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Sample Exercises

Chapter 6: Multibiometrics

1. List specific advantages of multibiometric systems compared to unibio-metric systems.

2. What are the advantages of score level fusion over fusion at the sensor,feature, and decision levels?

3. Consider the general minimum error rate Bayes decision rule in equation(6.6). Suppose that we assume the following: (i) the match scores of differ-

ent matchers are statistically independent, i.e.,p(s|j) =M

m=1p(sm|j),j =0, 1, and (ii) the prior probabilities of the genuine and impostor classesare equal, i.e., P(1) =P(0) = (1/2).

(a) Show that the minimum error rate Bayes decision rule can be sim-plified to the product rule.

(b) Let us further assume that the marginal posterior probabilities donot deviate dramatically from the prior probabilities for each class,i.e.,P(j |sm) = P(j)(1 + j,m), where j,m is a constant, 0 , else assign (s1

, s2

) to the impostorclass. Here, (,). Show that the above sum of scoresfusion rule will lead to a higher EER compared to the best individualmatcher. Explain why the above result is intuitively correct, thoughbiometric fusion is generally expected to lead to a better recognitionperformance than each of the individual matchers.

(d) Consider the weighted sum rule fusion that determines the fused scoreas follows: h(s1, s2) = (wg1(s1) + (1 w)g2(s2)), where w [0, 1].Let the fused score obtained using weighted sum rule be used in thedecision rule shown in (c). What should be the minimum value ofw,so that the EER of the multibiometric system is lower than the EERof the best individual matcher?

5. Lets1 and s2 be the match scores output by a fingerprint matcher and aface matcher, respectively. Suppose that the match scores have the follow-ing conditional distributions: p(s1|genuine) N(71, 9), p(s1|impostor) N(50, 9), p(s2|genuine) N(56, 9), and p(s2|impostor) N(50, 9), whereN(, 2) represents a Gaussian distribution with mean and standarddeviation 2. Further assume that the two matchers are statistically in-dependent. Show that the likelihood ratio-based fusion rule based on theabove match score distribution is equivalent to the weighted sum rule.(Hint: Apply logarithm to equation (6.13).)

6. Table 1 provides the match scores output by two iris matchers operatingin the identification mode. The match scores of the first matcher range

from 0 to 100, while the match scores from the second matcher rangefrom 0 to 1. What will be the decision output by a multibiometric systemthat employs the following fusion rules to combine the results of the twoiris matchers: (a) sum of scores fusion with min-max normalization, (b)highest rank method, (b) Borda count rank fusion method, and (d) logisticregression rank fusion method (with weights w1 = 0.25 and w2 = 0.75).

7. Why is it important to normalize the scores before performing a sum scorefusion? List the different factors that affect the choice of a normalizationscheme.

8. In the case of identification, is it always better to perform score level fusioncompared to rank level fusion. Construct an example where this is notthe case.

9. What is the effect of using of AND, OR, and Majority rules for decisionlevel fusion have on the FMR and FNMR of the recognition system?

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Table 1: Match scores output by two iris matchers operating in the identificationmode.

User Identity Match score from Match score fromIris Matcher 1 Iris Matcher 2

A 35 0.20B 15 0.40C 48 0.30D 77 0.35E 51 0.60F 38 0.10G 22 0.05H 49 0.45I 25 0.25J 54 0.18

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