Dynamic Score Selection for Fusion of Multiple Biometric ... - CiteSeerX

Dynamic Score Selection for Fusion of Multiple Biometric Matchers Roberto Tronci, Giorgio Giacinto and Fabio Roli Department of Electrical and Electronic Engineering University of Cagliari I-09123 Cagliari { roberto.tronci, giacinto, roli}@diee.unica.it

Abstract A biometric system for user authentication produces a matching score representing the degree of similarity of the input biometry with the set of templates for that user. If the score is greater than a prefixed threshold, then the user is accepted, otherwise the user is rejected. Typically the performance are evaluated in terms of the Receiver Operating Characteristic (ROC) curve, and the Equal Error Rate (EER). In order to increase the reliability of authentication through biometrics, the combination of different biometric systems is currently investigated by researchers. While a number of “fusion” algorithms have been proposed in the literature, in this paper we propose a theoretical analysis of a novel approach based on the “dynamic selection” of matching scores. Such a selector aims at choosing, for each user to be authenticated, just one of the scores produced by the different biometric systems available. We show that the “dynamic selection” of matching scores can provide a better ROC than those of individual biometric systems. Reported results on the FVC2004 dataset confirm the theoretical analysis, and show that the proposed “dynamic selection” approach is more effective when low quality scores are used.

1. Introduction Classifier ensemble approaches are widely used in many applications as they avoid the choice of the “best” classifier, and typically provide better performance than those provided by individual classifiers [7]. Ensemble approaches also allow “fusing” classifiers based on different input sources, so that complementary information can be exploited, and the resulting classifier is robust with respect to noise [7]. For this reason, they are widely used in security applications, such as biometric authentication systems, where the goal is to authorise the access to a protected resource by using one or more biometric traits to validate the

identity of the person. At present, there is an increasing interest in multi-biometrics, i.e. the combined use of different biometrics and/or processing algorithms, as in many application the performance attained by individual sensors or processing algorithms does not provide the necessary reliability [13]. Biometric authentication is performed by the so-called matchers, i.e. algorithms that compare the acquired biometric to those stored during the enrolment phase. The output of matchers is a matching score, i.e., a measure stating how much the acquired biometry is likely to be the stored biometry. When a threshold is set, users with a matching score that exceeds the threshold are accepted (i.e. assigned to the so-called genuine class), otherwise are rejected (i.e. assigned to the so-called impostor class). Combination of multiple biometric systems is typically performed at the score level, and different fusion techniques (e.g., average, product, sum, etc) have been successfully applied [6] [10]. “Dynamic selection” mechanisms have been proposed as an alternative to the combination by fusion techniques [3]. For each authentication task, “dynamic selection” chooses the most suited matching scores among those produced by the available systems. In [3] the authors proposed an ideal selection mechanism tailored to biometric systems, that allows attaining smaller error rates, i.e. the False Matching Rate (FMR, i.e. the percentage of impostors whose score is greater than the decision threshold) and the False Non-Matching Rate (FNMR, i.e. the percentage of genuines whose score is smaller than the decision threshold). As these errors vary according to the value of the chosen threshold, they are usually reported graphically in the Receiver Operating Characteristic (ROC) curve, where the value of 1 - FNMR is plotted against the value of FMR. To compare different ROC curves the Area Under the Curve (AUC) can be used: the greater the value of AUC, the better the performance. However, usually biometric systems are compared in terms of Equal Error Rate (EER), i.e., the point of ROC curve where F M R = F N M R, the smaller the EER, the better the performance.

In this paper we show that: i) for any value of FMR the ideal selector exhibits a value of FNMR smaller or equal than that of any matcher used in the ensemble; ii) the ideal selector always outperforms in terms of Area Under the ROC Curve (AUC) any linear combination of two matchers. In addition, we propose a novel algorithm based on the ideal selector, that allows attaining better performance in terms of EER and AUC than those of individual matchers, and fusion techniques. In particular the proposed algorithm is more effective when low quality scores are used. The rest of the paper is organised as follows: Section 2 presents the ideal score selector and the theoretical evaluation of its performance. Section 3 proposes the novel algorithm based on the ideal selector. The experimental results are presented in Section 4 and our conclusions are outlined in Section 5.

