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24• Fuzzy Systems
24• Fuzzy Systems Fuzzy Control Abstract | Full Text: PDF (379K) Fuzzy Image Processing and Recognition Abstract | Full Text: PDF (1029K) Fuzzy Information Retrieval and Databases Abstract | Full Text: PDF (136K) Fuzzy Model Fundamentals Abstract | Full Text: PDF (2041K) Fuzzy Neural Nets Abstract | Full Text: PDF (162K) Fuzzy Pattern Recognition Abstract | Full Text: PDF (280K) Fuzzy Statistics Abstract | Full Text: PDF (284K) Fuzzy Systems Abstract | Full Text: PDF (143K) Possibility Theory Abstract | Full Text: PDF (174K)
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Wiley Encyclopedia of Electrical and Electronics Engineering Fuzzy Control Standard Article Rainer Palm1 1Siemens AG, Munich, Germany Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W3504 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (379K)
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Abstract The sections in this article are Fuzzy Control Techniques The Fuzzy Controller as a Nonlinear Transfer Element Heuristic Control and Model-Based Control Cell Mapping Supervisory Control Adaptive Control About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.
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92
FUZZY CONTROL
FUZZY CONTROL Fuzzy control is a control approach which is based on the concept of fuzzy sets and fuzzy logic invented by Lotfi Zadeh in 1965 (1). Fuzzy sets are noncrisp or nonsharp sets or numbers, and fuzzy logic is a logic which deals with implications or IF THEN statements using noncrisp truth values. Fuzzy control deals with IF THEN statements or IF THEN rules, respectively, but in the sense of control commands like ‘‘IF temperature is LOW THEN change current of heater by a POSITIVE HIGH value.’’ In this rule LOW and POSITIVE HIGH are fuzzy terms which are not sharply described. With the help of rules like that, one can formulate the knowledge of an operator in a complex plant with the aim to introduce an automatic control of the plant or of parts of it. Another option is to build up an advisory system by means of a set of fuzzy rules that supports the human operator making decisions. Fuzzy control is not only useful when human operators come into play but also in existing automatic control loops. Here, the fuzzy controller is a nonlinear control element that is able to improve control performance and robustness of a plant. In automatic control it is often required to have a process model available for compensation of nonlinear system’s behavior and a corresponding feed forward control. For complex systems or plants it is therefore of advantage to use fuzzy system plant models in order to simplify both the identification and the control task. The following article deals with
common fuzzy control techniques seen both from the system’s and the controller’s point of view. A special part is attended to the nonlinear nature of fuzzy control. Aspects of heuristic and model based fuzzy control are dealt with and the main points of supervisory and adaptive control are discussed. Fuzzy control in the form of set of IF-THEN fuzzy rules was initiated by E. H. Mamdani when he started an investigation of fuzzy set theory–based algorithms for the control of a simple dynamic plant (2). Østergaard reported a fuzzy control application of a heat exchanger (3), and in 1982 Holmblad and Østergaard presented a cement kiln fuzzy controller (4). However, mainly due to the attention that Japan’s industry paid to the new control technology, it was not until the late 1980s that fuzzy control became more and more accepted in industry. The commonly used technique in industrial process control is the Proportional-Integral-Differential (PID) controller, and it is used in a variety of different control schemes (e.g., adaptive, gain scheduling, and supervisory control architectures). Today, processes and plants under control are so complex that PID controllers are not sufficient even though augmented with additional adaptive, gain scheduling, and supervisory algorithms. Although there is a large number of methods and theories (5) to cope with sufficiently complex control problems in the automation, robotics, consumer and industrial electronics, car, aircraft, and ship-building industries, the restrictions for applying these methods are either too strong or too complicated to be applied in a practically efficient and inexpensive manner. Therefore, control engineers are in a need of simpler process and plant models and controller design methods far removed from the sophisticated mathematical models available and their underlying rigorous assumptions. These simpler design methods should provide good performance characteristics, and they should be robust enough with regard to disturbances, parameter uncertainties, and unmodeled structural properties of the process under control. In connection with traditional control techniques, fuzzy control provides a variety of design methods that can cope with modern control problems. There are three main aspects of fuzzy controllers that go beyond the conventional controllers designed via traditional control methods: 1. The use of IF-THEN rules. 2. The universal approximation property. 3. The property of dealing with vague (fuzzy) values. The first aspect concerns the human operator’s knowledge and its heuristic experience for controlling a plant. This knowledge is formulated in terms of IF-THEN fuzzy rules. In the same way, the plant’s behavior can also be expressed by a set of IF-THEN fuzzy rules. The major problem is to identify the fuzzy rules and the regarding parameters such that the operator’s control actions and the systems’s response are sufficiently well described (6–8). Identification of this type of fuzzy rules can be done in two ways: 1. Knowledge acquisition via the use of interviewing techniques from the area of knowledge-based expert systems. This type of identification has been applied successfully to the control of Single Input/Single Output
J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.
FUZZY CONTROL
(SISO) plants and processes, but is difficult to apply and verify for Multi Input/Multi Output (MIMO) control problems. 2. Black box type of identification via the use of clustering, neural nets, and genetic algorithm–based techniques. In the latter approach one distinguishes between structure identification and parameter identification. Structure identification requires structural a priori knowledge about the system to be controlled (e.g., whether the system is assumed to be linear, and what the order of the system might be). If one has to identify a plant with only little structural knowledge, one has to use algorithms that learn from data. The result of structure identification is a set of fuzzy rules. Parameter identification deals with a proper parametrization, scaling, and normalization of physical signals. Parameter identification is a comparetively simple task and can be done by classical methods (e.g., Linear Quadratic (LQ) methods and related techniques). The second aspect, the universal approximation property, means that a fuzzy system with product-based rule firing, centroid defuzzification, and Gaussian membership functions can approximate any real continuous function on a compact set to arbitrary accuracy (9–11). However, in most cases the approximation of a finite state space by a finite number of fuzzy rules is required while using triangular or trapezoidal membership functions. In this case certain approximation errors must be accepted. The approximation property is due to the overlap of the membership functions from the IF parts of the set of fuzzy rules. Because of this overlap, every rule is influenced by its neighboring rules. The result is that every point in state space is approximated by a subset of fuzzy rules. The third aspect considers control tasks where the controller inputs are fuzzy values instead of being crisp variables. In contrast to classical controllers, fuzzy controllers (FC) can also deal with fuzzy values, and even the mixture of crisp and fuzzy values becomes possible. Fuzzy values are qualitative ‘‘numbers’’ obtained from different sources. One particular source is a qualitative statement of a human operator while controlling a plant, like ‘‘temperature is high.’’ Another source may originate from a sensor that provides information about the intensity of a physical signal with in a certain interval. Here, the intensity or distribution of the signal with respect to this interval is expressed by a membership function. This article is arranged as follows: The following section deals with fuzzy control techniques, including the design goal, the definition of a fuzzy region, and the most important FC techniques for systems and controllers. Then the article deals with the fuzzy controller as a nonlinear transfer element while the computational structure of a fuzzy controller, its transfer characteristics, and its nonlinearity are discussed. Different heuristic and model-based control strategies, such as the Mamdani controller, the sliding mode fuzzy controller, the cell mapping control strategy, and the Takagi Sugeno control strategy, are discussed, a short overview of supervisory control is provided; and finally the main aspects of adaptive fuzzy control are discussed. FUZZY CONTROL TECHNIQUES FC techniques can be divided into experiential (heuristic) and model-based techniques. The choice for a special FC tech-
93
nique depends on how the system to be controlled is described. Figure 1 shows the most important FC techniques dealing with systems and controllers. The following subsection deals with the design goal of fuzzy control. In a subsequent subsection the fuzzy region is defined. Finally, the individual FC techniques for systems and controllers are outlined. The Design Goal The objective of the design in fuzzy control can be stated as follows: 1. Stabilization. In stabilization control problems, a fuzzy controller, called a stabilizer, or regulator, is to be designed so that the state vector of the closed-loop system will be stabilized around a point (operating point, or a setpoint) of the state space. The asymptotic stabilization control problem is to find a control law in terms of a set of fuzzy rules such that, starting anywhere in a region around the setpoint xd, the state vector x of the closed-loop system goes to the setpoint xd, as t goes to infinity. 2. Tracking. In tracking control problems, a fuzzy controller is to be designed so that the closed-loop system output follows a given time-varying trajectory. The asymptotic tracking problem is to find a control law in terms of a set of fuzzy rules such that starting from any initial state x0 in a region around xd(t), the tracking error x(t) ⫺ xd(t) tends to 0 while the whole state vector remains bounded. Let us stress here that perfect tracking (i.e., when the initial states imply zero tracking error) is not possible. Therefore, the design objective of having asymptotic tracking cannot be achieved. In this case, one should aim at bounded-error tracking, with small tracking errors to be obtained for trajectories of particular interest. From a theoretical point of view, there is a relationship between the stabilization and the tracking control problems. Stabilization can be regarded as a special case of tracking where the desired trajectory is a constant. On the other hand, if, for example, we have to design a tracker for the open-loop system y¨ + f (y, ˙ y, u) = 0
(1)
so that e(t) ⫽ y(t) ⫺ yd(t) tends to zero, the problem is equivalent to the asymptotic stabilization of the system, e¨ + f (e, ˙ e, u, yd , y˙d , y¨d ) = 0
(2)
its state vector components being e and e˙. Thus the tracker design problem can be solved if one designs a regulator for the latter nonautonomous open-loop system. Performance. In linear control, the desired behavior of the closed-loop system can be systematically specified in exact quantitative terms. For example, the specifications of the desired behavior can be formulated in the time domain in terms of rise time and settling time, overshoot and undershoot, etc. Thus, for this type of control, one first postulates the quanti-
94
FUZZY CONTROL
Controllers
Systems
Figure 1. FC techniques. Collection of fuzzy control techniques for systems and controllers. Mamdani controllers can, e.g., be applied to systems described by differential equations.
Differential equations
Mamdani rules
Mamdani controllers (PD, PID, SMFC)
TS controllers
Relational equations
TS rules
Relational controllers
Predictive controllers
tative specifications of the desired behavior of the closed-loop system and then designs a controller that meets these specifications (for example, by choosing the poles of the closed-loop system appropriately). As observed in Ref. 12, such systematic specifications of the desired behavior of nonlinear closed-loop systems, except for those that can be approximated by linear systems, are not obvious at all because the response of a nonlinear system (open or closed loop) to one input vector does not reflect its response to another input vector. Furthermore, a frequency domain description of the behavior of the system is not possible either. The consequence is that in specifying the desired behavior of a nonlinear closed-loop system, one employs some qualitative specifications of performance, including stability, accuracy and response speed, and robustness. Stability. Stability must be guaranteed for the model used for design (the nominal model) either in a local or in a global sense. The regions of stability and convergence are also of interest. One should, however, keep in mind that stability does not imply the ability to withstand persistent disturbances of even small magnitude. This is so since the stability of a nonlinear system is defined with respect to initial conditions, and only temporary disturbances may be translated as initial conditions. Thus stability of a nonlinear system is different from stability of a linear system. In the case of a linear system, stability always implies the ability to withstand bounded disturbances when, of course, the system stays in its linear range of operation. The effects of persistent disturbances on the behavior of a nonlinear system are addressed by the notion of robustness. Accuracy and Response Speed. Accuracy and response speed must be considered for some desired trajectories in the region of operation. For some classes of systems, appropriate design methods can guarantee consistent tracking accuracy independent of the desired trajectory, as is the case in sliding mode control and related control methods. Robustness. Robustness reflects the sensitivity of the closed-loop system to effects that are neglected in the nominal model used for design. These effects can be disturbances, measurement noise, unmodeled dynamics, etc. The closedloop system should be insensitive to these neglected effects in the sense that they should not negatively affect its stability.
Hybrid controllers
We want to stress here that the aforementioned specifications of desired behavior are in conflict with each other to some extent, and a good control system can be designed only based on tradeoffs in terms of robustness versus performance, cost versus performance, etc. Fuzzy Regions In fuzzy control, a crisp state vector x ⫽ (x1, . . ., xn)T is a state vector the values of which are defined on the closed interval (the domain) X of reals. A crisp control input vector u ⫽ (u1, . . ., un)T is a control input vector the values of which are defined on the closed interval (the domain) U of reals. The set of fuzzy values of a component xi is called the term set of xi denoted as TXi ⫽ 兵LXi1, . . ., LXimi其 (e.g., NB, NM, NS, Z, PS, PM, PB with N negative, P positive, S small, M medium, B big). LXij is defined by a membership function 兰X 애Xij(x)/x. The term set of ui is likewise denoted as TUi ⫽ 兵LUi1, . . ., LUiki其. LUij is defined by a membership function 兰U 애Uij(u)/u. An arbitrary fuzzy value from TXi is denoted as LXi that can be any one of LXi1, . . ., LXimi. An arbitrary fuzzy value from TUi will be denoted as LUi and can be any one of LUi1, . . ., LUiki. A fuzzy state vector LX ⫽ (LX1, . . ., LXn)T denotes a vector of fuzzy values. Each component x1, . . ., xn of the state vector x takes a corresponding fuzzy value LX1, . . ., LXn, where LXi 僆 TXi. A fuzzy region LXi ⫽ (LX1i , . . ., LXni )T is defined as a fuzzy state vector for which there exists a contiguous set of crisp state vectors 兵x*其, each crisp state vector satisfying the given fuzzy state vector LXi to a certain degree different from 0. The fuzzy state space is defined as the set of all fuzzy regions LXi. Example Let x ⫽ (x1, x2)T, TX1 ⫽ 兵LX11, LX12, LX13其, and TX2 ⫽ 兵LX21, LX22, LX23其. Then the total number of different fuzzy state vectors is M ⫽ 9 and the corresponding state vectors are 1. LX 1 ⫽ (LX11, LX21)T 2. LX 2 ⫽ (LX11, LX22)T 3. LX 3 ⫽ (LX11, LX23)T 4. LX 4 ⫽ (LX12, LX21)T 5. LX 5 ⫽ (LX12, LX22)T 6. LX 6 ⫽ (LX12, LX23)T 7. LX 7 ⫽ (LX13, LX21)T
FUZZY CONTROL
8. LX 8 ⫽ (LX13, LX22)T 9. LX 9 ⫽ (LX13, LX23)T FC Techniques for Systems and Controllers In this subsection we deal with systems and controllers according to the scheme shown in Fig. 1. Given a model (heuristic or analytical) of the physical system to be controlled and the specifications of its desired behavior, design a feedback control law in the form of a set of fuzzy rules such that the closed-loop system exhibits the desired behavior. The general control scheme is shown in Fig. 2. Here, we have the following notations: x is the state vector (also controller input) xd is the desired state vector u is the control input vector (also controller output)
In the following we define two basic types of nonlinear control problems: namely, nonlinear regulation (stabilization) and nonlinear tracking (12). Then we will briefly discuss the specifications of desired behavior, such as performance, stability, and robustness, in the context of nonlinear control. Stabilization and Tracking. In general, the tasks of a control system can be divided into two basic categories: Heuristic System Models. When an analytical model of the plant is not available, the control design has to be carried out on the basis of qualitative modeling. This can be done either in terms of a set of Mamdani fuzzy rules or a fuzzy relation (6,13). A typical Mamdani rule of a continuous first-order system is x is PS AND u is NB THEN x˙ is NS
(3)
and for a discrete system RSi :
x˙ = A · x + B · u
IF x(k) is PS AND u(k) is NB THEN x(k + 1) is NS
A typical fuzzy relational equation of a discrete firstorder system is X (k + 1) = X (k) ◦ U (k) ◦ S
(4)
relating the state at time k ⫹ 1 to the state and control input at time k. S is the fuzzy relation. 폶 denotes the
On the other hand, a TS fuzzy rule consists of a fuzzy antecedent part and a consequent part consisting of an analytical equation. A typical TS rule for a first-order system is IF
x
Fuzzy controller FC
u
x = LX i
THEN x˙ = Ai · x + Bi · u
(6)
where LXi is the ith fuzzy region for x, and Ai and Bi are parameters corresponding to that region.
Mamdani Controller. A Mamdani controller works in the following way: 1. A crisp value is scaled into a normalized domain. 2. The normalized value is fuzzified with respect to the input fuzzy sets. 3. By means of a set of fuzzy rules, a fuzzy output value is provided. 4. The fuzzy output is defuzzified with the help of an appropriate defuzzification method (center of gravity, height method, etc.). 5. The defuzzified value is denormalized into a physical domain. A typical Mamdani controller is RCi :
Fuzzy or crisp system S or FS
Figure 2. General control structure.
x
IF
x = LX i
THEN u = LU i
(7)
where LUi is the corresponding fuzzy value for the control variable. Relational Controller. According to the description of the system in terms of a relational equation, a typical discrete fuzzy relational equation for a controller is U (k) = X (k) ◦ C
(8)
where X is the fuzzy state, U is the fuzzy control variable, and C is the fuzzy relation. A relational controller is another representation of a Mamdani Controller. Takagi Sugeno Controller. A typical TS controller is RCi :
xd
(5)
Fuzzy controllers can be classified as follows:
y =x
RSi : IF
relational composition (e.g., max-min composition). A fuzzy relation is another representation of a Mamdani fuzzy system. Analytical Systems Models. If an analytical model of the plant is available, then the system’s behavior can be described by a set of differential equations or by a set of so-called Takagi Sugeno fuzzy rules (TS rules) (8). A typical differential equation of an open-loop system is
RSi :
where the vectors x, xd, u, are continuous functions of time. For simplicity, the output vector y is set to be equal to the state vector x:
95
IF x = LX i
THEN u = Ki · x
(9)
where LXi is the ith fuzzy region for x, and Ki is the gain corresponding to that region. Predictive Controller. A special way of predictive fuzzy control was introduced by Yasunobu (14) for automatic train operation. It includes control rules for the time k to predict the behavior of the system for the next time-
96
FUZZY CONTROL
step k ⫹ 1. By means of a performance index J(k), which appears for a specific control action u(k), different features like velocity, riding comfort, energy saving, and accuracy of a stop gap are evaluated. By means of going through the whole range of possible control actions u(k), one obtains a range of corresponding performance indices J(k) from which the control action u(k ⫹ 1) with the highest performance index J(k) is applied to the plant. A typical predictive control rule is IF the performance index J(k) ⫽ LJi is obtained AND a control value u(k) is chosen to be LUi THEN the control value to be applied to the plant for the next timestep k ⫹ 1 is chosen to be u(k ⫹ 1) ⫽ LUi. A formal description is IF J(k) = LJ i AND u(k) = LU i
THEN u(k + 1) = LU i (10)
A further relationship to model predictive control (15) can be found in Refs. 16 and 17. Hybrid Controller. A hybrid controller is represented by a mixture of fuzzy controller and conventional controller. Fuzzy hybrid controllers are, e.g., applied for tuning conventional controllers and in adaptation schemes. Another application is the use of a nonlinear fuzzy mapping in nonlinear control tasks. A typical hybrid controller appears if the control law consists of a Mamdani controller Cfuzz and an analytical feedforward term Ccomp that compensates (e.g., statical or dynamical forces in a mechanical system): u = Cfuzz (xx, x d ) + Ccomp (xx )
(11)
where xd is the desired vector. Further information can be found in Ref. 18. THE FUZZY CONTROLLER AS A NONLINEAR TRANSFER ELEMENT A fuzzy logic controller defines a control law in the form of a static nonlinear transfer element (TE) due to the nonlinear nature of the computations performed by a fuzzy controller. However, the control law of a fuzzy controller is not represented in an analytic form, but by a set of fuzzy rules. The antecedent of a fuzzy rule (IF part) describes a fuzzy region in the state space. Thus one effectively partitions an otherwise continuous state space by covering it with a finite number of fuzzy regions and, consequently, fuzzy rules. The consequent of a fuzzy rule (THEN part) specifies a control law applicable within the fuzzy region from the IF part of the same fuzzy rule. During control with a fuzzy controller, a point in the
xd
e Scaling
Figure 3. The computational structure of a fuzzy controller. The arrangement of the blocks correspond to the sequence of computation.
x
state space is affected to a different extent by the control laws associated with all the fuzzy regions to which this particular point in the state space belongs. By using the operations of aggregation and defuzzification, a specific control law for this particular point is determined. As the point moves in the state space, the control law changes smoothly. This implies that a fuzzy controller yields a smooth nonlinear control law despite the quantization of the state space in a finite number of fuzzy regions. One goal of this section is to describe computation with a fuzzy controller and its formal description as a static nonlinear transfer element and thus provide the background knowledge needed for understanding control with a fuzzy controller. Furthermore, we show the relationship between conventional and rule-based transfer elements, thus establishing the compatibility between these two conceptually different, in terms of representation, types of transfer elements. The Computational Structure of a Fuzzy Controller A control law represented in the form of a fuzzy controller directly depends on the measurements of signals and is thus a static control law. This means that the fuzzy rule-based representation of a fuzzy controller does not include any dynamics, which makes a fuzzy controller a static transfer element, like a state controller. Furthermore, a fuzzy controller is, in general, a nonlinear static transfer element that is due to those computational steps of its computational structure that have nonlinear properties. In what follows we will describe the computational structure of a fuzzy controller by presenting the computational steps that it involves. The computational structure of a fuzzy controller consists of a number of computational steps and is illustrated in Fig. 3: 1. 2. 3. 4. 5.
Input scaling (normalization) Fuzzification of controller-input variables Inference (rule firing) Defuzzification of controller-output variables Output scaling (denormalization)
The state variables x1, x2, . . ., xn (or e, e˙, . . ., e(n⫺1)) that appear in the IF part of the fuzzy rules of a fuzzy controller are also called controller inputs. The control input variables u1, u2, . . ., um that appear in the THEN part of the fuzzy rules of a fuzzy controller are also called controller outputs. We will now consider each of the computational steps for the case of a multiple-input/single-output (MISO) fuzzy controller. The generalization to the case of multiple-input/multipleoutput fuzzy controller, where there are m controller outputs u1, u2, . . ., um instead of a single controller output u, can easily be done.
Fuzzification
Rule firing
Defuzzification
Denormalization
u
FUZZY CONTROL
Input Scaling. There are two principal cases in the context of input scaling: 1. The membership functions defining the fuzzy values of the controller inputs and controller outputs are defined off-line on their actual physical domains. In this case the controller inputs and controller outputs are processed only using fuzzification, rule firing, and defuzzification. For example, this is the case of a Takagi-Sugeno fuzzy controller. 2. The membership functions defining the fuzzy values of controller inputs and controller outputs are defined offline, on a common normalized domain. This means that the actual, crisp physical values of the controller inputs and controller outputs are mapped onto the same predetermined normalized domain. This mapping, called normalization, is done by appropriate normalization factors. Input scaling is then the multiplication of a physical, crisp controller input, with a normalization factor so that it is mapped onto the normalized domain. Output scaling is the multiplication of a normalized controller output with a denormalization factor so that it is mapped back onto the physical domain of the controller outputs. The advantage of the second case is that fuzzification, rule firing, and defuzzification can be designed independent of the actual physical domains of the controller inputs and controller outputs. To illustrate the notion of input scaling, let us consider, for example, the state vector e ⫽ (e1, e2, . . ., en)T ⫽ (e, e˙, . . ., e(n⫺1))T, where for each i, ei ⫽ xi ⫺ xdi. This vector of physical controller inputs is normalized with the help of a matrix Ne containing predetermined normalization factors for each component of e. The normalization is done as eN = Ne · e
(12)
with
Ne 1 0 Ne = .. . 0
0 Ne 2 .. . 0
... ... .. . ...
0 0 .. . Ne k
(13)
where Nei are real numbers and the normalized domain for e is, say, EN ⫽ [⫺a, ⫹a].
97
eN
e
eN
e
e N = Ne e e N = Ne e Figure 4. Normalization of the phase plane. Different normalization factors Ne and Ne˙ correspond to different slopes of the line Ne ⭈ e ⫹ Ne˙ ⭈ e˙ ⫽ 0.
fects the angle of a line that divides the phase plane into two semiplanes (see Fig. 4). Furthermore, we can see how the supports of the membership functions defining the fuzzy values of e and e˙ change because of the input scaling of these controller inputs (see Fig. 5). In the next three subsections on fuzzification, rule firing, and defuzzification, we consider only the case when the fuzzy values of the controller inputs and controller outputs are defined on normalized domains (e.g., EN and UN), and in this case we will omit the lower index N from the notation of normalized domains and fuzzy and crisp values. In the subsection on denormalization we will use the lower index N to distinguish between normalized and nonnormalized fuzzy and crisp values. Fuzzification. During fuzzification a crisp controller input x* is assigned a degree of membership to the fuzzy region from the IF part of a fuzzy rule. Let LEi1, . . ., LEin be some fuzzy values taken by the controller inputs e1, . . ., en in the IF part of the ith fuzzy rule RiC of a fuzzy controller; that is, these fuzzy values define the fuzzy region LEi ⫽ (LEi1, . . ., LEin)T. Each of the preceding fuzzy values, LEik is defined by a membership function on the same (normalized) domain of error E. Thus the fuzzy value LEik is given by the membership function 兰E 애LEik(ek)/ek. Let us consider now a particular normalized crisp controller input e ∗ = (e ∗1 , . . ., e ∗n )T
(16)
from the normalized domain E. Each e*k is a normalized crisp
Example Let e ⫽ (e1, e2)T ⫽ (e, e˙)T with e = x − xd
and e˙ = x˙d − x˙d
(14)
1 1
Then input scaling of e into eN and e˙ into e˙N yields eN = Ne · e and e˙N = Ne˙ · e˙
e
(15)
where Ne and Ne˙ are the normalization factors for e and e˙, respectively. In the context of a phase plane representation of the dynamic behavior of the controller inputs, the input scaling af-
1
–a e⋅
a eN e⋅ N
Figure 5. Change of the supports of the membership functions due to input scaling. Scaling normalizes different supports for e and e˙ to a common support for eN and e¨N.
98
FUZZY CONTROL
value obtained after the input scaling of the current physical controller input. The fuzzification of the crisp normalized controller input then consists of finding the membership degree of e*k in 兰E 애LEik(ek)/ek. This is done for every element of e*.
애i(e*) of the fuzzy region LEi is computed as
Example Consider the fuzzy rule RiC given as
Second, given the degree of satisfaction 애i(e*) of the fuzzy region LEi, the normalized controller output of the ith fuzzy rule is computed as
RiC : IF e = (PSe , NMe˙ )
THEN u = PMu
(17)
where PSe is the fuzzy value POSITIVE SMALL of the controller input e, NMe˙ is the fuzzy value NEGATIVE MEDIUM of the second controller input e˙, and PMu is the fuzzy value NEGATIVE MEDIUM of the single controller output u. The membership functions representing these two fuzzy values are given in Fig. 6. In this example we have e ⫽ (e, e˙)T and thus the IF part of the preceding rule represents the fuzzy region LEi ⫽ (PSe, NMe˙)T. Furthermore, let e* ⫽ a1 and e˙* ⫽ a2 be the current normalized values of the physical controller inputs e* and e˙*, respectively, as depicted in Fig. 6. Then from Fig. 6 we obtain the degrees of membership 애PSe(a1) ⫽ 0.3 and 애NMe˙(a2) ⫽ 0.65. Rule Firing. For a multi-input/single-output fuzzy controller, the ith fuzzy rule of the set of fuzzy rules has the form RiC : IF e = LE i
THEN u = LU i
(18)
where the fuzzy region LEi from the IF part of the preceding fuzzy rule is given as LEi ⫽ (LEi1, LEi2, . . ., LEin)T. Also, LEik denotes the fuzzy value of the kth normalized controller input ek that belongs to the term set of ek given as TEk ⫽ 兵LEk1, LUk2, . . ., LUkn其. Furthermore, LUi denotes an arbitrary fuzzy value taken by the normalized controller output u, and this fuzzy value belongs to the term set TU of u; that is, TU ⫽ 兵LU1, LU2, . . ., LUn其. Let the membership functions defining the fuzzy values from LEi and LUi be denoted by 兰E 애LEik(ek)/ek (k ⫽ 1, 2, . . ., n) and 兰U 애iLUi(u)/u, respectively. The membership function 兰U 애LUi(u)/u is defined on the normalized domain U, and the membership functions 兰E 애LEik(ek)/ek are defined on the normalized domain E. Given a controller input vector e* consisting of the normalized crisp values e*1 , . . ., e*n , first the degree of satisfaction
µ i (ee ∗ ) = min µLE i (e∗1 ), µLE i (e∗2 ), . . ., µLE i (e∗n ) 1
i
CLU = U
n
2
i ∗ e µCLU i (u)/u = min µ (e ), µLU i (u)/u
(19)
(20)
U
Thus the controller output of the ith fuzzy rule is modified by the degree of satisfaction 애i(e*) of the fuzzy region LEi and hence defined as the fuzzy subset CLUi ⫽ 兰U 애CLUi(u)/u of 兰U 애LUi(u)/u. That is,
∀u:µ CLU i (u) =
µLU i (u)
if µLU i (u) ≤ µ i ,
µLU i (u) = µ i (ee∗ ) otherwise
(21)
The fuzzy set CLUi ⫽ 兰U 애CLUi(u)/u is called the clipped controller output. It represents the modified version of the controller output 兰U 애LUi(u)/u from the ith fuzzy rule given certain crisp controller input e*1 , . . ., e*n . In the final stage of rule firing, the clipped controller outputs of all fuzzy rules are combined in a global controller output via aggregation: ∀u:µCU (u) = max(µCLU 1 , . . ., µCLU M )
(22)
where CU ⫽ 兰U 애CU(u)/u is the fuzzy set defining the fuzzy value of the global controller output. The type of rule firing described here is called max-min composition. Another type of composition can be found in Ref. 40. Defuzzification. The result of rule firing is a fuzzy set CU with a membership function 兰U 애CU(u)/u, as defined in Eq. (22). The purpose of defuzzification is to obtain a scalar value u from 애CU. The scalar value u is called a defuzzified controller output. This is done by the center of gravity method as follows. In the continuous case we have
µ
NB
NM
NS
1
Z
PS
PM
U
µCU (u) du
and for the discrete case
0.3
u= a1
(23)
PB
0.65
–a
µCU (u) · u du
U
u=
a2
U
µCU (u) · u du
a e e⋅
Figure 6. Fuzzification of crisp values e* and e˙*. Fuzzification of e* ⫽ a1 with respect to a fuzzy set NM is obtained by finding the crosspoint between a1 and the corresponding membership function NM.
