Computational Intelligence - August 2015 - 47
Then, a distribution of features is "meta-represented" by an IN
[24] induced by algorithm CALCIN [13]. A recent work [27]
has demonstrated specific advantages for an IN meta-representation including a significant dimensionality reduction as well
as a superior pattern recognition performance.
i
i!I
0.135
10
5.
9
6.
3
7.
3
3.
86
4.
6
2.
65
0
0
Recall that a population of features, which are induced from an
image pattern, can be represented by a (T1) IN. This section
investigates the representation of a class by a 3-D T1 IN (or a
3-D T2 IN) toward representing the distribution of T1 INs
used for inducing it.
For example, consider the seventeen trivial T2 INs
[C i, C i], i ! {1, f, 17} in Fig. 2(a). Fig. 2(b) displays the corresponding lattice join * [C i, C i] = [ O C i, iP
C i], I = {1, f, 17}.
!I
!
h Axis
B. Image Pattern Representation
1
x Axis
(a)
I
Note that any inclusion measure function v : F2 # F2 " [0, 1]
re s u l t s i n v ([C i, C i], [ O
C j, jP
C j]) = 1, i ! I = {1, f, 17}
!I
!
j
I
* [C i, C i] =
1
[ jO
C j, jP
C j] is a (2-D) T2 IN whose lower membership
!I
!I
0.8
according to Definition 2.1. The lattice join
upper membership function jP
C j has height hgt ( jP
C j ) = 1,
!I
!I
as shown in Fig. 2(b).
A disadvantage of a 2-D T2 IN is that it does not retain any
information regarding the distribution of INs used to induce it.
We will try to turn the aforementioned disadvantage into an
advantage by inducing "h-secondary membership functions" as
explained in Fig. 3(a), where such functions will be induced at
h = 0.135 and h = 1, respectively. Fig. 3(b) displays two h-secondary membership functions along the line through
h = 0.135, with supports [2.65, 3.86] and [6.3, 7.3], respectively; furthermore, Fig. 3(c) displays one h-secondary membership
function along the line through h = 1, with support [4.6, 5.9].
Fig. 4(a) displays the 3-D surface, which is induced from all the h
-secondary membership functions, truncated by a plane through
z = 0.3 parallel to the x - h plane. By definition, a zSlice is the
intersection of the latter surface with a plane through z ! [0, 1] parallel to the x - h plane. By construction, a zSlice includes two functions, namely primary membership functions, defined by the ends of the
supports of all the h-secondary membership functions on a zSlice.
For example, the zSlice for z = 0 of the surface calculated from the
INs in Fig. 2(a) is the (2-D) T2 IN shown in Fig. 2(b). Fig. 4(b) displays the two primary membership functions on the zSlice shown in
Fig. 4(a) for z = 0.3. The (truncated) surface shown in Fig. 4(a) is a
3-D T2 IN; whereas, the "T2 INs" shown in Fig. 1 are 2-D T2 INs.
Recall that the previously defined "(T1) INs" are alternatively called
2-D (T1) INs. Likewise, 3-D (T1) INs can be induced by computing h-secondary membership functions as detailed above.
0.6
0.4
0.2
0
0
2.65 3.86
6.3 7.3
10
x Axis
(b)
1
z Axis
function O C j has height hgt ( O C j) = 0.6471 and whose
j!I
j!I
z Axis
i!I
0.8
0.6
0.4
0.2
0
0
4.6 5.9
x Axis
(c)
10
Figure 3 (a) h-secondary membership functions will be induced at
h = 0.135 and h = 1, respectively. (b) Two h-secondary membership
functions at h = 0.135 with supports [2.65, 3.86] and [6.3, 7.3],
respectively. (c) One h-secondary membership function at h = 1 with
support [4.6, 5.9].
VI. Experiments and Results
A. Benchmark Datasets and Feature Extraction
In this section we provide experimental evidence regarding the
capacity of our proposed techniques in image pattern recognition applications. More specifically, we have dealt with image
pattern recognition as a classification problem as explained below.
We have employed the following four benchmark datasets.
1) YALE dataset [30]: It regards face recognition. It contains
165 (8-bit) images (320 × 243 pixels each) of 15 individuals (i.e., classes). More specifically, there are 11 images
august 2015 | IEEE ComputatIonal IntEllIgEnCE magazInE
47
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