2. Theoretical evaluation of ideal score selection The aim of multiple biometrics combined at the score level is to produce new scores whose distributions for genuine and impostor users exhibit a higher degree of separation than those produced by individual matchers. Under some hypothesis, and having defined a suitable separation measure, it has been shown that the larger separation between the distributions, the greater the AUC of the associated ROC [1]. Thus, by varying the decision threshold, a better tradeoff between the FMR, and the FNMR can be attained. The proposed selection algorithm is based on the above goal of combination at the score level. Let M = {M1 , M 2, . . . Mj . . . MN } be a set of N matchers and U = {ui } be the set of users, let also fj (·) be the function associated to matcher Mj that produces a score for each user ui , sji = fj (ui ). Let us denote with th a decision threshold so that users whose score is greater than th are assigned to the genuine class, while users whose score is smaller than th are assigned to the impostor class. Let us recall that for a given value of th, the two types of error, i.e. the FMR, and the FNMR are defined as follows:

for user ui is computed as follows:  max{sji } if ui is a genuine user si,∗ = min{sji } if ui is an impostor user

(3)

Let us now prove that the above defined ideal selector allows attaining smaller errors than those of the individual matchers. In particular, for any given value of FMR, the ideal selector provides a value of FNMR smaller than those provided by each individual matcher. Let us define smax = max [sj ] j

The following property holds for the distribution of the maximum of N random variables sj [12] P (smax ≤ th)

= P (s1 ≤ th; s2 ≤ th; . . .) ≤ ≤ P (sj ≤ th) , ∀j

By recalling the definition of FNMR, and the definition of the ideal selector that always select the maximum score for genuine users, the above property can be rewritten as follows F N M R∗ (th) ≤ F N M Rj (th) (4) Analogously, let smin = min [sj ] j

The following property holds for the distribution of the minimum of N random variables sj [12] P (smin > th)

= P (s1 > th; s2 > th; . . .) ≤ ≤ P (sj > th) , ∀j

Thus, the above property can be rewritten as follows: F M R∗ (th) ≤ F M Rj (th)

(5)

Let us now prove that for any value of FMR, the ideal selector exhibit a value of FNMR smaller than that provided by any individual matcher. Let th0 and th00 be two threshold values such that F M R∗ (th0 ) = F M Rj (th00 ) we shall prove that

F M Rj (th) = P (sj > th|sj ∈ impostor) F N M Rj (th) = P (sj ≤ th|sj ∈ genuine)

(1) F N M R∗ (th0 ) ≤ F N M Rj (th00 ) (2)

In order to produce distribution of genuine and impostor scores that allows attaining smaller errors than those of individual matchers, we have defined the ideal score selector as the ideal matcher selector that selects the maximum score for genuine users and the minimum score for impostor users [3, 4]. In other words the output si,∗ of the ideal selector

∀j

From equations (1) and (5), it is easy to see that th0 ≤ th00 , so that the proof can be subdivided into two cases: 1. th0 = th00 . This is the simplest case as from equation (4) we obtain F N M R∗ (th0 ) = F N M R∗ (th00 ) ≤ F N M Rj (th00 )

2. th0 < th00 . Equation (2) implies that F N M R∗ (th0 ) ≤ F N M R∗ (th00 ). By recalling equation (5), we also have that F M R∗ (th00 ) ≤ F M Rj (th00 ), consequently 0

00

Let us consider the linear combination flc (·) = f1 (·) + α · f2 (·), where the fused output is computed as follows: ξp = x1p + α · x2p ηq = yq1 + α · yq2