µCU (u)
(24)
U
Example Consider the normalized domain U ⫽ 兵1, 2, . . ., 8其 and let the fuzzy set CU be given as CU = {0.5/3, 0.8/4, 1/5, 0.5/6, 0.2/7}
(25)
FUZZY CONTROL
99
where µcu
x = input; y = output;
1
N = negative; P = positive; Z = zero; S = small; B = big
0.5
1
2
3
4
5 vcog
6
7
8
u
Figure 7. Defuzzification of a fuzzy controller output. Defuzzification of a fuzzy set 애CU is obtained by computing the u-coordinate of the center of gravity of the membership function.
Then the defuzzified controller output u is computed as (see also Fig. 7) u=
0.5 · 3 + 0.8 · 4 + 1 · 5 + 0.5 · 6 + 0.2 · 7 = 4.7 0.5 + 0.8 + 1 + 0.5 + 0.2
2. Shape and location of the corresponding membership functions are chosen so that they always overlap at the degree of membership 애X ⫽ 0.5 (see Fig. 8). 3. For the specific crisp controller input xin one obtains the degrees of membership 애XNS(xin) ⬎ 0 and 애XZ(xin) ⬎ 0, where the remaining degrees of membership 애XNB(xin), 애XPS(xin), and 애XPB(xin) are equal to zero. Hence, only rules R2 and R3 fire. The controller output set is computed by cutting the output set 애YPS at the level of 애XNS(xin) and 애YZ at 애XZ(xin). The resulting output membership function 애Y takes every rule into account, performing the union of the resulting output membership function 애YRi of each rule Ri (i ⫽ 1, . . ., 5) which means the maximum operation between them. 4. The crisp controller output y is obtained by calculating the center of gravity of the output set LY:
(26)
y= Denormalization. In the denormalization procedure the defuzzified normalized controller output uN is denormalized with the help of an off-line predetermined scalar denormalization factor N⫺1 u , which is the inverse of the normalization factor Nu. Let the normalization of the controller output be performed as uN = Nu · u
(27)
Then the denormalization of uN is u = Nu−1 · uN
+A −A
µY ( y) · y dy Ri
+A
−A
(29)
µY ( y) dy Ri
The cut operation (min operation), the max operation over all resulting fuzzy subsets LYRi, and the center of gravity are nonlinear operations that cause a nonlinear operating line between x and y. This seems to make a systematic design of a desired transfer function with the help of membership functions difficult. However, in the x domain there are operating points A1, A2, A3, A4, and A5 at which only one of the five
(28)
The choice of Nu essentially determines, together with the rest of the scaling factors, the stability of the system to be controlled. In the case of Takagi Sugeno fuzzy controllers, the preceding computational steps are performed on the actual physical domains of the controller inputs and outputs. Thus the computational steps of normalization and denormalization are not involved in the computational structure of a Takagi Sugeno fuzzy controller, which, in turn, eliminates the need for input and output scaling factors.
NB NS 1
1
Z µ
PS PB
Input set 0 –A = A1 A2 Crisp input
x 0
C2
A3 NB 1
NS
1
Z µ
C1 A4 +A= A1
PS
PB 1
The Transfer Characteristics
Output set
The way to obtain a specific input output transfer characteristics shows the following example (SISO): 0
1. Suppose there is a set of rules like
R1 : IF x = NB THEN y = PB R2 : IF x = NS THEN y = PS R3 : IF x = Z THEN y = Z R4 : IF x = PS THEN y = NS R5 : IF x = PB THEN y = NB
Support
0
0
y
a
–A
+A Crisp output center of gravity (c. o. g.)
Figure 8. Membership functions for input x and output y. The output membership function is obtained by clipping the output membership functions at the corresponding degrees of membership of the input.
100
FUZZY CONTROL
rules fires. At these operating points the center of gravity can be calculated more easily than for the intermediate points. The operating points A1, A2, A3, A4, and A5 form points in the x-y domain (see Fig. 9). The values of the transfer characteristic between the operating points may show a slight nonlinear behavior, but from a linear approximation (interpolation) between two operating points one obtains the relation between the supports of the input and output membership functions, on the one hand, and slopes required of the transfer characteristic, on the other hand. The Nonlinearity of the Fuzzy Controller In this subsection we will describe the sources of nonlinearity of the transfer characteristic of a fuzzy controller by relating them to particular computational steps. System theory distinguishes between two basic types of systems: linear and nonlinear. A system is linear if and only if it has both the additivity property and the scaling property; otherwise it is a nonlinear system. Additivity Property (Superposition Property). Let it be the case that y1 = f (x)
and y2 = f (z)
(30)
Then for the additivity property to hold, it is required that y1 + y2 = f (x + z)
(31)
f (x) + f (z) = f (x + z)
(32)
Hence, we obtain
Scaling Property (Homogeneity Property). Let it be the case that y = f (x)
(33)
Then for the scaling property to hold, it is required that α · y = f (α · x)
and α · f (x) = f (α · x)
Because of fuzzification and defuzzification, a fuzzy controller is in fact a crisp transfer element. This crisp TE has a nonlinear transfer characteristic because of the nonlinear character of fuzzification (when performed on nonlinear membership functions), rule firing, and defuzzification. The argument for this is that if one computational step within the computational structure of the TE is nonlinear, then the whole TE is nonlinear as well. Using the additivity and scaling properties of a linear system, we will now establish the linearity, or nonlinearity, of each computational step in the computational structure of a fuzzy controller with respect to these two properties. In what follows, without any loss of generality, we will use a single SISO fuzzy rule such as RC : IF
e = LE
THEN u = LU
(35)
where LE and LU are the fuzzy values taken by the normalized, single controller input e and the normalized, single controller output u, respectively. These two fuzzy values are determined by the membership functions 兰E 애LE(e)/e and 兰U 애LU(u)/u defined on the normalized domains E and U. Here again we only consider normalized domains, fuzzy and crisp values, and thus the lower index N will be omitted from the notation unless there is a need to distinguish between normalized and actual crisp and fuzzy values used within the same expression. Furthermore, let e*1 and e*2 be two normalized crisp controller inputs and u*1 and u*2 be the defuzzified controller outputs corresponding to these normalized controller inputs. Input Scaling and Output Scaling. Input scaling is linear because it simply multiplies each physical controller input e*1 and e*2 with a predetermined scalar Ne (normalization factor) to obtain their normalized counterparts e*1N and e*2N. Thus we have Ne · e∗1 + Ne · e∗2 = Ne · (e∗1 + e∗2 )
(36)
Furthermore, for a given scalar 움 we have (34)
α · Ne · e∗1 = Ne · (α · e∗1 )
(37)
Thus input scaling has the properties of additivity and scaling and is thus a linear computational step. The same is valid for output scaling since it uses N⫺1 e instead of Ne.
y 5/6a
Fuzzification. Let the membership function 兰E 애LE(e)/e defining the normalized fuzzy value LE be, in general, a nonlinear function (e.g., a triangular membership function). The fuzzification of e*1 and e*2 results in finding 애LE(e*1 ) and 애LE(e*2 ). Linearity requires
a c1 A1
A2
c2 A3
A4
A5
x
–a –5/6a
Figure 9. Transfer characteristic of a fuzzy controller. The transfer characteristic is a static input/output mapping of a fuzzy controller.
µLE (e∗1 ) + µLE (e∗2 ) = µLE (e∗1 + e∗2 )
(38)
The preceding equality cannot be fulfilled because the membership function 兰E 애LE(e)/e is, in general, nonlinear. Thus, fuzzification in the case of nonlinear membership functions is a nonlinear computational step. Rule Firing. Let the membership function 兰U 애LU(u)/u defining the normalized fuzzy value LU be, in general, a nonlin-
FUZZY CONTROL
ear function. Then the result of rule firing given the normalized crisp controller input e*1 will be ∀u:µCLU (u) = min(µLU (e∗1 ), µLU (u))
(39)
Similarly, for the normalized crisp controller input e*2 we obtain ∀u:µCLU (u) = min(µLE (e∗2 )µLU (u))
(40)
Linearity requires ∀u:µCLU (u) + µCLU (u) = min(µLE (e∗1 + e∗2 ), µLU (u))
HEURISTIC CONTROL AND MODEL-BASED CONTROL
• 兰U 애LU(u)/u is a nonlinear membership function. • 兰U 애⬘CLU(u)/u and 兰U 애⬙CLU(u)/u are nonlinear membership functions (usually defined as only piecewise linear functions). • the min-operator is nonlinear. Thus rule firing is a nonlinear computational step within the computational structure of a fuzzy controller. Defuzzification. Let defuzzification be performed with the center of gravity method. Furthermore, let u1 and u2 be the normalized defuzzified controller outputs obtained after defuzzification. That is, µCLU (u) · u du U , (42) u1 = µCLU (u) du
u2 =
U
U
µCLU (u) · u du
U
(43) µCLU (u) du
Linearity requires, however, (µCLU (u) + µCLU (u)) · u du U u1 + u2 = (µCLU (u) + µCLU (u)) du
However, in the case of a Takagi Sugeno FC-1, each single fuzzy rule is a linear TE for all controller inputs (state vectors) that belong to the center of the fuzzy region specified by the IF part of this rule. At the same time, for controller inputs outside the center of a fuzzy region, this same fuzzy rule is a nonlinear TE. Because of the latter, the set of all fuzzy rules of a Takagi Sugeno FC-1 defines a nonlinear TE. In the case of a Takagi Sugeno gain scheduler, we have that each fuzzy rule defines a linear TE everywhere in a given fuzzy region.
(41)
but the preceding equality does not hold because
101
Fuzzy control can be classified into the main directions heuristic fuzzy control and model-based fuzzy control. Heuristic control deals with plants that are unsufficiently described from the mathematical point of view, while model-based fuzzy control deals with plants for which a mathematical model is available. In this section we will describe the following control strategies: Mamdani control (MC) Sliding mode fuzzy control (SMFC) Cell mapping control (CM) Takagi Sugeno control (TS1) Takagi Sugeno control (TS2) with Lyapunov linearization The Mamdani Controller This type of fuzzy controller obtains its control strategy from expert knowledge. Since a model of the plant is not available, a simulation of the closed loop cannot be performed. Therefore, the control design is based on trial-and-error strategies, which makes the implementation of the fuzzy controller critical. The crucial point is that the behavior of the plant to be controlled is only reflected through the operator rules. However, from the control point of view this is not a satisfactory situation. Thus, one seeks methods to build qualitative models in terms of fuzzy rules. In the context of heuristic control, the so-called Mamdani control rules are used where both the antecedent and the consequent include fuzzy values. A typical control rule (operator rule) is
(44)
RCi : IF x = LX i
THEN u = LU i
(46)
U
However, the preceding equality cannot be fulfilled since instead of it we have µCLU (u) · u du (µCLU (u) · u du U U + (45) u1 + u2 = µCLU (u) du µCLU (u) du U
U
This shows that the nonlinearity of the computational step of defuzzification comes from the normalization of the products 兰U 애⬘CLU(u) ⭈ udu and 兰U 애⬙CLU(u) ⭈ udu. From all of the foregoing it is readily seen that a fuzzy controller is a nonlinear TE, its sources of nonlinearity being the nonlinearity of membership functions, rule firing, and defuzzification.
For a system with two state variables and one control variable, we have, for example, RCi : IF
x = PS AND x˙ = NB THEN u = PM
(47)
which can be rewritten into RCi :
IF
(x, x) ˙ T = (PS, NB)T
THEN u = PM
with
x = (x, x) ˙ T LX i = (PS, NB)T u=u LU i = PM
(48)
102
FUZZY CONTROL
Even if there is only a little knowledge about the system to be controlled, one has to have some ideas about the behavior of the system state vector x, its change with time x˙, and the control variable u. This kind of knowledge is structural and can be formulated in terms of fuzzy rules. A typical fuzzy rule for a system is RSi : IF x = LX i
THEN x˙ = LX˙ i (49)
AND u = LU i
e e PB
Z
PM
PS
RSi :
IF
THEN
(x, x) ˙ T = (PS, NB)T
AND u = PM
(x, ˙ x) ¨ T = (NM, PM)T
(50)
Z
PS PM PB
NS NS NM NM NB NB Z
NS NS NM NM NB
NM
P N PM PS PS Z NS NS NM Z S PM PM PS PS Z NS NS M PB PM PM PS PS Z NS B
NB
PB PB PM PM PS PS
PS Z NS
For the preceding system with two states and one control variable we have, for example,
NB NM NS
PS PS
Z
NS NS NM NM
— — — — — —
postitve negative zero small medium big
Z
Figure 10. A fuzzy controller in a diagonal form. Diagonal form means that the same fuzzy attributes appear along a diagonal.
with
x = (x, x) ˙ T LX i = (PS, NB)T x˙ = (x, ˙ x) ¨ T LX˙ i = (NM, PM)T u=u LU i = PM Once the qualitative system structure is known, one has to find the corresponding quantitative knowledge. Quantitative knowledge means the following: In general, both control rules and system rules work with normalized domains. The task is to map inputs and outputs of both the controller and the system to normalized domains. For the system, this task is identical with the identification of the system parameters. For the controller, this task is identical with the controller design (namely, to find the proper control gains). Sliding Mode Fuzzy Controller A typical Mamdani controller is the sliding mode fuzzy controller (SMFC) (19–21). Fuzzy controllers for a large class of second-order nonlinear systems are designed by using the phase plane determined by error e and change of error e˙ (22– 25). The fuzzy rules of these fuzzy controllers determine a fuzzy value for the input u for each pair of fuzzy values of error and change of error (that is, for each fuzzy state vector). The usual heuristic approach to the design of these fuzzy rules is the partitioning of the phase plane into two semiplanes by means of a sliding (switching) line. This means that the fuzzy controller has a so-called diagonal form (see Fig. 10). Another possibility is, instead of using a sliding line, to use a sliding curve like a time optimal trajectory (26). A typical fuzzy rule for the fuzzy controller in a diagonal form is IF
e = PS AND e˙ = NB THEN u = PS
(51)
where PS stands for the fuzzy value of error POSITIVE SMALL, NB stands for the fuzzy value change of error NEGATIVE BIG, and PS stands for the fuzzy value POSITIVE SMALL of the input. Each semiplane is used to define only negative or positive fuzzy values of the input u. The magnitude of a specific
positive/negative fuzzy value of u is determined on the basis of the distance 兩s兩 between its corresponding state vector e and the sliding line s ⫽ ⭈ e ⫹ e˙ ⫽ 0. This is normally done in such a way that the absolute value of the required input u increases/decreases with the increasing/decreasing distance between the state vector e and the sliding line s ⫽ 0. It is easily observed that this design method is very similar to sliding mode control (SMC) with a boundary layer (BL), which is a robust control method (12,27). Sliding mode control is applied especially to control of nonlinear systems in the presence of model uncertainties, parameter fluctuations, and disturbances. The similarity between the diagonal form fuzzy controller and SMC enables us to redefine a diagonal form fuzzy controller in terms of an SMC with BL and then to verify its stability, robustness, and performance properties in a manner corresponding to the analysis of an SMC with BL. In the following, the diagonal fuzzy controller is therefore called sliding mode fuzzy control (SMFC). However, one is tempted to ask here, What does one gain by introducing the SMFC type of controller? The answer is that SMC with BL is a special case of SMFC. SMC with BL provides a linear transfer characteristic with lower and upper bounds, while the transfer characteristic of an SMFC is not necessarily a straight line between these bounds, but a curve that can be adjusted to reflect given performance requirements. For example, normally a fast rise time and as little overshoot as possible are the required performance characteristics for the closed-loop system. These can be achieved by making the controller gains much larger for state space regions far from the sliding line than its gains in state space regions close to the sliding line (see Fig. 11). In this connection it has to be emphasized that an SMFC is a state-dependent filter. The slope of its transfer characteristic decides the convergence rate to the sliding line and, at the same time, the bandwidth of the unmodeled disturbances that can be coped with. This means that far from the sliding line higher frequencies are allowed to pass through than in the neighborhood of it. The other function of this state-dependent filter is given by the sliding line itself. That is, the velocity with which the origin is approached is determined by the slope of the sliding line s ⫽ 0. Because of the special form of the rule base of a diagonal form fuzzy controller, each fuzzy rule can be redefined in terms of the fuzzy value of the distance 兩s兩 between the state vector e and the sliding line and the fuzzy value of the input
FUZZY CONTROL
103
the rate with which the origin is approached. A fuzzy rule including this distance is of the form
µ
u1
IF
s = PS AND d = S THEN u = NS
(52)
u2 s1
s2
0
s
– du/ds Gain – du/ds1 – du/ds2 s1
s2
0
s
Figure 11. The adjustable transfer characteristic of an SMFC. The nonlinear input/output mapping of the SMFC provides a nonlinear gain for different input/output pairs.
u corresponding to this distance. This helps to reduce the number of fuzzy rules, especially in the case of higher-order systems. Namely, if the number of state variables is 2 and each state variable has m fuzzy values, the number of fuzzy rules of the diagonal form fuzzy controller is M ⫽ m2. For the same case, the number of fuzzy rules of an SMFC is only m. This is so because the fuzzy rules of the SMFC only describe the relationship between the distance to the sliding line and the input u corresponding to this distance, rather than the relationship between all possible fuzzy state vectors and the input u corresponding to each fuzzy state vector. Moreover, the fuzzy rules of an SMFC can be reformulated to include the distance d between the state vector e and the vector normal to the sliding line and passing through the origin (see Fig. 12). This gives an additional opportunity to affect e⋅
d e⋅ * s
sd =
e*
π
1 + λ2
e s
2
s = λ e + e⋅ = 0
Figure 12. The s and d parameters of an SMFC. s is the distance between the state and the line s ⫽ 0. d is the distance between the state and the line perpendicular to s ⫽ 0.
Despite of the advantages of an SMFC, it poses a number of problems the solutions of which can improve its performance and robustness. One such problem is the addition of an integrator to an SMFC in order to eliminate remaining errors in the presence of disturbances and model uncertainties. There are several ways to accomplish this. One option, for example, is to treat the integration term in the same manner as the other parameters of the IF part of the SMFC’s fuzzy rules. This and other available options will be described later in this article. Another problem is the so-called scaling of the SMFC parameters so that the domains on which their fuzzy values are defined are properly determined and optimized with respect to performance. This problem arises in the context of SMFC since the real physical domains of the SMFC parameters are normalized (i.e., their measured values are mapped on their respective normalized domains by the use of normalization factors). Thus a normalized input u is the result of the computation with SMFC. The normalized u is then consequently denormalized (i.e., mapped back on its physical domain) by the use of a denormalization factor. The determination of the proper scaling factors, via which the normalization and denormalization of the SMFC parameters is performed, is not only part of the design, but is also important in the context of adaptation and on-line tuning of the SMFC. The behavior of the closed-loop system ultimately depends on the normalized transfer characteristic (control surface) of the SMFC. This control surface is mainly determined by the shape and location of the membership functions defining the fuzzy values of the SMFC’s parameters. In this context one need pay attention to the following: 1. The denormalization factor for u influences most stability and oscillations. Because of its impact on stability, the determination of this factor has the highest priority in the design. 2. Normalization factors influence most of the SMFC sensitivity with respect to the proper choice of the operating regions for s and d. Therefore, normalization factors are second in priority in the design. 3. The proper shape and location of the membership functions and, with this, the transfer characteristics of the SMFC can influence positively the behavior of the closed-loop system in different fuzzy regions of the fuzzy state space provided that the operating regions of s and u are properly chosen through well-adjusted normalization factors. Therefore, this aspect is third in priority. A third problem is the design of SMFC for MIMO systems. The design for SISO systems can still be utilized, though some new aspects and restrictions come into play when this design is extended to the case of MIMO systems. First, we assume that the MIMO system has as many input variables ui as it has output variables yi. Second, we assume that the so-called matching condition holds (12). This condition constrains the so-called parametric uncertainties. These are, for example, imprecision on the mass or inertia of a mechanical
104
FUZZY CONTROL x2
outside the finite state space of interest is lumped together into one so-called sink cell. The state of the system of Eq. (53) while in the cell z is represented by the center point xc. Now a cell mapping C is defined by
Sink cell Regular cell xc(tk) z (tk)
x1
z (tk+1 ) = C (zz (tk ))
Image cell z (tk + 1)
Equilbrium cell z (tk ) = z (tk + 1)
Figure 13. Cell mapping principle. The state space is partitioned into a finite set of cells. Cell mapping deals with the transition behavior between cells.
system and inaccuracies on friction functions. Nonparametric uncertainties include unmodeled dynamics and neglected time delays. Let x˙ ⫽ f(x) ⫹ B ⭈ u, y ⫽ C ⭈ x be the nonlinear open-loop system to be controlled, where f is a nonlinear vector function of the state vector x, u is the input vector, B is the input matrix, y is the output vector, and C is the output matrix. Then the matching condition requires that the parametric uncertainties have to be within the range of the input matrix B. CELL MAPPING Cell mapping originates from a computational technique introduced by C. S. Hsu (28) that evaluates the global behavior and the stability of nonlinear systems. It is assumed that the computational (analytical) model of the system is available. Cell mapping was first applied to fuzzy systems by Chen and Tsao (29). The benefits of using cell mapping for fuzzy controlled systems are as follows: • Supporting of self-learning FC strategies • Creating of methodologies for the design of time optimal fuzzy controllers The basic idea of Hsu is as follows: Let a nonlinear system be described by the point mapping x (tk+1 ) = f (xx (tk ), u (tk ))
(54)
which is derived from the point mapping of Eq. (53) by computing the image of a point x(tk) and then determining the cell in which the image point is located. It is clear that not all points x(tk) in cell z(tk) have the same image cell z(tk⫹1). Therefore, only the image cell of the center x c(tk) is considered. A cell that maps to itself is called an equilibrium cell. All cells in the finite state space are called regular cells. The motivation for cell mapping is to obtain an appropriate sequence of control actions u(tk) that drive the system of Eq. (53) to an equilibrium while minimizing a predefined cost function. Therefore, every cell is characterized by the following: • The group number G(z) that denotes cells z belonging to the same periodic domain or domain of attraction • The step number S(z) that indicates the number of transitions needed to transmit from cell z to a periodic cell • The periodicity number P(z) that indicates the number of cells contributing to the periodic motion This characterization is introduced in order to find periodic motions and domains of attractions by a grouping algorithm. Applied to fuzzy control, it is evident that each cell describing the system’s behavior belongs to a corresponding fuzzy system rule. Furthermore, each cell describing a particular control action belongs to a corresponding fuzzy control rule. Smith and Comer developed a fuzzy cell mapping algorithm the aim of which is to calibrate (tune) a fuzzy controller on the basis of the cell state space concept (30). Each cell is associated with a control action and a duration, which map the cell to a minimum cost trajectory (e.g., minimum time). With a given cost function and a plant simulation model, the cell state space algorithm generates a table of desired control actions. The mapping from cell to cell is carried out by a fuzzy controller, which smoothes out the control actions while the transitions between the cells. The cell-to-cell mapping technique has been used to fine-tune a Takagi Sugeno controller (see Fig. 14) (31). Kang and Vachtsevanos developed a phase portrait assignment algorithm that is related to cell-to-cell mapping (32). In
(53)
where tk represent the discrete timesteps over which the point mapping occurs. It has to be emphasized that these timesteps need not to be uniform in duration. If one wants to create a map of the state space taking into account all possible states x and control vectors u, one obtains an infinite number of mappings even for finite domains for x and u, respectively. To simplify this mapping, the (finite) state space is divided into a finite number of cells (see Fig. 13). Cells are formed by partitioning the domain of interest of each axis xi of the state space into intervals of size si that are denoted by an integer valued index zi. Then a cell is an n-tuple (a vector) of intervals z ⫽ (z1, z2, . . ., zn)T. The remainder of the state space
Optimal control table LMS algorithm Input + –
Fuzzy controller
– + Output Plant
Figure 14. Cell mapping by Smith and Comer [Redrawn from Papa et al. 1995 (31)]. Cell mapping is used to fine-tune a Takagi Sugeno controller.
FUZZY CONTROL Search algorithm Optimal criteria Input + –
Fuzzy controller
105
Let the inputs measured be x*1 ⫽ 4 and x*2 ⫽ 60. From Fig. 16 we then obtain
Cell space
µX
Cell space Output
BIG
(x∗1 ) = 0.3 µX
BIG
(x∗2 ) = 0.35
and
Plant
Figure 15. Cell mapping by Kang and Vachtsevanos [Redrawn from Papa et al. 1995 (31)]. Cell mapping is used for construction of an optimal rule base from data.
µX
SMALL
(x∗1 ) = 0.7 µX
TS Model-Based Control Model-based fuzzy controller design starts from the mathematical knowledge of the system to be controlled (8,34). In this connection one is tempted to ask why one should use FC in this particular case while conventional control techniques work well. The reasons that apply FC in analytical known systems are as follows: 1. FC is a user-friendly and transparent control method because of its rule-based structure. 2. FC provides a nonlinear control strategy that is related to traditional nonlinear control techniques. 3. The nonlinear transfer characteristics of a fuzzy controller can be tuned by changing the shape and location of the membership functions so that adaptation procedures can be applied. 4. The approximation property of FC allows the design of a complicated control law with the help of only few rules. 5. Gain scheduling techniques can be transfered to FC. In this connection FC is used as an approximator between linear control laws.
y1 = 4 − 3 · 60 = −176 y2 = 4 + 2 · 4 = 12 So the two pairs corresponding to each rule are (0.3, ⫺176) and (0.35, 12). Thus, by taking the weighted normalized sum we get y=
R2 : if x1 is SMALL and x2 is BIG then y2 = 4 + 2 · x1 .
0.3 · (−176) + 0.35 · 12 = −74.77 0.3 + 0.35
This can be extended to differential equations in the following way: Let a fuzzy region LX i be described by the rule RSi :
IF x = LX i
THEN x˙ = A(xx i ) · x + B(xx i ) · u
(55)
This rule means that IF state vector x is in fuzzy region LXi THEN the system obeys the local differential equation x˙ ⫽ A(x i) ⭈ x ⫹ B(x i) ⭈ u. A summation of all contributing system rules provides the global behavior of the system. In Eq. (55) A(x i) and B(x i) are constant system matrices in the center of fuzzy region LX i that can be identified by classical identification procedures. The resulting system equation is x˙ =
n
wi (xx ) · (A(xx i ) · x + B(xx i ) · u )
(56)
i=1
where wi(x) 僆 [0, 1] are the normalized degrees of satisfaction of a fuzzy region LX i. The corresponding control rule (Takagi Sugeno FC1) is RCi :
Small
0.7 0
R1 : if x1 is BIG and x2 is MEDIUM then y1 = x1 − 3 · x2 .
min(0.7, 0.35) = 0.35
Furthermore, for the consequents of rules R1 and R2 we have
The description of the system starts from a fuzzy model of the system that uses both the fuzzy state space and a crisp description of the system. Let the principle of a Takagi Sugeno system be explained by the following example. Example Consider a TS system consisting of two rules with x1 and x2 as system inputs and y as the system output.
