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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
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  Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844Vol. 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 InformaticsUniversity of Craiova,Al.I. Cuza Street, No. 13, Craiova RO-200585, RomaniaE-mail: i_iancu@yahoo.com, nikyc@central.ucv.ro, mghindeanu@yahoo.com. Abstract: This paper presents an optimized method to reduce the points numberto be used in order to identify a person using fuzzy fingerprints. Two fingerprintsare similar if  n out of  N  points from the skin are identical. We discuss the criteriaused for choosing these points. We also describe the properties of fuzzy logic andthe classical methods applied on fingerprints. Our method compares two matchingsets and selects the optimal set from these, using a fuzzy reasoning system. Theadvantage of our method with respect to the classical existing methods consists in asmaller 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 levelsince 1960. The recognition of a fingerprint can be done with two methods: ”one-to-one” (verification)and ”one-to-many” (  : N  identification). The first method is applied when we have two fingerprintsand we want to verify if they belong to the same person. The second one is used when we have onefingerprint and we search it in a data base. The verification is much easier and faster because we have thetwo fingerprints and we just need to compare them. On the other hand, the identification implies moretime 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 pointsnamed ”minutiae”. These points are characterized by ridge ending (the abrupt end of a ridge), ridgebifurcation (a single ridge that divides in two ridges), delta (a Y-shaped ridge meeting), core (a U-turn inridge 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 thefingerprint quality. The matching of two fingerprints depends on the position and on the rotation. For thisreason, every fingerprint is represented, not only, as a group of points with two coordinates, but also, asa group of points with coordinates relative to other points. This allows obtaining an unique positioningof a point regarding to other three points. The three selected points must not be collinear. When twofingerprints 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 highprobability. Still, the cases when the belonging probability of a fingerprint is 0 (false) or 1(true) arerarely. In most of the cases, the probability will be a number p ∈ [ ,  ] . 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). Afuzzy value for an element x will express the degree of membership of  x in a set X  . It is essential torealize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon whileprobability is a mathematical model of randomness. Copyrightc  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. introducedthe fuzzy matching protocols [3]. Using these protocols, the information about the fingerprint we wantto identify (or verify) will not be revealed if no match is found. To describe the fuzzy private matchingproblem we will take a set of words X  = x  ...  x  N  where {  x i } are the letters. Two words X  = x  ...  x  N  and Y  = y  ...  y  N  match only if: n ≤ |{ k  : x k  = y k  |  ≤ k  ≤ N  }| and this relation is denoted with X  ≈ n Y  . Inthe subsequent we will name the set X  as the total set for selection . The input of the protocol will be twosets of words (  X  = X   ...  X  m for the client and Y  = Y   ... Y  s 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 { Y  i ∈ Y  | ∃  X  i ∈  X  : X  i ≈ n Y  i } ,where A ≈ n B means that the points A and B are very close. This set is, in fact, the intersection of thetwo input sets [1]. It was demonstrated that this protocol leads information about the input even if nomatch is found [1]. Another protocol, based on Freedman’s protocol, was presented in [1]. It uses σ  as acombination of  n different indices γ   , γ   ... γ  n and σ  (  X  ) = x γ   || ... ||  x γ  t  for a word X  . After the parametersand the public key are sent, the client constructs a polynomial representation of the points set: P σ  = (  x − σ  (  X   )) ∗ (  x − σ  (  X   )) ∗ ... ∗ (  x − σ  (  X  m )) Thisis afeedbackpolinomialvalueforasetoffingerprints. Thenhesends { P σ  } k   totheserver. Theserveranalyzes every received polynomial { P σ  } at the point σ  ( Y  i ) and computes { w σ  i } k   = { r  ∗ P σ  ( σ  ( Y  i ))+ Y  i } k   ,where r  is a random value. After all the calculations, the server sends { w σ  i } k   to the client. The clientwill decrypt all the messages and if  w σ  i matches with any word from X  then it is added to the outputset. { w σ  i } k   is a combination between fingerprint points value and the parameter which characterizes thecommon 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 vectorcomposed from  values between  and  . The vector is ordered and stable in size. The method usesEuclidean distance to find the matching. After estimating the block orientation, a curvature estimatoris designed for each pixel. Its maximal value