00

F N M R∗ (th ) ≤ F N M R∗ (th ) ≤ F N M Rj (th ) Thus we have proved that the proposed ideal selector always perform better than any matcher in the ensemble. By recalling that the Area Under the ROC Curve is computed as Z AU C = (1 − F N M R(th))dF M R(th) we can conclude that the AUC of the ideal selector is always greater than that of any matcher in the ensemble. In practise the AUC can be estimated by the WilcoxonMann-Whitney (WMW) statistic [9], as the integral value of the AUC is equivalent to the WMW statistic [5]. The AUC computed according to the WMW statistic is defined as follows. Given a matcher M , let us divide all the scores {si } obtained for all the {ui } users into two sets: {xp } the scores that belong to the genuine users and with {yq } the scores that belong to the impostor users. The AUC can be computed as: Pn+ Pn− p=1 q=1 I(xp , yq ) (6) AU C = n+ · n− where n+ is the number of genuine users and n− is the number of impostors, and the function I(xp , yq ) is1 :  1 xp > yq I(xp , yq ) = 0 xp < yq Moreover the AUC can be statistically interpreted as follows: given two randomly chosen users, one belonging to the set of the genuine users and belonging to the set of the impostor users, the AUC is the probability P (xp > yq ), i.e. the probability of correct pair-wise ranking [5]. The WMW statistics can be used to compare the AUC attained by the ideal selector with the AUC attained by a linear combiner. To this aim, let us consider two matchers, M1 and M2 , and all the possible pairs {{x1p , yq1 }, {x2p , yq2 }} obtained from these matchers. Let us divide these pairs into four subsets  Suv = (p, q)|I(x1p , yq1 ) = u and I(x2p , yq2 ) = v where u, v ∈ {0, 1}. Thus, S11 is made up of all the pairs where x1p > yq1 and x2p > yq2 , S00 is made up of all the pairs where x1p < yq1 and x2p < yq2 , S10 is made up of all the pairs where x1p > yq1 and x2p < yq2 , and S01 is made up of all the pairs where x1p < yq1 and x2p > yq2 . 1 for

discrete values I(xp , yq ) = 0.5 if xp = yq

The AUC attained by the linear combination, say AU Clc can be computed by estimating the contribution of the pairs of outputs belonging to each of the four subsets Suv , u, v ∈ {0, 1} [11]. All the pairs belonging to S11 do not depend on the value of α, as ξp > ηq is always verified, so that the contribution to the AU Clc from the pairs belonging to S11 is equal to c(S11 ) (where with c(S11 ) we denote the cardinality of the set S11 ). Similarly, it is easy to see that all the pairs belonging to S00 do not depend on the value of α, as ξp < ηq is always verified, so that the pairs belonging S00 give a nil contribution to the AU Clc . All the pairs belonging to S10 depend on α, and their contribution to the AU Clc is equal to c(S10 ) only if there is a value of α such that for all the pairs x1p + α · x2p > yq1 + α · yq2 . The same reasoning can be used to estimate the contribution to the AU Clc of pairs in S01 . It is worth noting that the value of α such that the contributions of S10 and S01 are equal respectively to c(S10 ) and c(S01 ) may not exists. Summing up, the maximum attainable value of AUC for the linear combination can be computed as follows: AU Clc =

c(S11 ) + c(S10 ) + c(S01 ) n + · n−

(7)

Let us now consider the ideal selector defined according to equation (3), whose outputs are: ϕp = max {x1p , x2p } ψq = min {yq1 , yq2 }

(8)

The AUC obtained with the ideal selector, say AU Csel , can be computed as follows. It is easy to see that for all the pairs belonging to S11 the following relation holds: ϕp > ψq . Thus the contribution to the AU Csel of S11 is equal to c(S11 ), as for the optimal linear combiner. By examining the pairs belonging to S00 we have to take into account two cases. One case is when ϕp and ψq come from the same matcher. Thus it follows that ϕp < ψq . The other case is when ϕp and ψq come from different matchers. In this case, two subcases have to be considered. If ϕp = x1p and ψq = yq2 , then the following majority chain holds yq1 > x1p > x2p . In addition, if x1p > yq2 holds, then ϕp > ψq . Let β01 be the ensemble of those pairs that satisfy the above relations: β01 = {(p, q) ∈ S00 |x1p > yq2 , ϕp = x1p , ψq = yq2 } It is easy to see that the contribution to the AUC of the pairs belonging to β01 is equal to c(β01 ). Analogously if ϕp = x2p

