The Concept of Application of Fuzzy Logic in Biometric
Authentication Systems
Anatoly Sachenko, Arkadiusz Banasik, and Adrian Kapczyński
Silesian University of Technology, Department of Computer Science and Econometrics,
F. D. Roosevelt 26-28, 41-800 Zabrze, Poland
sachenkoa@yahoo.com, arkadiusz.banasik@polsl.pl,
adrian.kapczynski@polsl.pl
Abstract. In the paper the key topics concerning architecture and rules of working of biometric
authentication systems were described. Significant role is played by threshold which constitutes
acceptance or rejection given authentication attempt. Application of elements of fuzzy logic
was proposed in order to define threshold value of authentication system. The concept was
illustrated by an example.
Keywords: biometrics, fuzzy logic, authentication systems.
1 Introduction
The aim of this paper is to present on the basis of theoretical foundations of fuzzy
logic and how to use it as a hypothetical, single-layered biometric authentication system. In the first part it will be provided biometric authentication systems primer and
the fundamentals of fuzzy logic. On that basis the idea of use of fuzzy logic in biometric authentication systems was formulated.
2 Biometric Authentication Systems Primer
Biometric authentication system is basically a system which identifies patterns and
carries out the objectives of authentication by identifying the authenticity of physical
or behavioral characteristics possessed by the user [5].
The logical system includes the following modules [1]: enrollment module and
identification or verification module.
The first module is responsible for the registration of user ID and the association of
this identifier with the biometric pattern (called biometric template), which is understood as a vector of vales presented in an appropriate form, as a result of processing
the collected by the biometric device, raw human characteristics.
The identification Module (verification) is responsible for carrying out the collection
and processing of raw biometric characteristics in order to obtain a biometric template,
which is compared with patterns saved by the registration module. Those modules cooperate with each other and carry out the tasks related to the collection of raw biometric
data, features extraction and comparison of features and finally decision making.
The session with biometric systems begins with taking anatomical or behavioral features by the biometric reader. Biometric reader generates in n-dimensional biometric
E. Corchado et al. (Eds.): CISIS 2008, ASC 53, pp. 274–279, 2009.
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data. Biometric data represented by the form of vector features are the output of signals
processing algorithms. Then, vector features are identified, and the result is usually
presented in the form of the match (called confidence degree). On the basis of their
relevance and value of the threshold (called threshold value) the module responsible
for decision making produce output considered as acceptance or rejection.
The result of the work of biometric system is a confirmation of the identity of the
user. In light of the existence of the positive population (genuine users) and negative
population (impostors), there are four possible outcomes:
•
•
•
•
Genuine user is accepted (correct response),
Genuine user is rejected (wrong response),
Impostor is accepted (wrong response),
Impostor is rejected (correct response.).
A key decision-making is based on the value of the threshold T, which determines the
classification of the individual characteristics of vectors, leading to a positive class
(the sheep population) and a negative class (the wolf population).
When threshold is increased (decreased) the likelihood of false acceptance decreases (increases) and the probability of false rejection increases (decreases). It is not
possible to minimize the likelihood of erroneous acceptance and rejection simultaneously. In empirical conditions it can be found that precise threshold value makes it
clear that confidence scores which are very close to threshold level, but still lower
than it, are rejected. It can be found that the use of fuzzy logic can be helpful mean of
reducing identified problem.
3 Basics of Fuzzy Logic
A fuzzy set is an object which is characterized by its membership function. That function is assigned to every object in the set and it is ranging between zero and one. The
membership (characteristic) function is the grade of membership of that object in the
mentioned set [4].
That definition allows to declare more adequate if the object is within a range of
the set or not; to be more precise the degree of being in range. That is a useful feature
for expressions in natural language, e.g. the price is around thirty dollars, etc. It is
obvious that if we consider sets (not fuzzy sets) it is very hard to declare objects and
their membership function.
The visualization of the example membership function S is presented on fig. 1 and
elaborated on eq. 1.
0
⎧
2
⎪
⎛ x−a⎞
⎪1 − 2⎜
⎟
⎪
⎝ c−a ⎠
s ( x; a, b, c) = ⎨
2
⎪1 − 2⎛⎜ x − c ⎞⎟
⎪
⎝c−a⎠
⎪
1
⎩
for
x≤a
for
a≤ x≤b
(1)
for
b≤x≤c
for
x≥c
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A. Sachenko, A. Banasik, and A. Kapczyński
1,2
1
μ(x)
0,8
0,6
0,4
0,2
0
0
a
2
4
6
b
8
c
10
Fig. 1. Membership function S applied to fuzzy set high level of security”
It is necessary to indicate the meaning of membership function [4]. The first possibility is to indicate similarity between object and the standard.
Another is to indicate level of preferences. In that case the membership function is
concerned as level of acceptance of an object in order to declared preferences.
And the last but not least possibility is to consider it as a level of uncertainty. In
that case membership function is concerned as a level of validity that variable X will
be equal to value x.
Another important aspect of fuzzy sets and fuzzy logic is possibility of fuzzyfication – ability to change sharp values into fuzzy ones and defuzzyfication as a process
of changing fuzzy values into crisp values. That approach is very useful in case of
natural language problems and natural language variables.
Fuzzyfication and defuzzyfication is also used in analysis of group membership. It
is the best way of presenting average values in order to whole set of objects I ndimensional space. This space may be a multicriteria analysis of the problem.
It is well known that there are a lot of practical applications of fuzzy techniques. In
many applications, people start with fuzzy values and then propagate the original
fuzziness all the way to the answer, by transforming fuzzy rules and fuzzy inputs into
fuzzy recommendations.
