模糊格计算扩展FAM神经分类器：人脸识别新方法
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"Lattice Computing Extension of the FAM Neural Classifier for Human Facial Expression Recognition"
本文探讨了FAM（Fuzzy ARTMAP）神经分类器的一种全新扩展——flrFAM，该扩展基于模糊格推理技术，旨在实现增量式实时学习和泛化。FAM首先通过参数优化训练阶段得到增强，然后具备处理部分有序（非数值）数据的能力，包括信息粒度。研究的重点在于区间数值（INs）数据，其中IN表示数据样本的分布。
flrFAM分类器被描述为一种模糊神经网络，能够从数据中诱导出描述性和灵活（可调整）的决策知识（规则）。这种能力使得它在处理如人类面部表情识别等复杂任务时具有优势。在基准数据集上，flrFAM分类器的性能得以验证。
新颖的特征提取方法以及知识表示基础是通过对数据进行模糊格分析来实现的。这种方法允许对数据的复杂模式进行更精细的捕获，尤其是对于面部表情这样的非结构化和多变的数据。模糊格提供了一种结构化的框架，可以在不确定性和模糊性共存的情况下进行推理。
Fuzzy ARTMAP（模糊自适应谐振理论映射）是一种自适应神经网络，主要用于分类任务，它结合了模糊逻辑和ART（自适应谐振理论）的优点，能够在线学习和适应新输入，同时避免过拟合。而在flrFAM中，引入的参数优化训练阶段有助于调整网络参数，提高学习效率和泛化能力。
部分有序数据处理是flrFAM的关键创新之一，这在传统神经网络中往往难以实现。通过处理部分有序数据，flrFAM能够更好地处理那些没有完全明确顺序关系但又存在某种关联性的数据，比如不同强度的面部肌肉活动。
在人类面部表情识别的应用中，flrFAM分类器能够从面部图像中提取特征，这些特征可能包括面部肌肉的动作、形状变化以及它们之间的相互作用。通过模糊规则，flrFAM可以理解和解析这些特征，从而识别出七种基本表情（快乐、悲伤、惊讶、愤怒、恐惧、厌恶和中立）。
总而言之，Lattice Computing Extension of the FAM Neural是一种先进的机器学习技术，特别适用于处理非结构化和复杂的数据，如面部表情识别。它通过模糊格推理和参数优化提升了FAM的性能，使其在实时学习和复杂数据处理方面具有显著优势。
1528 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 10, OCTOBER 2013
Note that, if a ∨ c > b ∧ d,then[a ∨ c, b ∧ d]=∅;in
words, if a ∨ c > b ∧ d, then we assume that the intersection
[a, b]∩[c, d] is the empty set (∅). We remark from [22] that a
preferable (in computing) representation for the least element
O
I1
=∅in lattice (I
1
, ⊆) is O
I1
=[I, O].
Consider a (strictly increasing) positive valuation function
v : L →[0, ∞); furthermore, consider a (strictly decreasing)
dual isomorphic function θ : L → L. Then, function v
1
:
L × L →[0, ∞) given by v
1
([a, b]) = v(θ(a)) + v(b)
is a positive valuation on lattice (L × L, ≥×≤) [25].
Furthermore, based on (1) and (2), two inclusion measures
σ
∩
: I
1
×I
1
→[0, 1] and σ
.
∪
: I
1
×I
1
→[0, 1] can be intro-
duced by σ
∩
(x, y) = σ
(x, x ∩ y) and σ
.
∪
(x, y) = σ
(x, y),
respectively, on the complete lattice (I
1
, ⊆), as will be shown
elsewhere.
Functions θ(.) and v(.) can be selected in different ways.
In the context of this paper, we select a pair of functions v(x)
and θ(x) so as to satisfy the equality “v
1
([x, x]) = v(θ(x)) +
v(x) = Constant” required by a “standard” FLR scheme [25],
[28], [29]. For instance, such pairs of functions v(x) and θ(x)
include, first, v(x) = px and θ(x) = Q − x,wherep, Q > 0,
x ∈[0, Q] and, second, v
s
(x) = A/(1 +e
−λ(x−μ)
) and
θ(x) = 2μ − x,whereA,λ ∈ R
+
0
, μ, x ∈ R. In particular, it
follows, first, v
1
([x, x]) = pQ and, second, v
1
([x, x]) = A.
C. Type-2 Intervals
A Type-2 interval is defined as an interval of Type-1
intervals. Consider the complete lattice (I
2
, ⊆) of Type-2
intervals on a complete lattice (L, ≤) of real numbers with
least and greatest elements O and I, respectively. Recall
that
[[a
1
, a
2
], [b
1
, b
2
]] ∩ [[c
1
, c
2
], [d
1
, d
2
]] =
[[a
1
, a
2
]
.
∪[c
1
, c
2
], [b
1
, b
2
]∩[d
1
, d
2
]],and
[[a
1
, a
2
], [b
1
, b
2
]]
.
∪[[c
1
, c
2
], [d
1
, d
2
]] =
[[a
1
, a
2
]∩[c
1
, c
2
], [b
1
, b
2
]
.