(x∗2 ) = 0.75
For the degree of satisfaction of R1 and R2, respectively, we obtain min(0.3, 0.75) = 0.3
this approach states and control variables are partitioned into different cell spaces. The x-cell space is recorded by applying a constant input to the system being simulated. Then, by means of a searching algorithm, the rule base of the funny controller is constructed such that asymptotic stability is guaranteed. This is performed by determining the optimal control actions regardless of from which cell the algorithm starts its search (see Fig. 15) (31). Hu, Tai, and Shenoi apply genetic algorithms to improve the searching algorithm using cell maps (33). The aim of this method is to tune a Takagi Sugeno controller.
MED
x1 = 4
IF x = LX i
THEN u = K(xx i ) · x
Big
Medium
(57)
Big
1
1
0.75 0.3
0.35
10
0
x2 = 60
100
Figure 16. Fuzzification procedure for a TS controller. The fuzzification procedure is the same as that for a Mamdani controller.
106
FUZZY CONTROL
and the control law for the whole state space is u=
n
wi (xx ) · K(xx i ) · x
(58)
i=1
Together with Eq. (56) one obtains the closed-loop system
x˙ =
n
wi (xx ) · w j (xx ) · (A(xx i ) + B(xx i ) · K(xx j )) · x
(59)
i, j=1
It has to be emphasized that a system described by a set of rules like Eq. (55) is nonlinear even in the vicinity of the center of the region. This is due to the fact that wi(x) depends on the state vector x. Even if wi(x) is a piecewise linear function of x, the product wi(x) ⭈ wj(x) ⭈ (A(x i) ⫹ B(x i) ⭈ K(x j)) ⭈ x in Eq. (58) will always be a nonlinear function. Model-Based Control with Lyapunov Linearization In the following we discuss the case when a mathematical model of the system to be controlled is available and the fuzzy controller is formulated in terms of fuzzy rules (8,21,34,35). In this case system and controller are formulated on different semantic levels. Let the system analysis starts from the mathematical model of the system x˙ = f (xx, u )
(60)
The control surface provides information about local and global properties of the controller. For example, the local gain for a specific state vector can be obtained by means of the tangential plane being attached to the corresponding point in state space. From this information one can conclude whether the controlled system is locally stable. Furthermore, one obtains a geometrical insight into how the control gain changes as the state trajectory moves in the state space. Another aspect is the following. To study the local behavior of the system around specific points in state space, we linearize the system around them and study the closed-loop behavior in the linearized region. Let, for example, the system of Eq. (60) be linearized around a desired state xd and a corresponding state vector ud: x˙ = f (xx d , u d ) + A(xx d , ud ) · (xx − x d ) + B(xx d , ud ) · (u u − ud ) (63) where
A(xx d , ud ) =
∂ f (xx, u ) ∂xx x
IF x = LX i
THEN u = LU i
(61)
To study stability, robustness, and performance of the closedloop system one has to bring system and controller onto the same semantic level. Thus, formally we translate the set of control rules into an analytical structure u = g (xx )
∂ f (xx, u ) u x ∂u
ud d ,u u
are Jacobians. An appropriate control law is
and let the fuzzy controller be formulated in terms of fuzzy rules (Takagi Sugeno FC1) RCi :
,u
d ,u d
and B(xx d , ud ) =
u = ud + K(xd ) · (xx − x d )
(64)
where K(xd) is the gain matrix. Since the system of Eq. (60) changes its behavior with the setpoint xd, the control law of Eq. (64) changes with the setpoint xd as well. To design the controller for the closed-loop system at any arbitrary point xd in advance, we approximate Eq. (63) by a set of TS fuzzy rules
(62)
where, in general, the function g(x) is a nonlinear control surface being a static mapping of the state vector x to the control vector u (see Fig. 17).
RSi : IF x d = LX i u − ud ) THEN x˙ = f (xx d , ud ) + A(xx i , ui ) · (xx − x d ) + B(xx i , ui ) · (u (65) The resulting system equation is (Takagi Sugeno FC2)
x˙ = f (xx d , ud ) +
n
A (xx i , ui ) · (xx − x d ) wi (xx d ) · (A
i=1
µ
(66)
u − ud )) + B (xx , u ) · (u i
i
This is a linear differential equation because the weights wi depend on the desired state vector xd instead of on x. The corresponding set of control rules is RCi : IF x d = LX i x2
x1
THEN u = ud + K(xx i ) · (xx − x d ) (67)
with the resulting control law u=
n
ud + K(xx i ) · (xx − x d ) wi (xx d ) · (u
(68)
i=1
Figure 17. Nonlinear control surface u ⫽ g(x1, x2). Nonlinear mapping is a translation of fuzzy rules into a numerical input/output relation.
Substituting Eq. (68) into Eq. (66), we obtain the equation for the closed-loop system
FUZZY CONTROL
107
be the model of the system and FC
u = g p (xx, p ) p
e xd
+ x
–
u
Controller g(x, p)
System f
x
(72)
be the control law, where p is a parameter vector that has to be determined by the supervisor. The subscript p means that with the change of p the structure of the control law may also change. Then the supervisory law can be written as p = h (xx, c )
Figure 18. Supervisory control. The supervisor changes controller parameters by means of input/output data u and x and desired values xd.
(73)
where c is the vector of conditions. For example, c = (|xx − x d | > K1; |x˙ d | < K2) T
x˙ = f (xx d , ud ) +
n
wi (xx d ) · w j (xx d ) · (A(xx i , ui )
i, j=1 i
i
(69)
j
+ B(xx , u ) · K(xx )) · (xx − x d )
IF x = LX i
Denoting A(x i, ui) ⫹ B(x i, ui) ⭈ K(x j) by Aij, asymptotic stability of x ⫺ xd is guaranteed if there exists a common positive definite matrix P such that the Lyapunov inequalities ATij P + PAi j < 0
(70)
hold, where Aij are Hurwitz matrices (34). With this result one is able to study the stability, robustness, and performance of the closed-loop system around an arbitrary setpoint xd just by considering the system at predefined operating points xi. SUPERVISORY CONTROL A commonly used control technique is supervisory control, which is a method to connect conventional control methods and so-called intelligent control methods (see Fig. 18). This control technique works in such a way that one or more controllers are supervised by a control law on a higher level. Applications to supervisory control for a milling machine and a steam turbine are reported in Refs. 36 and 37. Normally, the low-level controllers perform a specific task under certain conditions. These conditions can be • Keeping a predefined error between desired state and current state • Performing a specific control task (e.g., approaching a solid surface by a robot arm) • Being at a specific location of the state space Usually, supervisors intervene only if some of the predefined conditions fail. If so, the supervisor changes the set of control parameters or switches from one control strategy to another. Often, supervisory algorithms are formulated in terms of IF-THEN rules. Fuzzy IF-THEN rules avoid hard switching between set of parameters or between control structures. It is therefore useful to build fuzzy supervisors in the cases when ‘‘soft supervision’’ is required. A formal approach may be the following. Let x˙ = f (xx, u )
where K1 and K2 are constant bounds. The corresponding supervisory fuzzy rule is
(71)
AND c = LC i
THEN
p = pi
with LCi ⫽ (兩x ⫺ xd兩 ⬎ K1i; 兩x˙d兩 ⬍ K2i)T. Supervision is related to gain scheduling. The distinction between the two is that gain scheduling changes the controller gains with respect to a slowly time varying scheduling variable while the control structure is preserved (38–40). On the other hand, supervision can both change the control gains and the control structure and can deal with fast-changing system parameters as well (41). ADAPTIVE CONTROL Many dynamic systems have a known structure but uncertain or slowly varying parameters. Adaptive control is an approach to the control of such systems. Adaptive controllers, whether designed for linear or nonlinear systems, are inherently nonlinear. We distinguish between direct and indirect adaptive control methods. Direct adaptive methods start with sufficient knowledge about the system structure and its parameters. Direct change of controller parameters optimizes the system’s behavior with respect to a given criterion. In contrast, the basic idea of indirect adaptive control methods is to estimate the uncertain parameters of the system under control (or, equivalently, the controller parameters) on-line, and use the estimated parameters in the computation of the control law. Thus an indirect adaptive controller can be regarded as a controller with on-line parameter estimation. There do exist systematic methods for the design of adaptive controllers for the control of linear systems. There also exist adaptive control methods that can be applied to the control of nonlinear systems. However, the latter methods require measurable states and a linear parametrization of the dynamics of the system under control (i.e., that parametric uncertainty be expressed linearly in terms of a number of adjustable parameters). This is required in order to guarantee stability and tracking convergence. However, when adaptive control of nonlinear systems is concerned, most of the adaptive control methods can only be applied to SISO nonlinear systems. Since robust control methods are also used to deal with parameter uncertainty, adaptive control methods can be considered as an alternative and complimentary to robust control methods. In principle, adaptive control is superior to
108
FUZZY CONTROL
robust control in dealing with uncertainties in uncertain or slowly varying parameters. The reason for this is the learning behavior of the adaptive controller: Such a controller improves its performance in the process of adaptation. On the other hand, a robust controller simply attempts to keep a consistent performance. Furthermore, an indirect adaptive controller requires little a priori information about the unknown parameters. A robust controller usually requires reasonable a priori estimates of the parameter bounds. Conversely, a robust controller has features that an adaptive controller does not possess, such as the ability to deal with disturbances, quickly varying parameters, and unmodeled dynamics. In control with a fuzzy controller, there exist a number of direct adaptive control methods aimed at improving the fuzzy controller’s performance on-line. The FC’s parameters that can be altered on-line are the scaling factors for the input and output signals, the input and output membership functions, and the fuzzy IF-THEN rules. An adaptive fuzzy controller, its adjustable parameters being the fuzzy values and their membership functions, is called a self-tuning fuzzy controller. An adaptive fuzzy controller that can modify its fuzzy IFTHEN rules is called a self-organizing fuzzy controller. Detailed description of the design methods for these two types of direct adaptive fuzzy controllers can be found in Ref. 42. Descriptions of indirect adaptive fuzzy controllers can be found in Ref. 8. The methods for the design of a self-tuning fuzzy controller can be applied independent of whether its fuzzy IF-THEN rules are derived using model-based fuzzy control or a heuristic design approach and are thus applicable to the different types of fuzzy controllers. Since tuning and optimization of controllers is related to adaptive control, we use Fig. 19 to illustrate this relationship. In this scheme an adaptation block is arranged above the controller to force the closed-loop system to behave according to a parallel installed reference model. The task is to change the parameters of the controller by means of the adaptation block. Tuning or optimization is performed with the following steps: 1. Optimization criteria are needed that are sufficient for a relevant improvement of the behavior of the system
under control. One criterion mostly used is the integral criterion
T
u ) dt (eeT Qee + uT Ru
J=
(74)
0
where e ⫽ x ⫺ xd is the error, and Q, R are weighting matrices. Another performance criterion can be formulated by fuzzy rules; for example,
IF rise time = SMALL AND settling time = MEDIUM THEN performance = HIGH 2. The next point is to choose an appropriate optimization technique (e.g., gradient decent with constant searching step width, or Rosenbrock’s method with variable searching step widths). 3. A crucial point is to choose a tuning hierarchy (43) that considers the different impacts of the control parameters on stability, performance, and robustness of the closed-loop system: Tune the output scaling factors. Tune the input scaling factors. Tune the membership functions. BIBLIOGRAPHY 1. L. A. Zadeh, Fuzzy sets, Inf. Control, 8: 338–353, 1965. 2. E. H. Mamdani and S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Int. J. Man-Machine Studies, 7 (1): 1–13, 1975. 3. J. J. Østergaard, Fuzzy logic control of a heat exchanger process. In M. M. Gupta (ed.), Fuzzy Automata and Decision Processes, Amsterdam: North-Holland, 1977, pp. 285–320. 4. L. Holmblad and J. J. Østergaard, Control of a cement kiln by fuzzy logic. In M. M. Gupta and E. E. Sanchez (eds.), Fuzzy Information and Decision Processes, Amsterdam: North-Holland, 1982, pp. 389–399. 5. M. Vidyasagar, Nonlinear Systems Analysis, Englewood Cliffs, NJ: Prentice Hall, 1993. 6. W. Pedrycz, Fuzzy Control and Fuzzy Systems, 2nd revised ed., Research Studies, 1992. 7. W. Pedrycz, Fuzzy control engineering: Reality and challenges, IEEE Int. Conf. Fuzzy Syst. 1995, Fuzz-IEEE/IFES’95, Proc., Yokohama, March 1995, pp. 437–446. 8. T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modelling and control, IEEE Trans. Syst. Man Cybern., SMC-15: 1, 116–132, 1985.
Model
Optimization xd
+ e x
–
Controller g(x, p)
u
System f
⁄
Adaptation
–x
9. L. Wang, Fuzzy systems are universal approximators, IEEE Int. Conf. Fuzzy Syst. 1992, Fuzz-IEEE’92, Proc., San Diego, March 8–12, pp. 1163–1169.
+x
x
Figure 19. Adaptive control. Using a model of the system the scheme shows an indirect adaptation strategy. Direct adaptation works without an explicit model of the plant.
10. B. Kosko, Fuzzy systems as universal approximators, IEEE Int. Conf. Fuzzy Syst. 1992, Fuzz-IEEE’92, Proc., San Diego, March 8–12, 1992, pp. 1153–1162. 11. L. Koczy and S. Kovacs, Linearity and the cnf property in linear fuzzy rule interpolation, IEEE Int. Conf. Fuzzy Syst. 1994, FuzzIEEE’94, Proc., Orlando, June 26–29, 1994, pp. 870–875. 12. J-J. E. Slotine and W. Li, Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice-Hall, 1991.
FUZZY IMAGE PROCESSING AND RECOGNITION 13. R. M. Tong, Some properties of fuzzy feedback systems, IEEE Trans. Syst. Man Cybern., SMC-10: 327–330, 1980.
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35. U. Rehfuess and R. Palm, Design of Takagi-Sugeno controllers based on linear quadratic control, Proc. First Int. Symp. Fuzzy Logic, Zurich, Switzerland, May 26–27, 1995, pp. C10–C15.
14. S. Yasunobu and S. Miyamoto, Automatic train operation system by predictive fuzzy control. In M. Sugeno (ed.), Industrial Applications of Fuzzy Control, New York: Elsevier Science, 1985, pp. 1–18.
36. R. H. Haber et al., Two approaches for a fuzzy supervisory control system of a vertical milling machine, VI IFSA Congress, Sao Paulo, Brazil, 1995, pp. 397–400.
15. C. E. Garcı´a, D. M. Prett, and M. Morari, Model predictive control: Theory and practice—a survey, Automatica, 25 (3): 335– 348, 1989.
37. V. V. Badami et al., Fuzzy logic supervisory control for steam turbine prewarming automation, 3rd IEEE Int. Conf. Fuzzy Syst., Orlando, 1994, pp. 1045–1050.
16. J. Valente de Oliveira, Long-range predictive adaptive fuzzy relational control, Fuzzy Sets Syst., 70: 337–357, 1995.
38. W. J. Rugh, Analytical framework for gain scheduling, IEEE Control Syst. Mag., 11 (1): 79–84, 1991.
17. I. Scrjanc, K. Kavsek-Biasizzo, and D. Matko, Fuzzy predictive control based on fuzzy model, EUFIT ’96, Aachen Germany, 1996, pp. 1864–1869.
39. R. A. Nichols, R. T. Reichert, and W. J. Rugh, Gain scheduling for H-infinity controllers: A flight control example, IEEE Trans. Control Syst. Technol., 1: 69–79, 1993.
18. C. W. de Silva and A. G. J. MacFarlane, Knowledge-based control with applications to robots. Lecture Notes in Control and Information Sciences 123, Springer-Verlag, Berlin, 1989.
40. J. S. Shamma, Analysis and design of gain scheduled control systems, Ph.D. thesis No. LIDS-TH-1770, Lab. for Information and Decision Sciences, MIT, Cambridge, MA.
19. G.-C. Hwang and S.-C. Li, A stability approach to fuzzy control design for nonlinear systems, Fuzzy Sets Syst., 48: 279–287, 1992.
41. L-X Wang, Supervisory controller for fuzzy control systems that guarantees stability, 3rd IEEE Int. Conf. Fuzzy Syst., Orlando, 1994, pp. 1035–1039.
20. S. Kawaji and N. Matsunaga, Fuzzy control of VSS type and its robustness, IFSA’91 Brussels, July 7–12, 1991, preprints vol. ‘‘Engineering,’’ pp. 81–88. 21. R. Palm, Sliding mode fuzzy control, IEEE Int. Conf. Fuzzy Syst. 1992, Fuzz-IEEE’92, Proc., San Diego, March 8–12, 1992, pp. 519–526. 22. K. S. Ray and D. D. Majumder, Application of circle criteria for stability analysis of linear SISO and MIMO systems associated with fuzzy logic controller, IEEE Trans. Syst. Man Cybern., 14: 345–349, 1984. 23. K. S. Ray, S. Ananda, and D. D. Majumder, L-stability and the related design concept for SISO linear systems associated with fuzzy logic controller, IEEE Trans. Syst. Man Cyber., 14: 932– 939, 1992. 24. K. L. Tang and R. J. Mulholland, Comparing fuzzy logic with classical control designs, IEEE Trans. Syst. Cybern., SMC-17: 1085–1087, 1987. 25. B. A. M. Wakileh and K. F. Gill, Use of fuzzy logic in robotics, Computers in Industry, 10: 35–46, 1988. 26. S. M. Smith, A variable structure fuzzy logic controller with runtime adaptation, Proc. FUZZ-IEEE’94, Orlando, Florida, July 26– 29, 1994, pp. 983–988. 27. V. J. Utkin, Variable structure systems: A survey, IEEE Trans. Autom. Control, 22: 212–222, 1977. 28. C. S. Hsu, A theory of cell-to-cell dynamical systems, J. Appl. Mech., 47: 940–948, 1980. 29. Y. Y. Chen and T. C. Tsao, A description of the dynamical behavior of fuzzy systems, IEEE Trans. Syst. Man Cybern., 19: 745– 755, 1989. 30. S. M. Smith and D. J. Comer, An algorithm for automated fuzzy logic controller tuning, Proc. IEEE Int. Conf. Fuzzy Syst. 1992, pp. 615–622. 31. M. Papa, H-M. Tai, and S. Shenoi, Design and evaluation of fuzzy control systems using cell mapping, VI IFSA World Congress, Sao Paulo, Brazil, 1995, pp. 361–364. 32. H. Kang and G. Vachtsevanos, Nonlinear fuzzy control based on the vector field of the phase portrait assignment algorithm, Proc. Amer. Control Conf. 1990, pp. 1479–1484. 33. H-T. Hu, H-M. Tai, and S. Shenoi, Incorporating cell map information in fuzzy controller design, 3rd IEEE Int. Conf. Fuzzy Syst., Orlando, 1994, pp. 394–399. 34. K. Tanaka and M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets Syst., 45: 135–156, 1992.
42. D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control, 2nd ed., Berlin: Springer-Verlag, 1996. 43. R. Palm, Tuning of scaling factors in fuzzy controllers using correlation functions, Proc. FUZZ-IEEE’93, San Francisco, California, March 28–April 1, 1993, pp. 691–696.
RAINER PALM Siemens AG
FUZZY CONTROL. See FUZZY LOGIC; POSSIBILITY THEORY.
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Wiley Encyclopedia of Electrical and Electronics Engineering Fuzzy Image Processing and Recognition Standard Article Sankar K. Pal1 1Machine Intelligence Unit, Indian Statistical Institute, Calcutta, IN Copyright © 2007 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W3506.pub2 Article Online Posting Date: June 15, 2007 Abstract | Full Text: HTML PDF (1029K)
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Abstract This article describes various fuzzy set theoretic tools and explores their effectiveness in representing/describing various uncertainties that might occur in an image recognition system and the ways these uncertainties can be managed in making a decision. Some examples of uncertainties that often develop in the process of recognizing a pattern are given in the next section. The Image Ambiguity and Uncertainty Measures Section provides a definition of image and describes various fuzzy set theoretic tools for measuring information on grayness ambiguity and spatial ambiguity in an image. Concepts of bound functions and spect sets charactering the flexible in membership functions are discussed in Their applications to formulate some low level vision operations (e.g., enhancement, segmentation, skeleton extraction, and edge detection), whose outputs are crucial and responsible for the overall performance of a vision system, are then presented. Some real-life applications (e.g., motion frame analysis, character recognition, remote sensing image analysis, content-based image retrieval, and brain MR image segmentation) of these methodologies and tools are then described. Finally, conclusions and discussion are provided. Introduction Uncertainties in a Recognition System and Relevance of Fuzzy Set Theory Image Ambiguity and Uncertainty Measures Grayness Ambiguity Measures Flexibility in Membership Functions Some Examples of Fuzzy Image Processing Operations
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Some Applications Conclusions and Discussion Acknowledgment Keywords: pattern recognition; machine learning; fuzzy set theory; ambigvity; image processing; vision system About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.
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FUZZY IMAGE PROCESSING AND RECOGNITION
INTRODUCTION Pattern recognition and machine learning form a major area of research and development that encompasses the processing of pictorial and other non-numerical information obtained from interaction between science, technology, and society. A motivation for this spurt of activity in this field is the need for the people to communicate with computing machines in their natural mode of communication. Another important motivation is that scientists are also concerned with the idea of designing and making intelligent machines that can carry out certain tasks as we human beings do, the most salient outcome of which is the concept of future generation computing systems. The ability to recognize a pattern is an essential requirement for sensory intelligent machines. Pattern recognition is a must component of the so-called “Intelligent Control Systems,” which involve processing and fusion of data from different sensors and transducers. It is also a necessary function providing “failure detection,” “verification,” and “diagnosis task.” Machine recognition of patterns can be viewed as a two fold task, consisting of learning the invariant and common properties of a set of samples characterizing a class, and of deciding that a new sample is a possible member of the class by noting that it has properties common to those of the set of samples. Therefore, the task of pattern recognition by a computer can be described as a transformation from the measurement space M to the feature space F and finally to the decision space D. When the input pattern is a gray tone image, some processing tasks such as enhancement, filtering, noise reduction, segmentation, contour extraction, and skeleton extraction are performed in the measurement space to extract salient features from the image pattern, which is what is basically known as image processing. The ultimate aim is to make its understanding, recognition, and interpretation from the processed information available from the image pattern. Such a complete image recognition/interpretation system is called a vision system, which may be viewed as consisting of three levels, namely, low level, mid level, and high level, corresponding to M, F, and D with an extent of overlapping among them. In a pattern recognition or vision system, uncertainty can develop at any phase of the aforesaid tasks resulting from the incomplete or imprecise input information, the ambiguity/ vagueness in input image, the ill-defined and/or overlapping boundaries among the classes or regions, and the indefiniteness in defining/extracting features and relations among them. Any decision taken at a particular level will have an impact on all higher level activities. It is therefore required for a recognition system to have sufficient provision for representing these uncertainties involved at every stage, so that the ultimate output (results) of the system can be associated with the least uncertainty (and not be affected or biased very much by the earlier or lower level decisions).
UNCERTAINTIES IN A RECOGNITION SYSTEM AND RELEVANCE OF FUZZY SET THEORY Some of the uncertainties that one encounters often while designing a pattern recognition or vision (1, 2) system will be explained in this section. Let us consider, first of all, the problem of processing and analyzing a gray tone image pattern. A gray tone image possesses some ambiguity within the pixels because of the possible multivalued levels of brightness. This pattern indeterminacy is because of inherent vagueness rather than randomness. The conventional approach to image analysis and recognition consists of segmenting (hard partitioning) the image space into meaningful regions, extracting its different features (e.g., edges, skeletons, centroid of an object), computing the various properties of and relationships among the regions, and interpreting and/or classifying the image. As the regions in an image are not always crisply defined, uncertainty can occur at every phase of the aforesaid tasks. Any decision taken at a particular level will have an impact on all higher level activities. Therefore, a recognition system (or vision system) should have sufficient provision for representing the uncertainties involved at every stage (i.e., in defining image regions, its features and relations among them, and in their matching) so that it retains as much as possible the information content of the original input image for making a decision at the highest level. The ultimate output (result) of the system will then be associated with least uncertainty (and, unlike conventional systems, it will not be biased or affected very much by the lower level decisions). For example, consider the problem of object extraction from a scene. Now, the question is, “How can someone define exactly the target or object region in a scene when its boundary is ill-defined?” Any hard thresholding made for its extraction will propagate the associated uncertainty to the following stages, which might affect its feature analysis and recognition. Similar is the case with the tasks of contour extraction and skeleton extraction of a region. From the aforesaid discussion, it becomes therefore convenient, natural, and appropriate to avoid committing ourselves to a specific (hard) decision (e.g., segmentation/thresholding, edge detection, and skeletonization) by allowing the segments or skeletons or contours to be fuzzy subsets of the image, with the subsets being characterized by the possibility (degree) of a pixel belonging to them. Prewitt (3) first suggested that the results of image segmentation should be fuzzy subsets rather than ordinary subsets. Similarly, for describing and interpreting ill-defined structural information in a pattern, it is natural to define primitives (line, corner, curve, etc.) and relations among them using labels of fuzzy sets. For example, primitives that do not lend them-selves to precise definition may be defined in terms of arcs with varying grades of membership from 0 to 1 representing its belonging to more than one class. The production rules of a grammar may similarly be fuzzified to account for the fuzziness in physical relation among the primitives, thereby increasing the generative power of a grammar for syntactic recognition (4) of a pattern.
J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright © 2007 John Wiley & Sons, Inc.
2
Fuzzy Image Processing and Recognition
The incertitude in an image pattern may be explained in terms of grayness ambiguity, spatial (geometrical) ambiguity or both. Grayness ambiguity means “indefiniteness” in deciding a pixel as white or black. Spatial ambiguity refers to “indefiniteness” in shape and geometry (e.g., in defining centroid, sharp edge, perfect focusing, etc.) of a region. Another kind of uncertainty exists that my derive from the subjective judgment of an operator in defining the grades of membership of the object regions. This process is explained in the section on Flexibility in Membership Functions. Let us now consider the problem of determining the boundary or shape of a class from its sampled points or prototypes. Various approaches (5–7) are described in the literature that attempt to provide an exact shape of the pattern class by determining the boundary such that it contains (passes through) some of the sample points, which need not be true. It is necessary to extend the boundaries to some extent to represent the possible uncovered portions by the sampled points. The extended portion should have lower possibility to be in the class than the portions explicitly highlighted by the sample points. The size of the extended regions should also decrease with the increase of the number of sample points, which leads one to define a multivalued or fuzzy (with continuum grade of belonging) boundary of a pattern class (8, 9). Similarly, the uncertainty in classification or clustering of image points or patterns may develop from the overlapping nature of the various classes or image properties. This overlapping may result from fuzziness or randomness. In the conventional classification technique, it is usually assumed that a pattern may belong to only one class, which is not necessarily true. A pattern may have degrees of membership in more than one class. It is therefore necessary to convey this information while classifying a pattern or clustering a data set. In the following section, we explain various fuzzy set theoretic tools for image analysis (which were developed based on the realization that many of the basic concepts in pattern analysis, for example the concept of an edge or a corner, do not lend themselves to precise definition).
light of fuzzy set theory are available in Reference 10. Let us now explain the various image information measures (deriving from both fuzziness and randomness) and tools as well as their relevance to different operations for image processing and analysis. These measures are classified mainly in two groups, namely grayness ambiguity and spatial ambiguity. GRAYNESS AMBIGUITY MEASURES The definitions of some of the measures that were formulated to represent grayness ambiguity in an image X with dimension M × N and levels L (based on individual pixel as well as a collection of pixels) are listed below. rth Order Fuzzy Entropy:
where sir denotes the ith combination (sequence) of r pixels in X, k is the number of such sequences, and µ(sir ) denotes the degree to which the combination sir , as a whole, possesses some image property µ. Hybrid Entropy:
with
Here, µmn denotes the degree of “whiteness” of the (m,n)th pixel; Pw and Pb denote probability of occurrences of white (µmn = 1) and black (µmn = 0) pixels respectively; and Ew and Eb denote the average likeliness (possibility) of interpreting a pixel as white and black, respectively. Correlation:
IMAGE AMBIGUITY AND UNCERTAINTY MEASURES An L level image X (M × N) can be considered as an array of fuzzy singletons, each having a value of membership denoting its degree of possessing some property (e.g., brightness, darkness, edginess, blurredness, texture, etc.) In the notation of fuzzy sets, one may therefore write that
where µX (xmn ) or µmn denotes the grade of possessing such a property µ by the (m, n)th pixel. This property µ of an image may be defined using global information, local information, or positional information, or a combination there of, depending on the problem. Again, the aforesaid information can be used in a number of ways (in their various functional forms), depending on individuals opinion and/or the problem to his hand, to define a requisite membership function for an image property. Basic principles and operations of image processing and pattern recognition in the
with
Here, µ1mn and µ2mn denote the degree of possessing the properties µ1 and µ2 , respectively, by the (m, n)th pixel and C(µ1 , µ2 ) denotes the correlation between two such properties µ1 and µ2 (defined over the same domain). These expressions (eqs. 2–6) are the versions extended to the 2-D image plane from those defined (ll,12) for a fuzzy set. H r (X) gives a measure of the average amount of difficulty in taking a decision whether any subset of pixels of size r possesses an image property. Note that no probabilistic concept is needed to define it. If r = 1, H r (X) reduces to (non-normalized) entropy as defined by De Luca
Fuzzy Image Processing and Recognition
and Termini (13). Hhy (X), on the other hand, represents an amount of difficulty in deciding whether a pixel possesses a certain property µmn by making a prevision on its probability of occurrence it is assumed here that the fuzziness occurs because of the transformation of the complete white (0) and black pixels (1) through a degradation process, thereby modifying their values to lie in the intervals [0,0.5] and [0.5,1], respectively). Therefore, if µmn denotes the fuzzy set “object region”, then the amount of ambiguity in deciding µmn a member of object region is conveyed by the term hybrid entropy depending on its probability of occurrence. In the absence of fuzziness (i.e., with exact defuzzification of the gray pixels to their respective black or white version), Hhy reduces to the two-state classical entropy of Shannon (14), the states being black and white. As a fuzzy set is a generalized version of an ordinary set, the entropy of a fuzzy set deserves to be a generalized version of classical entropy by taking into account not only the fuzziness of the set but also the underlying probability structure. In that respect, µhy can be regarded as a generalized entropy such that classical entropy becomes its special case when fuzziness is properly removed. Note that equations (2) and (3) are defined using the concept of logarithmic gain function. Similar expressions using exponential gain function (i.e., defining the entropy of an n-state system) have been given by Pal and Pal (15– 18).
