Technical Report
`
`UCAM-CL-TR-482
`ISSN 1476-2986
`
`Number 482
`
`Computer Laboratory
`
`Biometric decision landscapes
`
`John Daugman
`
`January 2000
`
`i
`
`Apple 1029
`Apple v. USR
`IPR2018-00810
`
`15 JJ Thomson Avenue
`Cambridge CB3 0FD
`United Kingdom
`phone +44 1223 763500
`
`http://www.cl.cam.ac.uk/
`
`

`

`c(cid:2) 2000 John Daugman
`
`Technical reports published by the University of Cambridge
`Computer Laboratory are freely available via the Internet:
`
`http://www.cl.cam.ac.uk/TechReports/
`
`Series editor: Markus Kuhn
`
`ISSN 1476-2986
`
`ii
`
`

`

`Biometric Decision Landscapes
`
`John Daugman
`
`University of Cambridge
`
`The Computer Laboratory1
`
`Abstract
`
`This report investigates the “decision landscapes” that characterize several forms of biometric
`decision making. The issues discussed include: (i) Estimating the degrees-of-freedom associated
`with different biometrics, as a way of measuring the randomness and complexity (and therefore
`the uniqueness) of their templates. (ii) The consequences of combining more than one biometric
`test to arrive at a decision. (iii) The requirements for performing identification by large-scale
`exhaustive database search, as opposed to mere verification by comparison against a single
`template. (iv) Scenarios for Biometric Key Cryptography (the use of biometrics for encryption
`of messages). These issues are considered here in abstract form, but where appropriate, the
`particular example of iris recognition is used as an illustration. A unifying theme of all four
`sets of issues is the role of combinatorial complexity, and its measurement, in determining the
`potential decisiveness of biometric decision making.
`
`Keywords – Statistical decision theory, pattern recognition, biometric identification, combinatorial
`complexity, iris recognition, Biometric Key Cryptography.
`
`1 Yes/No Decisions
`
`Biometric identification fits squarely in the classical framework of statistical decision theory. This
`formalism emerged from work on statistical hypothesis testing1 in the 1920s - 1930s and on radar
`signal detection analysis2 in World War II, and its key elements are briefly summarized here in
`Figures 1 and 2. For decision problems in which prior probabilities are not known, error costs are
`not fixed, but posterior distributions are known, the formalism of Neyman and Pearson1 provides
`not only a mechanism for making decisions, but also for assigning confidence levels to such decisions
`and for measuring the overall “decidability” of the task.
`Yes/No pattern recognition decisions have four possible outcomes: either a given pattern is,
`or is not, in fact the target; and in either case, the decision made by the recognition algorithm
`may be either the correct or the incorrect one. In a biometric decision context the four possible
`outcomes are normally called False Accept (FA), Correct Accept (CA), False Reject (FR), and
`Correct Reject (CR). Obviously the first and third outcomes are errors (called Type I and Type
`II respectively), whilst the second and fourth outcomes are the ones sought. By manipulating the
`decision criteria, the relative probabilities of these four outcomes can be adjusted in a way that
`reflects their associated costs and benefits. These may be very different in different applications.
`In a customer context the cost of a FR error may exceed the cost of a FA error, whereas just the
`opposite may be true in a military context.
`It is important to note immediately the uselessness of either error rate statistic alone in char-
`acterizing performance. Any arbitrary system can achieve a FA rate of 0 (just by rejecting all
`candidates). Similarly it can achieve a FR rate of 0 (just by accepting all candidates). The notion
`of “decision landscape” is intended to portray the degree to which any improvement in one error
`rate must be paid for by a worsening in the other. This concept facilitates the definition of metrics
`quantifying the intrinsic decidability of a recognition problem, and this can be useful for comparing
`different biometric approaches and understanding their potential.
`
`1Cambridge CB2 3QG, England.
`
`tel +44 1223 334501 fax +44 1223 334679 john.daugman@CL.cam.ac.uk
`
`1
`
`