∪[d
1
, d
2
]].
We remark that a preferable representation for the least
element O
I2
=∅in lattice (I
2
, ⊆) is O
I2
=[[O, I],
[I, O]].
Consider a (strictly increasing) positive valuation function
v : L →[0, ∞) as well as a (strictly decreasing) dual
isomorphic function θ : L → L. Recall that function v
1
: L ×
L →[0, ∞) given by v
1
(a, b) = v(θ(a)) + v(b) is a positive
valuation. Furthermore, function θ
1
: L × L → L × L given
by θ
1
(a, b) = (b, a) is dual isomorphic. Therefore, function
v
2
: L×L×L×L →[0, ∞) given by v
2
([[a
1
, a
2
], [b
1
, b
2
]]) =
v(a
1
) +v(θ(a
2
)) +v(θ(b
1
)) +v(b
2
) is a positive valuation on
lattice (L × L × L × L, ≤×≥×≥×≤). In conclusion,
based on (1) and (2), inclusion measures σ
∩
: I
2
×I
2
→[0, 1]
and σ
.
∪
: I
2
× I
2
→[0, 1] can be introduced by σ
∩
(x, y) =
σ
(x, x ∩ y) and σ
.
∪
(x, y) = σ
(x, y), respectively, on the
complete lattice (I
2
, ⊆) of Type-2 intervals.
D. Type-1 Intervals’ Numbers
Consider the following definition.
Definition 2.1: A Type-1 IN is a function F :[0, 1]→I
1
which satisfies
F(0) = I
I1
h
1
≤ h
2
⇒ F(h
1
) ⊇ F(h
2
)
∀P ⊆ [0, 1] :∩
h∈P
F
(
h
)
= F
P
.
We will denote the set of INs by F
1
andequipitwithan
order relationship such that F G ⇔ (∀h ∈[0, 1]:
F(h) ⊆ G(h)). Furthermore, we will denote an IN by a capital
letter in italics, e.g., F
1
F = F(h) =[a
h
, b
h
], h ∈[0, 1].In
practice, an IN is interpreted as an information granule. It turns
out that (F
1
, ) is a complete lattice whose least element ∅ is
preferably represented as O
F1
= O(h) =[I, O], h ∈[0, 1].
Definition 2.1 implies that an IN can be represented by a set
of intervals, i.e., its interval representation. In addition, an IN
can, equivalently, be represented by a membership function;
i.e., the membership-function representation [25].
E. Type-2 Intervals’ Numbers
Another information granule of interest is an interval [U, W]
of Type-1 INs U and W , where interval [U, W] by definition
equals [U, W]
.
={X ∈ F
1
: U X W}. In the latter sense,
we say that X is encoded in [U, W].Interval[U, W] is called
Type-2 IN. It follows the complete lattice (F
2
, ) of Type-
2 INs. Recall that the least (empty) interval ∅ is preferably
represented in computing as O
F2
= O(h) =[[O, I], [I, O]],
where h ∈[0, 1]. A Type-2 IN will be denoted by a double-
line capital letter, e.g., F ∈ F
2
.
The lattice (F
2
, ) join operation is demonstrated in Fig. 1.
In particular, Fig. 1(a) shows the trivial Type-2 INs C
1
=
[C
1
, C
1
], C
2
=[C
2
, C
2
],andC
3
=[C
3
, C
3
].Fig.1(b)
displays the join C
1
C
2
=[C
1
C
2
, C
1
C
2
] in its
membership-function representation. Note that, since Type-1
INs C
1
and C
2
overlap, the Type-1 IN C
1
C
2
is not empty.
More specifically, it is (C
1
C
2
)(h) =∅,forh ∈[0, 0.6471];
nevertheless, for h ∈ (0.6471, 1],itis(C
1
C
2
)(h) =∅.
Fig. 1(c) displays the join C
1
C
2
in the (equivalent) interval
representation. Fig. 1(d) displays the join C
2
C
3
=[C
2
C
3
, C
2
C
3
] in its membership-function representation. Note
that, since Type-1 INs C
2
and C
3
do not overlap, the Type-1
IN C
2
C
3
is empty, i.e., (C
2
C
3
)(h) =∅,forallh ∈[0, 1].
We point out that there are similarities as well as differences
between Type-1 and 2 INs and Type-1 and 2 fuzzy sets [30].
Our interest here is on the inclusion measure σ
.
: F
2
×
F
2
→[0, 1], given as [22]
σ
.
(E
1
, E
2
) =
1
0
σ
.
∪
(E
1
(h), E
2
(h))dh. (3)
F. Extensions to More Dimensions
An N-tuple IN of Type-1/2 will be indicated by an “over
right arrow.” More specifically, a Type-1 IN will be denoted
by
−→
E = (E
1
,...,E
N
) ∈ (F
N
1
, ), whereas a Type-2 IN will
be denoted by
−→
E = (E
1
,...,E
N
) ∈ (F
N
2
, ).
The previous section has shown how to define inclusion
measure functions on lattice (F
2
, ). The latter functions
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