All these terms, which give an idea of “indefiniteness” or fuzziness of an image, may be regarded as the measures of average intrinsic information that is received when one has to make a decision (as in pattern analysis) to classify the ensembles of patterns described by a fuzzy set. H r (X) has the following properties: Pr 1: Hr attains a maximum if µi = 0.5 for all i. Pr 2: Hr attains a minimum if µi = 0 or 1 for all i. Pr 3: Hr > H∗r , where H∗r is the rth-order entropy of a sharpened version of the fuzzy set (or an image). r r Pr 4: Hr is, in general, not equal to H , where H is the rth-order entropy of the complement set. Pr 5: Hr ≤ Hr +1 when all µi ∈ [0.5,1]. H r ≥ H r+1 when all µi ∈ [0,0.5]. The “sharpened” or “intensified” version of X is such that µx∗ (xmn ) ≥ µx (xmn )
if
µx (xmn ) ≥ 0.5
µx∗ (xmn ) ≤ µx (Xmn ) if
µx (xmn ) ≤ 0.5
(8)
and
When r = 1, the property Pr 4 is valid only with the equal sign. The property Pr 5 (which does not occur for r = 1) implies that Hr is a monotonically non-increasing function of r for µi ∈ [0,0.5] and a monotonically nondecreasing function of r for µi ∈ [0.5,1] (when the “min” operator has been used to get the group membership value). When all µi values are the same, H 1 (X) = H 2 (X) = . . . = r H (X), which is because the difficulty in taking a decision regarding possession of a property on an individual is the
3
same as that of a group selected therefrom. The value of Hr would, of course, be dependent on the µi values. Again, the higher the similarity among singletons (supports), the quicker is the convergence to the limiting value of Hr . Based on this observation, an index of similarity of supports of a fuzzy set may be defined as S = H1 /H2 (when H2 = 0, H1 is also zero and S is taken as 1). Obviously, when µi ∈ [0.5,1] and the min operator are used to assign the degree of possession of the property by a collection of supports, S will lie in [0, 1] as Hr ≤ Hr+1 . Similarly, when µi ∈ [0,0.5], S may be defined as H2 /Hl so that S lies in [0, 1]. The higher the value of S, the more alike (similar) are the supports of the fuzzy set with respect to the fuzzy property µ. This index of similarity can therefore be regarded as a measure of the degree to which the members of a fuzzy set are alike. The details are available in Reference 19. Therefore, the value of first order fuzzy entropy (H1 ) can only indicate whether the fuzziness in a set is low or high. In addition, the value of Hr ,r > 1 also enables one to infer whether the fuzzy set contains similar supports (or elements). The similarity index thus defined can be successfully used for measuring interclass and intraclass ambiguity (i.e., class homogeneity and contrast) in pattern recognition and image processing problems. H1 (X) is regarded as a measure of the average amount of information (about the gray levels of pixels) that has been lost by transforming the classic pattern (two-tone) into a fuzzy (gray) pattern X. Further details on this measure with respect to image processing problems are available in References 10 and 20–22. It is to be noted that H 1 (X) reduces to zero whenever µmn is made 0 or 1 for all (m, n ), no matter whether the resulting defuzzification (or transforming process) is correct. In the following discussion, it will be clear how Hhy takes care of this situation. Let us now discuss some of the properties of Hhy (X). In the absence of fuzziness when MNPb pixels become completely black (µmn = 0) and MNPW pixels become completely white (µmn = 1), then Ew = Pw , Eb = Pb and Hhy boils down to the two-state classical entropy
the states being black and white. Thus Hhy reduces to Hc only when a proper defuzzification process is applied to detect (restore) the pixels. |Hhy − Hc | can therefore be treated as an objective function for enhancement and noise reduction. The lower the difference, the less the fuzziness associated with the individual symbol and the higher the accuracy in classifying them as their original value (white or black). (This property is lacking with the H1 (X) measure and the measure of Xie and Bedrosian (23), which always reduces to zero or some constant value irrespective of the defuzzincation process.) In other words, |Hhy − Hc | represents an amount of information that was lost by transforming a two-tone image to a gray tone. For a given Pw and Pb , (Pw + Pb = 1, 0 ≤ Pw , Pb , ≤ 1), of all possible defuzzifica-tions, the proper defuzzification of the image is the one for which Hhy is minimum.
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Fuzzy Image Processing and Recognition
and
For example, Hhy takes a constant value and becomes independent of Pw and Pb , which is logical in the sense that the machine is unable to make a decision on the pixels because all µmn values are 0.5. Spatial Ambiguity Measures Based on Fuzzy Geometry of Image Many of the basic geometric properties of and relationships among regions has been generalized to fuzzy subsets. Such an extension, called fuzzy geometry (24–28), includes the topological concept of connectedness, adjacency and surroundedness, convexity, area, perimeter, compactness, height, width, length, breadth, index of area coverage, major axis, minor axis, diameter, extent, elongatedness, adjacency, and degree of adjacency. Some of these geometrical properties of a fuzzy digital image subset (characterized by piece-wise constant membership function µX(xmn ), or simply µ are listed below with illustrations. These properties may be viewed as providing measures of ambiguity in the geometry (spatial domain) of an image. Compactness (24):
and
The length/breadth of an image fuzzy subset gives its longest expansion in the column/row direction. If µ is crisp, µmn = 0 or l, then length/breadth is the maximum number of pixels in a column/row. Comparing equations 17 and 18 with 15 and 16, we notice that the length/breadth takes the summation of the entries in a column/row first and then maximizes over different columns/rows, whereas the height/width maximizes first the entries in a column/row and then sums over different columns/rows. Index of Area Coverage (26, 27):
In the non-fuzzy case, the 10AC has a value of 1 for a rectangle (placed along the axes of measurement). For a circle, this value is πr 2 /(2r ∗ 2r) = π/4. 10AC of a fuzzy image represents the fraction (which may be improper also) of the maximum area (that can be covered by the length and breadth of the image) actually covered by the image. Again, note the following relationships.
and where
and
Here, a(µ) denotes area of µ, and p(µ), the perimeter of µ, is just the weighted sum of the lengths of the arcs A(i,j, k) (24) along which the region µ(i) and µ(j) meet, weighted by the absolute difference of these values. Physically, compactness means the fraction of maximum area (that can be encircled by the perimeter) actually occupied by the object. In the non-fuzzy case, the value of compactness is maximum for a circle and is equal to 1/4π. In the case of the fuzzy disc, where the membership value is only dependent on its distance from the center, this compactness value is ≥ 1/4π. Of all possible fuzzy discs, compactness is therefore minimum for its crisp version. Height and Width (24):
and
So, height/width of a digital picture is the sum of the maximum membership values of each row/column. Length and Breadth (26, 27):
When equality holds for equation (20), the object is either vertically or horizontally oriented. Similarly, major axis, minor axis, center of gravity, and density are also defined in Reference 27. Degree of Adjacency (27): The degree to which two crisp regions S and T of an image are adjacent is defined as
Here, d(p) is the shortest distance between p and q, q is a border pixel (BP) of T, and p is a border pixel of S. The other symbols have the same meaning as in the previous discussion. The degree of adjacency of two regions is maximum (= 1) only when they are physically adjacent (i.e., d(p) = 0) and their membership values are also equal [i.e., µ(p) = r(q)]. If two regions are physically adjacent, then their degree of adjacency is determined only by the difference of their membership values. Similarly, if the membership values of two regions are equal, their degree of adjacency is determined by their physical distance only. The readers may note the difference between equation (22) and the adjacency definition given in Reference 24. FLEXIBILITY IN MEMBERSHIP FUNCTIONS As the theory of fuzzy sets is a generalization of the classic set theory, it has greater flexibility to capture faithfully the various aspects of incompleteness or imperfection (i.e.,
Fuzzy Image Processing and Recognition
deficiencies) in information of a situation. The flexibility of fuzzy set theory is associated with the elasticity property of the concept of its membership function. The grade of membership is a measure of the compatibility of an object with the concept represented by a fuzzy set. The higher the value of membership, the less the amount (or extent) to which the concept represented by a set needs to be stretched to fit an object. As the grade of membership is both subjective and dependent on context, some difficulty of adjudging the membership value still remains. In other words, the problem is how to assess the membership of an element to a set, which is an issue where opinions vary, giving rise to uncertainties. Two operators, namely “Bound Functions” (29) and “Spectral Fuzzy Sets” (30), have been defined to analyze the flexibility and uncertainty in membership function evaluation. These operators are explained below along with their significance in image analysis and pattern recognition problems. Consider, for example, a “bright image,” which may be considered as a fuzzy set. It is represented by an S-type function that is a nondecreasing function of gray value. Now, the question is, “can any such nondecreasing function be taken to represent the above fuzzy set?” Intuitively, the answer is “no.” Bounds for such an S-type membership function µ have been reported (29) based on the properties of fuzzy correlation (11). The correlation measure between two membership functions µ1 and µ2 relates the variation in their functional values. The significance of the bound functions in selecting an S-type function µ for the image segmentation problem has been reported in detail in Reference 31. It has been shown that, for detecting a minimum in the valley region of a histogram, the window length w of the function µ: [0, w] → [0,1] should be less than the distance between two peaks around that valley region. The ability to make the fuzzy set theoretic approach flexible and robust will be demonstrated further in the next section. The concept of spectral fuzzy sets is used where, instead of a single unique membership function, a set of functions reflecting various opinions on membership elements is available so that each membership grade is attached to one of these functions. By giving due respect to all the opinions available for further processing, it reduces the difficulty (ambiguity) in selecting a single function. A spectral fuzzy subset F having n supports is characterized by a set or a band (spectrum) of r membership functions (reflecting r opinions) and may be represented as
5
The (dis)similarity between the concept of spectral fuzzy sets and those of the other tool such as probabilistic fuzzy set, interval-valued fuzzy set, fuzzy set, of type 2, or ultra fuzzy set (32–36) (which have also considered the difficulty in settling a definite degree of fuzziness or ambiguity), has been explained in Reference 30. The concept has been found to be significantly useful (30) in segmentation of ill-defined regions where the selection of a particular threshold becomes questionable as far as its certainty is concerned. In other words, questions may develop like, “where is the boundary?” or “what is the certainty that a level 1, say, is a boundary between object and background?” The opinions on these queries may vary from individual to individual because of the differences in opinion in assigning membership values to the various levels. In handling this uncertainty, the algorithm gives due respect to various opinions on membership of gray levels for object region, minimizes the image ambiguity d(= d1 + d2 ) over the resulting band of membership functions, and then makes a soft decision by providing a set of thresholds (instead of a single one) along with their certainty values. A hard (crisp) decision obviously corresponds to one with maximum d value (i.e. the level at which opinions differ most). The problems of edge detection and skeleton extraction (where incertitude occurs from ill-defined regions and various opinions on membership values) and any expert system-type application (where differences in experts’ opinions leads to an uncertainty) may also be similarly handled within this framework.
SOME EXAMPLES OF FUZZY IMAGE PROCESSING OPERATIONS Let us now describe some algorithms to show how the aforesaid information measures and geometrical properties can be incorporated in handling uncertainties in various operations (e.g., gray level thresholding, enhancement, contour detection and skeletonization by avoiding hard decisions, and providing output in both fuzzy and nonfuzzy (as a special case) versions). is to be noted that these low level operations (particularly image segmentation and object extraction) play a major role in an image recognition system. As mentioned earlier, any error made in this process might propagate to feature extraction and classification.
Enhancement in Property Domain
where r, the number of membership functions, may be called the cardinality of the opinion set. µiF (x j ) denotes the degree of belonging of Xj to the set F according to the ith membership function. The various properties and operations related to it have been defined by Pal and Das Gupta (30). The incertitude or ambiguity associated with this set is two-fold, namely ambiguity in assessing a membership value to an element (d1 ) and ambiguity in deciding whether an element can be considered to be a member of the set (d2 ).
The objective of enhancement techniques is to process a given image so that the result is more suitable than the original for a specific application. The term “specific” is, of course, problem-oriented. The techniques used here are based on the modification of pixels in the fuzzy property domain of an image (10, 20, 2l). The contrast intensification operator on a fuzzy set A generates another fuzzy set A = INT(A) in which the fuzziness is reduced by increasing the values of µA (xmn ) that are above 0.5 and decreasing those that are below it. Define this INT operator by a transformation T1 of the member-
6
Fuzzy Image Processing and Recognition
ship function µmn as T1 (µmn ) = T1 (µmn ) = 2µ2mn , 0 ≤ µmn ≤ 0.5 = T1 (µmn ) = 1 − 2(1 − µmn )2 , 0.5 ≤ µmn m = 1, 2, . . . M, n = 1, 2, . . . N
(21)
In general, each µmn in X (Eq. 1) may be modified to µmn to enhance the image X in the property domain by a transformation function Tr where µmn = Tr (µmn ) = Tr (µmn ), 0 ≤ µmn ≤ 0.5 (22) = Tr (µmn ), 0.5 ≤ µmn ≤ 1 r = 1, 2, . . . The transformation function Tr is defined as successive applications of T1 by the recursive relationship (20)
and T1 (Pmn ) represents the operator INT denned in equation (24). As r increases, the enhancement function (curve) in µmn − µmn plane tends to be steeper because of the successive application of INT. In the limiting case, as r →∞, Tr produces a two-level (binary) image. It is to be noted here that, corresponding to a particular operation of T , one can use any of the multiple operations of T , and vice versa, to attain a desired amount of enhancement. Similarly, some other enhancement functions can be used independently instead of those used in equation (24). The membership plane µmn for enhancing contrast around a cross-over point may be obtained from References 11 and 20.
where the position of cross-over points bandwidth, and hence the symmetry of the curve, are determined by the fuzzifiers Fe and Fd . When ªx = xmax (maximum level in X), µmn represents an S-type function. When ªx = any arbitrary level l,µmn represents a π-type function. After enhancement in the fuzzy property domain, the enhanced spatial domain xmn may be obtained from
where α is the value of µmn when xmn = 0. Note that the aforesaid method provides a basic module of fuzzy enhancement. In practice, one may use it with other smoothing, noise cleaning, or enhancement operations Tor resulting in desired outputs. An extension of this concept to enhance the contrast among various ill-defined regions using multiple applications of π and (1 − π) functions has been described in References 21 and 37 for edge detection of X-ray images. The edge detection operators involve max and min operations. Reference 38 demonstrates, in this regard, an attempt to use a relaxation (iterative) algorithm for fast image enhancement using various orders of S functions; convergence has also been analyzed. Fuzzy image enhancement technique has been applied by Krell et al. (39) for enhancing the quality of images taken by electronic postal imaging device needed by clinicians to verify the shape and the location of “therapy beam” with respect to the patients anatomy. Lukac et al. (40) performed cDNA microarray image processing using fuzzy vector filtering framework. Various other fuzzy enhancement operators have been developed to reduce degradation
in images (41–48). Reference 49 uses a fuzzy regularization approach to carry out blind image deconvolution. Recently, fuzzy techniques have also been used in impulse noise detection and reduction (50). Furthermore, the concept fuzzy transformation has been developed for low level image processing applications (51). Optimum Enhancement Operator Selection When an image is processed for visual interpretation, it is ultimately up to the viewers to judge its quality for a specific application and how well a particular method works. The process of evaluation of image quality therefore becomes subjective, which makes the definition of a wellprocessed image an elusive standard for comparison of algorithm performance. Again, it is customary to have an iterative process with human interaction to select an appropriate operator for obtaining the desired processed output. For example, consider the case of contrast enhancement using a nonlinear functional mapping. Not every kind of nonlinear function will produce a desired (meaningful) enhanced version. The questions that automatically develop are “Given an arbitrary image, which type of nonlinear functional form will be best suited without prior knowledge on image statistics (e.g., in remote applications like space autonomous operations where frequent human interaction is not possible) for highlighting its object?” and “Knowing the enhancement function, how can one quantify the enhancement quality for obtaining the optimal one?” Regarding the first question, even if the image statistics are given, it is possible only to estimate approximately the function required for enhancement and the selection of the exact functional form still needs human interaction in an iterative process. The second question, on the other hand, needs individual judgment, which makes the optimum decision subjective. The method of optimization of the fuzzy geometrical properties and entropy has been found (52) to be successful, when applied on a set of different images, in providing quantitative indices to avoid such human iterative interaction in selecting an appropriate nonlinear function and to make the task of subjective evaluation objective. Threshold Selection (Fuzzy Segmentation) Given an L level image X of dimension M × N with minimum and maximum gray values lmin and lmax , respectively, the algorithm for its fuzzy segmentation into object and background may be described as follows: Step 1: Construct the membership plane using the standard S function as
or
(depending on whether the object regions possess higher or lower gray values) with cross-over
Fuzzy Image Processing and Recognition
point b and band width b = b − a = c − b. Step 2: Compute the parameter I(X) where I(X) represents either grayness ambiguity or spatial ambiguity, as stated earlier, or both. Step 3: Vary b between lmin and lmax and select those b for which I(X) has local minima or maxima depending on I(X). (Maxima correspond to the correlation measure only.) Among the local minima/maxima, let the global one have a cross-over point at s.
The level s, therefore, denotes the cross-over point of the fuzzy image plane µmn , which has minimum grayness and/or geometrical ambiguity. The µmn plane then can be viewed as a fuzzy segmented version of the image X. For the purpose of nonfuzzy segmentation, we can take s as the threshold (or boundary) for classifying or segmenting an image into object and background. Faster methods of computation of the fuzzy parameters are explained in Reference 27. Note that w = 2b is the length of the window (such that [0, w] → [0,1]). that was shifted over the entire dynamic range. As w decreases, the possibility of detecting some undesirable thresholds (spurious minima) increases because of the smaller value of b. On the other hand, an increase in w results in a higher value of fuzziness and thus leads toward the possibility of losing some of the weak minima. The criteria regarding the selection of membership functions and the length of window (i.e., w) have been reported in References 29 and 31 assuming continuous functions for both histogram and membership function. It is shown that µ should satisfy the bound criteria derived based on the correlation flexibility in membership functions (section). Another way of handling this uncertainty using spectral fuzzy sets for providing a soft decision is explained in Referece 30. Let us now describe another way of extracting an object by minimizing higher order entropy (Eq. 2) of both object and background regions using an inverse π function as shown by the solid line in Fig. 1. Unlike the previous algorithm, the membership function does not need any parameter selection to control the output. Suppose s is the assumed threshold so that the gray level ranges [1, s] and [s + 1,L] denote, respectively, the object and background of the image X. The inverse π-type function to obtain µmn values of X is generated by taking the union of S[x; s − (L −s),s,L] and 1 − S(x; l,s, (s + s + s −1)], where S denotes the standard S function. The resulting function as shown by the solid line makes µ lie in [0.5,1]. As the ambiguity (difficulty) in deciding a level as a member of the object or the background is maximum for the boundary level s, it has been assigned a membership value of 0.5. Ambiguity decreases as the gray value moves away from s on either side. The µmn thus obtained denotes the degree of belonging of a pixel xmn to either object or background. As is not necessarily the mid point of the entire gray scale, the membership function may not be a symmetric one. Therefore, the task of object extraction is to:
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Step 1: Compute the rth-order fuzzy entropy of the object HOr and the background HBr considering only the spatially adjacent sequences of pixels present within the object and background, respectively. Use the “min” operator to get the membership value of a sequence of pixels. Step 2: Compute the total rth-order fuzzy entropy of the partitioned image as Hsr = HOr + HBr . Step 3: Minimize Hsr with respect to s to get the threshold for object background classification. Referring back to the section on Grayness Ambiguity Measures, it is seen that H 2 reflects the homogeneity among the supports in a set in a better way than H 1 does. The higher the value of r, the stronger is the validity of this fact. Thus, considering the problem of object–background classification, the improper selection of the threshold is more strongly reflected by Hr than H r−1 . The methods of object extraction (or segmentation) described above are all based on gray level thresholding. Another way of doing this task is by pixel classification. The details on this technique, using fuzzy c-means, fuzzy isodata, fuzzy dynamic clustering, and fuzzy relaxation, are available in References (2, 10, and 53–60). The fuzzy cmeans (FCM) algorithm is a well-known clustering algorithm used for pixel classification. Here, we describe it in brief. Fuzzy segmentation results in fuzzy partitions of X = {x1 , x2 , . . . , xn }, where X denotes a set of n unlabeled column vectors in RP (i.e., each element of X is a p-dimensional feature vector). A fuzzy c-partition (c is an integer, 1 ≤ c ≤ n) is the matrix U = [µik ], i = 1, 2, . . . , c, k = 1, . . . , n that satisfies the following constraints:
Here, the kth column of U represents membership values of xk to the c fuzzy subsets and µik = µi (xk ) denotes the grade of membership of xk in the ith fuzzy subset. The FCM algorithm searches the local minimum of the following objective function:
where U isafuzzy c-partition of X, · A is any inner product norm, V = {v1 , v2 , . . . , vc } is a set of cluster centers, vi ∈ R p , and m ∈ [1, ∞] is the weighting exponent on each fuzzy membership. For m > 1 and xk = vi for all i, k, it has been shown that Jm (U, V) may be minimized only if
and
The FCM algorithm, when Euclidian distance norm is considered, can only be used for hyperspherical clusters with approximately equal dimensions. To cope with clusters
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Fuzzy Image Processing and Recognition
having large variability in cluster shapes, densities, and the number of data points in each cluster, Gustafson and Kessel (61) used the scaled Mahalanobis distance in the FCM algorithms. By the use of a distance measure derived from maximum likelihood estimation methods, Gath and Geva (62) obtained an algorithm that is eifective even when the clusters are ellipsoidal in shape and unequal in dimension. As the value of c (i.e., the number of clusters) is not always known, several cluster validity criteria have been suggested in the literature to find the optimum number of clusters. These criteria include partition coefficient, classification entropy, properties coefficient, total within-class distance of clusters, total fuzzy hyper volume of clusters, and partition density of clusters (60–63). Generalizing the FCM algorithm further, Dave (64) proposed the fuzzy c shells (FCS) algorithm to search for clusters that are hyper ellipsoidal shells. One of its advanced versions is believed to be better than Hough transformation (in terms of memory and speed of computation) when used for ellipse detection. It is also shown (64) that the use of fuzzy memberships improves the ability to attain global optima compared with the use of hard membership. For the same purpose, Krishnapuram et al. (65) proposed another algorithm that is claimed to be less time consuming than that of Dave. For further information, readers may consult References 66–69. The article in Reference 66 describes a modified version of the FCM, which incorporates supervised training data. The article of Cannon et al. (67) describes an approach that reduces the computation required for the FCM, by using look up tables, by a factor of six. Another simplified form of FCM in this line is mentioned in Reference 68. The authors in Reference (69) have proposed a new heuristic fuzzy clustering technique and have referred to it as the Fuzzy J-Means (FJM). Soft decision making has been used to develop many other segmentation algorithms for various applications such as document image processing, ultrasound image processing, satellite image analysis, MR image analysis, and remote sensing (70–78). Algorithms for applications such as classification of MR brain images (79) and microcalcification detection (80) have been succesfully implemented.using fuzzy techniques. Before leaving this section, we mention the work in Reference (81), which defines the concept of fuzzy objects and describes algorithms for their extraction.
Contour Detection Edge detection is also an image segmentation technique where the contours/boundaries of various regions are extracted based on the detection of discontinuity in grayness. Here we present a method for fuzzy edge detection using an edginess measure based on H 1 (Eq. 2), which denotes an amount of difficulty in deciding whether a pixel can be 3 called an edge (19). Let Nxy be a 3× 3 neighborhood of a E pixel at (x,y). The edge–entropy Hxy of the pixel (x,y), giving a measure of edginess at (x,y), may be computed as follows. For every pixel (x,y), compute the average, maximum, and 3 minimum values of gray levels over Nxy . Let us denote the average, maximum, and minimum values by Avg, Max, and Min, respectively. Now define the following parameters.
A π-type membership function (Fig. 2) is then used to 3 compute µxy for all (x, y) ∈ Nxy such that µ(A) = µ(C) = 0.5 and µ(B) = 1. It is to be noted that µxy ≥ 0.5. Such a µxy , therefore, characterizes a fuzzy set “pixel intensity close 3 to its average value,” averaged over Nx,y . When all pixel 3 values over Nx,y are either equal or close to each other (i.e., they are within the same region), such a transformation will make all µxy = 1 or close to 1. In other words, if no edge exists, pixel values will be close to each other and the µ values will be close to one (1); thus resulting in a low value of H 1 . On the other hand, if an edge does exist 3 (dissimilarity in gray values over Nx,y ), then the µ values will be more away from unity; thus resulting in a high value 3 of H 1 . Therefore, the entropy H 1 over Nx,y can be viewed as E a measure of edginess (Hx,y ) at the point (x,y). The higher E the value of Hx,y , the stronger the edge intensity and the easier its detection. Such an entropy plane will represent the fuzzy edge detected version of the image. The proposed entropic measure is less sensitive to noise because of the use of a dynamic membership function based on a local neighborhood. The method is also not sensitive to the direction of edges. Other edginess measures and algorithms based on fuzzy set theory are available in References 10, 21, and 37.
Figure 1. Inverse π function (solid line) for computing object and background entropy.
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9
Figure 2. π Function for computing edge entropy.
