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
34 3. BAG OF FACIAL SOFT BIOMETRICS FOR HUMAN IDENTIFICATIONTable 3.3: Illustration of Example 2φ 1 φ 2 φ 3 F P(err|v)v 1 10 1 1 3 3/4v 2 4 4 4 3 3/4v 3 10 2 0 2 5/6Up to now the result corresponded to the case of specific realizations of v, where we saw thatthe probability of error for each realization of length n, was a function only of the realization ofF(v) which was a random variable describing the number of categories spanned by the specificgroup v. We now proceed to average over all such realizations v, and describe the overall probabilityof error. This analysis is better suited to evaluate an ensemble of distributed SBSs deployedover a large population. We henceforth focus on the interference limited setting 3 i.e., we make thesimplifying assumption that P φ = 0, φ > 0.Lemma 2 The probability of error averaged over all n-tuples v randomly drawn from a sufficientlylarge population, is given byand is dependent only on the first order statistics of F .E v [P(err|v)] = 1− E v[F(v)], (3.20)nProof: The proof follows directly from Lemma 1.□An example follows, related to the above.Example 3 Consider the case where the city of Nice installs throughout the city a number ofindependent SBSs 4 and is interested to know the average reliability that these systems will jointlyprovide, over a period of two months 5 . The result in Lemma 2 gives the general expression ofthe average reliability that is jointly provided by the distributed SBSs, indexed by n, for all n.Indexing bynsimply means that the average is taken over all cases where identification is relatedto a random set v of size n.We now proceed to establish the statistical behavior of F , including the meanE[F].Despite the fact that the probability of error in (3.24) is a function only of the first momentof F , our interest in the entire probability density function stems from our desire to be able tounderstand rare behaviors of F . More on this will be seen in the asymptotic analysis that willfollow.3. It is noted though that with increasing ρ, the probability of erroneous identification is, in real systems, expectedto increase. This will be considered in future work. Toward motivating the interference limited setting, we note thatsuch setting generally corresponds to cases where a very refined SBS allows forρto be substantially larger thann, thusresulting in a probability of interfence that is small but non negligible and which has to be accounted for.4. Independence follows from the assumption that the different SBSs are placed sufficiently far apart.5. In this example it is assumed that the number of independent SBSs and the time period are sufficiently large tojointly allow for ergodicity.
3.5.2 Analysis of interference patterns in SBSsGiven ρ and n < ρ, we are interested in establishing the probability P(F) that a randomlydrawn n-tuple of people will have F active categories out of a total of min(ρ,n) possible activecategories 6 . We here accept the simplifying assumption of uniform distribution of the observedsubjects over the categories ρ, i.e., thatP(v(i) ∈ S φ ) = 1 , ∀φ ∈ Φ, i ≤ n. (3.21)ρWe also accept that n < ρ. The following then holds.Lemma 3 Givenρandn, and under the uniformity assumption, the distribution ofF is describedbyF n−FP(F) =(ρ−F)!(n−F)! ∑ , (3.22)n i n−ii=1 (n−i)!(ρ−i)!where F can take values between 1 and n.Proof of Lemma 3: See Appendix A.Example 4 Consider the case where ρ = 9,n = 5,F = 3. Then the cardinality of the set of allpossible n-tuples that span F = 3 effective categories, is given by the product of the followingthree terms.– The first term is (ρ·(ρ−1)···(ρ−F +1)) = ρ!(ρ−F)!= 9·8·7 = 504 which describesthe number of ways one can pick whichF = 3 categories will be filled.– Having picked these F = 3 categories, the second term is (n·(n−1)···(n−F +1)) =n!(n−F)!= 5 · 4 · 3 = 60, which describes the number of ways one can place exactly onesubject in each of these picked categories.– We are now left with n−F = 2 subjects, that can be associated freely to any of the F = 3specific picked categories. Hence the third term is F n−F = 3 2 = 9 corresponding to thecardinality of {1,2,··· ,F} n−F .Motivated by Lemma 2, we now proceed to describe the first order statistics of F . The proofis direct.Lemma 4 Under the uniformity assumption, the mean ofF is given byE v [F(v)] =n∑FP(F) =F=1n∑∑ nF=1 i=1F n−F+1(ρ−F)!(n−F)!i n−i(n−i)!(ρ−i)!. (3.23)Remark 1 The event of no interference corresponds to the case whereF = n. Decreasing valuesof F nimply higher degrees of interference. An increasing ρ also results in reduced interference.Related cases are plotted in Figure 3.5.Finally, directly from the above, we have the following.6. Clarifying example: What is the statistical behavior of F that is encountered by a distributed set of SBSs in thecity of Nice?
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34 3. BAG OF <strong>FACIAL</strong> <strong>SOFT</strong> <strong>BIOMETRICS</strong> FOR HUMAN IDENTIFICATIONTable 3.3: Illustration <strong>of</strong> Example 2φ 1 φ 2 φ 3 F P(err|v)v 1 10 1 1 3 3/4v 2 4 4 4 3 3/4v 3 10 2 0 2 5/6Up to now the result corresponded to the case <strong>of</strong> specific realizations <strong>of</strong> v, where we saw thatthe probability <strong>of</strong> error for each realization <strong>of</strong> length n, was a function only <strong>of</strong> the realization <strong>of</strong>F(v) which was a random variable describing the number <strong>of</strong> categories spanned by the specificgroup v. We now proceed to average over all such realizations v, and describe the overall probability<strong>of</strong> error. This analysis is better suited to evaluate an ensemble <strong>of</strong> distributed SBSs deployedover a large population. We henceforth focus on the interference limited setting 3 i.e., we make thesimplifying assumption that P φ = 0, φ > 0.Lemma 2 The probability <strong>of</strong> error averaged over all n-tuples v randomly drawn from a sufficientlylarge population, is given byand is dependent only on the first order statistics <strong>of</strong> F .E v [P(err|v)] = 1− E v[F(v)], (3.20)nPro<strong>of</strong>: The pro<strong>of</strong> follows directly from Lemma 1.□An example follows, related to the above.Example 3 Consider the case where the city <strong>of</strong> Nice installs throughout the city a number <strong>of</strong>independent SBSs 4 and is interested to know the average reliability that these systems will jointlyprovide, over a period <strong>of</strong> two months 5 . The result in Lemma 2 gives the general expression <strong>of</strong>the average reliability that is jointly provided by the distributed SBSs, indexed by n, for all n.Indexing bynsimply means that the average is taken over all cases where identification is relatedto a random set v <strong>of</strong> size n.We now proceed to establish the statistical behavior <strong>of</strong> F , including the meanE[F].Despite the fact that the probability <strong>of</strong> error in (3.24) is a function only <strong>of</strong> the first moment<strong>of</strong> F , our interest in the entire probability density function stems from our desire to be able tounderstand rare behaviors <strong>of</strong> F . More on this will be seen in the asymptotic analysis that willfollow.3. It is noted though that with increasing ρ, the probability <strong>of</strong> erroneous identification is, in real systems, expectedto increase. This will be considered in future work. Toward motivating the interference limited setting, we note thatsuch setting generally corresponds to cases where a very refined SBS allows forρto be substantially larger thann, thusresulting in a probability <strong>of</strong> interfence that is small but non negligible and which has to be accounted for.4. Independence follows from the assumption that the different SBSs are placed sufficiently far apart.5. In this example it is assumed that the number <strong>of</strong> independent SBSs and the time period are sufficiently large tojointly allow for ergodicity.
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