`

`Statistical Decision Theory
`
`False Accept Rate
`
`Correct Reject Rate
`
`Correct Accept Rate
`
`False Reject Rate
`
`Accept if HD < Criterion
`
`Reject if HD > Criterion
`
`6
`
`5
`
`Authentics
`
`Impostors
`
`Criterion
`
`Probability Density
`
`3
`
`2
`
`4
`
`1
`
`0
`
`0.0
`
`0.1
`
`0.2
`
`0.3
`
`0.4
`0.5
`0.6
`Hamming Distance
`
`0.7
`
`0.8
`
`0.9
`
`1.0
`
`Figure 1: Decision landscape: general formalism for biometric decision making.
`
`Figure 1 illustrates the idea of the decision landscape. The two distributions represent the two
`states of the world, which are imperfectly separated. The abscissa is any metric of similarity or
`dissimilarity; in this case it happens to be Hamming Distance, which is the fraction of bits that
`differ between two binary strings. A decision about whether they are instances of the same pattern
`(albeit somewhat corrupted), or completely different patterns, is made by imposing some decision
`criterion for similarity as indicated by the dotted line. Similarity up to some Hamming Distance
`(0.4 in this case) is deemed sufficient for regarding the patterns as the same, but beyond that point,
`the patterns are declared to be different.
`The likelihoods that these are correct decisions, or not, correspond to the four stippled areas
`that lie under the two probability distributions on either side of the decision criterion. It is clear
`that moving the decision criterion to the right or left (becoming more liberal or more conservative)
`will change the relative likelihoods of the four outcomes. It is also clear that the “decidabiity” of a
`Yes/No decision problem is determined by how much overlap there is between the two distributions.
`The problem becomes more decidable if their means are further apart, or if their variances are
`smaller. One measure of decidability, although not the only possible one, is d(cid:2) (d-prime), defined
`as follows if the means of the two distributions are μ1 and μ2 and their two standard deviations
`are σ1 and σ2:
`|μ1 − μ2|
`(cid:2)
`(σ21 + σ22)
`
`
`(Note that d(cid:2) has the units of Z-score: distances are marked off in units of a conjoint standard
`deviation.) A shortcoming of the d(cid:2) statistic is that it ignores moments higher than second-order,
`and it becomes less informative if distributions depart significantly from modal form. Nevertheless,
`it can be a useful gauge for assessing different decision landscapes. It has the virtue of quantifying,
`in a single number, the intrinsic decidability of a decision task in a way that is independent of the
`chosen decision criterion. It assesses the degree of inevitable trade-off between the two error rates.
`Because it measures the separation between the two distributions defining the decision landscape,
`the higher it is, the better. In the schematic of Figure 1, d(cid:2) = 2.
`Let us name the two distributions PIm(x) and PAu(x), denoting respectively the probability
`densities of any measured dissimilarity x (such as a Hamming Distance) arising from two different
`biometric sources (“Impostor”), or from the same source (“Authentic”). Then the probabilities of
`each of the four possible decision outcomes FA, CR, CA, and FR are equal to the areas under these
`
`12
`
`d(cid:2) =
`
`(1)
`
`2
`
`

