : LX → LX are defined using some operators of the
fuzzy logic [4, 6, 9, 21].
L-Fuzzy Conc. An. for the Morph. Image and Signal Proc. 85
In general, there are two types of relevant operators in the Fuzzy Mathemat-
ical Morphology. One of them is formed by those obtained by using some pairs
(∗, I) of adjunct operators related by:
(α ∗ β ≤ ψ) ⇐⇒ (β ≤ I(α, ψ))
The other type are the morphological operators obtained by pairs (∗, I) re-
lated by a strong negation ′ : L → L:
α ∗ β = (I(α, β ′ ))′ , ∀(α, β) ∈ L × L
An example of one of these pairs that belongs to both types is the formed
by the t-norm and the implication of Lukasiewicz.
In this paper, we work taking as (X, +) the commutative group (Rn , +) or the
commutative group (Zn , +), and as (L, ≤,′ , I, ∗), the complete chain L = [0, 1]
or a finite chain as L = Ln = {0 = α1 , α2 , ..., αk−1 , 1} with the Zadeh negation
and (∗, I) the Lukasiewicz t-norm and implication.
We interpret the L-fuzzy sets A : X → L and S : X → L as n-dimensional
images in the space X = Rn (or n-dimensional digital images in the case of
X = Zn ).
In the literature, (see [4, 6, 18, 21]), erosion and dilation operators are intro-
duced associated with the residuated pair (∗, I) as follows:
If S : X → L is an image that we take as structuring element, then we
consider the following definitions associated with (L, X, S)
Definition 1. [6] The fuzzy erosion of the image A ∈ LX by the structuring
element S is the L-fuzzy set εS (A) ∈ LX defined as:
εS (A)(x) = inf{I(S(y − x), A(y))/y ∈ X} ∀x ∈ X
The fuzzy dilation of the image A by the structuring element S is the L-fuzzy
set δS (A) defined as:
δS (A)(x) = sup{S(x − y) ∗ A(y)/y ∈ X} ∀x ∈ X
Then we obtain fuzzy erosion and dilation operators εS , δS : LX → LX .
Moreover, it is verified:
Proposition 1. (1) If ≤ represents now the usual order in LX obtained by the
order extension in the chain L, then the pair (εS , δS ) is an adjunction in the
lattice (LX , ≤), that is:
δS (A1 ) ≤ A2 ⇐⇒ A1 ≤ εS (A2 )
(2) If A′ is the negation of A defined by A′ (x) = (A(x))′ , ∀x ∈ X and if S̆
represents the image associated with S such that S̆(x) = S(−x), ∀x ∈ X, then it
is verified:
εS (A′ ) = (δS̆ (A))′ , δS (A′ ) = (εS̆ (A))′ , ∀A, S ∈ LX
86 Cristina Alcalde, Ana Burusco and Ramón Fuentes-González
Proof. (1) Suppose that δS (A1 ) ≤ A2 . Then δS (A1 )(x) ≤ A2 (x) ∀x ∈ X. That
is, S(x − y) ∗ A1 (y) ≤ A2 (x) ∀(x, y) ∈ X × X.
From these inequalities and from the equivalence α ∗ β ≤ γ ⇔ β ≤ I(α, γ) :
A1 (y) ≤ I(S(x − y), A2 (x)), ∀(x, y) ∈ X × X
and interchanging x and y, we have:
A1 (x) ≤ I(S(y − x), A2 (y)), ∀(x, y) ∈ X × X
and consequently
A1 (x) ≤ inf{I(S(y − x), A2 (y))/y ∈ X}, ∀x ∈ X
That is: A1 (x) ≤ εS (A2 )(x), ∀x ∈ X that shows that A1 ≤ εS (A2 ).
We can prove the other implication in a similar way.
(2) Let be x ∈ X.
