Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844
Vol. V (2010), No. 4, pp. 525-531
Fingerprints Identification using a Fuzzy Logic System
I. Iancu, N. Constantinescu, M. Colhon
Ion Iancu, Nicolae Constantinescu, Mihaela Colhon
Department of Informatics
University of Craiova,
Al.I. Cuza Street, No. 13, Craiova RO-200585, Romania
E-mail: i_iancu@yahoo.com, nikyc@central.ucv.ro, mghindeanu@yahoo.com.
Abstract: This paper presents an optimized method to reduce the points number
to be used in order to identify a person using fuzzy fingerprints. Two fingerprints
are similar if n out of N points from the skin are identical. We discuss the criteria
used for choosing these points. We also describe the properties of fuzzy logic and
the classical methods applied on fingerprints. Our method compares two matching
sets and selects the optimal set from these, using a fuzzy reasoning system. The
advantage of our method with respect to the classical existing methods consists in a
smaller number of calculations.
Keywords: fuzzy models, fingerprint authentication, cryptographic signature model.
1 Introduction
Fingerprint identification is the most mature biometric method being implemented at an early level
since 1960. The recognition of a fingerprint can be done with two methods: ”one-to-one” (verification)
and ”one-to-many” (1 : N identification). The first method is applied when we have two fingerprints
and we want to verify if they belong to the same person. The second one is used when we have one
fingerprint and we search it in a data base. The verification is much easier and faster because we have the
two fingerprints and we just need to compare them. On the other hand, the identification implies more
time for extracting the fingerprint because there are needed much more details.
The fingerprints are not compared with images, they use a method based on characteristic points
named ”minutiae”. These points are characterized by ridge ending (the abrupt end of a ridge), ridge
bifurcation (a single ridge that divides in two ridges), delta (a Y-shaped ridge meeting), core (a U-turn in
ridge pattern), etc. All these features are grouped in three types of lines: line ending, line bifurcation and
short line. After the minutiae points are localized, a map with all their locations on the finger is created.
Every minutiae point has associated two coordinates (x,y), an angle for orientation and a measure for the
fingerprint quality. The matching of two fingerprints depends on the position and on the rotation. For this
reason, every fingerprint is represented, not only, as a group of points with two coordinates, but also, as
a group of points with coordinates relative to other points. This allows obtaining an unique positioning
of a point regarding to other three points. The three selected points must not be collinear. When two
fingerprints are compared, first are compared the relative coordinates. If this stage ends successfully,
these coordinates are transformed in 2D coordinates and verified.
After verifying the fingerprints, the result will tell us if they are from the same person with a high
probability. Still, the cases when the belonging probability of a fingerprint is 0 (false) or 1(true) are
rarely. In most of the cases, the probability will be a number p ∈ [0,1]. This fact leads to a fuzzy logic.
The values in fuzzy logic can range between 0 and 1 (1 is for absolute truth, 0 for absolute falsity). A
fuzzy value for an element x will express the degree of membership of x in a set X . It is essential to
realize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while
probability is a mathematical model of randomness.