Fuzzy Skeleton Extraction Let us now explain two methods for extracting the fuzzy skeleton (skeleton having an ill-defined boundary) of an object from a gray tone image without getting involved into its (questionable) hard thresholding. The first one is based on minimization of the parameter 1OAC (Eq. 19) or compactness (Eq. 12) with respect to α-cuts (α-cut of a fuzzy set A comprises all elements of X whose membership value is greater than or equal to α, 0 < α ≤ 1) over a fuzzy “core line” (or skeleton) plane. The membership value of a pixel to the core line plane depends on its property of possessing maximum intensity, and property of occupying vertically and horizontally middle positions from the -edges (pixels beyond which the membership value in the fuzzy segmented image becomes less than or equal to , > 0) of the object (82). If a nonfuzzy (or crisp) single-pixel-width skeleton is deserved, it can be obtained by a contour tracing algorithm (83) that takes into account the direction of contour. Note that the original image cannot be reconstructed, like the other conventional techniques of gray skeleton extraction (2–85) from the fuzzy skeleton obtained here. The second method is based on fuzzy medial axis transformation (FMAT) (28) using the concept of fuzzy disks. A fuzzy disk with center P is a fuzzy set in which membership depends only on the distance from P. For any fuzzy set f, a maximal fuzzy disk gP f ≤ f exists centered at every point P, and f is the sup of the gP f s. (Moreover, if f is fuzzy convex, so is every gP f , but not conversely.) Let us call a set Sf of points f-sufficient if every gP f ≤ gQ f for some set of Q in Sf ; evidently f is then the sup of the gQ f s. In particular, in a digital image, the set of Q’s at which gf is a (non-strict) local maximum is f-sufficient. This set is called the fuzzy medial axis of f, and the set of gQ f s is called the fuzzy medial axis transformation (FMAT) of f. These definitions reduce to the standard one if f is a crisp set. For a gray tone image X (denoting the non-normalized fuzzy “bright image” plane), the FMAT algorithm computes, first of all, various fuzzy disks centered at the pixels and then retains a few (as small as possible) of them, as designated by gQ’s, so that their union can represent the entire image X. That is, the pixel value at any point t can be obtained from a union operation, as t has membership value equal to its own gray value (i.e., equal to its nonnormalized membership value to the bright image plane) in one of those retained disks. Note that the above representation is redundant (i.e., some more disks can further be deleted without affecting the reconstruction). The redundancy in pixels (fuzzy disks) from the fuzzy medial axis output can be reduced by considf ering the criterion gP f (t) ≤ sup gQi (t), i = 1, 2, . . . instead f f of gP (t) ≤ gQ (t). In other words, eliminate many other
gP f s for which there exists a set of gQ f s whose sup is greater than or equal to gP f . Let RFMAT denote the FMAT after reducing its redundancy. The fuzzy medial axis is seen to provide a good skeleton of the darker (higher intensity) pixels in an image apart from its exact representation. FMAT of an image can be considered as its core (prototype) version for the purpose of image matching. It is to be mentioned here that such a representation may not be economical in a practical situation. The details on this feature and the possible approximation to make it practically feasible are available in Reference (86) Note that the membership values of the disks contain the information of image statistics. For example, if the image is smooth, the disk will not have abrupt change in its values. On the other hand, it will have abrupt change in case the image has salt and pepper noise or edginess. The concept of fuzzy MAT can therefore be used as spatial filtering (both high pass and low pass) of an image by manipulating the disk values to the extent desired and then putting them back while reconstructing the processed image. A gray-scale thinning algorithm is described in References 60 and 87 based on the concept of fuzzy connectedness between two pixels; the dark regions can be thinned without ever being explicitly segmented. SOME APPLICATIONS Here we provide a few applications of the methodologies and tools described before. Motion Frame Analysis and Scene Abstraction With rapid advancements in multimedia technology, it is increasingly common to have time-varied data like video as computer data types. Existing database systems do not have the capability of search within such information. It is a difficult problem to automatically determine one scene from another because no precise markers exist that identify where they begin and end. Moreover, divisions of scenes can be subjective, especially if transitions are subtle. One way to estimate scene transitions is to approximate the change of information between each of two successive frames by computing the distance between their discriminatory properties. A solution is provided in Reference 88 to the problem of scene estimation/abstraction of motion video data in the fuzzy set theoretic framework. Using various fuzzy geometrical and information measures (see image Ambiguity and uncertainty measures section) as image features, an algorithm is developed to compute the change of information in each of two successive frames to classify scenes/frames.
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Fuzzy Image Processing and Recognition
Frame similarity is measured in terms of weighted distance in fuzzy feature space. This categorization process of raw input visual data can be used to establish structure for correlation. The investigation not only attempts to determine the discrimination ability of the fuzziness measures for classifying scenes, but also enhances the capability of nonlinear, frame-accurate access to video data for applications such as video editing and visual document archival retrieval systems in multimedia environments. Such an investigation is carried out in NASA Johnson Space Center, Texas (88). A set of digitized videos of previous space shuttle missions obtained from NASA/JSC was used (Fig. 3). The scenes were named payload deployment, onboard astronaut, remote manipulator arm, and mission control room. Experiments were conducted for various combinations of uncertainty, orientation, and shape measures. As an illustration, Fig. 4 shows a result when entropy, compactness, length/height was considered as a feature set for computing distance between two successive frames. Here the abscissa represents the total number of frame distances in the sampled time series, and the ordinate is the compound distance value between two successive images. Each scene consists of six frames. Therefore, a change of scene occurs at every sixth index on the abscissa. The scene separation is denoted with vertical grid lines. The effectiveness of the aforesaid fuzzy geometrical parameter is also demonstrated (89) for recognizing overlapping fingerprints with a multilayer perceptron. In the last decade, substantial advancement has occurred in video and motion analysis using fuzzy sets. Recently, a new video shot boundary detection technique using fuzzy logic has been proposed in Reference 90. The authors in Reference 91 used fuzzy C-planes clustering to propose a motion estimation technique, which is an important block in most of the video processing systems. Other applications such as traffic handling (92) in video processing have also been implemented using fuzzy techniques. Handwritten Character Recognition Handwritten characters, like all patterns of human origin, are examples of ill-defined patterns. Hence, the recognition of handwritten characters is a very promising field for the application of pattern recognition techniques using the fuzzy approach. It has been claimed that the concept of vagueness underlying fuzzy theory is more appropriate for describing the inherent variability of such systems than the probabilistic concept of randomness. An important application of handwriting recognition is to build efficient man-machine interface for communicating with the computer by human beings. Several attempts have been made for handwritten character recognition in different languages. Here we mention a pioneering contribution of Kickert and Koppelaar (93), the subsequent developments based on their work, and then an attempt made for fuzzy feature description in this context. The 26 capital letters of the English alphabet constituting the set
are seen to be composed of the elements of the following set of “ideal” elements (93)
where ∈ is a null segment whose use will be explained shortly. Also, a set P exits of 11 ordered recognition routines capable of recognizing the “ideal” segments. Each element of P can be considered as a portion of a context-free grammar having productions of the form
with Vn being the non-terminal elements of the grammar. Each of the 11 recognition routines is applied sequentially to any unknown pattern S to be recognized as one of the members of L. Each routine attempts to recognize a given segment in a given structural context. If successful, the application of the rules in P results in a parsing of S as a vector of segments S = (x1 , x2 , . . . , xn ), where xi ∈ VT . Each letter, then, is defined by its vector of segments. Let us assume that the vectors are padded out with null segments ∈ so that all letters are defined by vectors of equal length. Each letter, therefore, can be defined as follows:
where
The element of fuzziness is introduced by associating with each segment ai ∈ VT a fuzzy set on the actual pattern space. With each ai is associated a fuzzy membership function µai so that, given a segment xi of a pattern S, µa j (xi ) is a measure of the degree to which the segment xi corresponds to the ideal segment aj . The recognition procedure is now simply explained. The sequence of recognition rules is executed, evaluating all possible parsings of the input pattern. For each letter Hk for which a parse can be made, the result is a sequence (xi , x2 , . . . , xn ) of segments. The membership of S in Hk is the intersection in the sense of fuzzy sets of the memberships of the segments xi
Finally, the pattern is recognized as letter Hm if
This approach was criticized by Stallings (94), who developed a Bayesian hypothesis-testing scheme for the same problem. Given a pattern S, hypothesis Hk is that the writer intended letter Hk Associated with each decision is a cost Cij , which is the cost of choosing Hi when Hj is true. The parsing of the pattern is performed as before. Only a probability is associated with each segment for a given letter. Regarding unknown densities, stallings (94) suggests the use of maximum likelihood tests. As both membership function and probability density functions are maps into the interval [0,1], the only difference is the use of min/max operators, where, the author argues, the “min” operator loses a lot of information and is drastically affected by one low value. The author claims that a though frequentistic
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Figure 3. A payload deployment sequence of four scenes as input data.
Figure 4. Distances between successive frames with feature set (entropy, compactness, length/height).
probabil-ity is not appropriate in dealing with pattern variability, subjective probability is perfectly suitable and more intuitively obvious than “grade of membership.” In a rejoinder (95), it is argued that fuzzy set theory is more flexible than is assumed in Reference 94, where all arguments are directed against a particular case (93). Recalling the idea of collectives (from property sets), where the arithmetic average replaces “min,” there remains little difference between the schemes in References 93 and 94. In a reply, Stalling insisted that the Bayesian approach is superior because offers a convenient way for assignment of costs to errors and gains to correct answers. For the recognition of handwritten English capital letters, the readers may also refer to the work described in Reference 96. Existing computational recognition methods use feature extraction to assign a pattern to a prototype class. Therefore, the recognition ability depends on the selection procedure. To handle the inherent uncertainties/imprecision in handwritten characters, Malaviya and
Peters (97) have introduced fuzziness factor in the definition of selected pattern features. The fuzzy features are confined to their meaningfulness with the help of a multistage feature aggregation, which can be combined in a set of linguistic rules that form the fuzzy rule-base for handwritten information recognition. Note that the concept of introducing fuzziness in the definition and extraction of features and in their relations is not new. A detailed discussion is available in Reference 61 and 98 by Pal and others, for extraction of primitives for X-ray identification and character recognition in terms of gentle, fair, and sharp curves. A similar interpretation of the shape parameters of triangle, rectangle, and quadrangle in terms of membership for “approximate isosceles triangles,” “approximate equilateral triangles,” “approximate right triangle,” and so on has also been made (99) for their classification in a color image. However, the work in Reference (97) is significant from the point that it has described many global, positional, and geometral features to account for the variabilities in
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patterns, which are supported with experimental results. To represent the uncertainty in physical relations among the primitives, the production rules of a formal grammar are fuzzified to account for the fuzziness in relation among the primitives, thereby increasing the generative power of a grammar. Such a grammar is called fuzzy grammar (100–102). It has been observed (98) that the incorporation of the element of fuzziness in defining “sharp,” “fair,” and “gentle” curves in the grammars enables one to work with a much smaller number of primitives. By introducing fuzziness in the physical relations among the primitives, it was also possible to use the same set of production rules and nonterminals at each stage, which is expected to reduce, to some extent, the time required for parsing in the sense that parsing needs to be done only once at each stage, unlike the case of the non-fuzzy approach, where each string has to be parsed more than once, in general, at each stage. However, this merit has to be balanced against the fact that the fuzzy grammars are not as simple as the corresponding nonfuzzy grammars. In recent times, the use of fuzzy theory in various kinds of recognition tasks has increased significantly. Complex fuzzy systems have been designed to recognize gestures (103) and describe relative positions in images (104). The authors in Reference (105) have extended the application of fuzzy logic to recognize olfactory (smell) signals. Detecting Man-Made Objects from Remote Sensing Images In a remotely sensed image, the regions (objects) are usually ill-defined because of both grayness and spatial ambiguities. Moreover, the gray value assigned to a particular pixel of a remotely sensed image is the average reflectance of different types of ground covers present in the corresponding pixel area (36.25m–36.25m for the Indian Remote Sensing (IRS) imagery). Therefore, a pixel may represent more than one class with a varying degree of belonging. A multivalued recognition system (6,7) formulated based on the concept of fuzzy sets has been used for detecting curved structure from IRS images (108). The system is capable of handling various imprecise inputs and in providing multiple class choices corresponding to any input. Depending on the geometric complexity (8, 9) and the relative positions of the pattern classes in the feature space, the entire feature space is decomposed into some overlapping regions. The system uses Zadeh’s compositional rule of inference (109) to recognize the samples. The recognition system is initially applied on an IRS image to classify (based on the spectral knowledge of the image) its pixels into six classes corresponding to six land cover types, namely pond water, turbid water, concrete structure, habitation, vegetation, and open space. The green and infrared band information, being sensitive than other band images to discriminate various land cover types, are used for the classification. The clustered images are then processed for detecting the narrow concrete structure curves. These curves include basically the roads and railway tracks. The width of such attributes has an upper bound, which was considered there
to be three pixels for practical reasons. So all the pixels lying on the concrete structure curves with width not more than three pixels were initially considered as the candidate set for the narrow curves. As a result of the low pixel resolutions (36.25m — 36.25m for IRS imagery) of the remotely sensed images, all existing portions of such real curve segments may not be reflected as concrete structures and, as a result, the candidate pixel set may constitute some broken curve segments. To identify the curves in a better extent, a traversal through the candidate pixels was used. Before the traversing process, one also needs to thin the candidate curve patterns so that a unique traversal can be made through the existing curve segments with candidate pixels. Thus, the total procedure to find the narrow concrete structure curves consists of three parts: 1) selecting the candidate pixels for such curves, 2) thinning the candidate curve patterns, and 3) traversing the thinned patterns to make some obvious connections between different isolated curve segments. The multiple choices provided by the classifier in making a decision are used to a great extent in the traversal algorithm. Some of the movements are governed by only the second and combined choices. After the traversal, the noisy curve segments (i.e., with insignificant lengths) are discarded from the curve patterns. The residual curve segments represent the skeleton version of the curve patterns. To complete the curve pattern, the concrete structure pixels lying in the eight neighboring positions corresponding to the pixels on the above obtained narrow curve patterns are now put back. This resultant image represents the narrow concrete structure curves corresponding to an image frame (108). The results are found to agree well with the ground truths. The classification accuracy of the recognition system (107, 108) is not only found to be good, but also its ability of providing multiple choices in making decisions is found to be very effective in detecting the road-like structures from IRS images. Content-Based Image Retrieval (CBIR) In the last few years, researchers have witnessed an upsurge of interest in content-based image retrieval (CBIR), which is a process of selecting similar images from a collection by measuring similarities between the extracted features from images themselves. Real-life images are inherently fuzzy because of several uncertainties developing in the imaging process. Moreover, measuring visual similarities between images highly depends on subjectivity of human perception of image content. As a result, fuzzy image processing for extracting visual features finds an important place in image pretrieval applications. Let us explain here, in brief, an investigation carried out in Reference 110 on image retrieval is based on fuzzy geometrical features. Here a fuzzy edge map is extracted for each image. Using the edge map, a fuzzy compactness vector is computed that is subsequently used for measuring the similarity between the query and the database image. The process involves extracting the possible edge candidates using the concept of Top and Bottom of the intensity surface of a smoothened image. The extracted edge candidates are assigned gradient membership value µm (P)
Fuzzy Image Processing and Recognition
within (0.0 to 1.0) computed from the pixel contrast ratio over a fixed window. The selected points are categorized as weak, medium, and strong edge pixels based on their gradient membership value µm (P). Multilevel thresholding is performed by using (α – cut) to segregate the edge pixels. Fuzzy edge maps snα consisting of different types of edge pixels are obtained from the candidate set sn by varying µm (P), from which the connected subsets snα as shown in Fig. 5(b) and (c) are obtained. Fuzzy compactness value is computed from the fuzzy edge map snα , obtained at different (α – cut) to index an image of the database. snα = {(P ∈ sn : µm (P) ≥ α)}
(32)
where 0.5 ≥ α ≥ 1. This geometrical feature is invariant to rotation translation and scaling by definition. It physically means the maximum area that can be encircled by the perimeter. The similarity between the feature vectors of two images are computed by the widely used Euclidean distance metric. The retrieval results are shown in Fig. 6. From the experimental results of Fig. 6, it is seen that images are retrieved with fairly satisfactory precision. Some other significant work on image retrieval are available in Reference (111–113). The authors in Reference (111) propose an image retrieval system using texture similarity, whereas the authors in Reference (112) present a novel information fusion approach for use in content-based image retrieval. Retrieval of color images has been investigated in Reference (113). Recently, similarity-based online feature selection was applied to bridge the gap between high level semantic concepts and low level visual features in content-based image retrieval (114). Note that all these methods mentioned above use fuzzy theory to handle various kinds of ambiguities. Segmentation of Brain Magnetic Resonance Image Image segmentation is an indispensable process in the visualization of human tissues, particularly during clinical analysis of magnetic resonance (MR) images. A robust segmentation technique based on fuzzy set theory for brain MR images is proposed in Reference (115). The method proposed in Reference (115) is based on a fuzzy measure to threshold the image histogram. The image is thresholded based on a criterion of similarity between gray levels. The second-order fuzzy correlation is used for assessing such a concept. The local information of the given image is extracted through a modified cooccurrence matrix. The technique proposed here consists of two linguistic variables bright, dark modeled by two fuzzy subsets and a fuzzy region on the gray level histogram. Each of the gray levels of the fuzzy region is assigned to both defined subsets one by one and the second-order fuzzy correlation using modified co-occurrence matrix is calculated. First, let us define two linguistic variables dark, bright modeled by two fuzzy subsets of X, denoted by A and B, respectively. The fuzzy subsets A and B are associated with the histogram intervals [xmin , x p ] and [xmax , xq ], respectively, where x p and xg are the final and initial gray-level limits for these subsets, and xmin and xmax are the lowest and highest gray levels of the image, respectively.
13
Next, we calculate CA (xmin : x p ) and CB (x p : xmax ), where CA (xmin : x p ) is the second-order fuzzy correlation of fuzzy subset A and its two-tone version and CB (x p : xmax ) is the second-order fuzzy correlation of fuzzy subset B and its two-tone version using modified co-occurrence matrix. The second-order fuzzy correlation can be expressed in the following way: 4
C(µ1 , µ2 ) = 1 −
L L i=1
j=1
[µ1 (i, j) − µ2 (i, j)]2 ti j Y1 + Y 2
(33)
where tij is the frequency of occurrence of the gray level i followed by j; that is, T = [tij ] is the modified co-occurrence matrix, which is given by
ti j =
a ∈ X,b ∈ ag
δ (1 + ||2 )
(34)
where b ∈ ag = {(m, n − 1), (m, n + 1), (m + 1, n), (m − 1, n), (m − 1, n − 1), (m − 1, n + 1), (m + 1, n − 1), (m + 1, n + 1)} =
1 max{|xm−1,n + xm−1,n+1 + xm,n + xm,n+1 − xm+1,n 4 − xm+1,n+1 − xm+2,n − xm+2,n+1 |, |xm,n−1 + xm,n + xm+1,n−1 + xm+1,n − xm,n+1 − xm,n+2 − xm+1,n+1 − xm+1,n+2 |} 1 δ={
if gray level value of a is i and that of b is j 0 otherwise
and Yk =
L L i=1
[2µk (i, j) − 1]2 ti j ; k = 1, 2
j=1
To calculate correlation between a gray-tone image and its two-tone version, µ2 is considered as the nearest two-tone version of µ1 . That is, µ2 (x) = {
0 if µ2 (x) ≤ 0.5 1 otherwise
(35)
As the key of the proposed method is the comparison of fuzzy correlations, we have to normalize those measures, which is done by computing a normalizing factor α according to the following relation: α=
CA (xmin : x p ) CB (xq : xmax )
(36)
To obtain the segmented version of the gray-level histogram, we add to each of the subsets A and B a gray-level xi picked up from the fuzzy region and form two fuzzy subsets A and B that are associated with the histogram intervals [xmin , xi ] and [xi , xmax ], where x p < xi < xq . Then we calculate CA (xmin : xi ) and CB (xi : xmax ). The ambiguity of the gray value of xi is calculated as follows: A(xi ) = 1 −
|CA (xmin : xi ) − α · CB (xi : xmax )| (1 + α)
(37)
Finally, applying this procedure for all gray levels of the fuzzy region, we calculate the ambiguity of each gray level.
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Figure 5. (a) Original image. Fuzzy edge map for candidates with (b) µm P ≥ 0.6 (c) µm P ≥ 0.8.
Figure 6. Retrieved result (from fuzzy edge map), (a) general purpose database (b) logo retrieval from (USPTO) database, with top left image as the query image.
The process is started with xi = x p + 1, and xi is incremented one by one until xi < xq . In other words, we calculate the ambiguity by observing how the introduction of a gray level xi of the fuzzy region affects the similarity measure among gray levels in each of the modified fuzzy subsets A and B . The ambiguity A is maximum for the gray level xi in which the correlations of two modified fuzzy subsets are equal. The threshold level (T) for segmentation corresponds to gray value with maximum ambiguity A. That is, A(T ) = max arg{A(xi )}; ∀ x p < xi < xq
(38)
As an example, we explain the merits of the proposed method in Figs. (7) and (8). Figure (7) shows the original MR images and their gray-value histograms, whereas Fig. 8 represents the fuzzy second-order correlations CA (xmin : xi ) and CB (xi : xmax ) of two modified fuzzy subsets A and B with respect to the gray level xi of the fuzzy region and the ambiguity of each gray level xi . The value of α is also given here. Figure (8) depicts the segmented image of the proposed method. The thresholds are determined according to the strength of ambiguity.
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Figure 7. Original image and corresponding histogram.
Figure 8. Correlations of two fuzzy subsets, measure of ambiguity and segmented image (proposed).
Other Advances Over the years, applications of fuzzy theory in image processing and recognition has developed extensively in many other domains. Fuzzy morphology is a tool that has received considerable attention among researchers in the field of image processing (116, 117). Bloch (118, 119) used fuzzy theory to define spatial positioning of objects in images. A fuzzy error diffusion method has been proposed in Reference (120) to perform dithering to hide quantization errors in images. Zahlmann et al. (121) applied a hybrid fuzzy image, processing system to assess the damage to the blood vessels in the retina because of diabetis. A fractal coding scheme using a fuzzy image metric has been proposed in Reference (122). Adaptive schemes of digital watermarking in images and videos using fuzzyadaptive resonance theory (fuzzy-ART) classifier has been given in Reference 123. In Reference 124, fuzzy theory has been used to represent the uncertain location of a normal Eucledian point, and its application in doppler image sequence processing has been demonstrated. Fuzzy theory
have also been used in intelligent Web image retrieval purposes (125–127). In Reference 126, an image search engine named (STRICT) has been designed using fuzzy similarity measures. The authors in Reference 127 combine fuzzy text and image retrieval techniques to present a comprehensive image search engine. For an image, the histogram, which gives the frequency (probability) of occurrence of each gray value, and the co-occurrence matrix, which gives the frequency (jointprobability) of occurrence of two gray vajues seperated by a specific distance, are the first- and second-order statistics. In Reference 128, the authors used fuzzy theory to explain the inherent imprecision in the gray values of an image and defined the first- and second-order fuzzy statistics of digital images, namely, fuzzy histogram and fuzzy co-occurrence matrix, respectively. Fuzzy theory has also been used in various other applications such as automatic taxget detection and tracking, stereovision matching, urban structure detection in synthetic aperture radar (SAR) images, and image reconstrution (129–132).
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Fuzzy Image Processing and Recognition
CONCLUSIONS AND DISCUSSION The problem of image processing and recognition under fuzziness and uncertainty has been considered. The role of fuzzy logic in representing and managing the uncertainties in these tasks was explained. Various fuzzy set theoretic tools for measuring information on grayness ambiguity and spatial ambiguity in an image were listed along with their characteristics. Some examples of image processing operations (e.g., segmentation, skeleton extraction, and edge detection), whose outputs are responsible for the overall performance of a recognition (vision) system, were considered to demonstrate the effectiveness of these tools in providing both soft and hard decisions. The significance of retaining the gray information in the form of class membership for a soft decision is evident. Uncertainty in determining a membership function in this regard and the tools for its management were also stated. Finally, a few real-life applications of these methodologies are described. In conclusion, gray information is expensive and informative. Once it is thrown away, there is no way to get it back. Therefore, one should try to retain this information as long as possible throughout the decision-making tasks for its full use. When it is required to make a crisp decision at the highest level, one can always throw away or ignore this informaion. Most of the algorithms and tools described here were developed by the author with his colleagues. Processing of color images has not been considered here. Some significant results on color image information and processing in the notion of fuzzy logic are available in References 133– 137. Note that fuzzy set theory has led to the development of the concept of soft computing as a foundation for the conception and design of a high Machine IQ (MIQ) system. The merits of fuzzy set theory have also been integrated with those of other soft computing tools (e.g., artificial neural networks, genetic algorithms, and rough sets) with a hope of building more efficient processing and recognition systems. ACKNOWLEDGMENT The author acknowledges Mr. Debashis Sen for his assistance in preparing the manuscript. BIBLIOGRAPHY 1. Gonzalez, R. C.; Wintz, P. Digital Image Processing, 2nd ed.; Addison-Wesley: Reading, MA, 1987. 2. Rosenfeld, A.; Kak, A. C. Digital Picture Processing; Academic Press: New York, 1982. 3. Prewitt, J. M. S. Object Enhancement and Extraction, In Picture Processing and Psycho-Pictorics, Lipkin, B. S.;A., Rosenfeld Eds.; Academic Press: New York, 1970, pp 75–149. 4. Fu, K. S. Syntactic Pattern Recognition and Application; Academic Press: London, 1982. 5. Murthy, C. A. On Consistent Estimation of Classes in R2 in the Context of Cluster Analysis;Ph.D Thesis, Indian Statistical Institute, Calcutta, 1988.
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SANKAR K. PAL Machine Intelligence Unit, Indian Statistical Institute, Calcutta, IN
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Wiley Encyclopedia of Electrical and Electronics Engineering Fuzzy Information Retrieval and Databases Standard Article Frederick E. Petry1 1Tulane University, New Orleans, LA Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W3508 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (136K)
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Abstract The sections in this article are Fuzzy Databases Fuzzy Information Retrieval Future Directions About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.
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FUZZY INFORMATION RETRIEVAL AND DATABASES
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predicates:
X θ Y ≡ ⊥ if X or Y is null and θ is ⊥ ∈ S ≡ ⊥ for any set S S ⊇ {⊥} ≡ ⊥ for any set S
FUZZY INFORMATION RETRIEVAL AND DATABASES Information processing and management has become one of the topics that has stimulated great interest over the past several years. The technological advances in databases and retrieval systems and the ability to access such data over the Internet has focused developments in this area. Information systems are designed to model, store, and retrieve large amounts of information effectively. From a developmental point of view, the management of unstructured information (texts), on one hand, and structured information (formatted data representing factual business information), on the other, have given rise to two different lines of research and products: information retrieval systems and database management systems. Being able to naturally handle the imprecision and vagueness that we experience in the real world of information systems is very desirable. Fuzzy set theory has proven to be a very powerful tool to handle this sort of uncertainty in many areas. In information systems, the two main issues in which uncertainty should be reflected are the representation scheme and the querying mechanism; these are discussed here. FUZZY DATABASES The earliest attempt to represent inexact data in databases was the introduction of the concept of null values by Codd (1). The first extensions of the relational data model that incorporated nonhomogeneous domain sets did not use fuzzy set theory. Rather, they attempted to represent null values and intervals. The ANSI/X3/SPARC report of 1975 (2), for instance, notes more than a dozen types of null. At one end of the spectrum, null means completely unknown. For example, a null value in the current salary of an employee could mean the actual value is any one of the permissible values for the salary domain set. Without resorting to fuzzy measures, a user can specify some information about a value that further restricts it. A subset or range of values of the domain set may be described within which the actual attribute value must lie. The user or the system (via functional dependencies) may specify subsets or subranges within which the actual value must not lie. Yet another option is to label null values in a manner that requires distinct nulls in different portions of the database to have a particular actual value relationship (usually equality) if they have the same label. The semantics of the null value range from ‘‘unknown’’ (e.g., the current salary of an employee) to ‘‘not applicable’’ (e.g., subassembly number of a part that is not a subassembly) to ‘‘does not exist’’ (e.g., middle name of a person). These last two meanings, however, are not related to uncertainty. Codd proposes a three-value logic using T, F, and ⬜ (null in the sense of unknown) in conjunction with the following
There is a problem. Because of the variety of meanings possible for null values, they cannot discriminate well enough (i.e., they are ‘‘overloaded’’ in the programming language sense). Two possible solutions are to maintain multiple nulls or to provide semantic interpretation external to the database. Range Values Approach As discussed, it is possible to have a variety of nulls with different semantics. However, these are not adequate to represent the possibility of a range of values. For example, we may not know exactly the age of a house, but we know it is in the range of 20 to 30 years. So we have an interval of values and know one is correct but do not necessarily know exactly which one. An early development in this area by Grant (3) extended the relational model to allow range values. Basically three types of values are allowed: a single number for the case of complete information; a pair of numbers (i, j) [i ⱕ j] for the case of partial information in the form of a range of possible values; and finally a null value in the case of no information. To deal with comparisons of such values for purposes of defining relations and relational operators, true and maybe predicates are defined where the maybe predicate means that it is true or maybe true. For example, consider a relation R with three tuples. For an attribute Years, the values for each tuple are: 15; 8; (20,30). It is definite that 15 僆 R, but it is not certain if 25 is in R, so we have 25 僆M (maybe an element of) R. Note that, by the definition of the maybe predicate, we also have 15 僆M R. The basis of the relational model is set theoretic, so we can view a relation as a set of tuples. In a set there should not normally be duplicate elements, and the issue of elimination of duplicate tuples plays a significant role in inexact and imprecise models of data. For several fuzzy database models, the elimination of redundant tuples requires careful consideration. In the case of a range of values, we can see some of the issues that will arise in the case of fuzzy databases. In particular, duplicate tuples are allowed because, even if they appear to be identical, they may actually stand for different values (i.e., have different interpretations). Consider the possibility of the tuple (20,30) appearing twice in the preceding relation R. In one case, it may stand for the actual value 25 and in the other, 28. If the set of possible interpretations for this range comprises the 11 values: 20, 21, . . ., 30, then there can be at most 11 occurrences of the tuple value (20,30) without violating the ‘‘no duplicate tuples’’ rule. Lipski’s Generalized Approach to Uncertainty Lipski (4) proposed a more general approach. He does not, for instance, assume that null means that a value is completely unknown. Given that there may be labeled or restricted value nulls, let 储Q储 denote all real-world objects that a query Q could represent. Let T be a database object and 储T 储 be all
J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.