εS (A′ )(x) = inf{I(S(y − x), A′ (y))/y ∈ X} = inf{I(S̆(x − y), A′ (y))/y ∈ X}
= inf{(S̆(x − y) ∗ A(y))′ /y ∈ X} = (sup{(S̆(x − y) ∗ (A(y))/y ∈ X})′
= (δS̆ (A)(x))′ = (δS̆ (A))′ (x)
The second equality is proved analogously. ⊔
⊓
✸ ❘ ✁✂✄☎✆✝ ❜ ✄✇ ✝ ❜✆✄♦ ✄♦ ✆t☎ ✞
The erosion and dilation operators given in Definition 1 are used to construct
the basic morphological filters: the opening and the closing (see [4, 6, 18, 21]).
Definition 2. The fuzzy opening of the image A ∈ LX by the structuring ele-
ment S ∈ LX is the fuzzy subset γS (A) that results from the composition of the
erosion εS (A) of A by S followed by its dilation:
γS (A) = δS (εS (A)) = (δS ◦ εS )(A)
The fuzzy closing of the image A ∈ LX by the structuring element S ∈ LX
is the fuzzy subset φS (A) that results from the composition of the dilation δS (A)
of A by S followed by its erosion:
φS (A) = εS (δS (A)) = (εS ◦ δS )(A)
It can be proved that the operators γS and φS are morphological filters, that
is, they preserve the order and they are idempotent:
A1 ≤ A2 =⇒ (γS (A1 ) ≤ γS (A2 )) and (φS (A1 ) ≤ φS (A2 ))
L-Fuzzy Conc. An. for the Morph. Image and Signal Proc. 87
γS (γS (A)) = γS (A), φS (φS (A)) = φS (A), ∀A ∈ LX , ∀S ∈ LX
Moreover, these filters verify that:
γS (A) ≤ A ≤ φS (A) ∀A ∈ LX , ∀S ∈ LX
Analogous results that those obtained for the erosion and dilation operators
can be proved for the opening and closing:
Proposition 2. If A′ is the negation of A defined by A′ (x) = (A(x))′ ∀x ∈ X,
then:
γS (A′ ) = (φS̆ (A))′ , φS (A′ ) = (γS̆ (A))′ ∀A, S ∈ LX
Proof. γS (A′ ) = δS (εS (A′ )) = δS ((δS̆ (A))′ ) = (εS̆ (δS̆ (A)))′ = (φS̆ (A))′ .
The other equality can be proved in an analogous way. ⊔
⊓
Since the operators γS and φS are increasing in the complete lattice (LX , ≤),
by Tarski’s theorem, the respective fixed points sets are not empty. These fixed
points will be used in the following definition:
Definition 3. An image A ∈ LX is said to be S-open if γS (A) = A and it is
said to be S-closed if φS (A) = A.
These S-open and S-closed sets provide a connection between the Fuzzy
Mathematical Morphology and the Fuzzy Concept Theory, as we will see next.
For that purpose, given the complete chain L that we are using, and a com-
mutative group (X, +), we will associate with any fuzzy image S ∈ LX , the
fuzzy relation RS ∈ LX×X such that:
RS (x, y) = S(x − y), ∀(x, y) ∈ X × X
′
It is evident that RS ′ = RS and, if RSop represents the opposite relation of
RS , then RSop = RS̆ .
In agreement with this last point, we can redefine the erosion and dilation as
follows:
εS (A)(x) = inf{I(RS (y, x), A(y))/y ∈ X}
= inf{I(RSop (x, y), A(y))/y ∈ X}, ∀x ∈ X
δS (A)(x) = sup{RS (x, y) ∗ A(y)/y ∈ X}, ∀x ∈ X
With this rewriting, given the structuring element S ∈ LX , we can interpret
the triple (L, X, S) as an L-fuzzy context (L, X, X, RS′ ) where the sets of objects
and attributes are coincident. The incidence relation RS′ ∈ LX×X is at the same
time the negation of an interpretation of the fuzzy image by the structuring
element S.