Copyright c⃝ 2006-2010 by CCC Publications
526 I. Iancu, N. Constantinescu, M. Colhon
2 State of the Art
Two fingerprints are similar if n out of N points match. To verify this, Freedman et al. introduced
the fuzzy matching protocols [3]. Using these protocols, the information about the fingerprint we want
to identify (or verify) will not be revealed if no match is found. To describe the fuzzy private matching
problem we will take a set of words X = x1 ...xN where {xi} are the letters. Two words X = x1 ...xN and
Y = y1 ...yN match only if: n ≤ |{k : xk = yk | 1 ≤ k ≤ N}| and this relation is denoted with X ≈n Y . In
the subsequent we will name the set X as the total set for selection. The input of the protocol will be two
sets of words (X = X1 ...Xm for the client and Y = Y1 ...Ys for the server) and the parameters m,s,N and
n. While the output of the server is empty, the output of the client will be a set {Yi ∈ Y |∃Xi ∈ X : Xi ≈n Yi},
where A ≈n B means that the points A and B are very close. This set is, in fact, the intersection of the
two input sets [1]. It was demonstrated that this protocol leads information about the input even if no
match is found [1]. Another protocol, based on Freedman’s protocol, was presented in [1]. It uses σ as a
combination of n different indices γ1,γ2 ...γn and σ(X) = xγ1||... ||xγt for a word X . After the parameters
and the public key are sent, the client constructs a polynomial representation of the points set:
Pσ = (x − σ(X1))∗(x − σ(X2))∗...∗(x − σ(Xm))
This is a feedback polinomial value for a set of fingerprints. Then he sends {Pσ }k2 to the server. The server
analyzes every received polynomial {Pσ } at the point σ(Yi) and computes {wσi }k2 = {r ∗ Pσ (σ(Yi))+Yi}k2 ,
where r is a random value. After all the calculations, the server sends {wσi }k2 to the client. The client
will decrypt all the messages and if wσi matches with any word from X then it is added to the output
set. {wσi }k2 is a combination between fingerprint points value and the parameter which characterizes the
common information between a base set and the current set of collected values.
A particular scheme of fingerprint authentication describes a method which is not based on the minu-
tiae points [12], but by the texture of the finger, called FingerCode. Such a FingerCode is a vector
composed from 640 values between 0 and 7. The vector is ordered and stable in size. The method uses
Euclidean distance to find the matching. After estimating the block orientation, a curvature estimator
is designed for each pixel. Its maximal value is, in fact, the morphological searched center. Using a
properly tuned Gabor filter ( [11, 13]) we can catch ridges and valleys from the fingerprint. The Fin-
gerCode is computed as the average absolute deviation from the mean of every sector of each image.
Error-correction for the FingerCode would never be efficient enough to recognize a user. In [12], the
method proposed uses a secret d +1− letter word , which correspond to the d +1 coefficients of a poly-
nomial p of degree d. The public key will be extract from (F, p), where F is the FingerCode. Then, we
choose n random point of p. These points will be hidden like in a fuzzy commitment scheme. To find the
polynomial p, each point is decoded. If at least d +1 points are decoded then p can, also, be retrieved.
A method based on the minutiae points and, also, on the pattern of the finger was presented in [4]. All
the ridges that cross a line (x,y) where x and y are minutiae points are counted. Then, are presented all the
possible combinations of three minutiae points and the ridges crossing that line. Such a combinations’
list needs C3n entries, where n is the number of minutiae points. This method is more complex because
before all the calculations are done we need to identify the minutiae points and then combine them.
3 Our Method
3.1 System Description
A commercial fingerprint-based authentication system requires a very low False Reject Rate (FRR)
for a given False Accept Rate (FAR) where FAR is the probability that the system will incorrectly identify
and FRR is the probability of failure in identification.
Fingerprints Identification using a Fuzzy Logic System 527
Our method is, also, based on the minutiae points of the fingerprints. We can identify at least 40
minutiae points on a fingerprint, depending on its quality. In general, the number of the minutiae points
varies from 0 to 100. All the methods mentioned above can be applied to a fingerprint verification. But,
for an identification we need an algorithm with a low level of complexity because the data bases used
in practice have millions of fingerprints. To reduce the search time and complexity, we first propose to
classify the fingerprints, and then, to identify the input fingerprints only in one subset of the data base.
To choose the right subset the fingerprint is matched at a coarse level to one of the existing types. After
that, it is matched at a finer level to all the fingerprints of the subset. The FBI in the United States
recognize eight different types of patterns [5]. For example, we have an input fingerprint and we want
to identify it in a data base with 15000 entries. We will take the minim number of minutiae points, 40.
If no classification is made we have to do at least 40 × 15000 = 600000 operations. But, if we use a
classification with eight types (each subset has the same number of fingerprints 15000/8 = 1875) we
will have at least (8+1875)×40 = 75320 calculations. This is because we will first compare the input
fingerprint with each group and after that it will be compared with each element of the chosen group.