`

`two probability distributions on either side of the chosen decision criterion C:
`(cid:3) C
`(cid:3) 1
`(cid:3) C
`(cid:3) 1
`
`P (F A) =
`
`P (CR) =
`
`P (CA) =
`
`0
`
`C
`
`0
`
`PIm(x)dx
`
`PIm(x)dx
`
`PAu(x)dx
`
`(2)
`
`(3)
`
`(4)
`
`(5)
`
`C
`It is clear that these four probabilities separate into two pairs that must sum to unity, and two
`pairs that are governed by inequalities:
`
`P (F R) =
`
`PAu(x)dx
`
`(6)
`
`(7)
`
`(8)
`
`(9)
`
`P (CA) + P (F R) = 1
`
`P (F A) + P (CR) = 1
`
`P (CA) > P (F A)
`
`P (CR) > P (F R)
`
`Decision
`Strategy
`Curve
`
`Liberal
`
`Conservative
`
`More conservative:
`Lower Hamming Distance Criterion
`
`More liberal:
`Raise Hamming Distance Criterion
`
`0.5
`False Accept Rate
`
`1.0
`
`1.0
`
`Correct Accept Rate
`
`0.5
`
`0.0
`
`0.0
`
`Figure 2: The Neyman-Pearson (ROC) decision strategy curve.
`
`Manipulation of the decision criterion C in the integrals (2) - (5) in order to implement different
`decision strategies appropriate for the costs of either type of error in a given application, is illus-
`trated schematically in Figure 2. Such a decision strategy diagram, sometimes called a Receiver
`Operating Characteristic or Neyman-Pearson curve, plots P (CA) from (4) against P (F A) from (2)
`as a locus of points. Each point on such a curve represents a different decision strategy as specified
`by a different choice for the operating criterion C, as was indicated schematically in Figure 1.
`Inequality (8) states that the Neyman-Pearson strategy curve shown in Figure 2 will always lie
`above the diagonal line. Clearly, strategies that are excessively liberal or conservative correspond
`to sliding along the curve towards either of its extremes. Irrespective of where the decision criterion
`is placed along this continuum (hence how liberal or conservative one wishes to be in a particular
`application), the overall power of a pattern recognition method may be gauged by how bowed the
`ROC curve is. The length of the short line segment in Figure 2 is monotonically related to the
`
`3
`
`

`

`quantity d(cid:2) defined in (1). Ideally one would like a biometric that generates a decision landscape
`whose ROC curve is extremely bowed, reaching as far as possible into the upper left corner of
`Figure 2, since reaching that limit corresponds to achieving a Correct Accept rate of 100% whilst
`keeping the False Accept rate at 0%.
`The decision landscape concept that has been illustrated in Figures 1 and 2 and quantified by
`equations (1) - (9) can be applied to any biometric, regardless of its chosen template or its measure of
`similarity. Such dual histogram plots should always be provided for a biometric, together with some
`measure of “separability” or “decidability” such as d(cid:2) in order to describe the decision landscape
`more fully than merely citing a FA and FR rate, either of which can always be brought to 0. It is
`unfortunate that such fuller portrayals of a biometric’s decision landscape are rarely provided.
`
`2 Decisions from Combined Biometrics
`
`In this section we investigate the consequences of combining two or more biometric tests into an
`enhanced test. There is a common and intuitive assumption that the combination of different tests
`must improve performance, because “surely more information is better than less information.” On
`the other hand, a different intuition suggests that if a strong test is combined with a weaker test,
`the resulting decision landscape is in a sense averaged, and the combined performance will lie some-
`where between that of the two tests conducted individually (and hence will be degraded from the
`performance that would be obtained by relying solely on the stronger test).
`
`There is truth in both intuitions. The key to the apparent paradox is that when two tests are
`combined, one of the resulting new error rates (FA or FR depending on the combination rule used)
`becomes better than that of the stronger of the two tests, while the other error rate becomes worse
`even than that of the weaker of the tests.
`If the two biometric tests differ significantly in their
`power, and each operates at its own cross-over point where P (F A) = P (F R), then combining
`them actually results in significantly worse performance than relying solely on the one, stronger,
`biometric.
`
`Notation: Two hypothetical independent biometric tests will be considered, named 1 and 2.
`For example, 1 might be voice-based verification, and 2 might be fingerprint verification. Each
`biometric test is characterized by its own pair of error rates at a given operating point, denoted as
`the error probabilities P1(F A), P1(F R), P2(F A), and P2(F R):
`P1(F A) = probability of a False Accept using Biometric 1 alone.
`P1(F R) = probability of a False Reject using Biometric 1 alone.
`P2(F A) = probability of a False Accept using Biometric 2 alone.
`P2(F R) = probability of a False Reject using Biometric 2 alone.
`
`There are two possible ways to combine the outcomes of the two biometric tests when forming
`the conjoint (“enhanced”) decision: the Subject may be required to pass both of the biometric
`tests, or he may be accepted if he can pass at least one of the two tests. These two cases define the
`disjunctive and conjunctive rules:
`Rule A: Disjunction (“OR” Rule) – Accept if either test 1 or test 2 is passed.
`Rule B: Conjunction (“AND” Rule) – Accept only if both tests 1 and 2 are passed.
`
`We can now calculate False Accept and False Reject error rates of the combined biometric, both
`for disjunctive (Rule A) and conjunctive (Rule B) combinations of the two tests. These new error
`probabilities will be denoted: PA(F A), PA(F R), PB(F A), and PB(F R).
`
`4
`
`

