Chapter 4
Soft Computing Based Theory and
Techniques
4.1 Introduction
In many multimedia data mining applications, it is often required to make
a decision in an imprecise and uncertain environment. For example, in the
application of mining an image database with a query image of green trees,
given an image in the database that is about a pond with a bank of earth and a
few green bushes, is this image considered as a match to the query? Certainly
this image is not a perfect match to the query, but, on the other hand, it is
also not an absolute mismatch to the query. Problems like this example, as
well as many others, have intrinsic imprecision and uncertainty that cannot be
neglected in decision making. Traditional intelligent systems fail to solve such
problems, as they attempt to use Hard Computing techniques. In contrast,
a Soft Computing methodology implies cooperative activities rather than au-
tonomous ones, resulting in new computing paradigms such as fuzzy logic,
neural networks, and evolutionary computation. Consequently, soft comput-
ing opens up a new research direction for problem solving that is difficult to
achieve using traditional hard computing approaches.
Technically, soft computing includes specific research areas such as fuzzy
logic, neural networks, genetic algorithms, and chaos theory. Intrinsically, soft
computing is developed to deal with pervasive imprecision and uncertainty of
real-world problems. Unlike traditional hard computing, soft computing is
capable of tolerating imprecision, uncertainty, and partial truth without loss
of performance and effectiveness for the end user. The guiding principle of soft
computing is to exploit the tolerance for imprecision, uncertainty, and partial
truth to achieve a required tractability, robustness, and low solution cost. We
can easily come to the conclusion that precision has a cost. Therefore, in
order to solve a problem with an acceptable cost, we need to aim at a decision
with only the necessary degree of precision, not exceeding the requirements.
In soft computing, fuzzy logic is the kernel. The principal advantage of

static or dynamic systems, were originally developed as a parallel computa-
tional model. A very important advantage of these networks is their adaptive
capability, where “learning by example” replaces the traditional “program-
ming” in problem solving. Another important advantage is the intrinsic par-
allelism that allows fast computations. Artificial neural networks are a viable
computational model for a wide variety of problems, including pattern classi-
fication, speech synthesis and recognition, curve fitting, approximation, image
compression, associative memory, and modeling and control of non-linear un-
known systems, in addition to the application of multimedia data mining. The
third advantage of artificial neural networks is the generalization capability,
which allows correct classification of new patterns. A significant disadvantage
of artificial neural networks is their poor interpretability. One of the main
criticisms addressed to neural networks concerns their black box nature.
Evolutionary computing is a revolutionary paradigm for optimization. One
component of evolutionary computing — genetic algorithms — studies the al-
© 2009 by Taylor & Francis Group, LLC
Soft Computing Based Theory and Techniques 145
Table 4.1: Comparative characteristics of the components of soft computing.
Reprint from [8]
c
2001 World Scientific.
Fuzzy sets Artificial neu-
ral networks
Evolutionary
computing,
Genetic algo-
rithms
Weakness Knowledge
acquisition;
Learning

the computational speed of genetic algorithms is typically low.
Table 4.1 summarizes the comparative characteristics of the different paradigms
of soft computing. For each paradigm of soft computing, there are appropriate
problems where this paradigm is typically applied.
4.3 Fuzzy Set Theory
In this section, we give an introduction to fuzzy set theory, fuzzy logic, and
their applications in multimedia data mining.
4.3.1 Basic Concepts and Properties of Fuzzy Sets
DEFINITION 4.1 Let X be a classic set of objects, called the universe,
with the generic elements denoted as x. The membership of a classic subset
© 2009 by Taylor & Francis Group, LLC
146 Multimedia Data Mining
FIGURE 4.1: Fuzzy set to characterize the temperature of a room.
A of X is often considered as a characteristic function µ
A
mapped from X to
{0,1} such that
µ
A
(x) =

1 iff x ∈ A
0 iff x /∈ A
where {0,1} is called a valuation set; 1 indicates membership while 0 indicates
non-membership.
If the valuation set is allowed to be in the real interval [0,1], A is called a
fuzzy set. µ
A
(x) is the grade of membership of x in A:
µ

2
b
)
which is defined by three parameters, a, b, and c. Figure 4.2 summarizes
the graphical and analytical representations of frequently used membership
functions.
An appropriate construction of the membership function for a specific fuzzy
set is the problem of knowledge engineering [125]. There are many methods for
an appropriate estimation of a membership function. They can be categorized
as follows:
1. Membership functions based on heuristics
2. Membership functions based on reliability concepts with respect to the
specific problem
3. Membership functions based on a certain theoretic foundation
4. Neural networks based construction of membership functions
The following rules which are common and valid in the classic set theory
also apply to fuzzy set theory.
• De Morgan’s law:
A ∩ B = A ∪ B
and
A ∪ B = A ∩ B
• Associativity:
(A ∪ B) ∪ C = A ∪ (B ∪ C)
and
(A ∩ B) ∩ C = A ∩ (B ∩ C)
• Commutativity:
A ∪ B = B ∪ A
and
A ∩ B = B ∩ A
• Distributivity:

• T (P ∧ Q) = min(T (P ), T (Q))
• T (P ∨ (P ∧ Q)) = T (P )
• T (P ∧ (P ∨ Q)) = T (P )
• T (¬(P ∧ Q)) = T (¬P ∨ ¬Q)
• T (¬(P ∨ Q)) = T (¬P ∧ ¬Q)
It is shown that multi-valued logic is the fuzzification of the traditional
propositional calculus (in the sense of the extension principle). Here each
proposition P is assigned a normalized fuzzy set in [0,1]; i.e., the pair {µ
P
(0), µ
P
(1)}
is interpreted as the degree of false or true, respectively. Since the logical con-
nectives of the standard propositional calculus are functionals of truth, i.e.,
they are represented as functions, they can be fuzzified.
Let A and B be fuzzy sets of the subsets of the non-fuzzy universe U ; in
fuzzy set theory it is known that A is a subset of B iff µ
A
≤ µ
B
, i.e., ∀x ∈ U,
µ
A
(x) ≤ µ
B
(x).
In fuzzy set theory, great attention is paid to the development of fuzzy
conditional inference rules. This is connected to the natural language under-
standing where it is necessary to have a certain number of fuzzy concepts;
therefore, we must ensure that the inference of the logic is made such that the

called the degree of membership, to feature vectors of each image block in the
feature space. As a result, the feature vector of a block belongs to multiple
regions with different degrees of membership as opposed to the classic region
representation, in which a feature vector belongs to exactly one region. We
first discuss the fuzzy representation of the texture feature, and then discuss
that of the shape feature.
We take each region as a fuzzy set of blocks. In order to propose a unified
approach consistent with the fuzzy color histogram representation described in
Section 2.4.5.2, we again use the Cauchy function to be the fuzzy membership
function, i.e.,
µ
i
(f) =
1
1 + (
d(f,
ˆ
f
i
)
σ
)
α
(4.1)
where f ∈ R
k
is the texture feature vector of each block with k as the di-
mensionality of the feature vector;
ˆ
f

ˆ
f
i
is the average
texture feature vector of region i.
With this block membership function, the fuzzified texture property of re-
gion i is represented as

f
i
T
=

f ∈U
T
fµ
i
(f) (4.3)
where U
T
is the feature space composed of texture features of all blocks.
Based on the fuzzy membership function µ
i
(f) obtained in a similar fashion,
we also fuzzify the p-th order inertia as the shape property representation of
region i as:
l(i, p) =

f ∈U
S

S
.
4.4 Artificial Neural Networks
Historically, in order to “simulate” the biological systems to make non-
symbolic computations, different mathematical models were suggested. The
artificial neural network is one such model that has shown great promise and
thus attracted much attention in the literature.
4.4.1 Basic Architectures of Neural Networks
Neurons represent a special type of nervous cells in the organism, having
electric activities. These cells are mainly intended for the operative control
of the organism. A neuron consists of cell bodies, which are enveloped in the
membrane. A neuron also has dendrites and axons, which are its inputs and
outputs. Axons of neurons join dendrites of other neurons through synaptic
contacts. Input signals of the dendrite tree are weighted and added in the
© 2009 by Taylor & Francis Group, LLC
152 Multimedia Data Mining
FIGURE 4.3: Mathematical model of a neuron. Reprint from [8]
c
2001
World Scientific.
cell body and formed in the axon, where the output signal is generated. The
signal’s intensity, consequently, is a function of a weighted sum of the input
signal. The output signal is passed through the branches of the axon and
reaches the synapses. Through the synapses the signal is transformed into a
new input signal of the neighboring neurons. This input signal can be either
positive or negative, depending upon the type of the synapses.
The mathematical model of the neuron that is usually utilized under the
simulation of the neural network is represented in Figure 4.3. The neuron
receives a set of input signals x
1

© 2009 by Taylor & Francis Group, LLC
Soft Computing Based Theory and Techniques 153
FIGURE 4.4: Linear function. Reprint from [8]
c
2001 World Scientific.
FIGURE 4.5: Binary function. Reprint from [8]
c
2001 World Scientific.
• sigmoid function (see Figure 4.6),
y =
1
1 + exp
−I
The totality of the neurons, connected with each other and with the envi-
ronment, forms the neural network. The input vector comes to the network by
activating the input neurons. A set of input signals x
1
, x
2
, ..., x
n
of a network’s
neurons is called the vector of the input activeness. Connection weights of
neurons are represented in the form of a matrix W , the element w
ij
of which
is the connection weight between the i-th and the j-th neurons. During the
network functioning process, the input vector is transformed into output one;
i.e., a certain information processing is performed. The computational power
© 2009 by Taylor & Francis Group, LLC

2001 World
Scientific.
networks).
In feed-forward networks the neurons of each layer receive signals either
from the environment or from neurons of the previous layer and pass their
outputs either to the environment or to neurons of the next layer (see Fig-
ure 4.9). In recurrent networks (Figure 4.10) neurons of a particular layer
may also receive signals from themselves and from other neurons of the layer.
Thus, unlike non-recurrent networks, the values of the output signals in a re-
current neural network may be determined only if (besides the current value
of the input signals and the weights of the corresponding connections) there
is information available about values of the outputs of the neurons in the pre-
vious step of the time. This means that such a network possesses elements of
memory that allow it to keep information about the outputs’ state from some
time interval. That is why recurrent networks can model the associative mem-
ory. The associative memory is content-addressable. When an incomplete or
a corrupted vector comes to such a network, it can retrieve the correct vector.
A non-recurrent (feed-forward) network has no feedback connections. In
this network topology neurons of the i-th layer receive signals from the en-
vironment (when i = 1) or from the neurons of the previous layer, i.e., the
(i − 1)-th layer (when i > 1), and pass their outputs to the neurons of the
next (i + 1)-th layer or to the environment (when i is the last layer).
The hierarchical non-recurrent network may be single-layer or multi-layer.
A non-recurrent network containing one input and one output layer, respec-
tively, usually is called a single-layer network. The input layer serves to dis-
tribute signals out of all the inputs of a neuron to all the neurons of the output
layer. Neurons of the output layer are the computing units (i.e., they compute
© 2009 by Taylor & Francis Group, LLC
156 Multimedia Data Mining
FIGURE 4.9: A feed-forward neural network. Reprint from [8]