As we can see, the calculations are reduced to only 12,5%. The classification of the fingerprints is
preferred to have more than three types of subsets. This is because a higher accuracy is achieved. Such
a classification, also, helps to reduce the number of calculations with a higher percentage.
3.2 Fuzzy Mathematical Background
A fuzzy set A in X is characterized by its membership function:
µA : X → [0,1]
where µA(x) ∈ [0,1] represents the membership degree of the element x in the fuzzy set A. We will work
with membership functions represented by trapezoidal fuzzy numbers. Such a number N = (m,m,α,β )
is defined as
µN (x) =
0 f or x < m − α
x − m + α
α f or x ∈ [m − α,m]
1 f or x ∈ [m,m]
m + β − x
β f or x ∈ [m,m + β ]
0 f or x > m + β
The rules are represented by fuzzy implications. Let X and Y be two variables whose domains are U and
V , respectively. The rule
i f X is A then Y is B
is represented by its conditional possibility distribution ( [14], [15]) πY /X :
πY /X (v,u) = µA(u)→ µB(v), ∀u ∈ U, ∀v ∈ V
where → is an implication operator ( [2]) and µA and µB are the membership functions of the fuzzy sets
A and B, respectively. One of the most important implication is Lukasiewicz implication [2], IL(x,y) =
min(1− x + y,1).
3.3 Proposed Fuzzy Logic System
Fuzzy control provides a formal methodology for representing, manipulating and implementing hu-
man’s heuristic knowledge about how to control a system. In a fuzzy logic controller, the expert knowl-
edge is of the form
528 I. Iancu, N. Constantinescu, M. Colhon
IF (a set o f conditions are satis f ied) T HEN (a set o f consequences are in f erred)
where the antecedents and the consequences of the rules are associated with fuzzy concepts (linguistic
terms). The most known systems are: Mamdani, Tsukamoto, Sugeno and Larsen which work with crisp
data as inputs. A Mamdani type model which works with interval inputs is presented in [10].
In this paper we use a version of Fuzzy Logic Control (FLC) system from [9] in fingerprints identi-
fication. This version is characterized by:
• the linguistic terms (or values), that are represented by trapezoidal fuzzy numbers
• Lukasiewicz implication, which is used to represent the rules
• the crisp control action of a rule, computed by Middle-of-Maxima method
• the overall crisp control actions, computed by discrete Center-of-Gravity.
We assume that the facts can be given by crisp data, intervals and/or linguistic terms and a rule is
characterized by:
• a set of linguistic variable A, having as domain an interval IA = [aA,bA]
• nA linguistic values A1,A2,...,AnA for each linguistic variable A
• membership function µ0Ai(x) for each value Ai, where i ∈ {1,2,...,nA} and x ∈ IA.
According to the structure of a FLC, the following steps are necessary in order to work with our
system.
Firing levels
We consider an interval input [a,b] with aA ≤ a < b ≤ bA. The membership function of Ai is modified
( [10]) by membership function of [a,b] as follows
∀x ∈ IA, µAi(x) = min(µ
0
Ai(x), µ[a,b](x))
where
µ[a,b](x) =
{
1 i f x ∈ [a,b]
0 otherwise
It is obvious that, any t-norm T can be used instead of min (see, for instance, [6–8]).
The firing level, generated by the input interval [a,b], corresponding to the linguistic value Ai is given
by:
µAi = max{µAi(x)|x ∈ [a,b]}.
The firing level µAi , generated by a linguistic input value A
′
i is
µAi = max{min{µ
0
Ai(x), µA′i (x)}|x ∈ IA}.
The firing level µAi , generated by a crisp value x0 is µ
0
Ai
(x0).
Fuzzy inference
We consider a set of fuzzy control rules
Ri : i f X1 is A
1
i and ... and Xr is A
r
i then Y is Ci
where the variables X j, j ∈ {1,2,...,r}, and Y have the domains U j and V, respectively. The firing levels
of the rules, denoted by {αi}, are computed by
αi = T (α1i ,...,α
r
i )
where T is a t-norm and α ji is the firing level for A
j
i , j ∈ {1,2,...,r}. The conclusion inferred from the
rule Ri, using the Lukasiewicz implication is
C ′i (v) = I(αi,Ci(v)),∀v ∈ V.