`

`Rule A: Disjunction – Accept if either test 1 or test 2 is passed.
`
`If Rule A (the “OR” Rule) is used to combine the two tests 1 and 2, a False Reject can only
`occur if both tests 1 and 2 produce a False Reject. Thus the combined probability of a False
`Reject, PA(F R), is the product of its two probabilities for the individual tests:
`
`PA(F R) = P1(F R)P2(F R)
`
`(10)
`
`(clearly a lower probability than for either test alone). But the probability of a False Accept when
`using this Rule, which can be expressed as the complement of the probability that neither test 1
`nor 2 produces a False Accept, is higher than it was for either test alone:
`PA(F A) = 1 − [1 − P1(F A)] [1 − P2(F A)]
`= P1(F A) + P2(F A) − P1(F A)P2(F A)
`
`(11)
`(12)
`
`Rule B: Conjunction – Accept only if both tests 1 and 2 are passed.
`
`If Rule B (the “AND” Rule) is used to combine the two tests 1 and 2, a False Accept can only
`occur if both tests 1 and 2 produce a False Accept. Thus the combined probability of a False
`Accept, PB(F A), is the product of its two probabilities for the individual tests:
`
`PB(F A) = P1(F A)P2(F A)
`
`(13)
`
`(clearly a lower probability than for either test alone). But the probability of a False Reject when
`using this Rule, which can be expressed as the complement of the probability that neither test 1
`nor 2 produces a False Reject, is higher than it was for either test alone:
`PB(F R) = 1 − [1 − P1(F R)] [1 − P2(F R)]
`= P1(F R) + P2(F R) − P1(F R)P2(F R)
`
`(14)
`(15)
`
`Example: Combination of two hypothetical biometric tests, one stronger than the other.
`
`Suppose weaker Biometric 1 operates with both of its error rates equal to 1 in 100, and sup-
`pose stronger Biometric 2 operates with both of its error rates equal to 1 in 1,000. Thus if 100,000
`verification tests are conducted with impostors and another 100,000 verification tests are conducted
`with authentics, Biometric 1 alone should make a total of 2,000 errors, whereas Biometric 2 alone
`should make a total of only 200 errors. But what happens if the two biometrics are combined to
`make an “enhanced” test?
`
`If the “OR” Rule were followed in this same batch of tests, the combined biometric should make
`1,099 False Accepts and 1 False Reject, for a total of 1,100 errors. If instead the “AND” Rule is
`followed, the combined biometric should make 1,099 False Rejects and 1 False Accept, thus again
`producing a total of 1,100 errors. Either method of combining the two biometric tests produces 5.5
`times more errors than if the stronger of the two tests had been used alone.
`
`Conclusion: A strong biometric is better used alone than in combination with a weaker one...
`when both are operating at their cross-over points. To reap any benefit from the combination,
`equations (10) - (15) show that the operating point of the weaker biometric must be shifted to
`satisfy the following criteria: If the “OR” Rule is to be used, the False Accept rate of the weaker
`test must be made smaller than twice the cross-over error rate of the stronger test. If the “AND”
`Rule is to be used, the False Reject rate of the weaker test must be made smaller than twice the
`cross-over error rate of the stronger test.
`
`5
`
`

