Multimodal fusion decision logic system

[0001] This application claims the benefit of priority to U.S. provisional patent application Ser. No. 60/643,853, filed on Jan. 14, 2005.

[0002] The United States government provided finding related to this invention via US Army, under contract W911NF-04-C-0067, and the United States government may have certain rights in the invention.

[0003] The present invention relates to the use of multiple biometric modalities and multiple biometric matching methods in a biometric identification system. Biometric modalities may include (but are not limited to) such methods as fingerprint identification, iris recognition, voice recognition, facial recognition, hand geometry, signature recognition, signature gait recognition, vascular patterns, lip shape, ear shape and palm print recognition.

[0004] Algorithms may be used to combine information from two or more biometric modalities. The combined information may allow for more reliable and more accurate identification of an individual than is possible with systems based on a single biometric modality. The combination of information from more than one biometric modality is sometimes referred to herein as “biometric fusion”.

[0005] Reliable personal authentication is becoming increasingly important. The ability to accurately and quickly identify an individual is important to immigration, law enforcement, computer use, and financial transactions. Traditional security measures rely on knowledge-based approaches, such as passwords and personal identification numbers (“PINs”), or on token-based approaches, such as swipe cards and photo identification, to establish the identity of an individual. Despite being widely used, these are not very secure forms of identification. For example, it is estimated that hundreds of millions of dollars are lost annually in credit card fraud in the United States due to consumer misidentification.

[0006] Biometrics offers a reliable alternative for identifying an individual. Biometrics is the method of identifying an individual based on his or her physiological and behavioral characteristics. Common biometric modalities include fingerprint, face recognition, hand geometry, voice, iris and signature verification. The Federal government will be a leading consumer of biometric applications deployed primarily for immigration, airport, border, and homeland security. Wide scale deployments of biometric applications such as the US-VISIT program are already being done in the United States and else where in the world.

[0007] Despite advances in biometric identification systems, several obstacles have hindered their deployment. Every biometric modality has some users who have illegible biometrics. For example a recent NIST (National Institute of Standards and Technology) study indicates that nearly 2 to 5% of the population does not have legible fingerprints. Such users would be rejected by a biometric fingerprint identification system during enrollment and verification. Handling such exceptions is time consuming and costly, especially in high volume scenarios such as an airport. Using multiple biometrics to authenticate an individual may alleviate this problem.

[0008] Furthermore, unlike password or PIN based systems, biometric systems inherently yield probabilistic results and are therefore not fully accurate. In effect, a certain percentage of the genuine users will be rejected (false non-match) and a certain percentage of impostors will be accepted (false match) by existing biometric systems. High security applications require very low probability of false matches. For example, while authenticating immigrants and international passengers at airports, even a few false acceptances can pose a severe breach of national security. On the other hand false non matches lead to user inconvenience and congestion.

[0009] Existing systems achieve low false acceptance probabilities (also known as False Acceptance Rate or “FAR”) only at the expense of higher false non-matching probabilities (also known as False-Rejection-Rate or “FRR”). It has been shown that multiple modalities can reduce FAR and FRR simultaneously. Furthermore, threats to biometric systems such as replay attacks, spoofing and other subversive methods are difficult to achieve simultaneously for multiple biometrics, thereby making multimodal biometric systems more secure than single modal biometric systems.

[0010] Systematic research in the area of combining biometric modalities is nascent and sparse. Over the years there have been many attempts at combining modalities and many methods have been investigated, including “Logical And”, “Logical Or”, “Product Rule”, “Sum Rule”, “Max Rule”, “Min Rule”, “Median Rule”, “Majority Vote”, “Bayes' Decision”, and “Neyman-Pearson Test”. None of these methods has proved to provide low FAR and FRR that is needed for modern security applications.

[0011] The need to address the challenges posed by applications using large biometric databases is urgent. The US-VISIT program uses biometric systems to enforce border and homeland security. Governments around the world are adopting biometric authentication to implement National identification and voter registration systems. The Federal Bureau of Investigation maintains national criminal and civilian biometric databases for law enforcement.

[0012] Although large-scale databases are increasingly being used, the research community's focus is on the accuracy of small databases, while neglecting the scalability and speed issues important to large database applications. Each of the example applications mentioned above require databases with a potential size in the tens of millions of biometric records. In such applications, response time, search and retrieval efficiency also become important in addition to accuracy.