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real-world objects it could represent. These are also known as external and internal interpretations. Assume a relation EMPLOYEE with domains NAME and AGE. The database object T ⫽ [Bob 30–35] could represent six real-world objects (one for each year in the age range). A query Q places each database object in one of three categories.
T ∈ {surely set} T ∈ {possible set} T ∈ {eliminated set}
if Q ⊃ T if Q ∩ T = if Q ∩ T =
For instance, the query, EMPLOYEE [AGE ⬎ 32], places T in the possible set, while EMPLOYEE [AGE ⬎ 25] EMPLOYEE [AGE ⬍ 40] places T in the surely set. The first two categories are also known as the lower value 储Q储* and upper value 储Q储*, and these limiting interpretations are characterized in this approach. A number of relationships that assist in evaluating this sort of query have been developed. It should be noted that because the representation of inexact data is sufficiently generalized, it becomes intimately related to the uncertainty data modeling using fuzzy sets, which we will be describing shortly. Statistical and Probabilistic Databases The main work in the area of statistical approaches is that of Wong (5) in which he handles a large class of uncertainty cases by statistical inference. This formulation approaches the uncertainty of the real-world data by assuming an ideal world of perfect information to which the incomplete data may be statistically compared. The prior information from this comparison is represented either as a distortion function or a conditional distribution. Missing and combined attributes can be dealt with by distortion functions. The more direct method of dealing with uncertainty and incompleteness is to specifically use a probabilistic data model, and the most completely developed approach is that in which probabilities are associated with the values of the attributes (6). In this model, because each stochastic attribute is treated as a discrete probability distribution function, the probabilities for each attribute (in a tuple) must be normalized (sum to 1.0). However, it may be difficult to ascertain exact probabilities for all possible domain values. As a result, they developed the concept of missing probabilities to account for such incompletely specified probability distributions. It permits the probabilistic model to capture uncertainty in data values as well as in the probabilities. When updating or entering data into a probabilistic relation, it is not necessary to have all information before some tuple can be entered, allowing a natural use of such uncertain information. Fuzzy Databases Models of Imprecision The relational model has been the dominant database model for a considerable period of time, and so it was naturally used by researchers to introduce fuzzy set theory into databases. Much of the work in the area has been in extending the basic model and query languages to permit the representation and retrieval of imprecise data. A number of related issues such as functional dependencies, security, and implementation considerations have also been investigated (7).
Two major approaches have been proposed for the introduction of fuzziness in the relational model. The first one uses the principle of replacing the ordinary equivalence among domain values by measures of nearness such as similarity relationships (8), proximity relationships (9), and distinguishability functions (10). The second major effort involves a variety of approaches that directly use possibility distributions for attribute values (11,12). There have also been some mixed models combining these approaches (13,14). We can also characterize these approaches relative to their extensions of the relational model. As we have seen in capturing incompleteness or uncertainty, it is necessary to extend the basic relational model by using non-first normal forms. In the first approach using nearness measures, the imprecision of the actual data values is implicit, using a separate relation or table for the similarity or proximity relationship. Generally with the use of possibility distributions, most approaches have some imprecise description of the data explicitly or directly represented in the basic attribute values of the relation. We characterize these approaches as being either homogeneous or heterogeneous representations. The distinguishing characteristic of an ordinary relational database (or ordinary databases of other forms) is the uniformity or homogeneity of the represented data (15). For each domain, there is a prescribed set of values from which domain values may be selected. Furthermore, each element of the domain set is of the same structure (e.g., integers, real numbers, or character strings). With the use of similarity or proximity relationships, the imprecision in domain values is implicit, and so the representation remains homogeneous. These approaches are thus closer to ordinary crisp relational models and can be shown to have properties that closely follow those of conventional relational models. To more directly represent uncertainty within the domain values themselves requires departure from homogeneity of representation. These models based on possibility theory provide the ability to model more forms of uncertainty. As would be expected from the increased power of representation, there is a tradeoff in more complexity of implementation. The more complex extensions of the basic relational model lead us to classify them using a heterogeneous representation. This is just a matter of degree, and some approaches may be more heterogeneous than others. Membership Values Models. The simplest form for a fuzzy database is the attachment of a membership value (numeric or linguistic) to each tuple. This permits maintenance of homogeneous data domains and strongly typed data sets. However, the semantic content of the fuzzy membership domain is used during query processing. We will consider examples that illustrate two distinct semantics for the membership domain. In the first relation, Investment_Sites, we have tuples with attributes of [site-id, classification, membership value]: 兵[12, residential-1, 1.0], [14, residential-2, 0.7], [79, light-commercial, 0.85], . . .其. The membership value here denotes the degree to which the tuple belongs within the relation (16). The second example is the relation Resume_Analysis, which represents the analysis criteria of potential employees: 兵[physics, science, 1.0], [botany, science, 0.7], [statistics, analysis, 0.8], . . .其. In the relation, the membership value denotes the strength of the dependency between the key attribute, Subject, and the attribute Classification (17).
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Similarity-Based Fuzzy Models. In the late 1970s, Buckles and Petry (8) were the first to use similarity relationships in a relational model. Their approach attempted to generalize the concept of null and multiple-valued domains for implementation within an operational environment consistent with the relational algebra. In fact, the nonfuzzy relational database is a special case of their fuzzy relational database approach. For each domain j in a relational database, a domain base set Dj is understood. Domains for fuzzy relational databases are either discrete scalars or discrete numbers drawn from either a finite or infinite set. An example of a finite scalar domain is a set of linguistic terms. For example, consider a set of terms that can be used for subjective evaluation of a patient’s health: 兵critical, severe, poor, so-so, average, good, excellent其. The fuzzy model uses a similarity relationship to allow the comparison of these linguistic terms. The domain values of a particular tuple may also be single scalars or numbers (including null) or a sequence of scalars or numbers. Consider, for example, the assessments made in the triage database to permit ranking of patient treatment. If we include linguistic descriptions of the severity of patients and combine these with procedure time estimates, we have tuples in the relation such as: 兵[p1, 兵so-so, average其, 兵20, 30其], [p2, poor, 兵20, 50其], [p3, 兵poor, severe其, 兵80–120其], . . .其 The identity relation used in nonfuzzy relational databases induces equivalence classes (most frequently singleton sets) over a domain D, which affects the results of certain operations and the removal of redundant tuples. The identity relation is replaced in this fuzzy relational database by an explicitly declared similarity relation (18) of which the identity relation is a special case. A similarity relation s(x, y) for given domain D is a mapping of every pair of elements in the domain onto the unit interval [0, 1] with the following three properties, x, y, z 僆 D: 1. Reflexive: sD(x, x) ⫽ 1 2. Symmetric: sD(x, y) ⫽ sD(y, x) 3. Transitive: sD(x, z) ⱖ Max(Min[sD(x, y), sD(y, z)]) Next the basic concepts of fuzzy tuples and interpretations must be described. A key aspect of most fuzzy relational databases is that domain values need not be atomic. A domain value di, where i is the index of the attribute in the tuple, is defined to be a subset of its domain base set Di. That is, any member of the power set may be a domain value except the null set. Let P(Di) denote the power set of Di ⫺ . A fuzzy relation R is a subset of the set cross product P(D1) ⫻ P(D2) ⫻ ⭈ ⭈ ⭈ ⫻ P(Dm). Membership in a specific relation r is determined by the underlying semantics of the relation. For instance, if D1 is the set of major cities and D2 is the set of countries, then (Paris, Belgium) 僆 P(D1) ⫻ P(D2)—but it is not a member of the relation A (capital-city, country). A fuzzy tuple t is any member of both r and P(D1) ⫻ P(D2) ⫻ ⭈ ⭈ ⭈ ⫻ P(Dm). An arbitrary tuple is of the form ti ⫽ [di1, di2, . . ., dim] where dij 債 Dj. An interpretation 움 ⫽ [a1, a2, . . ., am] of a tuple ti ⫽ [di1, di2, . . ., dim] is any value assignment such that aj 僆 dij for all j. In summary, the space of interpretations is the set cross product D1 ⫻ D2 ⫻ ⭈ ⭈ ⭈ ⫻ Dm. However, for any particular
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relation, the space is limited by the set of valid tuples. Valid tuples are determined by an underlying semantics of the relation. Note that in an ordinary relational database, a tuple is equivalent to its interpretation. Some aspects of the max-min transitivity in a similarity can cause difficulty in modeling the relationship between domain elements. It can be difficult to formulate the transitive property of the relationship correctly. Furthermore at some 움 level, domain elements only weakly related can be forced together in a merged set of retrieved values. The essential characteristic that produces the desirable properties of uniqueness and well-defined operations is partitioning of the attribute domains by the similarity relationship. Shenoi and Melton (9) show how to use proximity relations (nontransitive) for the generation of partitions of domains. The fuzzy relational model is extended by replacing similarity relations with proximity relations on the scalar domains. Recall that a proximity relation P(x, y) is reflexive and symmetric but not necessarily transitive. This can also be related to a more generalized approach to equivalence relations for a fuzzy database model (19). Possibility Theory-Based Database Models. In the possibility theory-based approach (11,20), the available information about the value of a single-valued attribute A for a tuple t is represented by a possibility distribution 앟A(t) on D 傼 兵e其 where D is the domain of the attribute A and e is an extra-element that stands for the case when the attribute does not apply to t. The possibility distribution 앟A(t) can be viewed as a fuzzy restriction of the possible value of A(t) and defines a mapping from D 傼 兵e其 to [0, 1]. For example, the information ‘‘Paul has considerable experience’’ (앟e(p)) will be represented by (᭙d 僆 D): πe( p) (e) = 0 and πe( p) (d) = µc (d) Here 애c is a membership function that represents the vague predicate ‘‘considerable’’ in a given context, such as the number of years of experience or the number of years of education. It is important to notice that the values restricted by a possibility distribution are considered as mutually exclusive. The degree 앟A(t)(d) rates the possibility that d 僆 D is the correct value of the attribute A for the tuple t. Note that 앟A(t)(d) ⫽ 1 only means that d is a completely possible value for A(t), but it does not mean that it is certain that d is the value of A for the tuple (or in other words that d is necessarily the value of A for t), unless ∀d = d, πA(t ) (d ) = 0 Moreover, the possibility distribution 앟A(t) should be normalized on D 傼 兵e其 (i.e., ᭚d 僆 D such that 앟A(t)(d) ⫽ 1 or 앟A(t)(e) ⫽ 1). This means that it must be the case that at least one value of the attribute domain is completely possible or that the attribute does not apply. The following null value situations may be handled in this framework: 1. Value of A for t is completely unknown: ᭙d 僆 D, 앟A(t)(d) ⫽ 1, 앟A(t)(e) ⫽ 0. 2. The attribute A does not apply for the tuple t: ᭙d 僆 D, 앟A(t)(d) ⫽ 0, 앟A(t)(e) ⫽ 1.
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3. It is not clear whether situation 1 or 2 applies: ᭙d 僆 D, 앟A(t)(d) ⫽ 1, and 앟A(t)(e) ⫽ 1.
respect to a relation r having domain sets D1, D2, . . ., Dm, each factor Vj must be
Thus, such an approach is able to represent, in a unified manner, precise values (represented by singletons), null values, and ill-known values (imprecise ones represented by crisp sets or vague ones represented by fuzzy sets). In this approach, multiple-valued attributes can be formally dealt with in the same manner as single-valued ones, provided that possibility distributions defined on the power set of the attribute domains rather than on the attribute domains themselves are used. Indeed, in the case of multiple-valued attributes, the mutually exclusive possibilities are represented by subsets of values.
1. a domain element a, a 僆 Dj, where Dj is a domain set for r, or 2. a domain element modified by one or more linguistic modifiers (e.g., NOT, VERY, MORE-OR-LESS).
Possibility and Necessity Measures. If two values a and b are described by their respective possibility distributions 앟a and 앟b, then they can be compared according to the extension principle (21). This leads to two degrees, expressing the extent to which the values possibly and necessarily satisfy the comparison relation. For equality, these degrees are given by
poss(a = b) = supx,y (min(πa (x), πb ( y), µ = (x, y))) nec(a = b) = 1 − supx,y (min(πa (x), πb ( y), µ = (x, y))) = infx,y (max(1 − πa (x), 1 − πb ( y), µ = (x, y))) Of course, when a and b are precisely known, these two degrees collapse (and take their value in 兵0, 1其) because there is no uncertainty. Otherwise, the fact that two attribute values (in the same tuple or in two distinct tuples) are represented by the same possibility distribution does not imply that these values must be equal. For instance, if John’s experience is ‘‘considerable’’ and Paul’s experience is also ‘‘considerable,’’ John and Paul may still have different amounts (e.g., years) of experience. This point is just a generalization of what happens with null values (if John’s experience and Paul’s experience are completely unknown, both are represented by a null value, whatever its internal representation, even though their years of experience are potentially distinct). The equality of two incompletely known values must be made explicit and could be handled in the relational model in extending the notion of marked nulls. Querying Fuzzy Relational Databases In systems that are relationally structured and use fuzzy set concepts, nearly all developments have considered various extensions of the relational algebra. Its syntactic structure is modified to the extent that additional specifications are required. Use of the relational calculus with a similarity model has also been studied (22). The relational calculus provides a nonprocedural specification for a query and can be extended more easily to a higher-level query language. Similarity-Based Querying. To illustrate the process of query evaluation for similarity databases, we examine a generalized form of Boolean queries that may also be used to retrieve information (23). The details of query evaluation can be seen more easily in this sort of query. A query Q (ai, ah, . . ., ak) is an expression of one or more factors combined by disjunctive or conjunctive Boolean operators: Vi op Vh op ⭈ ⭈ ⭈ op Vk. In order to be well formed with
The relation r may be one of the original database relations or one obtained as a result of a series of fuzzy relational algebra operations. Fuzzy semantics apply to both operators and modifiers. An example query is MORE-OR-LESS big and NOT VERY VERY heavy where ‘‘big’’ is an abbreviation of the term (SIZE ⫽ big) in a relation having domain called SIZE. The value ‘‘heavy’’ is likewise an abbreviation. The linguistic hedge VERY can be interpreted as CON(F), concentration, and MORE-OR-LESS as DIL(F), dilation. A membership value of a tuple in a response relation r is assigned according to the possibility of its matching the query specifications. Let a 僆 Dj be an arbitrary element. The membership value 애a(b), b 僆 Dj, is defined based on the similarity relation sj(a, b) over the domain. The query Q( ⭈ ) induces a membership value 애Q(t) for a tuple t in the response r as follows: 1. Each interpretation I ⫽ [a⬘1, a⬘2, . . ., a⬘m] of t determines a value 애aj (a⬘j ) for each domain element aj, of Q (ai, ah, . . ., ak). 2. Evaluation of the modifiers and operators in Q( ⭈ ) over the membership values 애aj (a⬘j ) yields 애Q(I), the membership value of the interpretation with respect to the query. 3. Finally, 애Q(t) ⫽ maxI of t兵애Q(I)其. In short, the membership value of a tuple represents the best matching interpretation. The response relation is then the set of tuples having nonzero membership values. In practice, it may be more realistic to consider only the tuple with the highest value. Possibility-Based Framework for Querying. There are several approaches for querying relational databases where some incompletely known attribute values are represented by possibility distributions. One may distinguish between an approach that is set in a pure possibilistic framework (11) (approximate reasoning under uncertainty) and others that do not use such a strict theoretic framework (24–26). According to the possibilistic view (11), when a condition applies to imperfectly known data, the result of a query evaluation can no longer be a single value. Because the precise values of some attributes for some items are not known, the fact that these items do or do not satisfy the query (to some degree) may be uncertain. This is why the two degrees attached to two points of view are used: the extent to which it is possible (resp. certain) that the condition is satisfied. From the possibility distributions 앟A(t) and a subset P (ordinary or fuzzy), one can compute the fuzzy set ⌸P (resp. NP) of the items whose A-value possibly (resp. necessarily) satis-
FUZZY INFORMATION RETRIEVAL AND DATABASES
fies the condition P. The membership degrees of a tuple t to ⌸P and NP are, respectively, given by (27)
µP (t) = (P; A(t)) = supd∈D min(µP (d), πA(t ) (d)) µNP (t) = N(P; A(t)) = 1 − (P; A(t)) = 1 − supd∈D∪{e} min(µP (d), πA(t ) (d)) = infd∈D∪{e} max(µP (d), 1 − πA(t ) (d)) ⌸(P; A(t)) estimates to what extent at least one value restricted by 앟A(t) is compatible with P, and N(P; A(t)) estimates to what extent all the values more or less possible for A(t) are included in P. It can be shown that ⌸P and NP always satisfy the inclusion relation ⌸P 傶 NP (i.e., ᭙t, 애NP(t) ⱕ 애⌸P(t)), provided that 앟A(t) is normalized. If John’s age and the fuzzy predicate ‘‘middle-aged’’ are represented according to a possibility distribution, the evaluation of the condition: John’s age ⫽ ‘‘middle-aged’’ is based on the computation of the values: min(π ja (u), µma (u)) and max(1 − π ja (u), µma (u)) Thus, in case of incomplete information, it is possible to compute the set of items that more or less possibly satisfy an elementary condition and to distinguish the items that more or less certainly satisfy this condition. FUZZY INFORMATION RETRIEVAL Information retrieval systems (IRS) are concerned with the representation, storage, and accessing of a set of documents. These documents are often in the form of textual information items or records of variable length and format, such as books and journal articles (28). The specific aim of an IRS is to evaluate users’ queries for information based on a content analysis of the documents stored in the archive. In response to a user query, the IRS must identify what documents deal with the information being requested via the query and retrieve those that satisfy the query. Fuzzy IR models have been defined to overcome the limitations of the crisp Boolean IR model so as to deal with 1. discriminated (and possibly ranked) answers reflecting the variable relevance of the documents with respect to queries 2. imprecision and incompleteness in characterizing the information content of documents 3. vagueness and incompleteness in the formulation of queries Fuzzy extended Boolean models constitute a superstructure of the Boolean model by means of which existing Boolean IRSs can be extended without redesigning them completely. The softening of the retrieval activity in order to rank the retrieved items in decreasing order of their presumed relevance to a user query can greatly improve the effectiveness of such systems. This objective has been approached by extending the Boolean models at various levels. 1. Fuzzy extension of document representation—The aim here is to provide more specific and exhaustive repre-
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sentations of the documents’ information content in order to lower the imprecision and incompleteness of the Boolean indexing. This is done by incorporating significance degrees, or index term weights, in the representation of documents (29). 2. Fuzzy generalization of Boolean query language—The objective here is to render the query language more expressive and natural than crisp Boolean expressions in order to capture the vagueness of user needs as well as simplify the user system interaction. This is carried out at two levels. The first is through the definition of more expressive, as well as soft, selection criteria that allow the specification of different importance levels of the search terms. Query languages based on numeric query term weights with different semantics have been presented as an aid to define more expressive selection criteria (30,31). Also, an evolution of these approaches introduced linguistic query weights specified by fuzzy variables (e.g., important or very important) to express different levels of importance for query terms (32). Incorporating fuzzy representations for documents in a Boolean IRS is a sufficient condition to improve the system with the ranking ability. As a consequence of this extension, the exact matching applied by a Boolean system can be softened to a partial matching mechanism, evaluating, for each document, the anticipated degree of satisfaction of the document with regard to a user’s query. The value thus generated is called a retrieval status value (RSV) and is used as the basis for ranking the documents. This ranking is used for retrieval and display of those documents. Fuzzy knowledge-based IRS models have been defined to index and retrieve documents in specific subject areas. To date, it has been found that IRSs are not adequate to deal with general collections. Reference 33 uses rules to represent semantic links between concepts; the nature of the links (e.g., synonymous terms, broader terms, narrower terms) and the strength of the links (represented by weights) are stored in the knowledge base and are defined by experts in the field. This is used to expand the query evaluation, by applying an inference process that allows one to find information that the user did not explicitly request but that is deemed ‘‘likely’’ to be of interest. Fuzzy Indexing Procedures In an information retrieval system, the generation of a representation of each document’s subject content is called indexing. The basic problem is to capture and synthesize the meaning of a document written in natural language. In defining an indexing procedure (which can be either manual or automatic), one must first consider retrieval performance, via a document representation that allows the IRS to be able to retrieve all the relevant documents and none of the nonrelevant documents in response to a user query and then also consider exhaustivity (describing fully all aspects of a document’s contents). The Boolean retrieval model can be associated with automatic text indexing. This model provides a crisp representation of the information content of a document. A document is
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formally represented by the set of its index terms: R(d) = {t t ∈ T, F (d, t) > 0} for d ∈ D in which the indexing (membership) function F correlating terms and documents is restricted to 兵0, 1其. Of course, F(d, t) ⫽ 1 implies the presence of term t in document d; and F(d, t) ⫽ 0 implies the absence of the term in the document. To improve the Boolean retrieval with a ranking ability, the Boolean representation has been extended within fuzzy set theory by allowing the indexing function F to take on values in the unit interval [0, 1]. Here, the index term weight F(d, t) represents the degree of significance of the concept as represented by term t in document d. This value can be specified between no significance [F(d, t) ⫽ 0] and full significance [F(d, t) ⫽ 1] and allows a ranking of the retrieval output, providing improved user satisfaction and system performance. Consequently, a document is represented as a fuzzy set of terms R(d) = {t, µd(t ) t ∈ T}
for d ∈ D
in which 애d(t) ⫽ F(d, t). This implies that F is a fuzzy set membership function, measuring the degree to which term t belongs to document d (34). Through this extension, the retrieval mechanism can compute the estimated relevance of each document relative to the query, expressed by a numeric score called a retrieval status value. The RSV denotes how well a document seemingly satisfies the query (35,36). The definition of the criteria for an automatic computation of F(d, t) is a crucial aspect; generally this value is defined on the basis of statistical measurements with the aim of optimizing retrieval performance. Fuzzy Associations. Another concept linked to automatic indexing to enhance the retrieval of documents is that based on fuzzy associations, named fuzzy associative information retrieval models (37–40). These associative information retrieval models work by retrieving additional documents that are not indexed by the terms in a given query but are indexed by other terms, associated descriptors, that are related to the query terms. Fuzzy association in information retrieval generally refers to the use of fuzzy thesauri where the links between terms are weighted to indicate strength of association. Moreover, this notion includes generalizations such as fuzzy pseudothesauri (41) and fuzzy associations based on a citation index (42). Ogawa et al. (43) propose a keyword connection matrix to represent similarities between keywords so as to reduce the difference between relationship values initially assigned using statistical information and a user’s evaluation. Generally, a fuzzy association between two sets X ⫽ 兵x1, . . ., xm其 and Y ⫽ 兵y1, . . ., yn其 is formally defined as a fuzzy relation: f : X × Y → [0, 1] By varying the semantics of the sets X and Y in information retrieval, different kinds of fuzzy associations can be derived.
Fuzzy Querying Two factors have been independently taken into account to extend the Boolean query language, making the selection criteria more powerful, and softening and enriching the aggregation operators. First, consider the basic query processing model. The main aim in extending the selection criteria is to provide users with the possibility of specifying differing importances of terms in order to determine which documents should be relevant. This has been achieved by preserving the Boolean structure of the query language and by associating with each term a numeric value to synthesize importance. Now, let’s define for Q ⫽ 兵a set of user queries for document其, a(q, t): Q ⫻ T 씮 [0, 1], where a(q, t) is the importance of term t in describing the query q and is called a query term weight. It is here that one begins to introduce problems in terms of maintaining the Boolean lattice (44). Because of that, certain mathematical properties can be imposed on F, but more directly on a and on the matching procedure. Moreover, there is a problem in developing a mathematical model that will preserve the semantics (i.e., the meaning) of the user query. The weight a can be interpreted as an importance weight, as a threshold, or as a description of the ‘‘perfect’’ document. Let g: [0, 1] ⫻ [0, 1] 씮 [0, 1] [i.e., g(F, a) is the RSV for a query q of one term t, with query weight a, with respect to a given document d, which has index term weight F(d, t) for the same term t]. This function g can be interpreted as the evaluation of the document in question along the dimension of the term t if the actual query has more than one term. It has been suggested that terms be evaluated from the bottom up, evaluating a given document against each term in the query and then combining those evaluations according to the query structure (45). Reference 46 shows that this criterion for a g function, called separability (47), preserves a homomorphism between the document evaluations for singleterm queries and the document evaluations for complex Boolean queries. A first formulation of the g function treats the a values as relative importance weights; for example, one could specify g ⫽ F*a. However, this can lead to problems, such as when using an AND (44). In this case, a very small value of a for one of the terms in an AND query will dominate the min function and force a decision based on the least important (smallest a) term, which is just the opposite of what is desired by the user. This problem is precisely what prompted some researchers to consider g functions that violate separability (31,48). To achieve consistency in the formalization of weighted Boolean queries, some approaches do not maintain all the properties of the Boolean lattice: Kantor (49) generates a mathematical formulation of the logical relationships between weighted queries, using a vapid query with all zero weights. FUTURE DIRECTIONS Several specialized aspects not covered in this article are of increasing research importance. Fuzzy functional dependencies relate to several issues for fuzzy databases including database design and integrity management (50,51). The actual
FUZZY INFORMATION RETRIEVAL AND DATABASES
application of uncertainty in deployed database systems is following two directions. The first is the addition of uncertainty in object oriented databases (52,53). This is due to newer developments in object-oriented databases and their inherent capabilities such as encapsulated methods. Another direction is that of fuzzy-front end querying (54,55). This approach allows a general use with existing databases and also permits fuzzy querying of crisp data. A good general survey of some of the issues in these directions is (56).
BIBLIOGRAPHY 1. E. Codd, Extending the database relational model to capture more meaning, ACM Trans. Database Sys., 4 (2): 156–174, 1979. 2. Anonymous, American National Standards Institute study group on database management: Interim report, ACM SIGMOD Rec., 7: 25–56, 1975. 3. J. Grant, Incomplete information in a relational database, Fundamenta Informaticae, 3 (4): 363–378, 1980. 4. W. Lipski, On semantic issues connected with incomplete information databases, ACM Trans. Database Syst., 4 (3): 262–296, 1979. 5. E. Wong, A statistical approach to incomplete information in database systems, ACM Trans. Database Systems, 7 (4): 479–488, 1982. 6. D. Barbara, H. Garcia-Molina, and D. Porter, The management of probabilistic data, IEEE Trans. Knowl. Data Eng., 4 (4): 487– 502, 1992. 7. F. Petry, Fuzzy Databases: Principles and Applications, Boston: Kluwer, 1996. 8. B. Buckles and F. Petry, A fuzzy model for relational databases, Fuzzy Sets and Systems, 7 (3): 213–226, 1982. 9. S. Shenoi and A. Melton, Proximity relations in fuzzy relational databases, Fuzzy Sets and Systems, 31 (2): 287–296, 1989. 10. M. Anvari and G. Rose, Fuzzy relational databases, in J. Bezdek (ed.), The Analysis of Fuzzy Information Vol. II, Boca Raton FL: CRC Press, 1987, pp. 203–212. 11. H. Prade and C. Testemale, Generalizing database relational algebra for the treatment of incomplete/uncertain information and vague queries, Information Sci., 34 (2): 115–143, 1984.