`

`3 Identification Decisions
`
`We now compare the requirements of performing a verification (a one-to-one comparison against a
`single stored template), versus performing an identification (a one-to-many comparison against all
`the enrolled templates in some database containing N impostors). If the presenting template is in
`fact one of the enrolled templates in the database, the probability of a False Reject when it comes
`up is obviously the same as in the verification trial. But we are interested now in how much more
`strenuous the demands against getting a single False Accept need to be, in the case of identification
`trials involving N other templates.
`
`Notation: Let P1 = probability of a False Accept in a verification trial. Let PN = probabil-
`ity of a False Accept in identification trials after an exhaustive search through a database of N
`unrelated templates. We wish to calculate this.
`Clearly the probability of not getting a False Accept in any given comparison is (1 − P1). This
`must happen N independent times, and so the probability of it not occurring in any of those N
`comparisons is (1 − P1)N . Thus the probability of making at least one False Accept among those
`N comparisons is one minus that probability. This is an extremely demanding relationship:
`PN = 1 − (1 − P1)N
`
`(16)
`
`Example: Consider a biometric verifier that achieves a 99.9% Correct Rejection performance in
`one-to-one verification trials. Thus P1 = 0.001, as per equation (7). How will it perform when
`searching through a database of unrelated templates?
`
`Using equation (16), we see that for the following sizes of databases with N unrelated templates,
`these will be the probabilities PN that this biometric makes at least one False Accept:
`
`Database Size False Accept probability
`N = 200
`PN = 18%
`N = 2,000
`PN = 86%
`N = 10,000
`PN = 99.995%
`
`Once the enrolled database size reaches only about 7,000 persons, this biometric actually becomes
`more likely (99.91%) to produce a False Accept in identification trials than it is to produce a Cor-
`rect Reject (99.9%) in verification trials.
`
`Conclusion:
`Identification is vastly more demanding than verification, and even for moderate
`database sizes, merely “good” verifiers are of no use as identifiers. Observing the approximation
`that PN ≈ N P1 for small P1 << 1
`N << 1, when searching a database of size N an identifier needs
`to be roughly N times better than a verifier to achieve comparable odds against a False Accept.
`
`It has been suggested that this fundamental problem might be overcome by “binning” or “fil-
`tering” the database into smaller subsets, thereby reducing the problematical exponent N in (16).
`For example, fingerprints might be pre-classified into three standard types as whorls, loops, or
`arches, and then each search could be restricted just to databases of about one-third the full size.
`But this strategem creates its own problems:
`
`1. Binning is equivalent to “combining multiple tests,” as analyzed in the previous section. The
`additional test here is the classification operation, which itself has its own P (F R), P (F A),
`ROC curve, and d(cid:2). For a Subject to be correctly identified, he must pass both the biometric
`matching test, and any pre-sorting test which classified his prints. Proponents of such com-
`bined schemes often ignore the effect on the other error rate when contemplating the benefits
`
`6
`
`

