FACIAL SOFT BIOMETRICS - Library of Ph.D. Theses | EURASIP
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FACIAL SOFT BIOMETRICS - Library of Ph.D. Theses | EURASIP

FACIAL SOFT BIOMETRICS - Library of Ph.D. Theses | EURASIP FACIAL SOFT BIOMETRICS - Library of Ph.D. Theses | EURASIP

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36 3. BAG OF FACIAL SOFT BIOMETRICS FOR HUMAN IDENTIFICATION120Expected value of F, for N=1,2,...,ρ, ρ = 20,50,100,12010080mean of F604020ρ=120ρ=100ρ=50ρ=2000 20 40 60 80 100 120NFigure 3.5: E v [F] for ρ = 20,50,100,120, n ∈ [3,4,...,ρ]. We note that for ρ sufficiently largerthan n, then E v [F] ≈ n.Theorem 1 In the described operational setting of interest, under the interference limited anduniformity assumptions, the probability of error averaged over all possible n-tuples v, that isprovided by an SBS endowed with ρ categories, is given byE v [P(err)] = 1−F n−F+1(ρ−F)!(n−F)!n ∑ . (3.24)n i n−ii=1 (n−i)!(ρ−i)!Proof of Theorem 1: The proof is direct from Lemma 2 and from (3.23).Related examples are plotted in Figure 3.6.□0.180.160.14Average P(err) for N=[3,4,...,ρ] ρ = 20,50,100,120ρ=120ρ=100ρ=50ρ=20average Prob of error0.120.10.080.060.040.0200 20 40 60 80 100 120NFigure 3.6: E v [P(err)] for ρ = 20,50,100,120, n ∈ [3,4,...,ρ].We proceed to explore scaling laws of SBS employed for human identification.3.5.3 Asymptotic bounds on subject interferenceIn this section we seek to gain insight on the role of increasing resources (increasing ρ) inreducing the subject interference experienced by an SBS. Specifically we seek to gain insighton the following practical question: if more funds are spent towards increasing the quality of

an SBS by increasing ρ, then what reliability gains do we expect to see? This question is onlypartially answered here, but some insight is provided in the form of bounds on the different subjectinterferencepatterns seen by an SBS. The asymptotic bounds simplify the hard to manipulateresults of Lemma 3 and Theorem 1, and provide insightful interpretations. A motivating exampleis presented before the result.Example 5 Consider an SBS operating in the city of Berlin, where for a specific n, this systemallows for a certain average reliability. Now the city of Berlin is ready to allocate further funds,which can be applied towards doubling the number of categories ρ that the system can identify.Such an increase can come about, for example, by increasing the number and quality of sensors,which can now better identify more soft-biometric traits. The natural question to ask is how thisextra funding will help to improve the system? The bounds, when tight, suggest that doubling ρ,will result in a doubly exponential reduction in the probability that a specific degree of interferencewill occur.Further clarifying examples that motivate this approach are given in Section 3.5.3.1.The following describes the result.Lemma 5 Letnh := limρ→∞ ρ , (3.25)define the relative throughput of a soft biometrics system, and let F := fn, 0 ≤ f ≤ 1. Then theasymptotic behavior of P(F) is bounded asProof of Lemma 5: See Appendix A.3.5.3.1 Interpretation of bounds1− lim logP(f) ≥ 2−h(1+f). (3.26)ρ→∞ ρlogρLemma 5 bounds the statistical behavior of P(F) in the high ρ regime (for large values ofρ). To gain intuition we compare two cases corresponding to two different relative-throughputregimes. In the first case we ask that n is close to ρ, corresponding to the highest relativethroughputof r = 1, and directly get from (3.26) that d(h,f) := 2 − h(1 + f) = d(1,f) =1 − f, 0 < f < 1. In the second case we reduce the relative-throughput to correspond to thecase where n is approximately half of ρ (h = 1/2), which in turn gives d(h,f) = d( 1 2 ,f) =32 − f 2 , 0 < f < 1/2. As expected d(1 2 ,f) > d(1,f), ∀f ≤ 1 2 .Towards gaining further insight, let us use this same example to shed some light on howLemma 5 succinctly quantifies the increase in the probability that a certain amount of interferencewill occur, for a given increase in the relative-throughput of the soft biometrics system. To seethis, consider the case where there is a deviation away from the typical f = h by some smallfixed ɛ, to a new f = h − ɛ, and note that the value of ɛ defines the extend of the interference 7 ,because a larger ɛ implies a smaller f, and thus a reduced F for the same n. In the high relativethroughputcase of our example, we have that f = h−ɛ = 1−ɛ, and thus that d(1,1 −ɛ) = ɛ,which implies that the probability of such deviation (and of the corresponding interference) is inthe order of ρ −ρd(1,1−ɛ) = ρ −ρɛ . On the other hand, in the lower relative-throughput case wheref = h−ɛ = 1 2 −ɛ, we have thatd(1 2 , 1 2 −ɛ) = 5 4 + ɛ 2, which implies that the probability of the samedeviation in the lower throughput setting is in the order of ρ −ρd(1 2 ,1 2 −ɛ) = ρ −ρ(5 4 +ɛ 2 ) 0.

36 3. BAG OF <strong>FACIAL</strong> <strong>SOFT</strong> <strong>BIOMETRICS</strong> FOR HUMAN IDENTIFICATION120Expected value <strong>of</strong> F, for N=1,2,...,ρ, ρ = 20,50,100,12010080mean <strong>of</strong> F604020ρ=120ρ=100ρ=50ρ=2000 20 40 60 80 100 120NFigure 3.5: E v [F] for ρ = 20,50,100,120, n ∈ [3,4,...,ρ]. We note that for ρ sufficiently largerthan n, then E v [F] ≈ n.Theorem 1 In the described operational setting <strong>of</strong> interest, under the interference limited anduniformity assumptions, the probability <strong>of</strong> error averaged over all possible n-tuples v, that isprovided by an SBS endowed with ρ categories, is given byE v [P(err)] = 1−F n−F+1(ρ−F)!(n−F)!n ∑ . (3.24)n i n−ii=1 (n−i)!(ρ−i)!Pro<strong>of</strong> <strong>of</strong> Theorem 1: The pro<strong>of</strong> is direct from Lemma 2 and from (3.23).Related examples are plotted in Figure 3.6.□0.180.160.14Average P(err) for N=[3,4,...,ρ] ρ = 20,50,100,120ρ=120ρ=100ρ=50ρ=20average Prob <strong>of</strong> error0.120.10.080.060.040.0200 20 40 60 80 100 120NFigure 3.6: E v [P(err)] for ρ = 20,50,100,120, n ∈ [3,4,...,ρ].We proceed to explore scaling laws <strong>of</strong> SBS employed for human identification.3.5.3 Asymptotic bounds on subject interferenceIn this section we seek to gain insight on the role <strong>of</strong> increasing resources (increasing ρ) inreducing the subject interference experienced by an SBS. Specifically we seek to gain insighton the following practical question: if more funds are spent towards increasing the quality <strong>of</strong>

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