[0013] The present invention includes a method of deciding whether a data set is acceptable for making a decision. For example, the present invention may be used to determine whether a set of biometrics is acceptable for making a decision about whether a person should be allowed access to a facility. The data set may be comprised of information pieces about objects, such as people. Each object may have at least two types of information pieces, that is to say the data set may have at least two modalities. For example, each object represented in the database may by represented by two or more biometric samples, for example, a fingerprint sample and an iris scan sample. A first probability partition array (“Pm(i,j)”) may be provided. The Pm(i,j) may be comprised of probability values for information pieces in the data set, each probability value in the Pm(i,j) corresponding to the probability of an authentic match. Pm(i,j) may be similar to a Neyman-Pearson Lemma probability partition array. A second probability partition array (“Pfm(i,j)”) may be provided, the Pfm(i,j) being comprised of probability values for information pieces in the data set, each probability value in the Pfm(i,j) corresponding to the probability of a false match. Pfm(i,j) may be similar to a Neyman-Pearson Lemma probability partition array.

[0014] A method according to the invention may identify a no-match zone. For example, the no-match zone may be identified by identifying a first index set (“A”), the indices in set A being the (i,j) indices that have values in both Pfm(i,j) and Pm(i,j). A second index set (“Z∞”) may be identified, the indices of Z∞ being the (i,j) indices in set A where both Pfm(i,j) is larger than zero and Pm(i,j) is equal to zero. FAR_Z∞ may be determined, where FAR_Z∞=1−Σ_(i,j)∈Z∞P_fm(i,j). FAR_Z∞ may be compared to a desired false-acceptance-rate (“FAR”), and if FAR_Z∞ is greater than the desired false-acceptance-rate, than the data set may be rejected for failing to provide an acceptable false-acceptance-rate. If FAR_Z∞ is less than or equal to the desired false-acceptance-rate, then the data set may be accepted, if false-rejection-rate is not important.

[0015] If false-rejection-rate is important, further steps may be executed to determine whether the data set should be rejected. The method may further include identifying a third index set ZM∞, the indices of ZM∞ being the (i,j) indices in Z∞ plus those indices where both Pfm(i,j) and Pm(i,j) are equal to zero. A fourth index set (“C”) may be identified, the indices of C being the (i,j) indices that are in A but not ZM∞. The indices of C may be arranged such that

[0016] $\frac{{P_{\mathrm{fm}}\left(i,j\right)}_k}{{P_m\left(i,j\right)}_k}>=\frac{{P_{\mathrm{fm}}\left(i,j\right)}_{k+1}}{{P_m\left(i,j\right)}_{k+1}}$
to provide an arranged C index. A fifth index set (“Cn”) may be identified. The indices of Cn may be the first N (i,j) indices of the arranged C index, where N is a number for which the following is true: FAR_Z∞∪CN=1−Σ_(i,j)∈Z∞P_fm(i,j)−Σ_(i,j)∈CNP_fm(i,j)≦FAR . The FRR may be determined, where FRR=Σ_(i,j)∈CNP_m(i,j), and compared to a desired false-rejection-rate. If FRR is greater than the desired false-rejection-rate, then the data set may be rejected, even though FAR∞ is less than or equal to the desired false-acceptance-rate. Otherwise, the data set may be accepted.

[0017] In another method according to the invention, the false-rejection-rate calculations and comparisons may be executed before the false-acceptance-rate calculations and comparisons. In such a method, a first index set (“A”) may be identified, the indices in A being the (i,j) indices that have values in both Pfm(i,j) and Pm(i,j). A second index set (“Z∞”) may be identified, the indices of Z∞ being the (i,j) indices of A where Pm(i,j) is equal to zero. A third index set (“C”) may be identified, the indices of C being the (i,j) indices that are in A but not Z∞ The indices of C may be arranged such that

[0018] $\frac{{P_{\mathrm{fm}}\left(i,j\right)}_k}{{P_m\left(i,j\right)}_k}>=\frac{{P_{\mathrm{fm}}\left(i,j\right)}_{k+1}}{{P_m\left(i,j\right)}_{k+1}}$
to provide an arranged C index, and a fourth index set (“Cn”) may be identified. The indices of Cn may be the first N (i,j) indices of the arranged C index, where N is a number for which the following is true: FAR_Z∞∪CN=1−Σ_(i,j)∈Z∞P_fm(i,j)−Σ_(i,j)∈CNP_fm(i,j)≦FAR. The FRR may be determined, where FRR=Σ_(i,j)∈CNP_m(i,j), and compared to a desired false-rejection-rate. If the FRR is greater than the desired false-rejection-rate, then the data set may be rejected. If the FRR is less than or equal to the desired false-rejection-rate, then the data set may be accepted, if false-acceptance-rate is not important. If false-acceptance-rate is important, then the FAR_Z∞, may be determined, where FAR_Z∞=1−Σ_{(i,j) ÅZ∞}P_fm(i,j). The FAR_Z∞ may be compared to a desired false-acceptance-rate, and if FAR_Z∞ is greater than the desired false-acceptance-rate, then the data set may be rejected even though FRR is less than or equal to the desired false-rejection-rate. Otherwise, the data set may be accepted.

[0019] The invention may also be embodied as a computer readable memory device for executing any of the methods described above.