and ψq = yq1 , then the following majority chain holds yq2 > x2p > x1p . In addition, if x2p > yq1 holds, then ϕp > ψq . Let β10 be the ensemble of those pairs that verify the above relations: β10 = {(p, q) ∈ S00 |x2p > yq1 , ϕp = x2p , ψq = yq1 } It is easy to see that the contribution to the AUC of the pairs belonging to β10 is equal to c(β10 ). For the subsets S10 and S01 using majority chains it is easy to see that the following relation holds for every pair. ϕp > ψq . As a consequence the contribution of S10 and S01 to the AUC is always c(S10 ) + c(S01 ), while for the optimal linear combination this is an upper bound. Summing up, the AUC of the ideal selector can be computed as follows: c(S11 ) + c(S10 ) + c(S01 ) + c(β10 ) + c(β01 ) AU Csel = n+ · n− (9) By comparing equations (7) and (9), it easy to see that AU Csel ≥ AU Clc

(10)

3. An algorithm for Dynamic Score Selection In Section 2 we have introduced the ideal selector. We have shown that it provides a better ROC than those provided by individual matchers. Such an ideal selector, however, is based on the knowledge of the state of nature of each user, i.e. if the user is a genuine or an impostor. In this section we introduce an algorithm for Dynamic Score Selection (DSS), that selects the scores according to equation (3), where the most likely state of nature for each user is estimated according to a confidence measure. It is worth noting that the proposed algorithm doesn’t provide a label for each user, as the estimated state of nature is used only to select one of the available scores provided by the matchers in the ensemble. From the literature it is clear that is difficult to assign probabilistic meaning to the matching scores [13], mainly because usually the number of users is small, and so a small set of scores is available. On the other hand, it is possible to build a vector space where, for each user, the vector components are the scores assigned to that user by the ensamble of matchers. Thus, given an ensamble made up of N matchers, for each user ui , we use the following feature vector vi = {s1i , s2i . . . sji . . . sN i }. In this way a classifier can be trained on this N -dimensional vector space, using a training set of genuine and impostor users. The classifier can than be used to estimate the state of nature of the users to be authenticated. After the state of nature is computed, the user’s score is selected according to the idea of the algorithm of the ideal score selector. Summing up, the proposed algorithm for Dynamic Score Selection (DSS) is made up of the following steps:

1. Train a classifier C in the N -dimensional vector space built using a set of training scores 2. Classify the user to be authenticated with the classifier C 3. Select the score ssel based on the output class of classifier C as follows  maxj (sji ) if class = genuine ssel = minj (sji ) if class = impostor

4. Experimental results Experiments have been performed using the scores produced by a large number of matchers during the third Fingerprint Verification Competition (FVC2004) [8] [2]. The competitors were divided in two categories Open and Light. The Open category is composed by 41 matching algorithms, while the Light category is composed by 21 algorithms with restricted computing and memory usage. The fingerprint images consists of four different databases, three acquired with different sensors and one created with a synthetic fingerprint generator. For each sensor and for each matching algorithm, a set of scores is available related to authentication attempts of genuine users and authentication attempts of impostors. For the details on how the scores where obtained and normalised, the reader is referred to [8]. This database is not freely available, so our algorithms were executed at the Biometric Systems Lab (University of Bologna, Italy) which organises the competition. In order to create training sets for the DSS algorithm and the trainable fusion rules used for comparison, we had randomly divided the set of users into four subsets of the same size, where each of the four subsets has been used for training, while the remaining three subsets have been used for testing. Performance are than assessed by averaging over these four experiments. We computed an exhaustive multi-algorithmic experiment for each sensor, combining two matchers at a time. We selected the matchers to be combined by using the following two measures: the EER and the d’. The d0 is a measure of separability of the distributions of genuine and impostor scores, derived from Signal Detection Theory [1]. |µGen − µImp | d0 = q 2 σ2 σGen + Imp 2 2 the greater the d0 , the better the performance. We sorted the pairs of matchers both by the mean value of d0 and by the mean value of EER computed for each pair. Then, for each sensor, we considered the first ten pairs with the largest value of d0 , and the first ten pairs with the largest