Fingerprints Identification using a Fuzzy Logic System 529
Figure 1: Conclusion obtained with Lukasiewicz implication
Defuzzification
The fuzzy output C ′i of the rule Ri is transformed in a crisp output zi using the Middle-of-Maxima
operator. The crisp value z0 associated to a conclusion C ′ inferred from a rule having the firing level α
and the conclusion C represented by the fuzzy number (mC,mC,αC,βC) is:
z0 =
mC + mC +(1− α)(βC − αC)
2
The overall crisp control action is computed by the discrete Center-of-Gravity method: if the number of
fired rules is N then the final control action is
z0 =
(
N∑
i=1
αizi
)
/
N∑
i=1
αi
where αi is the firing level and zi is the crisp output of the i-th rule.
4 An application in fingerprint identification
For the proposed FLC we consider rules with two inputs and one output. The input variables are
σ1 = {xi|xi ∈ X} and σ2 = {x j|x j ∈ X,xi ̸= x j,∀i, j} where X is the set defined in Section 2. By σ1 we
represent the values set of basic input data and σ2 is a user data to be evaluated and authenticated. These
values sets will be denoted by S1 and S2 respectively. The σ set values denote the optimal points set
to be used for the fuzzy matching authentication, according with S output variable. The fuzzy rule-base
consists of
R1: If S1 is Low and S2 is Very Low then S is Low
R2: If S1 is Very Low and S2 is Low then S is Low
R3: If S1 is Very Low and S2 is Very Low then S is Very Low
R4: If S1 is Very Low and S2 is Very Low then S is Low
R5: If S1 is Very Low and S2 is Middle then S is Middle
R6: If S1 is Very Low and S2 is Middle then S is Low
R7: If S1 is High and S2 is Very High then S is High
R8: If S1 is Very High and S2 is High then S is High
R9: If S1 is Very High and S2 is Very High then S is High
R10: If S1 is Low and S2 is Very High then S is Middle
The value Very Low for a variable S1, S2 or S represents a minimum degree of trust while the value Very
High represents the maximum degree. There are five linguistic values for every variable
{Very Low, Low, Middle, High, Very High}.
We consider the universes of discourse [0,10]. The membership functions corresponding to the lin-
guistic values are represented by the following trapezoidal fuzzy numbers:
530 I. Iancu, N. Constantinescu, M. Colhon
• for the variable S1: {(0,1,0,1.5),(3,4,1,0.5),(5,6,1,0.5),(7.5,8.5,3,1),(9.5,10,0.5,0)}
• for the variable S2: {(0,1.5,0,1),(2.5,4,0.5,0.5),(5,6,2,0),(6.5,8,1.5,0),(8.5,10,0.5,0)},
• for the variable S: {(0,1,0,1.5),(2.5,4,0.5,0.5),(5,7,2.5,0.5),(8,8.5,2,0.5),(9.5,10,1,0)}
We consider the following interval input values: [1.5,2.2] for S1 and [3.2,4.2] for S2. The positive firing
levels corresponding to the linguistic values of the input variable S1 are
µVeryLow = 0.666, µLow = 0.2
and the positive firing levels corresponding to the linguistic values of the input variable S2 are:
µLow = 1, µMiddle = 0.6
The fired rules and their firing levels, computed with t-norm Product T (x,y) = xy, are:
R2 with firing level α2 = 0.666,
R5 and R6 with α5 = α6 = 0.3996.
The fired rules give the following crisp values as output:
z2 = 3.25, z5 = 5.3996, z6 = 3.25;
then the overall crisp control action is
z0 = 3.836.
These values represent the matching approach for every subset points which are candidate to be in the
final set, and are computed using a fuzzy merging comparison between selection sets σ1 and σ2. The
optimal selection set (which has less points) is represented by the output variable σ .