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20. H. Prade, Lipski’s approach to incomplete information databases restated and generalized in the setting of Zadeh’s possibility theory, Information Systems, 9 (1): 27–42, 1984. 21. L. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1 (1): 3–28, 1978. 22. B. Buckles, F. Petry, and H. Sachar, A domain calculus for fuzzy relational databases, Fuzzy Sets and Systems, 29 (4): 327–340, 1989. 23. B. Buckles and F. Petry, Query languages for fuzzy databases, in J. Kacprzyk and R. Yager (eds.), Management Decision Support Systems Using Fuzzy Sets and Possibility Theory, Koln GR: Verlag TUV Rheinland, 1985, pp. 241–252. 24. M. Umano, Retrieval from fuzzy database by fuzzy relational algebra, in E. Sanchez and M. Gupta (eds.), Fuzzy Information, Knowledge Representation and Decision Analysis, New York: Pergamon, 1983, pp. 1–6. 25. Y. Takahashi, A fuzzy query language for relational database, IEEE Trans. Syst. Man. Cybern., 21 (6): 1576–1579, 1991. 26. H. Nakajima, T. Sogoh, and M. Arao, Fuzzy database language and library—fuzzy extension to SQL, Proc. Second International Conference on Fuzzy Systems, Los Alamitos, CA: IEEE Computer Society Press, 1993, pp. 477–482. 27. D. Dubois and H. Prade (with the collaboration of H. Farreny, R. Martin-Clouaire, and C. Testemale), Possibility Theory: An Approach to Computerized Processing of Uncertainty, New York: Plenum, 1988. 28. G. Salton and M. J. McGill, Introduction to Modern Information Retrieval, New York: McGraw-Hill, 1983. 29. T. Radecki, Fuzzy set theoretical approach to document retrieval, Information Processing and Management, 15 (5): 247–260, 1979. 30. D. A. Buell and D. H. Kraft, A model for a weighted retrieval system, J. Amer. Soc. Information Sci., 32 (3): 211–216, 1981. 31. A. Bookstein, Fuzzy requests: An approach to weighted Boolean searches, J. Amer. Soc. Information Sci., 31 (4): 240–247, 1980. 32. G. Bordogna and G. Pasi, A fuzzy linguistic approach generalizing Boolean information retrieval: A model and its evaluation, J. Amer. Soc. Information Sci., 44 (2): 70–82, 1993. 33. D. Lucarella, Uncertainty in information retrieval: An approach based on fuzzy sets, Ninth Annual Int. Phoenix Conference on Computers and Communications, Los Alamitos CA: IEEE Computer Society Press, 1990, pp. 809–814.
12. M. Zemankova and A. Kandel, Implementing imprecision in information systems, Information Sci., 37 (1): 107–141, 1985.
34. L. J. Mazlack and L. Wonboo, Identifying the most effective reasoning calculi for a knowledge-based system, IEEE Trans. Syst. Man. Cybern., 23 (5): 404–409, 1993.
13. E. Rundensteiner, L. Hawkes, and W. Bandler, On nearness measures in fuzzy relational data models, Int. J. Approximate Reasoning 3 (4): 267–298, 1989.
35. D. A. Buell, A general model of query processing in information retrieval systems, Information Processing and Management, 17 (5): 236–247, 1981.
14. J. Medina, O. Pons, and M. Vila, Gefred: A generalized model to implement fuzzy relational databases, Information Sci., 47 (5): 234–254, 1994.
36. C. V. Negoita, On the notion of relevance in information retrieval, Kybernetes, 2 (3): 112–121, 1973.
15. B. Buckles and F. Petry, Uncertainty models in information and database systems, J. Information Sci.: Principles and Practice, 11 (1): 77–87, 1985. 16. C. Giardina, Fuzzy databases and fuzzy relational associative processors, Technical Report, Hoboken NJ: Stevens Institute of Technology, 1979. 17. J. Baldwin, Knowledge engineering using a fuzzy relational inference language, Proc IFAC Symp. on Fuzzy Information Knowledge Representation and Decision Analysis, pp. 15–21, 1983. 18. L. Zadeh, Similarity relations and fuzzy orderings, Information Sci., 3 (3): 177–200, 1971. 19. S. Shenoi and A. Melton, An extended version of the fuzzy relational database model, Information Sci., 51 (1): 35–52, 1990.
37. S. Miyamoto, Fuzzy sets in Information Retrieval and Cluster Analysis. Boston: Kluwer, 1990. 38. S. Miyamoto, Two approaches for information retrieval through fuzzy associations, IEEE Trans. Syst. Man. Cybern., 19 (1): 123– 130, 1989. 39. E. Neuwirth and L. Reisinger, Dissimilarity and distance coefficients in automation-supported thesauri, Information Systems, 7 (1): 54–67, 1982. 40. T. Radecki, Mathematical model of information retrieval system based on the concept of fuzzy thesaurus, Information Processing and Management, 12 (5): 298–317, 1976. 41. S. Miyamoto and K. Nakayama, Fuzzy information retrieval based on a fuzzy pseudothesaurus. IEEE Trans. Syst. Man. Cybern., 16 (2): 237–243, 1986.
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42. K. Nomoto et al., A document retrieval system based on citations using fuzzy graphs, Fuzzy Sets and Systems, 38 (2): 191–202, 1990. 43. Y. Ogawa, T. Morita, and K. Kobayashi, A fuzzy document retrieval system using the keyword connection matrix and a learning method, Fuzzy Sets and Systems, 39 (2): 163–179, 1991. 44. D. H. Kraft and D. A. Buell, Fuzzy sets and generalized Boolean retrieval systems, Int. J. Man-Machine Studies, 19 (1): 45–56, 1983. 45. S. C. Carter and D. H. Kraft, A generalization and clarification of the Waller-Kraft wish-list, Information Processing and Management, 25 (1): 15–25, 1989. 46. M. Bartschi, An overview of information retrieval subjects, Computer, 18 (5): 67–74, 1985. 47. W. G. Waller and D. H. Kraft, A mathematical model of a weighted Boolean retrieval system, Information Processing and Management, 15 (3): 235–245, 1979. 48. R. R. Yager, A note on weighted queries in information retrieval systems, J. Amer. Soc. Information Sci., 38 (1): 47–51, 1987. 49. P. B. Kantor, The logic of weighted queries, IEEE Trans. Syst. Man. Cybern., 11 (12): 151–167, 1981. 50. G. Chen, E. Kerre, and J. Vandenbulcke, A computational algorithm for the FFD transitive closure and a complete axiomatization of fuzzy functional dependency, Int. J. of Intelligent Systems, 9 (3): 421–440, 1994. 51. P. Saxena and B. Tyagi, Fuzzy functional dependencies and independencies in extended fuzzy relational database models, Fuzzy Sets and Systems, 69 (1): 65–89, 1995. 52. P. Bosc and O. Pivert, SQLf: A relational database language for fuzzy querying, IEEE Trans. Fuzzy Syst., 3 (1): 1–17, 1995. 53. J. Kacprzyk and S. Zadrozny, FQUERY for ACCESS: Fuzzy querying for Windows-based DBMS, in P. Bosc and J. Kacprzyk (eds.) Fuzziness in Database Management Systems, Heidelberg GR: Physica-Verlag, 1995, pp. 415–435. 54. R. George et al., Uncertainty management issues in the objectoriented data model, IEEE Trans. Fuzzy Syst., 4 (2): 179–192, 1996. 55. V. Cross, R. DeCaluwe, and N. VanGyseghem, A perspective from the Fuzzy Object Data Management Group, Proc. 6th Int. Conf. on Fuzzy Systems, Los Alamitos, CA: IEEE Computer Society Press, 1997, pp. 721–728. 56. V. Cross, Fuzzy information retrieval, J. Intelligent Information Syst., 11 (3): 115–123, 1994.
FREDERICK E. PETRY Tulane University
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Wiley Encyclopedia of Electrical and Electronics Engineering Fuzzy Model Fundamentals Standard Article Fabrizio Russo1 1University of Trieste, Italy Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W3501 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (2041K)
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Abstract The sections in this article are Fuzziness and Uncertainty Fuzzy Sets Arithmetic of Fuzzy Numbers Fuzzy Relations Fuzzy Aggregation Connectives Linguistic Variables and Fuzzy Systems Parameterized Membership Functions About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.
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FUZZY MODEL FUNDAMENTALS The concept of the fuzzy set was introduced in 1965 by Zadeh (1). After this important event, a large number of theoretical contributions were proposed and the formal framework of fuzzy set theory grew fast. For several years, fuzzy models were mainly devoted to specific problems in the areas of pattern recognition and decision-making (2, 3). In the mid-1980s, the successful development of fuzzy controllers opened up new vistas in the application of fuzzy models to engineering problems. Rule-based approaches emerged, in particular, as a powerful and general methodology for information processing. As a result, fuzzy systems became very attractive and the number of applications increased very rapidly in different fields (4–6). During the first half of the 1990s, important relationships with artificial neural networks were established. Fuzzy and neural techniques were presented from a common perspective (7), and new structures able to combine the advantages of fuzzy and neural paradigms were proposed (8–16). Fuzzy set computing is now a well-established problemsolving technology which aims at replacing (or improving) classical methods in a growing number of research and application areas including control systems, pattern recognition, data classification, signal processing, and low-level and high-level computer vision (17–28). The aim of this article is not to provide a thorough description of all concepts of fuzzy models. There is a large body of fuzzy literature devoted to this purpose. This article rather aims at presenting an up-to-date selection of most useful concepts from an electronic engineering perspective. For this reason, theoretical aspects and mathematical formalism will be kept to a minimum.
FUZZINESS AND UNCERTAINTY One of the key features of fuzzy models is their ability to deal with the uncertainty which typically affects physical systems and human activities. Unlike classical methods which resort to a crisp Yes/No approach, fuzzy models adopt a gradual approach which deals with degrees (or grades) of certainty. Let us focus on a simple example. If we observe the object A depicted in Fig. 1, we can easily see that it represents a square. How do we describe the object B in the same figure? It is more or less a square. It does not belong to the (crisp) class of squares, because it possesses round corners. However, it may partially belong to a fuzzy class of squares. Its degree of membership could be, for example, 0.8 (where unity denotes full membership). Conversely, object D is more or less a circle. It does not belong to the (crisp) class of circles, because it possesses straight lines. However, it may belong to a fuzzy class of circles to a certain extent. Depending on their shapes, all objects in Fig. 1 possess degrees of membership to both fuzzy classes. This simple example also highlights the difference between fuzziness and probability. This important subject has been addressed by different authors in the literature (3–17). It suffices here to observe that probability is related to the occurrence of events, whereas fuzziness is not. Again, let us focus on the object B in Fig. 1. A sentence like “Is it
probably a square?” is quite inappropriate to address the uncertainty which affects our process of characterizing the object. The object is not exactly a square. It is more or less a square. Fuzzy concepts represent the basis of human thinking and decision-making. Sentences are very often characterized by vagueness and linguistic imprecision. As an example, if we are driving a car, we could act according to the following statement: “If the speed is low and the vehicle ahead is more or less far away, then moderately increase the speed.” Despite their vague appearance, fuzzy concepts represent a powerful way to condense information about real life. The great success of fuzzy models is the result of combination of the following key features: 1. Effectiveness in representing the knowledge about a problem 2. Effectiveness in processing this knowledge by adopting a numerical framework FUZZY SETS A fuzzy set can be considered a generalization of a classical (“crisp”) set. In classical set theory, the degree of membership of an element to a set is either zero (no membership) or unity (full membership). The membership of an element to a crisp set, say A, is described by the characteristic function χA :
No partial membership is allowed. Fuzzy set theory permits us to deal with partial membership. A fuzzy set F is indeed represented as a set of ordered pairs (2):
where U is the universe of discourse (i.e., the collection of objects where the fuzzy set is defined) and µF (x) is the membership function that maps U to the real interval [0, 1]:
For each element x ∈ U, the function µF (x) yields a real number which represents the degree (or grade) of membership of x to the fuzzy set F (0 ≤ µF (x) ≤ 1). As an example, let us consider the fuzzy set: F = numbers close to 4. A possible membership function µF describing this fuzzy set is represented in Fig. 2. It can be observed that the maximum degree of membership is obtained for x = 4: µF (4) = 1. The closer the number to 4 the more is the membership to F. On the contrary, a number very different from 4 is assigned a low (or zero) membership degree, as it should be. The difference between fuzzy and crisp sets is graphically highlighted in the same figure which shows the characteristic function χA of the crisp set A = real numbers between 3 and 5. According to the “crisp” nature of set A, we observe a hard transition from full membership to no membership and vice versa. As a second example, let U = {0, 1, 2, . . . , 255} be the set of integers ranging from 0 to 255. Such a universe may
J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright © 2007 John Wiley & Sons, Inc.
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Fuzzy Model Fundamentals
Figure 1. Example of crisp and fuzzy classes.
Figure 2. Example of membership and characteristic functions. Figure 3. Example of fuzzy sets dark (DK), medium (MD), and bright (BR).
represent the set of possible gray levels (or luminances) of a digitized image, as represented in Fig. 3 (0 = black, 255 = white). Let us define three fuzzy sets labeled dark (DK), medium (MD), and bright (BR) by means of the membership functions µDK , µMD , and µBR depicted in the same figure. It is worth pointing out that almost all pixel luminances possess a nonzero degree of membership to more than one fuzzy set. For example, if we choose a pixel luminance x = 135 as shown in Fig. 3, we have µDK (x) = 0.08, µMD (x) = 0.91 and µBR (x) = 0.24. The concept of membership function plays a key role in fuzzy modeling. Indeed, properties and operators dealing with fuzzy sets can be easily defined in terms of membership functions. The use of linguistic labels to identify fuzzy sets is also quite common. Linguistic labels are often associated with simple operations which change or modify the “shape” of a fuzzy set. Complement of a Fuzzy Set. The complement F¯ of fuzzy set F is described by the membership function:
The linguistic label which is usually adopted is “NOT.” As an example, the membership function of fuzzy set NOT DARK is represented in Fig. 4. Union of Fuzzy Sets. The union Fun = F1 ∪ F2 of fuzzy sets F1 and F2 is described by the membership function:
The commonly used linguistic label is “OR.” The membership function of fuzzy set DARK OR MEDIUM is shown in Fig. 5. Intersection of Fuzzy Sets. The intersection Fint = F1 ∩ F2 of fuzzy sets F1 and F2 is described by the membership function:
Fuzzy Model Fundamentals
3
in Fig. 7 too. Other fuzzy modifiers can be found in Refs. 2 and 17. We previously used Eq. (2) to generically represent a fuzzy set F. When the universe U is continuous, the following expression is also adopted in the fuzzy literature (17): Figure 4. Complement of fuzzy set DARK.
On the contrary, when U is discrete, fuzzy set F is often expressed in the following form:
Figure 5. Union of fuzzy sets DK and MD.
Figure 6. Intersection of fuzzy sets DK and MD.
Figure 7. Examples of concentration and dilation.
The associated label is “AND.” The membership function of fuzzy set DARK AND MEDIUM is represented in Fig. 6. It should be noted that the above definitions are generalizations of the corresponding definitions for crisp sets. Linguistic Modifiers. Linguistic modifiers (also called linguistic hedges) operate on membership functions in order to modify the meaning of the corresponding fuzzy set. Two popular modifiers are described here. Concentration is a modifier that operates on the membership function of a fuzzy set F in order to decrease values smaller than unity. A commonly used definition (2) is:
(Remember that 0 ≤ µF (x) ≤ 1.) The typical linguistic label is “VERY.” The membership function of fuzzy set VERY DARK is depicted in Fig. 7. Dilation is a modifier that operates on the membership function of a fuzzy set F in order to increase values smaller than unity. A typical definition is yielded by the following relationship:
The associated label is “MORE OR LESS.” The membership function of fuzzy set MORE OR LESS DARK is represented
Of course, integral and summation symbols in the above expressions do not mean integration and arithmetic addition. They are used to denote the collection of all elements x ∈ U. The slash symbol is also typically adopted to associate x with the corresponding degree of membership. Let us introduce some specific terminology. Support. The support of a fuzzy set F on the universe U is the crisp set S(F) formed by the elements having nonzero degree of membership:
Crossover Point. The crossover point of a fuzzy set F is an element xc with membership degree µF (xc ) = 0.5. Fuzzy Singleton. A fuzzy singleton is a fuzzy set whose support is a single element x with µF (x) = 1. Normal Fuzzy Set. A fuzzy set F is said to be normal if maxx∈U {µF (x)} = 1. α-Level Set. The α-level set (α-cut) of fuzzy set F is the crisp set defined by the following relationship (2):
The strong α-level set is defined as:
A more general definition resorts to the concept of αlevel set. A fuzzy set is convex if all its α-level sets are convex (as crisp sets). Convex Fuzzy Set. A fuzzy set F is said to be convex (4) if its support is a set of real numbers and the following relation applies for all x ∈ [x1 , x2 ] over any interval [x1 , x2 ]:
Extension Principle. The extension principle is commonly used to generalize crisp mathematical concepts to fuzzy sets (2). Let F be a fuzzy set on U and let y = f(x) denote a function from U to V (f:U → V). By extending the function f, the fuzzy set f(F) of V is defined as follows (4):
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Fuzzy Model Fundamentals
Figure 9. Example of extended addition.
Figure 8. Example of a fuzzy relation.
The fuzzy set f(F) is also expressed by
A simple example is depicted in Fig. 8.
ARITHMETIC OF FUZZY NUMBERS Fuzzy Numbers. A fuzzy number is a normal and convex fuzzy set such that (2): 1. 2.
Only one element (called the mean value) has membership degree equal to unity. Its membership function is piecewise continuous.
In practice the above definition is often modified in order to include trapezoid-shaped fuzzy sets. Fuzzy arithmetic resorts to the extension principle in order to extend algebraic operations from crisp to fuzzy numbers. Since computational efficiency is an element of paramount importance for many applications, a simplified representation of a fuzzy number, called “LR representation,” is often adopted. A fuzzy number is of LR type (2) if its membership function is defined by means of two reference functions L (left) and R (right):
Figure 10. (a) Signal corrupted by impulse noise. (b) Result of fuzzy filtering.
A fuzzy interval is symbolically denoted by (xm , x m , α, β)LR . As mentioned above, the extension principle is used to extend some algebraic operations to fuzzy numbers. Let F1 and F2 be two fuzzy numbers of LR type: F1 = (xm1 , α1 , β1 )LR , F2 = (xm2 , α2 , β2 )LR . The following relations can be used to define extended addition and subtraction (2)
where xm is the mean value and α(α > 0) and β(β > 0) are called the left and right spreads, respectively. A fuzzy number of LR type is symbolically denoted by (xm , α, β)LR . The choice of functions L(u) and R(u) depends on the context. A fuzzy interval of LR type is very similarly defined by the membership function: As an example, let us consider the fuzzy numbers: “about 5” = (5, 3, 3)LR and “about 10” = (10, 3, 3)LR . In order to give fuzzy numbers a triangular shape, we adopt the following reference functions: L(u) = R(u) = max{0, 1 − u}. The result yielded by the extended addition (5, 3, 3)LR + (10, 3, 3)LR = (15, 6, 6)LR is depicted in Fig. 9.
Fuzzy Model Fundamentals
FUZZY RELATIONS A fuzzy relation R between sets U and V is a fuzzy set characterized by a membership function µR :U × V → [0, 1] and is expressed by (2):
As an example, let U = V be a set of real numbers. The relation: “x is much larger than y” can be described by the membership function:
If sets U and V represent finite sets U = {x1 , x2 , . . . , xm } and V = {y1 , y2 , . . . , yn }, a fuzzy relation R can be described by an m × n matrix (4):
where ai,j = µR (xi , yj ) represents the strength of association between a pair of elements.
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the real interval [0, 1]. Fuzzy aggregation connectives can be grouped into the following classes (13,16,29): 1. Union connectives 2. Intersection connectives 3. Compensative connectives Union Connectives The simplest aggregation connective of union type is the mentioned “Max” operator. A useful generalization is represented by the family of union aggregators defined by Yager (30):
It can be observed that limp→∞ yU (µ1 , µ2 , . . . , µn ) = max(µ1 , µ2 , . . . , µn ). Thus, the range of this connective is between max and unity. In this respect, this aggregation connective is more optimistic than the MAX operator (13). By varying the value of parameter p from zero to +∞, different aggregation strategies can be realized. Intersection Connectives
Composition. Let R1 and R2 be two fuzzy relations defined in different product spaces:
The simplest aggregation connective of intersection type is the very popular “min” operator. A useful generalization is represented by the family of intersection aggregators defined by (30)
The above relations can be combined by means of the operation “composition.” A variety of methods have been proposed in the literature (2). For example, we could be interested in combining the relations R1 (patients, symptoms) and R2 (symptoms, diseases) in order to discover relationships between patients and diseases. The so-called max–min composition yields a resulting fuzzy relation described as follows:
It can be observed that limp→∞ yI (µ1 , µ2 , . . . , µn ) = min(µ1 , µ2 , . . . , µn ). Thus, the range of this connective is between min and zero. This aggregation connective is more pessimistic than the min operator. As in the previous case, different aggregation strategies can be realized by suitably varying the value of parameter p. Compensative Connectives
The max–∗ composition is a more general definition of composition (2). It is defined by the following membership function:
FUZZY AGGREGATION CONNECTIVES Minimum and maximum operators represent the simplest way to aggregate different degrees of membership. More sophisticated choices are available in the literature. They resort to fuzzy aggregation connectives. Fuzzy aggregation connectives are (possibly nonlinear) functions that map a set of membership (or certainty) values µ1 , µ2 , . . . , µN to
Compensative connectives can be categorized into the following classes depending on their aggregation structure: 1. Mean operators 2. Hybrid operators Mean Connectives. A mean connective is a mapping m: [0, 1] × [0, 1] → [0, 1] such that 1. m(µ1 , µ2 ) ≥ m(µ3 , µ4 ) if µ1 ≥ µ3 and µ2 ≥ µ4 2. min(µ1 , µ2 ) ≤ m(µ1 , µ2 ) ≤ max(µ1 , µ2 ) A useful mean connective is the generalized mean (31). By using this connective, different degrees of certainty (or criteria) can be suitably weighted in order to take care of their
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Fuzzy Model Fundamentals
relative importance:
where n i=1 wi = 1. It is worth pointing out that this connective yields all values between min and max by varying the parameter p between p → −∞ and p → +∞. Hybrid Connectives. Hybrid connectives combine outputs of union and intesection operators (29). This combination is generally performed using a multiplicative or an additive model as follows:
where yU and yI denote the outputs of union and intersection operators. The degree of compensation between these components depends on the value of the parameter γ. The multiplicative γ-model proposed by Zimmermann and Zysno (32) adopts union and intersection components based on products:
If x is low and y is medium, then z is large. The typical IF–THEN structure of a fuzzy rule includes a group of antecedent clauses which define conditions and a consequent clause which identifies the corresponding action. In general, fuzzy systems adopt rules to map fuzzy sets to fuzzy sets (7). Many engineering applications, however, require techniques which map scalar inputs to scalar outputs. We can address this issue by adding an input fuzzifier and an output defuzzifier to the classical model (17). The result is a very important class of fuzzy systems which are able to map scalar inputs to one (or more) scalar output(s). Since the successful application of these systems is playing a key role in the widespread diffusion of fuzzy techniques, we shall decribe their structure in details. Let us consider a fuzzy system which maps M input variables x1 , x2 , . . . , xM to one output variable y by means of N fuzzy rules R1, R2, . . . , RN. Such a system can be expressed in the following form:
where n i=1 wi = n and 0 ≤ γ ≤ 1. The additive γ-model is defined by
The additive γ-model adopting Yager’s union and intersection is defined by
LINGUISTIC VARIABLES AND FUZZY SYSTEMS As mentioned in the section entitled “Fuzzy Set,” fuzzy models permit us to express concepts in a way that is very close to human thinking. In fact, linguistic labels can be associated with fuzzy sets in order to form sentences like “the pixel luminance is very bright,” “the voltage is low,” “the temperature is high,” and so on. In this respect, quantities such as pixel luminance, voltage, and temperature can be interpreted as linguistic variables—that is, variables whose values are words or sentences (17). For example, the linguistic variable pixel luminance can be decomposed into a set of terms such as dark, medium, and bright (Fig. 3) which correspond to fuzzy sets in its universe of discourse. Fuzzy rules permit us to express a processing strategy in a form that mimics human decision making. For example:
where Ai,j (1 ≤ i ≤ M, 1 ≤ j ≤ N) is the fuzzy set associated with the ith input variable in the jth rule and Bj is the fuzzy set associated with the output variable in the same rule. The set of fuzzy rules as a whole is called a rulebase. Since the fuzzy rulebase contains the necessary information to process the data, it represents the knowledge base of the system. The knowledge base is numerically processed by the fuzzy inference mechanism. For a given set of input data, the inference mechanism evaluates the degrees of activation of the component rules and then combines their resulting effects. More precisely, let λj be the degree of activation (or satisfaction) of the jth rule. This degree can be evaluated by using the following relation:
where µAi , j denotes the membership function of fuzzy set Ai,j . It should be noticed that the choice of an intersection connective to aggregate membership degrees depends on the presence of the “AND” for combining the antecedent clauses in each fuzzy rule. Of course, different aggregation connectives (see the section entitled “Fuzzy Aggregation Connectives”) can be adopted depending on the specific problem. The degree of activation λj yields the following effect on fuzzy set Bj which identifies the consequent action of the jth rule. A new fuzzy set B j is generated, whose member-
Fuzzy Model Fundamentals
7
ship function is defined by
Two different inference schemes are commonly used. If the correlation-product inference is adopted (7), symbol “∗” denotes the product operator. If, on the other hand, the correlation-minimum inference is chosen, symbol “∗” denotes the minimum operator. Fuzzy sets B j (j = 1, . . . , N) are then combined in order to obtain a resulting fuzzy set B. If we resort to the union, the corresponding membership function µB (u) is yielded by
If we adopt the additive model (7), on the contrary, we obtain
where K is a scaling factor that limits the degree of membership to unity. As a final step, we want to derive a scalar value from the fuzzy set B. A very popular technique is the so-called “centroid” or “center of gravity” method which yields the output y as follows:
where V denotes the support of fuzzy set B. (If this support is discrete, summation should replace the integral symbol. Of course, integral and summation symbols here denote integration and arithmetic addition.) Notice that if we adopt the additive scheme, we can evaluate the output y by means of the centroids y j of the component fuzzy sets B j :
where
Let us adopt correlation-product inference. Relations (42) and (43) become:
Figure 11. Fuzzy sets positive (PO), zero (ZE), and negative (NE).
Thus, we can express relation (41) as follows:
Relation (46) is very attractive from the point of view of computational efficiency. In fact, the component terms wj and yj do not depend on λj . If all consequent fuzzy sets Bj have the same shape, i.e., wj = w(j = 1, . . . , N), we finally obtain
In this case, the final output only depends on the degrees of activation of fuzzy rules and on the centroids of the original consequent fuzzy sets. Let us consider a simple example. Let {sk } be the digitized signal in the range [0, L − 1] depicted in Fig. 10(a). This signal represents a staircase waveform corrupted by impulse noise. Suppose we want to design a filter able to reduce (or possibly cancel) the noise pulses (33). Let sk be the sample to be processed at the time k. Let k−1 = sk − sk−1 and k+1 = sk − sk+1 be the amplitude differences between this element and the neighboring samples sk−1 and sk+1 , respectively. In order to estimate the noise amplitude nk , we may use the following fuzzy system:
where PO (positive), ZE (zero), and NE (negative) are triangular fuzzy sets represented in Fig. 11. The first fuzzy rule (R1) aims at detecting a positive noise pulse (i.e., a noise pulse whose amplitude is higher than the one of the neighborhood). The second fuzzy rule (R2) aims at detecting a negative noise pulse (i.e., a noise pulse whose amplitude is lower than the one of the neighborhood). The third fuzzy rule (R3) deals with the absence of any noise pulse (i.e., with the case of an uncorrupted sample). Formally, we have A1,1 = PO, A2,1 = PO, A1,2 = NE, A2,2 = NE, A1,3 = ZE, A2,3 = ZE, B1 = PO, B2 = NE, B3 = ZE. The degrees of activation λ(k) 1 , λ(k) 2 , λ(k) 3 of three rules at the time k are evaluated by
8
Fuzzy Model Fundamentals
Figure 12. Fuzzy sets zero (ZE) and nonzero (NZ).
