Designing Classi ers for Fusion-Based Biometric Veri cation in .NET

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Designing Classi ers for Fusion-Based Biometric Veri cation
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Second-order probability: P(ui1 = 1, ui2 = 1|H1 ) P(ui1 = 1|H1 )P(ui2 = 1|H1 ), i1 = i2 , 1 i1 , i2 N. / Third-order probability: P(ui1 = 1, ui2 = 1, ui3 = 1|H1 )
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P(uij = 1|H1 ), (4.7)
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/ / i1 = i2 = i3 , 1 i1 , i2 , i3 N. kth-order probability: P(ui1 = 1, . . . , uik = 1|H1 )
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P(uij = 1|H1 ), P(uij = 1|H1 ),
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/ ij = il , 1 j, l, k, 1 ij , il N. We can quantify the conditional dependence through correlation coef cients of the normalized decisions. The normalized decisions with zero mean and unit variance, zh (i), are given by zh (i) = ui P(ui = 1|Hh ) , P(ui = 1|Hh )(1 P(ui = 1|Hh )) h = 0, 1. (4.8)
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The second and higher-order correlation coef cients of these normalized variables are de ned as follows. Second-order coef cient: h (i1 , i2 ) = E (zh (i1 )zh (i2 )) , i1 = i2 , 1 i1 , 12 N, h = 0, 1 / (4.9)
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u P(u = 1|H ) u P(u = 1|H ) i1 i2 i2 h h i |Hh . = E 1 2 P(uij = 1|Hh )(1 P(uij = 1|Hh ))
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Third-order coef cient: h (i1 , i2 , i3 ) = E (zh (i1 )zh (i2 )zh (i3 )) , i1 = i2 = i3 , 1 i1 , i2 , i3 N, h = 0, 1 / / 3 uij P(uij = 1|Hh ) j=1 |Hh . (4.10) = E 3 P(uij = 1|Hh )(1 P(uij = 1|Hh ))
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4.3 Analysis of Or rule Fusion
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kth-order coef cient: h (i1 , i2 , i3 , . . . , ik ) = E (zh (i1 )zh (i2 )zh (i3 ) zh (ik )) , ij = il , 1 j, l, k, 1 ij , il N. / (4.11)
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The sign of the kth-order authentic correlation coef cients at favorable authentic conditional dependence are obtained as follows by taking into account the desired inequalities in Eq. (4.7). For the sake of convenience in demonstration, let qj = 1 . P(uij = 1|H1 )(1 P(uij = 1|H1 )) (4.12)
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At favorable authentic conditional dependence, the sign of the second-order authentic correlation coef cient of the normalized decisions is negative, as demonstrated below. 1 (i1 , i2 ) = q1 q2 E ui1 ui2
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uim P(uin = 1|H1 ) |H1
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m=1 n=1,n = m /
+ q1 q2
P(uij = 1|H1 )
= q1 q2 E ui1 ui2 |H1 P(ui1 = 1|H1 )P(ui2 = 1|H1 )
since E ui1 ui2 |H1
P(uij = 1|H1 ) from Eq. (4.7)
(4.13)
The sign of the third-order authentic correlation coef cient is positive at favorable conditional dependence, as demonstrated below. 3 1 (i1 , i2 , i3 ) = q1 q2 q3 E(ui1 ui2 ui3 |H1 ) + 2 P(uij = 1|H1 )
j=1 3 3 3
q1 q2 q3 E(ui1 ui2 ui3 |H1 )+2
2 j=1 m=1,m = j n=1,n = m,j / /
E(uij uim |H1 )P(uin = 1|H1 )
P(uij = 1|H1 ) 3
j=1 j=1
P(uij = 1|H1 )
since E ui1 ui2 |H1
P(uij = 1|H1 ) from Eq. (4.7)
since E ui1 ui2 ui3 |H1
P(uij = 1|H1 ) from Eq. (4.7). (4.14)
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Designing Classi ers for Fusion-Based Biometric Veri cation
Following the same procedure, it can be shown that even-order authentic correlation coef cients are negative and odd-order authentic correlation coef cients are positive at favorable conditional dependence for the Or rule. For the two classi er case, the second-order correlation coef cient has the same sign as the Q statistic, which should be negative for authentics and has an upper limit of 1 for the optimal authentic conditional dependence, thus agreeing with our previous analysis of two classi er Or fusion [24]. It should be noted that the above constraints on the sign of the authentic correlation coef cients is only a suf cient condition for favorable conditional dependence. Since the probability of detection [Eq. (4.6)] is composed of all the kth-order probabilities, k = 2, 3, . . . , N, the relative weightage of these terms plays a role in the favorable conditional dependence for the Or rule. It is possible that there is favorable conditional dependence for the Or rule even when some of the even-order correlation coef cients are positive and some of the odd-order correlation coef cients are negative. The role of conditional dependence on the false acceptance rate (FAR) of the Or rule is now considered. For the Or rule fusion of N classi ers, the PFA is given by PFA = 1 P u = 0 0 . . . 0 |H0 . (4.15)