value of EER. In addition, we also considered the last 10 pairs with the smallest value of d0 , and EER. We compared the ideal selector and the DSS algorithm (Section 2, and 3, respectively), to the optimal linear combiner (see Section 2), the Mean rule, the Product rule, and a linear combination whose weights are computed by the Linear Discriminant Analysis (LDA). For the DSS we used as classifiers the k-Nearest Neighbour, the Quadratic Bayes and Parzen Windows. The performance of the ideal linear combiner are estimated by performing an exhaustive search on the value of the combination weight α, using values between 0 and 100 with a step of 0.01. Results are assessed in terms of the EER and the AUC. Tables 1 and 2 present the results of the performed experiments on the Open dataset. When the performance of the pair of single matchers is good (i.e., large d0 and small EER) typically the Mean rule and the linear combination based on the LDA provide better performance than those provided by the DSS. When the performance of the pair of single matchers is weak (i.e., small d0 and large EER) the DSS outperforms the other combination rules. In particular DSS always outperforms the optimal linear combiner in terms of EER. Table 1. Open dataset, d0 sorted. Mean and standard deviation among 4 sensors and 10 pairs of matchers. large d’

EER

AUC

Average Single Ideal Selector Optimal Linear Mean Product LDA DSS kNN DSS QB DSS Parzen

0.0308 (±0.0130) 0.0046 (±0.0056) 0.0154 (±0.0081) 0.0183 (±0.0098) 0.0302 (±0.0164) 0.0161 (±0.0090) 0.0224 (±0.0118) 0.0231 (±0.0127) 0.0228 (±0.0121)

0.9882 (±0.0068) 0.9993 (±0.0017) 0.9970 (±0.0040) 0.9965 (±0.0041) 0.9854 (±0.0088) 0.9937 (±0.0079) 0.9885 (±0.0063) 0.9886 (±0.0062) 0.9883 (±0.0062)

small d’

EER

AUC

Average Single Ideal Selector Optimal Linear Mean Product LDA DSS kNN DSS QB DSS Parzen

0.3302 (±0.0545) 0.0486 (±0.0492) 0.2665 (±0.1020) 0.2777 (±0.0851) 0.2797 (±0.0948) 0.2749 (±0.0700) 0.2502 (±0.0912) 0.2564 (±0.0876) 0.2516 (±0.0922)

0.7174 (±0.0541) 0.9805 (±0.0200) 0.8123 (±0.0792) 0.7952 (±0.0758) 0.7782 (±0.0920) 0.7984 (±0.0765) 0.8075 (±0.0875) 0.7995 (±0.0830) 0.8075 (±0.0882)

Tables 3 and 4 present the results of the performed experiments on the Light dataset. When the performance of

Table 2. Open dataset, EER sorted. Mean and standard deviation among 4 sensors and 10 pairs of matchers. large EER

EER

AUC

Average Single Ideal Selector Optimal Linear Mean Product LDA DSS kNN DSS QB DSS parzen

0.3464 (±0.0396) 0.0576 (±0.0505) 0.2898 (±0.0768) 0.2927 (±0.0653) 0.3077 (±0.0679) 0.2886 (±0.0579) 0.2743 (±0.0688) 0.2798 (±0.0599) 0.2731 (±0.0720)

0.6987 (±0.0378) 0.9760 (±0.0221) 0.7891 (±0.0526) 0.7777 (±0.0578) 0.7435 (±0.0618) 0.7760 (±0.0510) 0.7854 (±0.0695) 0.7745 (±0.0551) 0.7853 (±0.0678)

small EER

EER

AUC

Average Single Ideal Selector Optimal Linear Mean Product LDA DSS kNN DSS QB DSS parzen

0.0219 (±0.0111) 0.0038 (±0.0049) 0.0133 (±0.0090) 0.0140 (±0.0094) 0.0202 (±0.0120) 0.0150 (±0.0098) 0.0175 (±0.0108) 0.0176 (±0.0120) 0.0175 (±0.0108)

0.9914 (±0.0052) 0.9991 (±0.0017) 0.9968 (±0.0038) 0.9966 (±0.0039) 0.9896 (±0.0068) 0.9935 (±0.0074) 0.9909 (±0.0061) 0.9914 (±0.0062) 0.9910 (±0.0060)

the pair of single matchers is good, the Mean rule provides better performance than those provided by the other combination rules. On the other hand, when the performance of the pair of single matchers is weak, the DSS always outperforms the other combination rules, and in particular the optimal linear combiner. Among all the combination rules, the Product rule provided the worst performance, while the DSS exhibits its effectiveness when the performance of the pairs of matchers are weak. Finally, reported results show that the ideal score selector always outperforms the optimal linear combiner, thus confirm that the selection strategy is an alternative to linear combination strategies.