`

`for just one of the error rates. In this case, the benefit of reducing the FA probability (by
`reducing the database search size N) is paid for by an increase in the FR probability, since
`any misclassification of a print into the wrong bin must cause a failure to match. It has been
`reported3 in tests of four AFIS systems that fingerprint binning classification error rates are
`in the range of 1% to 5%. Such large increases in failures to match are unacceptable.
`
`2. Even if there were no binning errors, the argument for its benefits requires that P (F A)
`within a given class (e.g. whorl prints) is no higher than across mixed classes. If instead,
`as would certainly be the case for matching methods such as optical Moire-pattern analysis,
`loop prints were only confused with other loops, and whorls with whorls, then there would
`be no performance benefit from binning. The size of each search database would be smaller,
`but the probability of false matches within each class would be proportionately higher, since
`all of the potential false matching prints remain present in the (now smaller) bin.
`
`3. Even if there were no binning errors, and even if the within-bin false match rate were no
`higher than the across-bin rate, the potential benefit of binning is limited by the number of
`bins. Their number determines the factor by which the search database size can be reduced.
`As the numerical examples illustrate, a reduction factor such as 3 or 10 is utterly insufficient.
`
`The only real key to surviving equation (16) when performing large-scale database searches, in
`which perhaps many millions of templates must be compared in seeking to make an identification
`while avoiding false matches, is to ensure that P1 is sufficiently small. For a search database size
`of N, the single-trial FA probability must be significantly smaller than 1/N. For example, when
`being compared against a database of 10 million different templates, an “innocent” Subject can
`only feel 99% certain that he won’t be falsely matched with any of them if the single-comparison
`false match probability is no greater than P1 = 10−9: 1 in a billion.
`
`4 The Degrees-of-Freedom in Biometric Feature Sets
`
`Achieving such demanding confidence levels against false matches when attempting identification
`decisions against large search databases is only possible if the combinatorics of the biometric are
`sufficiently vast. The combinatorial complexity of a biometric test can be gauged by its number of
`degrees-of-freedom, which is essentially the number of independent dimensions of variation, or the
`number of independent yes/no questions that the biometric decision is based upon. For biometrics
`that do not compare lists of distinct features but rather use a simple analogue measure such as
`correlation, the number of degrees-of-freedom is the number of independent data that can be
`resolved, as limited by (among other things) the imaging quality and the intrinsic auto-correlation
`within each pattern. This metric of data size divided by its mean correlation length corresponds to
`Hartley’s4 classic definition of the number of degrees-of-freedom in a signal, and to the number of
`resolvable cells in the Information Diagram proposed by Gabor5.
`Clearly, the larger the number of underlying features that are compared, the less chance that
`two unrelated templates might just by chance agree in them all. But it also becomes less likely
`that even their source will be able to produce a perfect match to them all. Therefore it is the
`combinatorial question of how likely it is that some proportion of the features will be matched by
`chance by different people, and some proportion will fail to be matched even by the same person,
`that really determines the shape of the decision landscape. The goal of biometric feature encoding
`is to maximize the number of degrees-of-freedom that will belong to the “impostors” distribution
`in Figure 1, while minimizing the number that will belong to the “authentics” distribution. Thus
`for example, in face recognition, one would like to choose feature dimensions that vary the most
`among different individuals, but that do not vary when a given individual changes expression, pose
`angle, hairstyle, age, etc.
`The number of degrees-of-freedom contained in a wide range of biometrics can be estimated by
`counting the number of yes/no questions that they ask (e.g. “is this minutia from fingerprint A
`
`7
`
`

`

`Disagreement Among Unrelated IrisCodes (British Database)
`
`Solid Curve: Binomial of N=244, p=0.5
`f(x=m/N) = p^(m) q^(N-m) N! / (m! (N-m)! )
`
`326,028 comparisons of different eyes
`
`mean = 0.499, stnd.dev. = 0.032
`
`Degrees of Freedom: 244
`
`18000
`
`14000
`
`10000
`Count
`
`6000
`
`02000
`
`0.0
`
`0.1
`
`0.2
`
`0.3
`
`0.6
`0.5
`0.4
`Hamming Distance
`
`0.7
`
`0.8
`
`0.9
`
`1.0
`
`Figure 3: Disagreement among unrelated iris patterns in a British database, providing an estimate
`of the number of degrees-of-freedom in IrisCodes.
`
`also in B?”, or, “does the phase of this local patch in iris A agree with the local phase in B?”), and
`then measuring the dimension of the subspace of these questions that are independent.
`If the thresholds for such yes/no questions can be adjusted so that each has the same probability
`p of being true, then the proportion of them that are true when unrelated templates are compared
`will be binomially distributed with parameters (p, N), where N is the size of the subset of questions
`posed that are independent. (If all the yes/no questions were independent, then N would equal
`their number.) The probability that some fraction x = m/N of the N independent questions may
`agree in their answers just by chance between unrelated templates will be distributed according to
`the density function f(x):
`
`f(x = m/N) =
`
`pm(1 − p)(N−m)
`
`N!
`m!(N − m)!
`(17)
`(cid:4)
`p(1 − p)/N.
`In
`The mean of such a distribution is μ = p, and its standard deviation is σ =
`general it is asymmetrical unless it happens that p = 0.5, in which case it resembles a Gaussian
`apart from being discrete rather than continuous, and having strictly finite support (the unit
`interval for x = m/N in this case.) Although the classic example of a random variable that is
`binomially distributed is a series of independent Bernoulli trials (N independent coin-tosses with
`fixed probability p), Viveros6 et al. have pointed out that correlated Bernoulli trials are also
`distributed as such families of functions but with reduction in N.
`An illustration of this principle is seen in Figure 3, which shows the data from 326,028 exhaustive
`pairwise comparisons among 808 different iris patterns. Each IrisCode contains 2,048 bits of phase
`data, but these are strongly correlated because of iris radial structure. The fraction of disagreeing
`bits, or Hamming Distance, between every pair of different IrisCodes is plotted in the histogram,
`and the solid curve is equation (17) with parameters p and N = p(1 − p)/σ2 measured from the
`mean and standard deviation of the data. The quality of the fit between the data and equation (17)
`
`8
`
`