5 Conclusions
Among all the biometric techniques, the identification based on fingerprints is used in the most
applications. The uniqueness of the fingerprint can be determinate by the pattern of ridges and the
minutiae points. For identifying an input fingerprint, the proposed method uses a fuzzy classification of
the data. The proposed system is much more efficient than the FLC presented in [8]. This is because, in
order to reduce the necessary points number, we find the minutiae points by using a fuzzy logic reasoning
system which compare two points sets matching values. In a practical application, it is recommended to
use the proposed system with a set of implications and aggregate the results given by every implication,
in order to obtain the overall output; in this way can be obtained a stronger base for more accurate results
of our system. We intend to use these results in a future work, by maping their relative placement on the
finger, and comparing all its points with the ones of the fingerprints for the right subset.
Bibliography
[1] L. Chmielewski and J. H. Hoepman, Fuzzy Private Matching (Extended Abstract), ARES ’08: Pro-
ceedings of the 2008 Third International Conference on Availability, Reliability and Security, IEEE
Computer Society, pp. 327–334, 2008.
[2] E. Czogola and J. Leski, On equivalence of approximate reasoning results using different interpreta-
tions of fuzzy if-then rules, Fuzzy Sets and Systems, vol. 117, no. 2, pp. 279–296, 2001.
Fingerprints Identification using a Fuzzy Logic System 531
[3] M. Freedman, K. Nissim and B. Pinkas, Efficient private matching and set intersection, Advances in
Cryptology, EUROCRYPT 2004, Springer-Verlag, pp. 1–19, 2004
[4] R. S. Germain, A. Califano and S. Colville, Fingerprint Matching Using Transformation Parameter
Clustering, IEEE Comput. Sci. Eng., vol. 4, no. 4, pp. 42–49, 1997.
[5] M. R. Hawthorne, Fingerprints. Analysis and Understanding, CRC Press, 2009
[6] I. Iancu, T -norms with threshold, Fuzzy Sets and Systems. International Journal of Soft Computing
and Intelligence, vol. 85, no. 1, pp. 83–92, 1997.
[7] I. Iancu, Operators with n-thresholds for uncertainty management, Journal of Applied Mathematics
& Computing, Springer Berlin, vol. 19, no. 1-2, pp. 1–17, 2005.
[8] I. Iancu, Generalized Modus Ponens Using Fodor’s Implication and T -norm Product with Threshold,
International Journal of Computers, Communications & Control (IJCCC), vol. 4, no. 4, pp. 330–343,
2009.
[9] I. Iancu, Extended Mamdani Fuzzy Logic Controller, The Fourth IASTED International Conference
on Computational Intelligence CI 2009, ACTA Press, vol. 5, pp. 143–149, 2009.
[10] F. Liu, H. Geng and Y. Q. Zhang, Interactive Fuzzy Interval Reasoning for Smart Web Shopping,
Applied Soft Computing, Elsevier, vol. 5, no. 4, pp. 433–439, 2005.
[11] D. G. Radojevic, Fuzzy Set Theory in Boolean Frame, International Journal of Computers, Com-
munications & Control (IJCCC), vol. 3, no. 5, pp. 121–131, 2008.
[12] V. V. T. Tong, H. Sibert, J. Lecoeur and M. Girault, Biometric fuzzy extractors made practical: a
proposal based on Finger Codes, Advances in Biometrics, Springer Berlin / Heidelberg, pp. 604–613,
2009.
[13] T. Vesselenyi, S. Dzitac, I. Dzitac and M.J. Manolescu, Fuzzy and Neural Controllers for a Pneu-
matic Actuator, International Journal of Computers, Communications & Control (IJCCC), vol. 4,
no. 2, pp. 375–387, 2007.
[14] L. A. Zadeh, A theory of approximate reasoning, Machine Intelligence 9, Elsevier, pp. 149–194,
1979.
[15] L. A. Zadeh, Fuzzy sets as a basis for a theory of a possibility, Fuzzy Sets and Systems, vol. 100,
pp. 9–34, 1999.