Computational Intelligence - May 2016 - 20
In order to obtain the a-cuts (volumes), we follow the
inductive learning paradigm employed in the theory of
Conceptual Spaces [31], [32]. Considering a conceptual space,
consisting of a crisp color space C plus a suitable metric d in
C, the following issues are required to define a fuzzy color Cu
representing a certain color term C:
❏ A crisp color r, called the positive prototype, which is fully
representative of the color term C, i.e., whose membership
to the fuzzy color must be one.
❏ A set of volumes VCu = {V1, f, Vq} with q $ 2 and
Vi 1 Vi + 1 61 # i # q - 1 corresponding to some a cuts of Cu plus its suppor t. In par ticular, let
X Cu = {a 1, f, a q} 1 [0, 1] with 1 = a 1 2 a 2 2 g 2 a q = 0
be a set of levels, Vi corresponds to the a i -cut of Cu for
1 # i 1 q (hence, V1 is the core) while Vq is the support.
❏ An interpolation mechanism for determining the membership
function Cu : C " [0, 1] on the basis of the positive prototype
and the set of volumes that define the a -cuts and the support.
The methodology proposed in [33] to define both the set of
volumes VCu i associated to a fuzzy color Cu i and the membership function is summarized in the following:
1) In order to define a fuzzy color Cu i, a representative color
corresponding to the color term Ci (called positive prototype) and a Voronoi tessellation are required. To obtain that
tessellation, prototypes of color terms distinct to Ci (called
negative prototypes) are needed. In this sense the tessellation is composed of one cell that contains colors of the
category Ci, while the rest of the cells contain colors that
do not belong to Ci. Therefore, let R+ and R- be the set of
positive and negative prototypes respectively. For learning a
color term Ci modeled by means of a fuzzy color Cu i, the
set R +i = {r i} containing a single positive prototype, and a
collection of negative prototypes R -i = {nr i1, f, nr ki }
such that R +i + R -i = Q are considered.
2) For obtaining the 0.5-cut of the fuzzy color Cu i, we propose to calculate a Voronoi tessellation of the conceptual
space C = RGB for the set of prototypes R i = R +i , R -i
(positive prototype and the set of negative prototypes,
respectively). The (crisp) Voronoi cell V i corresponding to
the positive prototype r i defines the 0.5-cut of the fuzzy
color Cu i . This interpretation will provide fuzzy sets consistent with the natural criterion of assigning degrees
greater than or equal to 0.5 to colors that are closer to
the prototype of the cell than other prototypes (colors of
the cell) and lower degrees to the rest (with except for
the points that are in the own border of the cells).
3) The volumes corresponding to the core and the support
of Cu i (V i1 and V iq respectively) are obtained by two scalings of V i centered in r considering two scaling factors m
and ml . The volume V i is "reduced" by means of a scaling
with m ! [0, 1] to obtain V i1, while it is "enlarged" with
i
ml ! [1, 2] to obtain V q .
4) To obtain the membership function of Cu i a linear interpolation is carried out. Let Cu i with positive prototype r i
a n d X Cu i = {a 1, f, a q} 1 [0, 1], w i t h q $ 2 a n d
20
IEEE ComputatIonal IntEllIgEnCE magazInE | may 2016
1 = a 1 2 a 2 2 g 2 a q = 0, be a set of levels, and let
VCu i = {V i1, f, V iq} w i t h q $ 2 a n d V j 1 V j + 1
61 # j # q - 1 be a set of volumes corresponding to the
u i with a j ! X Cu i \ {a q}, plus its support defined
a j -cuts of C
by V iq . Let S ij 61 # j 1 q be the surface that limits the volume V ij of VCu i . A membership function is defined as follows:
Cu i (c) = f (c; r i, VCu i, X Cu i )
(2)
for each crisp color c, where
f (r i; r i, VCu i, X Cu i ) = 1
(3)
and
f (c; r i, VCu i, X Cu i) =
aj
! X Cu i 6c ! S ij
(4)
and for the rest of crisp colors the membership degree is
determined by linear interpolation in the segment of infinity length originating in r i and passing through c, that we
denote r i c + , as follows: let S ij + r i c + = {s j} be the
intersection point between the surface S ij and the segment
r i c+ . Let d (a, b) be the Euclidean distance between the
crisp colors a and b. Then:
❏ If d (r i, c) # d (r i, s 1) then (the point is inside the core)
f (c; r i, VCu i, X Cu i) = 1
i
(5)
i
❏ If d (r , c) $ d (r , s q) then (the point is outside the support)
f (c; r i, VCu i, X Cu i ) = 0
(6)
❏ If d (r i, s j) # d (r i, c) # d (r i, s j + 1) with 1 # j # q - 1
then (the point is between Sj and S j + 1)
f (c; r i, VCu i, X Cu i) =
a j + 1 + (a j - a j + 1)
#e
d (r i, s j + 1) - d (r i, c)
o
d (r i, s j + 1) - d (r i, s j)
(7)
D. Relation between Parameters and Properties
It is possible to obtain different types of fuzzy color spaces
depending on the use of positive R+, and negative R- prototypes and the scaling factors m and ml . Thereby, some interesting conditions can be specified for obtaining particular types of
fuzzy color spaces when R +C = {r 1, f, r m}, R -C = , mi = 1 R -i
and 1 # m + ml # 2:
+
u is a disjoint space.
❏ If R C \ {r i} 3 R -i 61 # i # m then C
This condition allows us to obtain fuzzy color spaces that
represent disjoint colors. Thus, the color terms represented
in this type of spaces are discriminant, i.e., they essentially
represent exclusive color concepts (but not precluding fuzzy
boundaries with other colors).
+
u is a covering space.
❏ If R i- 3 R C \ {r i} 61 # i # m then C
This condition allows us to obtain fuzzy color spaces to
ensure that all crisp colors are represented by at least one
color term with membership degree greater than 0.
+
u is a partition space.
❏ If R i- = R C \ {r i} 61 # i # m then C
This condition allows us to obtain a fuzzy color space by
specifying only the set of positive prototypes.
Table of Contents for the Digital Edition of Computational Intelligence - May 2016
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