However, there is one well known exception to this general feature of fuzzy technique applications. One of the main applications of fuzzy techniques is intelligent
control. In fuzzy control, the objective is not so much to provide an advise to the
expert, but rather to generate a single (crisp) control value uc that will be automatically applied by the automated controller.
To get this value it is necessary to use the fuzzy control rules to combine the membership functions of the inputs into a membership function μ(u) for the desired control
u. This function would be a good output if the result will be an advise to a human
expert. However, our point o concern is in generating a crisp value for the automatical
controller, we must transform the fuzzy membership function μ(u) into a single value
uc. This transformation from fuzzy to crisp is called defuzzification [3].
One of the most widely used defuzzification technique based on centroids is centroid defuzzification. It is based on minimizing the mean square difference between
The Concept of Application of Fuzzy Logic
277
the actual (unknown) optimal control u and the generated control uc produced as a
result. In this least square optimization it is possible to weigh each value u with the
weight proportional to its degree of possibility μ(u). The resulting optimization
problem [2]:
∫ μ (u ) ⋅ (u − u )
c
2
du → min
uc
(2)
can be explicitly solved if there is a possibility of differentiation the corresponding
objective function by uc and equate the resulting value to 0. The result of it is a
formula [2]:
uc =
∫ u ⋅ μ (u )du
∫ μ (u )du
(3)
This formula is called centroid defuzzification because it describes the ucoordinate of the center of mass of the region bounded by the graph of the membership function μ(u).
As it was mentioned before fuzzy sets and fuzzy logic is a way to cope with qualitative and quantitative problems. That possibility is a great advantage of presented
approach and it is very commonly used in different fields. That provides us a possibility of using its mechanisms in many different problems and it usually gives reasonable solutions.
4 The Use of Fuzzy Logic in Biometric Authentication Systems
There are two main characteristics of biometric authentication system: false acceptance rate and false rejection rate. The false acceptance rate can be defined as relation
of number of accepted authentication attempts to number of all attempts undertaken
by impostors. Authentication attempt is successful only if confidence score resulted
from comparison of template created during enrolment process with template created
from current authentication attempts eqauls or is greater than specified threshold
value. The threshold value can be specified apriori basing on theoretical estimations
or can be defined based on requirements from given environment. For example for
high security environments the importance of false acceptance errors is greater than of
false rejection errors; for low security environments the situation is quite opposite.
Threshold is the parameter which defines the levels of false acceptance and false
rejection errors. One of the most popular approach assumes that threshold value is
chosen at level were false acceptance rate equals false reject rate.
In our paper we consider the theoretical biometric authentication system with five
levels of security: very low, low, medium, high, very high and a main error considered
is only a false acceptance error. The security level is associated with a given level of
false acceptance error and the values of false acceptance rates are obtained emipirically
from given set of biometric templates. Obtained false acceptance rates are function of
threshold value which is expressed precisely as a value from range 0 to 100.
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A. Sachenko, A. Banasik, and A. Kapczyński
In biometric systems the threshold value can be set globally (for all users) or individually. From perspective of security officer responsible for effictient work of whole
biometric system the choice of indivual thresholds requires setting as many thresholds
as number of users enrolled in the system.
We propose the use of fuzzy logic as a mean of more natural expression of accepted level of false acceptance errors. In our approach the first parameter considered
is the value of false accept rate and basing on which we the level of security is determined which is finally transformed into threshold value.
Step-by-step proposed procedure consists of three steps.
In fist step for a given value of false acceptance rate we determine the value of the
membership functions for given level of security. If we consider five levels of false
acceptance rate, e.g. 5%, 2%, 1%, 0.5% and 0.1% than for each of those levels we can
calculate the values of membership functions to one of five levels of security: very
low, low, medium, high and very high (see fig. 2).
μ
Fig. 2. Membership functions for given security levels (S). Medium level was distinguished.
In second step through appropriate application of fuzzy rules we receive a result of
fuzzy request of the threshold value. Those rules are developed in order to obtain an
answer to a question about the relationship between threshold (T) and the specified
level of security (S):
Rule
Rule
Rule
Rule
Rule
1:
2:
3:
4:
5:
IF
IF
IF
IF
IF
“S
“S
“S
“S
“S
is very low” THEN “T is very low”
level is low” THEN “T is low”
is medium” THEN “T is medium”
level is high” THEN “T is high”
is very high” THEN “T is very high”.
In third step we apply defuzzyfication during which the fuzzy values are transformed
based on specified values of the membership functions and point a fuzzy centroid of
given values.
Our approach was depicted on fig. 3.
For example if we assume that the false acceptance rate is 0.1 and is fuzzified and
belongs to "S is high" with value of the membership function of 0.2 and belongs to "S
is very high" with value of the membership function of 0.8.
Then, based on rule 4 and rule 5 we can see that threshold value is high with the
value of membership function equaled to 0.2 and threshold value is very high at the
The Concept of Application of Fuzzy Logic
279
Fig. 3. Steps of fuzzy reasoning applied in biometric authentication system
value of membership function equaled to 0.8. The modal established at the threshold
value of membership functions shall be: 10 (very low level), 30 (low level), 50 (medium), 70 (high level), 90 (very high level).
The calculation of fuzzy centroid is carried out by the following calculation:
T = 90 ⋅ 0.8 + 70 ⋅ 0.2 = 86
(3)
In this example, for the value of false acceptance rate of 0.1, the threshold value
equals to 86.
5 Conclusions
The development of this concept in the use of fuzzy logic to determine the threshold
value associated with a given level of security provides an interesting alternative to
the traditional concept to define threshold values in biometric authentication systems.
References
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