`

`is extraordinary. The calculated value of N is 244, based upon the measurement that μ = 0.499
`and σ = 0.032, and this tells us that among the 2,048 bits computed in an IrisCode, effectively only
`244 of them are independent. Hence the number of degrees-of-freedom in IrisCodes acquired with
`this particular camera focus quality and imaging resolution was 244. The solid curve was computed
`using Stirling’s formula to calculate the large factorials in equation (17):
`N! ≈ eN ln(N )−N + 1
`(18)
`which errs by less than 1% in estimating N! for N ≥ 9. The presence of large factorials causes the
`binomial probability density to attenuate extremely rapidly in its tails, which in turn means that
`the cumulative in equation (2) for False Accept probability becomes infinitesimally small when N,
`the number of degrees-of-freedom in the biometric, is sufficiently large.
`
`ln(2πN )
`
`2
`
`Disagreement Among Unrelated IrisCodes (American Database)
`
`Solid Curve: Binomial of N=244, p=0.5
`f(x=m/N) = p^(m) q^(N-m) N! / (m! (N-m)! )
`
`482,600 comparisons of different eyes
`
`30000
`
`25000
`
`20000
`
`mean = 0.498, stnd.dev. = 0.032
`
`Degrees of Freedom: 244
`
`15000
`Count
`
`10000
`
`5000
`
`0
`
`0.0
`
`0.1
`
`0.2
`
`0.3
`
`0.6
`0.5
`0.4
`Hamming Distance
`
`0.7
`
`0.8
`
`0.9
`
`1.0
`
`Figure 4: Disagreement among unrelated iris patterns in an American database, providing an
`estimate of the number of degrees-of-freedom in IrisCodes.
`
`Whereas Figure 3 presented data from 326,028 comparisons among iris patterns acquired in
`Britain, Figure 4 presents 482,600 such comparisons among an entirely different gallery of iris
`images acquired on a different optical platform in America. It is once again perfectly described by
`a binomial, as the solid curve illustrates. But because the optical systems used were quite different,
`it is only coincidental that the measured number of degrees-of-freedom is again 244, exactly the
`same as for the data in Figure 3. The actual number can be lower or higher than this (as many
`as 266 have been reported in one study7 using the author’s algorithms8), because variations in
`imaging focus and resolution affect the auto-correlation within each image.
`It would be possible to estimate and compare the number of degrees-of-freedom for different
`biometrics, using this mapping of biometric feature set comparisons into an equivalent number of
`Bernoulli trials as described above. Moreover, through this expression of the decision processes as
`an ensemble of elementary yes/no questions, the different biometrics would all acquire the benefits
`
`9
`
`

