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

32 3. BAG OF FACIAL SOFT BIOMETRICS FOR HUMAN IDENTIFICATION3.5 Reliability and scaling lawsIn this section we seek to provide insightful mathematical analysis of reliability of general softbiometric systems (SBSs), as well as to study error events and underlying factors. Furthermorewe will proceed to articulate related closed form expressions for the single and averaged SBS behaviorby concisely describing the asymptotic behavior of pertinent statistical parameters that areidentified to directly affect performance. Albeit its asymptotic and mathematical nature, the approachaims to provide simple expressions that can yield insight into handling real life surveillancesystems.Along with the general setting from section 3.1, in which we referred to the randomly extractedset ofnpeople, out of which one person is picked for identification, we here introduce the notationof v for such a n-tuple of people. Furthermore we denote by v(i), i = 1,...,n the i-th candidatebelonging to the specific group v.3.5.1 Error events, interference, and effective categoriesLet the randomly chosen subject for identification, belong in category φ ∈ Φ. The SBS firstproduces an estimates ̂φ of φ, and based on this estimate, tries to identify the chosen subject, i.e.,tries to establish which candidate in v corresponds to the chosen subject. An error occurs whenthe SBS fails to correctly identify the chosen subject, confusing him or her with another candidatefrom the current n-tuple v. An error can hence certainly occur when the category is incorrectlyestimated 1 , i.e., when ̂φ ≠ φ, or can possibly occur when the chosen subject v(i) interferes withanother subject v(j) from the authentication group v, i.e., when the chosen subject is essentiallyindistinguishable to the SBS from some other candidates inv.. We recall that interference occurs,whenever two or more subjects belong in the same category.For a given v, let S φ ⊂ v be the set of subjects in v that belong in a specific category φ.Furthermore let S 0 denote the set of people in v who do not belong in any of the categories inΦ. We here note that no subject can simultaneously belong to two or more categories, but alsonote that it is entirely possible that |S φ | = 0, for some φ ∈ Φ. Hence an error is caused due toestimation noise (resulting in ̂φ ≠ φ), due to interference (subject indistinguishable from othersubjects in v), or when the chosen candidate belongs in S 0 (system is not designed to recognizethe subject of interest).For a given v, letF(v) := |{φ ∈ Φ : |S φ | > 0}|denote the number of effective categories, i.e., the number of (non-empty) categories that fullycharacterize the subjects in v. For notational simplicity we henceforth write F to denote F(v),and we let the dependence on v be implied.3.5.1.1 The role of interference on the reliability of SBSsTowards evaluating the overall probability of identification error, we first establish the probabilityof error for a given set (authentication group) v. We note the two characteristic extremeinstances of F(v) = n and F(v) = 1. In the first case, the random n-tuple v over which identificationwill take place, happens to be such that each subject inv belongs to a different category, inwhich case none of the subjects interferes with another subject’s identification. On the other hand,the second case corresponds to the (unfortunate) realizations ofv where all subjects inv fall under1. this possibility will be addressed later on