is, in fact, the morphological searched center. Using aproperly 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], themethod proposed uses a secret d  +  − letter word  , which correspond to the d  +  coefficients of a poly-nomial p of degree d  . The public key will be extract from ( F  ,  p ) , where F  is the FingerCode. Then, wechoose n random point of  p . These points will be hidden like in a fuzzy commitment scheme. To find thepolynomial p , each point is decoded. If at least d  +  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]. Allthe ridges that cross a line (  x ,  y ) where x and y are minutiae points are counted. Then, are presented all thepossible combinations of three minutiae points and the ridges crossing that line. Such a combinations’list needs C   n entries, where n is the number of minutiae points. This method is more complex becausebefore 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 527Our method is, also, based on the minutiae points of the fingerprints. We can identify at least 40minutiae points on a fingerprint, depending on its quality. In general, the number of the minutiae pointsvaries 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 usedin practice have millions of fingerprints. To reduce the search time and complexity, we first propose toclassify 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. Afterthat, it is matched at a finer level to all the fingerprints of the subset. The FBI in the United Statesrecognize eight different types of patterns [5]. For example, we have an input fingerprint and we wantto 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  ×  =  operations. But, if we use aclassification with eight types (each subset has the same number of fingerprints / =  ) wewill have at least (  +  ) ×  =  calculations. This is because we will first compare the inputfingerprint 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 , %. The classification of the fingerprints ispreferred to have more than three types of subsets. This is because a higher accuracy is achieved. Sucha 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: µ   A : X  → [ ,  ] where µ   A (  x ) ∈ [ ,  ] 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 µ   N  (  x ) =   for x < m − α   x − m + α α  for x ∈ [ m − α  , m  ]  for x ∈ [ m , m  ] m + β  −  x β  for x ∈ [ m , m + β   ]  for 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 if X is A thenY is B is represented by its conditional possibility distribution ( [14], [15]) π  Y  /  X  : π  Y  /  X  ( v , u ) = µ   A ( u ) → µ   B ( v ) , ∀ u ∈ U  , ∀ v ∈ V  where → is an implication operator ( [2]) and µ   A and µ   B are the membership functions of the fuzzy sets  A and B , respectively. One of the most important implication is Lukasiewicz implication [2], I   L (  x ,  y ) = min (  −  x +  y , ) . 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 of conditions are satisfied  ) THEN  ( a set of consequences are inferred  ) where the antecedents and the consequences of the rules are associated with fuzzy concepts (linguisticterms). The most known systems are: Mamdani, Tsukamoto, Sugeno and Larsen which work with crispdata 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: ã the linguistic terms (or values), that are represented by trapezoidal fuzzy numbers ã Lukasiewicz implication , which is used to represent the rules ã the crisp control action of a rule , computed by Middle-of-Maxima method ã 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 ischaracterized by: ã a set of linguistic variable A , having as domain an interval I   A = [ a  A , b  A  ] ã n  A linguistic values A  ,  A  ,...,  A n  A for each linguistic variable A ã membership function µ    A i (  x ) for each value A i , where i ∈ { ,,..., n  A } and x ∈ I   A .According to the structure of a FLC, the following steps are necessary in order to work with oursystem. Firing levels We consider an interval input [ a , b  ] with a  A ≤ a < b ≤ b  A . The membership function of  A i is modified( [10]) by membership function of  [ a , b  ] as follows ∀  x ∈ I   A , µ   A i (  x ) = min ( µ    A i (  x ) , µ  [ a , b  ] (  x )) where µ  [ a , b  ] (  x ) =   if x ∈ [ a , b  ]  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 A i is givenby: µ   A i = max  { µ   A i (  x ) |  x ∈ [ a , b  ] } . The firing level µ   A i , generated by a linguistic input value A ′ i is µ   A i = max  { min  { µ    A i (  x ) , µ   A ′ i (  x ) }|  x ∈ I   A } . The firing level µ   A i , generated by a crisp value x  is µ    A i (  x  ) . Fuzzy inference We consider a set of fuzzy control rules  R i : if X   is A  i and  ... and X  r  is A r i then Y is C  i where the variables X   j , j ∈ { ,,..., r  } , and Y  have the domains U   j and V, respectively. The firing levelsof the rules, denoted by { α  i } , are computed by α  i = T  ( α   i ,..., α  r i ) where T  is a t-norm and α   ji is the firing level for A  ji , j ∈ { ,,..., r  } . The conclusion inferred from therule R i , using the Lukasiewicz implication is C  ′ i ( v ) = I  ( α  i , C  i ( v )) , ∀ v ∈ V  .
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