We will use this representation as L-fuzzy context to prove the most impor-
tant results that connect both theories:
88 Cristina Alcalde, Ana Burusco and Ramón Fuentes-González
Theorem 1. Let (L, X, S) be the triple associated with the structuring element
′ ′
S ∈ LX . Let (L, X, X, RS ) be the L-fuzzy context whose incidence relation RS ∈
LX×X is the negation of the relation RS associated with S. Then the operators
erosion εS and dilation δS en (L, X, S) are related to the derivation operators
′
DRS′ and DR′op in the L-Fuzzy context (L, X, X, RS ) by:
S
εS (A) = DRS′ (A′ ) ∀A ∈ LX
δS (A) = (DR′op (A))′ ∀A ∈ LX
S
Proof. Taking into account the properties of the Lukasiewicz implication, for
any x ∈ X, it is verified that:
′
εS (A)(x) = inf{I(RS (y, x), A(y))/y ∈ X} = inf{I(A′ (y), RS (y, x))/y ∈
′
X} = DRS′ (A )(x)
Analogously,
δS (A)(x) = sup{RS (x, y) ∗ A(y)/y ∈ X} = sup{(I(RS (x, y), A′ (y)))′ /y ∈ X} =
′
(inf{I(RS (x, y), A′ (y))/y ∈ X})′ = (inf{I(A(y), RS (x, y))/y ∈ X})′ =
′op
(inf{I(A(y), RS (y, x))/y ∈ X})′ = ((DR′op (A))(x))′ = (DR′op (A))′ (x). ⊔
⊓
S S
As a consequence, we obtain the following result which proves the connection
between the outstanding morphological elements and the L-fuzzy concepts:
Theorem 2. Let be S ∈ LX and let be RS ∈ LX×X its associated relation. The
following propositions are equivalent:
′
1. The pair (A, B) ∈ LX ×LX is an L-fuzzy concept of the context (L, X, X, RS ),
′
where RS (x, y) = S ′ (x − y) ∀(x, y) ∈ X × X.
2. The pair (A, B) ∈ LX × LX is such that the negation A′ of A is S-open
(γS (A′ ) = A′ ) and B is the S-erosion of A′ (that is, B = εS (A′ )).
3. The pair (A, B) ∈ LX × LX is such that B is S-closed (φS (B) = B) and A
is the negation of the S-dilation of B (that is, A = (δS (B))′ ).
Proof.
1 =⇒ 2) Let be S ∈ LX and RS ∈ LX×X its associated relation. Let us
′
consider an L-fuzzy concept (A, B) of the L-fuzzy context (L, X, X, RS ) in
′
which RS is the negation of RS . Then, it is verified that B = DRS′ (A) and
A = DR′op (B), and, by the previous proposition, εS (A′ ) = DRS′ (A) = B.
S
Moreover, it is fulfilled that γS (A′ ) = δS (εS (A′ )) = δS (B) = (DR′op (B))′ =
S
A′ which proves that A′ is S-open.
2 =⇒ 3) Let us suppose that the hypothesis of 2 are fulfilled. Then, φS (B) =
εS (δS (B)) = εS (δS (εS (A′ ))) = εS (γS (A′ )) = εS (A′ ) = B, which proves that
B is S-closed. On the other hand, from the hypothesis B = εS (A′ ) can be
deduced that δS (B) = δS (εS (A′ )) = γS (A′ ), and consequently, taking into
account that A′ is S-open, that δS (B) = A′ , and finally, A = (δS (B))′ .
L-Fuzzy Conc. An. for the Morph. Image and Signal Proc. 89
3 =⇒ 1) Let (A, B) be a pair fulfilling the hypothesis of 3. Let us consider the
′
L-fuzzy context (L, X, X, RS ). Then, by the previous theorem we can deduce
that (DR′op (B)) = (δS (B))′ = A. On the other hand, applying the previous
S
theorem and the hypothesis, DRS′ (A) = εS (A′ ) = εS (δS (B)) = φS (B) = B,
which finishes the proof. ⊔
⊓
Let us see now some examples.
Example 1. Interpretation of some binary images as formal concepts.
In the referential set X = R2 , if ① = (x1 , x2 ) ∈ R2 , w is a positive number
and if S is the structuring binary image
S = {(x1 , x2 ) ∈ R2 /x21 + x22 ≤ w2 }
then the associated incidence relation RSc ⊂ R2 × R2 is such that:
①RSc ② ⇐⇒ ((x1 − y1 )2 + (x2 − y2 )2 > w2 )
which is irreflexive and transitive. The pair (A, B) showed in Figure 1 is a concept
of the context (R2 , R2 , RSc ), because γS (Ac ) = Ac and B = εS (Ac ).