5. Conclusions In this paper a theoretical evaluation of score selection mechanism is presented. In particular the aim of score selection is to provide smaller errors than individual matchers, and consequently to obtain a better ROC and a greater AUC. We also show that the proposed ideal selection attains a greater AUC than that of the optimal linear combiner. Reported experiments showed that the proposed DSS algorithm outperforms the other fusion techniques when low

Table 3. Light dataset, d0 sorted. Mean and standard deviation among 4 sensors and 10 pairs of matchers.

Table 4. Light dataset, EER sorted. Mean and standard deviation among 4 sensors and 10 pairs of matchers.

large d’

EER

AUC

large EER

EER

AUC

Average Single Ideal Selector Optimal Linear Mean Product LDA DSS kNN DSS QB DSS Parzen

0.0406 (±0.0132) 0.0119 (±0.0079) 0.0259 (±0.0119) 0.0272 (±0.0118) 0.0341 (±0.0146) 0.0310 (±0.0119) 0.0316 (±0.0150) 0.0285 (±0.0140) 0.0314 (±0.0150)

0.9842 (±0.0062) 0.9972 (±0.0025) 0.9921 (±0.0048) 0.9917 (±0.0048) 0.9845 (±0.0074) 0.9899 (±0.0055) 0.9866 (±0.0069) 0.9882 (±0.0064) 0.9868 (±0.0069)

Average Single Ideal Selector Optimal Linear Mean Product LDA DSS kNN DSS QB DSS parzen

0.3487 (±0.0646) 0.1029 (±0.0933) 0.2299 (±0.1481) 0.2348 (±0.1420) 0.4522 (±0.0787) 0.2228 (±0.1548) 0.1822 (±0.0942) 0.1925 (±0.0925) 0.1756 (±0.0854)

0.7004 (±0.0728) 0.9082 (±0.0914) 0.8298 (±0.1412) 0.8183 (±0.1342) 0.5953 (±0.0610) 0.8259 (±0.1450) 0.8394 (±0.0964) 0.8315 (±0.0875) 0.8458 (±0.0862)

small d’

EER

AUC

small EER

EER

AUC

Average Single Ideal Selector Optimal Linear Mean Product LDA DSS kNN DSS QB DSS Parzen

0.3186 (±0.0843) 0.1070 (±0.0881) 0.2008 (±0.1634) 0.2287 (±0.1602) 0.4086 (±0.0849) 0.2095 (±0.1656) 0.1590 (±0.1072) 0.1594 (±0.1109) 0.1609 (±0.0924)

0.7063 (±0.0788) 0.9012 (±0.0871) 0.8367 (±0.1488) 0.8125 (±0.1419) 0.5985 (±0.0839) 0.8288 (±0.1495) 0.8538 (±0.1071) 0.8559 (±0.1047) 0.8507 (±0.0904)

Average Single Ideal Selector Optimal Linear Mean Product LDA DSS kNN DSS QB DSS parzen

0.0409 (±0.0131) 0.0144 (±0.0077) 0.0285 (±0.0119) 0.0307 (±0.0113) 0.0382 (±0.0146) 0.0354 (±0.0114) 0.0358 (±0.0134) 0.0310 (±0.0129) 0.0353 (±0.0135)

0.9833 (±0.0063) 0.9968 (±0.0025) 0.9911 (±0.0052) 0.9905 (±0.0052) 0.9818 (±0.0079) 0.9877 (±0.0059) 0.9841 (±0.0070) 0.9867 (±0.0064) 0.9845 (±0.0069)

quality scores are used.

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