`

`Phase-Quadrant Iris Demodulation Code
`
`Im
`
`[0, 1]
`
`[1, 1]
`
`Re
`
`[0, 0]
`
`[1, 0]
`
`Figure 5: Iris encoding by phase demodulation with complex-valued 2D Gabor wavelets.
`
`of having factorial (binomial-class) tails, i.e. rapidly attenuating densities, instead of exponential
`or flatter ones. This in turn would facilitate the execution of decision making by a simple test of
`statistical independence8: a Subject becomes highly likely to pass a test of statistical independence
`when compared with any other person’s biometric template, but to fail this same test of statistical
`independence against his own.
`
`5 Decision Landscape for Iris Recognition
`
`Iris patterns are encoded into “IrisCodes” by a process of phase demodulation8 employing a two-
`dimensional generalization9 of complex Gabor wavelets. These can represent10 a textured pattern
`by an ensemble of phasors in the complex plane. In an IrisCode, each phasor angle (Figure 5) is
`quantized into just the complex quadrant in which it lies for each local patch (r0, θ0) of the iris, and
`this operation is repeated all across the iris, at many different scales (α, β, ω) of analysis. Such local
`phase quantization is performed by evaluating the real and the imaginary parts of the following
`integral expression:
`(cid:3)
`(cid:3)
`
`e−iω(θ0−φ)e−(r0−ρ)2/α2e−(θ0−φ)2/β2I(ρ, φ)ρdρdφ
`
`ρ
`
`φ
`
`(19)
`
`where the raw image data is given in a pseudo-polar coordinate system I(ρ, φ) following spatial
`localization of an iris using methods described in earlier papers8. Bits in an IrisCode are set on
`the basis of whether the real and imaginary parts of (19) are positive or negative. These form
`the elementary “yes/no questions” that underlie decision making with iris patterns. All currently
`available systems for iris recognition are based upon the author’s algorithms as described here, in
`licensed executable code.
`The ability of phase demodulation to extract enough degrees-of-freedom from iris patterns to
`permit their reliable recognition is summarized in Figure 6. This is based upon the same set of
`482,600 iris comparisons as shown in Figure 4 but after rotations to each of 7 angles (to correct for
`head tilt and for imaging through a pan/tilt mirror), keeping only the best value. This selection
`of the lowest Hamming Distance among 7 rotated comparisons skews the impostors distribution to
`
`10
`
`

`

`45000
`
`35000
`
`25000
`
`15000
`
`05000
`
`d’ = 14.10
`
`482,600 different-eye comparisons
`(min = 0.345)
`
`solid curve = equation (20)
`
`6,300 same-eye comparisons
`(max = 0.313)
`
`5001000150020002500300035004000
`
`0
`
`Count
`
`0.0
`
`0.1
`
`0.2
`
`0.3
`
`0.6
`0.5
`0.4
`Hamming Distance
`
`0.7
`
`0.8
`
`0.9
`
`1.0
`
`Figure 6: Decision landscape for iris recognition using the same (American) gallery as used in
`Figure 4, but allowing several image rotations to find each closest fit. The histograms show 482,600
`different-eye comparisons, and 6,300 same-eye comparisons. The solid curve is equation (20).
`
`the left, but its functional form remains perfectly described by the appropriate density expression
`(plotted as the solid curve) which is
`fn(x) = nf(x) [1 − F (x)]n−1
`
`(20)
`
`where f(x) is the simple binomial density function given in equation (17), F (x) is its cumulative
`from 0 to x, and n = 7 rotations. The overal decidability for iris recognition in this decision
`landscape is measured at d(cid:2) = 14.1 as defined by equation (1).
`We can now compute P (F A), the probability of False Accept in single comparisons, using the
`impostors distribution on the right side of Figure 6 as PIm(x) in equation (2). The lowest Hamming
`Distance actually observed in this empirical series of 482,600 comparisons after 7 rotations each
`was 0.345. But because of the principled reasons making f(x) in Figure 4 a binomial (17), we may
`just integrate the expression in (20) for fn(x) (plotted as the solid curve in Figure 6) from x = 0
`up to various possible decision criteria C as per (2) in ord