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http://www.cubs.buffalo.eduPattern RecognitionClassification costBiometric system errorsROC curvehttp://www.cubs.buffalo.eduBayesian classification• Bayes classification rule: classify x to the class which has biggest posterior probability)|( xwPiiw2121:?)|()|( wwxwPxwP>• Bayes classification rule minimizes the total probability of misclassification. Cost of errors.• Errors happen when samples of class 1 are incorrectly classified to belong to class 2, and samples of class 2 are classified to belong to class 1.• The cost of making these errors can be different : 1λ- the cost of misclassifying samples of class 12λ- the cost of misclassifying samples of class 2http://www.cubs.buffalo.eduTotal cost (or risk) of classificationClassification algorithm splits feature space into two decision regions:1R- samples in this region are classified as being in class 12R- samples in this region are classified as being in class 2∫2)|(1Rdxwxp- the proportion of samples of class 1 being classified as class 2∫1)|(2Rdxwxp- the proportion of samples of class 2 being classified as class 1∫2)|()(11RdxwxpwP- the proportion of all input samples being class 1 but classified as being in class 2∫1)|()(22RdxwxpwP- the proportion of all input samples being class 2 but classified as being in class 1∫∫+=12)|()()|()(222111RRdxwxpwPdxwxpwPCostλλ-total costhttp://www.cubs.buffalo.eduBayesian classificationBayesian classifier is an optimal classifier minimizing total classification cost. Such classifier is possible only if we have full knowledge about class distributions.If then classify as class 1. )|()()|()(222111wxpwPwxpwPλλ>xIf then classify as class 2. )|()()|()(222111wxpwPwxpwPλλ≤xAlternatively, assuming non-zero terms, the class assignment is based ontesting whether or)()()|()|(112221wPwPwxpwxpλλ≤)()()|()|(112221wPwPwxpwxpλλ>Decision surface separates two decision regions.)()()|()|(112221wPwPwxpwxpλλ=)|()|(21wxpwxp- likelihood ratio)()()()|()|(112221wPwPwxpwxpλλ<>- likelihood ratio testhttp://www.cubs.buffalo.eduBiometric Application Types Verification System (1:1) Claim is made (enrollee identity)  User’s biometric is matched only with stored biometric of claimed enrollee The decision to accept claim is made using only one matching score Identification System (1:N) No claim about identity is made  User’s biometric is matched with stored biometrics of all enrolled persons The highest matching score determines the most probable enrollee The decision about accepting identification attempt is made based on the matching score for that enrollee (and optionally using other matching scores too)Screening  Matching against a watch list Opposite of verificationhttp://www.cubs.buffalo.eduEach verification attempt has two possibilities:1. Genuine event - input biometrics and stored biometrics from claimed identity belong to the same person.2. Impostor event - input biometrics is different from claimed identity biometrics.Errors in Verification Systems)event genuine|()( spspgen=The scores produced by matching algorithm will have distributions:)eventimpostor |()( spspimp=http://www.cubs.buffalo.eduErrors in Verification SystemsFAR and FRR are determined by the decision rule – accept or reject results of recognition.Usually FAR and FRR are defined using some threshold:)eventimpostor |()()(θθθ>==∫∞sPdsspFARimp)event genuine|()()(θθθ<==∫∞−sPdsspFRRgenhttp://www.cubs.buffalo.eduErrors in Verification Systemshttp://www.cubs.buffalo.edu-2 -1 0 1 2 3Scores00.511.52Probability)|( impxp)|( genxp)(tFAR )(tFRRt)(tFAR)(tFRRPerformance of Biometric Matchershttp://www.cubs.buffalo.eduEstimating FAR and FRR}impostor is |{}impostor is ,|{)|()(iiiiitxxxxtxxdximpxptFAR>≈=∫>In contrast to estimating pdf, FAR and FRR are easily estimated:}genuine is |{}genuine is ,|{)|()(iiiiitxxxxtxxdxgenxptFRR<≈=∫<http://www.cubs.buffalo.eduROC CurveROC curve connects and curves.)(θFAR )(θFRRNote that they both use same at the same time, so we are able to construct such plot.θhttp://www.cubs.buffalo.eduTypes of ROC CurveTaking and instead of and is reasonable if they are small.))(log(θFAR))(log(θFRR)(θFAR)(θFRRhttp://www.cubs.buffalo.eduTypes of ROC Curvehttp://www.cubs.buffalo.eduUsing ROC Curvehttp://www.cubs.buffalo.eduComparing ROC Curveshttp://www.cubs.buffalo.eduTrade-offsSelection of the operating point in a particular application is a trade-off between security and convenience.http://www.cubs.buffalo.eduIn Bayesian framework we want to minimize total cost:)()()genuine|()genuine()impostor|()impostor(θθθθFRRPCFARPCsPPCsPPCCostgenFRimpFAFRFA+=<+>=Correct setting of in verification application requires estimating θ)genuine(),impostor(,,21PPCCUsing FAR and FRRhttp://www.cubs.buffalo.eduConsider the problem of deploying biometric matcher for an amusement park admission)(99.)(2.)(99.1)(01.20)()(θθθθθθFRRFARFRRFARFRRPCFARPCCostgenFRimpFA×+×=××+××=+=Example20$=FAC- cost of accepting impostor to the park%1=impP- probability of impostor attempts1$=FRC- cost of rejecting genuine user%99=genP- probability of genuine attemptshttp://www.cubs.buffalo.edu)(99.)(2.θθFRRFARCost×+×=Face matcher ‘C’ better minimizes costhttp://www.cubs.buffalo.eduIf we had more impostor attempts, say , then matcher ‘ri’ would get lower cost)(9.)(2θθFRRFARCost×+×=%10=impPhttp://www.cubs.buffalo.eduComparing ROC CurvesArea under ROC curve (1-FRR vs FAR) represents the probability that random genuine score is higher than random impostor score.http://www.cubs.buffalo.eduComparing ROC Curves22nmnmdσσµµ+−=′Compare match and non-match score densities by d-prime method:http://www.cubs.buffalo.eduComparing ROC Curves)()(θθFARFRREER==Equal Error Rate (EER):at such as)()(θθFARFRR=θ)()(minθθθFARFRRTER+=Minimum Total Error Rate (TER):http://www.cubs.buffalo.eduErrors in Identification SystemsN people are enrolled in the database. The recognition algorithm performs N matchings with output scores:Nsss>>>...21(the scores are ordered by magnitude, but not by people id)The decision algorithm usually considered: • Accept class 1 if • Reject otherwiseNsss>>>>... and


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UB CSE 666 - Pattern Recognition

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