Suppose we adopt correlation-product inference and the additive model. Since all fuzzy sets have the same shape, the output is yielded by relation (47). We observe, in particular, that the centroids have the following values (Fig. 11): yPO = L − 1, yZE = 0 and yNE = −L + 1. Thus, we have
Figure 13. Resulting edge map.
data. In general, neuro-fuzzy models can be successfully adopted to find the most appropriate rulebase for a given application. PARAMETERIZED MEMBERSHIP FUNCTIONS
Let s k = sk − nk be the output of the filter. The result of the application is shown in Fig. 10(b). As a second example, let us consider the digitized image in Fig. 3. Let xi,j be the pixel luminance at location (i, j). Let i,j −1 = xi,j − xi,j −1 and i−1,j = xi,j − xi−1,j be the luminance differences between this element and the neighboring pixels at locations (i, j − 1) and (i − 1, j), respectively. Let us suppose we want to detect edges in the image—that is, possible object borders (34). Our goal is to produce another image (called “edge map”) where dark pixels denote uniform regions and bright pixels denote possible object contours. In order to perform this task, we define a pair of fuzzy rules as follows:
where yi,j is the luminance of the pixel at location (i, j) in the edge map. Zero (ZE) and nonzero (NZ) are fuzzy sets in the interval [−L + 1, L − 1] (Fig. 12). White (WH) and black (BL) are fuzzy singletons centered on L − 1 and zero. We can evaluate the degrees of activation λ(i,j ) 1 and λ(i,j ) 2 by using simple intersection and union aggregators:
Fuzzy systems are powerful tools for data processing. However, it is not always necessary to express fuzzy reasoning in form of rules. Sometime one (ore more) parameterized fuzzy sets suffice. As an example, let us consider the filtering of Gaussian noise in digital images. It is known that noise having Gaussian-like distribution is very often encountered during image acquisition. Our goal is to reduce the noise without (significantly) blurring the image details. A simple idea is to adopt a fuzzy weighted mean filter for this purpose (39, 40). Again, let us suppose we deal with digitized images having L gray levels (typically L = 256). Let xi,j be the pixel luminance at location (i, j) in the noisy image and let i+m,j+n = xi,j − xi+m,j+n be the luminance difference between this element and the neighboring pixel at location (i+m, j+n). The output yi,j of the fuzzy weighted mean filter is defined by the following relationships: yi, j =
N N
wi+m, j+n xi+m, j+n
(55)
m=−N n=−N
wi+m, j+n = N m=−N
µSM (i+m, j+n )
N
n=−N
µSM (i+m, j+n )
(56)
where µSM (u) is the membership function of fuzzy set small. Let us define this set by resorting to a bell-shaped parameterized function: The output yi,j is yielded by
The result is shown in Fig. 13. Fuzzy inference schemes different from that described above are also possible. As an example, the well-known Takagi–Sugeno Model (4, 35) found wide application in the design of fuzzy controllers. More sophisticated approaches are also available in the literature (36–38). In any case an appropriate choice of fuzzy sets and rules plays a key role in determining the desired behavior of a fuzzy system. If we adopt parameterized membership functions, we can try to acquire the optimal fuzzy set shapes from a set of training
u 2 µSM (u) = exp{−( ) } c
(57)
A graphical representation of µSM (u) is depicted in Fig. 14 for three different values of the parameter c (u≥0). According to (58–59), the algorithm performs a weighted mean of the luminance values in a (2N+1) × (2N+1) window around xi,j . The weights are chosen according to a simple fuzzy model: small luminance differences (possibly) denote noise, while large luminance differences denote object contours. Thus, when i+m,j+n is small, the corresponding wi+m,j+n is large and vice versa. As a result, the processing gradually excludes pixel luminances that are different from xi,j in order to preserve image details. The value of the parameter c mainly depends upon the variance of the Gaussian noise. Typically, this value is chosen so that a suitable per-
Fuzzy Model Fundamentals
9
Figure 14. Graphical representation of the membership function µSM (u) for three different values of the parameter c.
formance index is maximized, for example, the well-known peak signal-to-noise ratio (PSNR), which is defined as:
PSNR = 10log 10 ( i
i
j j
(L − 1)2
(yi, j − si, j )2
)
(58)
where si,j and yi,j denote the pixel luminances of the original noise-free image and the filtered image, respectively, at location (i,j). This procedure is briefly depicted in Fig. 15. An example of processed data is also reported in Fig. 16. We generated the picture in Fig. 16a by adding Gaussian noise with variance σ 2 =100 to the original noise-free image. The result of the application of the fuzzy filter is reported in Fig. 16b (N=2). Details of the noisy and the processed images are respectively depicted in Fig 16c and 16d for visual inspection. The noise reduction is apparent, especially in the uniform regions of the image. According to our previous observation, we chose the parameter value that gives the maximum PSNR (Fig. 17). Larger values would increase the image blur, smaller values would leave some noise unprocessed. It is worth pointing out that we can define the same filtering operation by resorting to the concept of fuzzy relation. In an equivalent way, we can formally define the weights of the filter as follows: wi+m, j+n = N m=−N
µEQ (xi, j , xi+m, j+n )
N
n=−N
µEQ (xi, j , xi+m, j+n )
(59)
where mEQ (u,v) is the parameterized membership function that describes the fuzzy relation “u is equal to v”: µEQ (u, v) = exp{−(
u−v 2 ) } c
(60)
A graphical representation of mEQ (u,v) is shown in Fig. 18 (c=40). The fuzzy weighted mean filter is not the only available scheme for reducing Gaussian noise. Other approaches are possible (40). For example, we can adopt fuzzy models to estimate the noise amplitude gi,j and then subtract it from
Figure 16. (a) Image corrupted by Gaussian noise, (b) filtered image, (c) detail of the noisy image, (d) detail of the filtered image.
10
Fuzzy Model Fundamentals
Figure 15. Block diagram of the procedure for parameter tuning.
> xi,j−1 , xi,j > xi−1,j , xi,j < xi,j−1 , xi,j < xi−1,j . As a result, µDI (xi,j , xi,j−1 ) ≈1 and/or µDI (xi,j , xi−1,j ) ≈1, and the output of the edge detector is yi,j ≈ L−1. Conversely, in the presence of an uniform region, we have xi,j ≈ xi,j−1 and xi,j ≈ xi−1,j . Thus the output becomes yi,j =0, as it should be. The parameters b and c control the actual behavior of the edge detector. Large values of these parameters can be chosen to decrease its sensitivity to fine details and to noise.
Figure 17. Filtering performance.
the pixel luminance xi,j , as follows: 1 (xi, j − xi+m, j+n ) µSI (xi, j , xi+m, j+n ) 8 1
gi, j =
1
(61)
m=−1n=−1
yi, j = xi, j − gi, j
(62)
where µSI (u,v) is the parameterized membership function of fuzzy relation “u is similar to v”. A possible definition is given by the following relationship (Fig. 19): 1 5c − |u − v| |u − v|0 i, j=1,2,n
N(i, j, c) ≥ α
c∈C
where N(i, j,c) is the response of the neuron located at the (i, j) and considered (placed) in context c from a certain family of contexts C . Some other criteria could be also anticipated; for example, one may request that the linguistic descriptions are well separated, meaning that their corresponding activation regions in the map are kept almost disjoint. FUZZY NEURAL COMPUTING STRUCTURES The examples discussed in the preceding section have revealed a diversity of approaches taken towards building neural network–fuzzy architectures. Taking this into account, we distinguish between two key facets one should take into account in any design endeavor: • Architectural • Temporal These properties are exemplified in the sense of the plasticity and explicit knowledge representation of the resulting neural network–fuzzy structure. The strength of the interaction itself can vary from the level at which the technology of fuzzy sets and neurocomputing are loosely combined and barely coexist to the highest one where there emerges a genuine fusion between the technologies.
FUZZY NEURAL NETS
171
Architectures of Fuzzy–Neural Network Systems
Classes of Fuzzy Neurons
The essence of the architectural interaction of fuzzy sets and neural networks is visualized in Fig. 4. This point has already been made clear through the studies in the previous section. By and large, the role of fuzzy sets gets more visible at the input and output layers of any multilayer structure of the network. The input and output layers are much more oriented toward the capturing the semantics of data rather than focusing on pure numeric processing.
We elaborate on the three models that are representative of most of the existing hybrid architectures as developed in the setting of fuzzy sets and neurocomputing. The first fuzzy set–oriented construct proposed by Lee and Lee (11) was contrasted with the generic model of the neuron as discussed by McCulloch and Pitts as a binary device. Let us recall that the basic binary neuron has n excitatory inputs (e1,e2,. . .,en) and m inhibitory inputs (i1,i2,. . .,im). The firing of the neuron is binary: y is set to 1 if all inhibitory inputs are set to 0 and the sum of all excitatory inputs exceeds a threshold level,
Temporal Aspects of Interaction in Fuzzy–Neural Network Systems
The temporal aspects of interaction arise when dealing with the various levels of intensity of learning, Fig. 4. Again the updates of the connections are much more vigorous at the hidden layers—we conclude that their plasticity (that is an ability to modify the values of the connections) is higher than the others situated close to the input and output layers.
The main generalization proposed by Lee and Lee (11) was to consider that an activity of the neuron is continuous. More specifically, the output of the neuron is one of the positive numbers ui in [0, 1], i ⫽ 1,2, . . ., p, which means that the output (y) reads as
FUZZY NEUROCOMPUTING—AN ARCHITECTURAL FUSION OF FUZZY AND NEURAL NETWORK TECHNOLOGY
y=
In this section we concentrate on a certain category of hybrid processing in which the neurons combine a series of features that are essential to neural networks and symbolic processing. In fact, this is one of the approaches among these reported in the literature (9,10,11). Very often these basic constructs (fuzzy neurons) are exploited as generic building blocks in the development of fuzzy–neural network architectures.
ui
if the neuron is firing
0
otherwise
The firing rules are also restated accordingly: 1. All inhibitory inputs are set to 0. 2. The sum of excitatory inputs must be equal or greater than a threshold T.
Knowledge representation
Numeric processing and learning
Knowledge representation
Fuzzy sets
Neurocomputing
Fuzzy sets
Preprocessing
ei > T
i=1
Postprocessing
Plasticity
Number of layer
Figure 4. Architectural and temporal synergy of fuzzy set constructs and neural networks. Architectural level: fuzzy sets contribute to the construction of preprocessing and postprocessing modules (input and output layer) and are aimed at knowledge representation whereas numeric processing and learning occurs at the level of the hidden layers of the entire architecture. Temporal interactions: most parametric learning occurs at the level of hidden layers which exhibit high plasticity.
172
FUZZY NEURAL NETS
The original study (11) illustrates the application of such fuzzy neurons to a synthesis of fuzzy automata. An idea proposed by Buckley and Hayashi (12) is to develop a fuzzy neuron in the sense that its connections are viewed as fuzzy sets (more precisely, fuzzy numbers); similarly we consider the inputs to be fuzzy numbers. The underlying formula of the neuron generalizes from the pure numeric neurons and is expressed as
Y = f
n
!
Wi Xi +
i=1
Here the connections and bias and inputs are fuzzy numbers. Moreover, the operations (summation and product) are viewed in terms of fuzzy set operations. Here that the output of the neuron is a fuzzy number. All the operations in the expression that follows are carried out via the extension principle. To illustrate the relevant calculations, let us consider the fuzzy neuron with two inputs (X1 and X2). In light of the extension principle, the output of the fuzzy neuron reads as Y( y) = sup{min[X1 (x1 ),X2 (x2 ),W1 (w1 ),W2 (w2 ),(ϑ )]} where the supremum in the above expression is taken over all the arguments satisfying the nonlinear constraint
y= f
2
!
wi xi − i + ϑ
i=1
The idea developed by Bortolan (13) makes the previous concept of the fuzzy neuron more computationally attractive by restricting the form of the input fuzzy sets as well as the connections of the neuron to trapezoidal fuzzy sets T (x;a,b,c,d), see Fig. 5. Such piecewise membership functions represent uncertain variables. This drastically reduces computational overhead. The pertinent version of a well-known 웃 learning algorithm is covered in Ref. 13. Fuzzy Logic Neurons The main rationale behind this choice (14,15,16) is that the resulting neural networks effortlessly combine learning capabilities with the mechanism of knowledge representation in its explicit manner. The neurons are split into two main categories, namely aggregative and referential processing units. We discuss here aggregative neurons. In what follows, we denote t-novus by t (or T). S-novus will be denoted by s (or S).
a
b
c
d
Figure 5. An example of a trapezoidal fuzzy number: a and d denote a lower and upper bound of the linguistic concept. The elements situated in-between b and c belong to the concept at degree 1 and are indistinguishable.
The n-input OR processing unit is governed by the expression y = OR(x; w) namely n
y=
S(w tx ) i=1
i
i
The inputs of the neuron are described by x while the vector w summarizes its connections. The computations of the output (y) rely on the use of some triangular norms (s and t norm). Observe that if w ⫽ 1 then y ⫽ OR(x1,x2,. . .,xn) so that the neuron reduces to a standard OR gate encountered in digital logic. The AND neuron is described in the following form y = AND(x; w) or equivalently n
y=
T(w sx ) i=1
i
i
Note that the composition operation used here uses the s and t norm in a reversed order. An important class of fuzzy neural networks concerns an approximation of mappings between the unit hypercubes (namely, from [0,1]n to [0,1]m or [0,1] for m ⫽ 1). These mappings are realized in a logic-based format. To fully comprehend the fundamental idea behind this architecture, let us note some very simple yet powerful concepts form the realm of two-valued systems. The well-known Shannon’s theorem states that any Boolean function 兵0,1其n 씮 兵0,1其 can be uniquely represented as a logical sum (union) of minterms (a so-called SOM representation) or, equivalently, a product of some maxterms (known as a POM representation). By minterm we mean an AND combination of all the input variables of this function; they could appear either in a direct or complemented (negated) form. Similarly, the maxterm consists of the variables that now occur in their OR combination. A complete list of minterms and maxterms for Boolean functions of two variables consists of the expressions x1 AND x2 , x1 AND x2 , x1 AND x2 , x1 AND x2 for minterms x1 OR x2 , x1 OR x2 , x1 OR x2 , x1 OR x2 for maxterms From a functional point of view, the minterms can be identified with the AND neurons while the OR neurons can be used to produce the corresponding maxterms. It is also noticeable that the connections of these neurons are restricted to the two-valued set 兵0,1其, therefore making these neurons two-valued selectors. Taking into account the fundamental representation of the Boolean functions, two complementary (dual) architectures are envisioned. In the first case, the network includes a single hidden layer that is constructed with the aid of the AND neurons and the output layer consisting of the OR neurons (SOM version of the network). The dual type of the network is of the POM type in which the hidden layer consists of some OR neurons while the output layer is formed by the AND neurons.
FUZZY PATTERN RECOGNITION
Two points are worth making here that contrast between the logic processors (LP) in their continuous and two-valued versions: 1. The logic processor represents or approximates data. For the Boolean data, assuming that all the input combinations are different, we are talking about a representation of the corresponding Boolean function. In this case the POM and SOM versions of the logic processors for the same Boolean function are equivalent. 2. The logic processor used for the continuous data approximates a certain unknown fuzzy function. The equivalence of the POM and SOM types of the obtained logic processors is not guaranteed at all. Moreover, the approximation exhibits an inherent logical flavor not necessarily leading to the same approximation accuracy as achieved for ‘‘classic’’ neural networks. This should not be regarded as a shortcoming, as in return we obtain some essential transparency of the neural architecture that could be easily interpreted in the form of ‘‘if–then’’ statements—the most evident enhancement of the architecture in an attempt to alleviate the black box nature inherent of most of the neural networks. CONCLUSIONS Fuzzy sets and neurocomputing are two supplementary technologies. The two-way integration is not only possible but highly beneficial. The knowledge-based faculties are well handled by the technology of fuzzy sets, while the learning activities are chiefly addressed by neural networks. Interestingly, there are a number of new constructs combining the ideas stemming from fuzzy sets and neural networks. We have investigated various levels of synergy and proposed a consistent classification of the systems emerging as an outcome of the symbiosis of these two technologies. BIBLIOGRAPHY 1. D. E. Rumelhart and J. L. McLelland, Parallel Distributed Processing, Cambridge, MA: MIT Press, 1986. 2. L. A. Zadeh, Fuzzy sets and information granularity. In: M. M. Gupta, R. K. Ragade, and R. R. Yager (eds.), Advances in Fuzzy Set Theory and Applications, Amsterdam: North Holland, 1979, pp. 3–18. 3. W. Pedrycz, Selected issues of frame of knowledge representation realized by means of linguistic labels, Int. J. Intelligent Systems, 7:155–170, 1992. 4. S. Chen, C. F. N. Cowan, and P. M. Grant, Orthogonal least squares learning algorithm for radial basis function networks, IEEE Trans. Neural Networks, 2:302–309 1991. 5. J. Moody and C. Darken, Fast learning networks of locally-tuned processing units, Neural Computing, 1:281–294, 1989. 6. D. Dubois and H. Prade, Possibility Theory—An Approach to Computerized Processing of Uncertainty, New York: Plenum Press, 1988. 7. F. M. Silva and L. B. Almeida, Acceleration techniques for the back-propagation algorithm. In Lecture Notes in Computer Science, Berlin: Springer-Verlag, 1990, Vol. 412, pp. 110–119. 8. J. M. Keller and D. J. Hunt, Incorporating fuzzy membership functions into the perceptron algorithm, IEEE Trans. Pattern Anal. Mach. Intell., PAMI-7:693–699, 1985.
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9. H. Ishibuchi, R. Fujioka, and H. Tanaka, An architecture of neural networks for input vectors of fuzzy numbers. Proc. IEEE Int. Conf. Fuzzy Systems (FUZZ-IEEE ’92), San Diego, March 8–12, 1992, pp. 1293–1300. 10. I. Requena and M. Delgado, R-FN: A model of fuzzy neuron, Proc. 2nd Int. Conf. Fuzzy Logic Neural Networks (IIZUKA ’92), Iizuka, Japan, July 17–22, 1992, pp. 793–796. 11. S. C. Lee and E. T. Lee, Fuzzy neural networks, Math Biosci. 23:151–177, 1975. 12. J. Buckley and Y. Hayashi, Fuzzy neural networks: A survey, Fuzzy Sets Systems 66:1–14, 1994. 13. G. Bortolan, Neural networks for the processing of fuzzy sets. In: M. Marinaro and P. G. Morasso (eds.), Proceeding of the International Conference on Artificial Neural Networks, London: SpringerVerlag, 1994, pp. 181–184. 14. W. Pedrycz, Fuzzy neural networks and neurocomputations. Fuzzy Sets Systems 56:1–28, 1993. 15. W. Pedrycz, Fuzzy Sets Engineering. Boca Raton, FL: CRC, 1995. 16. W. Pedrycz and A. F. Rocha, Fuzzy-set based models of neurons and knowledge-based networks, IEEE Trans. Fuzzy Systems, 1:254–266, 1993.
WITOLD PEDRYCZ University of Manitoba
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Wiley Encyclopedia of Electrical and Electronics Engineering Fuzzy Pattern Recognition Standard Article James C. Bezdek1 and Ludmila Kuncheva2 1University of West Florida, Pensacola, FL 2University of Wales, Bangor, Sofia, Bulgaria Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W3505 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (280K)
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Abstract The sections in this article are Pattern Recognition: Data, Label Vectors, and Measures of Similarity Fuzzy Cluster Analysis Fuzzy Classifier Design Feature Analysis Remarks on Applications of Fuzzy Pattern Recognition Keywords: c-means clustering models; classifier design; feature analysis; fuzzy clustering; fuzzy models; k-nearest neighbor classifier; label vectors; nearest prototype classifier; partitions of data About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.
file:///N|/000000/0WILEY%20ENCYCLOPEDIA%20OF%20EL...ICS%20ENGINEERING/24.%20fuzzy%20systems/W3505.htm17.06.2008 15:58:54
FUZZY PATTERN RECOGNITION
173
FUZZY PATTERN RECOGNITION Fuzzy sets were introduced by Zadeh (1) to represent nonstatistical uncertainty. Suppose you must advise a driving student when to apply the brakes of a car. Would you say ‘‘begin braking 74.2 feet from the crosswalk’’? Or would you say ‘‘apply the brakes pretty soon’’? You would choose the second instruction because the first one is too precise to be implemented. So, precision can be useless, while vague directions can be interpreted and acted upon. Fuzzy sets are used to endow computational models with the ability to recognize, represent, manipulate, interpret, and use (act on) nonstatistical imprecision. Conventional (crisp) sets contain objects that satisfy precise properties. The set H ⫽ 兵r 僆 ᑬ兩6 ⱕ r ⱕ 8其 is crisp. H can be described by its membership function, 1 6≤r≤8 mH (r) = 0 otherwise Since mH maps all real numbers onto the two points 兵0, 1其, crisp sets correspond to 2-valued logic; every real number either is in H or is not. Consider the set F of real numbers that are close to seven. Since ‘‘close to seven’’ is fuzzy, there is not a unique membership function for F. Rather, the modeler must decide, based on the potential application and imprecise properties of F, what mF should be. Properties that seem plausible for this F include: (1) normality (mF(7) ⫽ 1); (2) unimodality (only mF(7) ⫽ 1); (3) the closer r is to 7, the closer mF(r) is to 1, and conversely; and (4) symmetry (numbers equally far left and right of 7 should have equal memberships). Infinitely many functions satisfy these intuitive constraints. For example, 2 m1F(r) ⫽ e⫺(r⫺7) and m2F(r) ⫽ 1/(1 ⫹ (r ⫺ 7)2). Notice that no J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.
174
FUZZY PATTERN RECOGNITION
physical entity corresponds to F. Fuzzy sets are realized only through membership functions, so it is correct to call mF the fuzzy set F, even though it is a function. Formally, every function m: X 哫 [0, 1] could be a fuzzy subset of any set X, but functions like this become fuzzy sets when and only when they match some intuitively plausible semantic description of imprecise properties of the objects in X. A question that continues to spark much debate is whether or not fuzziness is just a clever disguise for probability. The answer is no. Fuzzy memberships represent similarities of objects to imprecisely defined properties; probabilities convey information about relative frequencies. Another common misunderstanding is that fuzzy models are offered as replacements for crisp or probabilistic models. But most schemes that use fuzziness use it in the sense of embedding: Conventional structure is preserved as a special case of fuzzy structure, just as the real numbers are a special case of the complex numbers. Zadeh (2) first discussed models that had both fuzziness and probability. A recent publication about this is special issue 2(1) of the IEEE Transactions on Fuzzy Systems, 1994. PATTERN RECOGNITION: DATA, LABEL VECTORS, AND MEASURES OF SIMILARITY There are two major approaches to pattern recognition: numerical (3) and syntactic (4). Discussed here are three areas of numerical pattern recognition for object data: clustering, classifier design, and feature analysis. The earliest reference to fuzzy pattern recognition was Bellman et al. (5). Fuzzy techniques for numerical pattern recognition are now fairly mature. Reference 6 is an edited collection of 51 papers on this subject that span the development of the field from 1965 to 1991. Object data are represented as X ⫽ 兵x1, . . ., xn其, a set of n feature vectors in feature space ᑬp. The jth object is a physical entity such as a fish, medical patient, and so on. Column vector xj is the object’s numerical representation; xkj is its kth feature. There are four types of class labels—crisp, fuzzy, probabilistic and possibilistic. Let integer c denote the number of classes, 1 ⬍ c ⬍ n. Define three sets of label vectors in ᑬc as follows:
between 0 and 1 and are constrained to sum to 1. If y is a label vector for some z 僆 ᑬp generated by, say, the fuzzy cmeans clustering method, y is a fuzzy label for z. If y came from a method such as maximum likelihood estimation in mixture decomposition, it would be a probabilistic label for z. Npc ⫽ [0, 1]c ⫺ 兵0其 is the unit hypercube in ᑬc, excluding the origin. Vectors such as y ⫽ (0.4, 0.2, 0.7)T are possibilistic label vectors in Np3. Labels in Npc. are produced, for example, by possibilistic clustering algorithms (7) and neural networks (8). Most pattern recognition models are based on statistical or geometrical properties of substructure in X. Two key concepts for describing geometry are angle and distance. Let A be any positive-definite p ⫻ p matrix. For vectors x, v 僆 ᑬp, v xx , v A = x T Av √ xx A = x T Axx
(4) (5)
and δA (xx, v ) = xx − v A =
N f c = {yy ∈ N pc :
c
yi = 1}
(1) (2)
i=1
Nhc = {yy ∈ N f c : yi ∈ {0, 1}∀i} = {ee1 , e 2 , . . ., e c }
(6)
In Eq. (7) I is the p ⫻ p identity matrix. Equations (8) and (9) n (xk ⫺ v) use the covariance matrix of X, M ⫽ cov(X) ⫽ ⌺k⫽1 n xk /n. D is the diagonal matrix (xk ⫺ v)T /n, where v ⫽ ⌺k⫽1 extracted from M by deletion of its off-diagonal entries. A second family of commonly used lengths and distances are the Minkowski norm and norm metrics:
p q 1/q x xxq = j ,
q≥1
q 1/q p δq (xx, v ) = xx − v q = x j − v j ,
(10)
c times
0} N pc = {yy ∈ c :yi ∈ [0, 1]∀ i, yi > 0∃ i} = [0, 1]c − {0
(xx − v )T A(xx − v )
are the inner product, norm (length), and norm metric (distance) induced on ᑬp by A. The most important instances of Eq. (6), together with their common names and inducing matrices, are xx − v I = (xx − v )T (xx − v ) Euclidean, A = I (7) xx − v D −1 = (xx − v )T D−1 (xx − v ) Diagonal, A = D−1 (8) −1 T −1 xx − v M −1 = (xx − v ) M (xx − v ) Mahalanobis, A = M (9)
j=1
let [0, 1]c = [0, 1] × · · · × [0, 1], ! "
(3)
In Eq. (1) 0 is the zero vector in ᑬc. Note that Nhc 傺 Nfc 傺 Npc. Nhc is the canonical (unit vector) basis of Euclidean cspace.
Three are commonly used: p xx − v 1 = x j − v j
q≥1
(11)
City block (1-norm); q = 1
j=1
(12) 2 1/2 p xx − v 2 = x j − v j
Euclidean (2-norm); q = 2
j=1
f
e i = (0, 0, . . ., 1 , . . ., 0)T ,
j=1
i
the ith vertex of Nhc, is the crisp label for class i, 1 ⱕ i ⱕ c. Nfc, a piece of a hyperplane, is the convex hull of Nhc. The vector y ⫽ (0.1, 0.6, 0.3)T is a label vector in Nf3; its entries lie
xx − v ∞
(13)
$ # = max x j − v j 1≤ j≤ p
Sup or Max norm; q → ∞
(14)
FUZZY PATTERN RECOGNITION
Equations (7) and (13) both give the Euclidean norm metric, the only one in both of the inner product and Minkowski norm metric families. FUZZY CLUSTER ANALYSIS
Crisp c-partitions of X obtained this way are denoted by UH ⫽ [H(U1) . . . H(Un)]. Example 1. Let O ⫽ 兵o1 ⫽ peach, o2 ⫽ plum, o3 ⫽ nectarine其, and let c ⫽ 2. Typical 2-partitions of O are as follows:
Object
This field comprises three problems: tendency assessment, clustering and validation. Given an unlabeled data set X, is there substructure in X? This is clustering tendency—should you look for clusters at all? Very few methods—fuzzy or otherwise—address this problem. Jain and Dubes (9) discuss some formal methods for assessment of cluster tendency, but most users begin clustering without checking the data for possible tendencies. Why? Because it is impossible to guess what structure your data may have in p dimensions, so hypothesis tests cast against structure that cannot be verified are hard to interpret. The usefulness of tendency assessment lies with its ability to rule out certain types of cluster structure. Different clustering algorithms produce different partitions of X, and it is never clear which one(s) may be most useful. Once clusters are obtained, how shall we pick the best clustering solution (or solutions)? This is cluster validation (4,5,9,10). Brevity precludes a discussion of this topic here. Clustering (or unsupervised learning) in unlabeled X is the assignment of (hard or fuzzy or probabilistic or possibilistic) label vectors to the 兵xk其. Cluster substructure is represented by a c ⫻ n matrix U ⫽ [U1 . . . Uk . . . Un] ⫽ [uik], where Uk denotes the kth column of U. A c-partition of X belongs to one of three sets: M pcn = {U ∈ cn : U k ∈ N pc ∀k}
(15)
M f cn = U ∈ M pcn : U k ∈ N f c ∀k; 0