❳
❇ ❂ ❡❙ ✭ ❈ ✮
❆
❋♦r♠❛❧ ✁✂♥❝✄♣t ☎✆✱✝✞ ✟ ✠ ✡ ☛
Fig. 1. A formal concept of the context (❘2 , ❘2 , RSc ).
Example 2. Interpretation of some open digital signals as fuzzy concepts.
It is known that the erosion εS (A) of an image A by a binary structuring
element S can be rewritten in terms of infimum of the traslations of A by elements
of S [27]: ^
εS (A) = A−s where Ak (x) = A(x − k)
s∈S
90 Cristina Alcalde, Ana Burusco and Ramón Fuentes-González
If X ⊆ Z and L = {0, 0.1, 0.2, ..., 0.9, 1} then, the maps A : X → L can be
interpret as 1-D discrete signals. In Figure 2 there are some examples of discrete
signals.
The signal in Fig 2(c) is the erosion of A′ in Fig 2(b), using a line segment
of three pixels as a structuring element, the middle pixel being its origin (S is a
crisp set).
▲ ✑ ❆ ❞ ◆ ❡❢
✎✏✾ ❏❑▼
✌✍✽ ●❍■
☛☞✼ ❉❊❋
✠✡✻ ❅❇❈
✞✟✺ ❂❃❄
✆✝✹ ✿❀❁
✄☎✸ ✲✳✴
✁✂✷ ✮✯✰
✱✶ ✫✬✭
✵ ✪
✒ ✓ ✔ ✕ ✖ ✗✘ ✙✚ ✛✜ ✢✣ ✤✥ ✦✧ ★✩ ❖ P ◗ ❘ ❙ ❚❯ ❱❲ ❨❩ ❬❭ ❪❫ ❴❵ ❛❜
❳ ❝
(a) Discrete signal A as an L-Fuzzy set (b) Negation A′ of the discrete signal A
➌ ❷ ➍
⑨⑩❶
⑥⑦⑧ ➎➏
➐➑ ➑
③④⑤
✇①②
t✉✈
qrs
♥♦♣
❦❧♠
❤✐❥
❣
❸ ❹ ❺ ❻ ❼ ❽❾ ❿➀ ➁➂ ➃➄ ➅➆ ➇➈ ➉➊
➋
(c) Discrete signal B = εS (A′ )
Fig. 2. Discrete signals
Here, we canValso find the erosions in terms of intersections of image trasla-
tions: εS (A′ ) = {A′−1 , A′0 , A′+1 }.
It can be proved that A′ verifies γS (A′ ) = A′ . So, with A in Fig 2 the pair
(A, B) with B = εS (A′ ) is a fuzzy concept of the context (L, Z, Z, RSc ) with the
crisp incidence relation xRSc y ⇔ (x − y) ∈
/ S, (that is, |x − y| > 1).
L-Fuzzy Conc. An. for the Morph. Image and Signal Proc. 91
✭ ✱ ✮
Fig. 3. L-Fuzzy concept (A, B) of the L-Fuzzy context (L, ❩, ❩, RSc )
✹ ❈ ✁✂✄☎✆✝ ✁✆ ❛✁✞ ❋☎✉☎✟✠ ✇ ✟♦
The main results of this work show an interesting relation between the L-fuzzy
Concept Analysis and the Fuzzy Mathematical Morphology that we want to
develop in future works. So, we can apply the algorithms for the calculus of
L-fuzzy concepts in Fuzzy Mathematical Morphology and vice versa.
On the other hand, we are extending these results to other type of operators
as other implications, t-norms, conjunctive uninorms etc... and to some L-fuzzy
contexts where the objects and the attributes are not related to signal or images.
❆✂♦✁ ✇✄✠✞✡✠☛✠✁✉✆
This work has been partially supported by the Research Group “Intelligent Sys-
tems and Energy (SI+E)” of the Basque Government, under Grant IT519-10,
and by the Research Group “Adquisición de conocimiento y minerı́a de datos,
funciones especiales y métodos numéricos avanzados” of the Public University
of Navarra.
❘✠☞✠✟✠✁✂✠✆
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