[0039] Preliminary Considerations: To facilitate discussion of the invention it may be beneficial to establish some terminology and a mathematical framework. Biometric fusion may be viewed as an endeavor in statistical decision theory [1] [2]; namely, the testing of simple hypotheses. This disclosure uses the term simple hypothesis as used in standard statistical theory: the parameter of the hypothesis is stated exactly. The hypotheses that a biometric score “is authentic” or “is from an impostor” are both simple hypotheses.

[0040] In a match attempt, the N-scores reported from N<∞ biometrics is called the test observation. The test observation, x∈S, where S⊂R^N (Euclidean N-space) is the score sample space, is a vector function of a random variable X from which is observed a random sample (X₁,X₂, . . . ,X_N). The distribution of scores is provided by the joint class-conditional probability density function (“pdf”):
ƒ(x|θ)=ƒ(χ₁,χ₂, . . . ,χ_N|θ)  (1)
where θ equals either Au to denote the distribution of authentic scores or Im to denote the distribution of impostor scores. So, if θ=Au, Equation (1) is the pdf for authentic distribution of scores and if θ=Im it is the pdf for the impostor distribution. It is assumed that ƒ(x|θ)=ƒ(χ₁, χ₂, . . . ,χ_N|θ)>0 on S.

[0041] Given a test observation, the following two simple statistical hypotheses are observed: The null hypothesis, H₀, states that a test observation is an impostor; the alternate hypothesis, H₁, states that a test observation is an authentic match. Because there are only two choices, the fusion logic will partition S into two disjoint sets: R_Au⊂S and R_Im⊂S, where R_Au∩R_Im=Øand R_Au∪R_Im=S. Denoting the compliment of a set A by A^C, so that R_Au^C=R_Im and R_Im^C=R_Au.

[0042] The decision logic is to accept H₁ (declare a match) if a test observation, x, belongs to R_Au or to accept H₀ (declare no match and hence reject H₁) if x belongs to R_Im. Each hypothesis is associated with an error type: A Type I error occurs when H₀ is rejected (accept H₁) when H₀ is true; this is a false accept (FA); and a Type II error occurs when H₁ is rejected (accept H₀) when H₁ is true; this is a false reject (FR). Thus denoted, respectively, the probability of making a Type I or a Type II error follows:

[0043] $\begin{array}{cc}\begin{array}{c}\mathrm{FAR}=P_{\mathrm{Type}I}\\ =P\left(x\in R_{\mathrm{Au}}|\mathrm{Im}\right)\\ ={\int }_{R_{\mathrm{Au}}}f\left(x|\mathrm{Im}\right)dx\end{array}& \left(2\right)\\ \begin{array}{c}\mathrm{FRR}=P_{\mathrm{Type}\mathrm{II}}\\ =P\left(x\in R_{\mathrm{Im}}|\mathrm{Au}\right)\\ ={\int }_{R_{\mathrm{Im}}}f\left(x|\mathrm{Au}\right)dx\end{array}& \left(3\right)\end{array}$
So, to compute the false-acceptance-rate (FAR), the impostor score's pdf is integrated over the region which would declare a match (Equation 2). The false-rejection-rate (FRR) is computed by integrating the pdf for the authentic scores over the region in which an impostor (Equation 3) is declared. The correct-acceptance-rate (CAR) is 1-FRR.

[0044] The class-conditional pdf for each individual biometric, the marginal pdf, is assumed to have finite support; that is, the match scores produced by the i^th biometric belong to a closed interval of the real line, γ_i=[α_i,ω_i], where −∞<α_i<ω_i<+∞. Two observations are made. First, the sample space, S, is the Cartesian product of these intervals, i.e., S=γ₁×γ₂× . . . ×γ_N. Secondly, the marginal pdf, which we will often reference, for any of the individual biometrics can be written in terms of the joint pdf:

[0045] $\begin{array}{cc}{f}_i\left(x_i|\theta \right)=\underset{{\gamma }_j\forall j\ne i}{\int \int }\cdots \int f\left(x|\theta \right)dx& \left(4\right)\end{array}$
For the purposes of simplicity the following definitions are stated:

• • [0046] Definition 1: In a test of simple hypothesis, the correct-acceptance-rate, CAR=1−FRR is known as the power of the test.
  • [0047] Definition 2: For a fixed FAR, the test of simple H₀ versus simple H₁ that has the largest CAR is called most powerful.
  • [0048] Definition 3: Verification is a one-to-one template comparison to authenticate a person's identity (the person claims their identity).
  • [0049] Definition 4: Identification is a one-to-many template comparison to recognize an individual (identification attempts to establish a person's identity without the person having to claim an identity).
    The receiver operation characteristics (ROC) curve is a plot of CAR versus FAR, an example of which is shown in FIG. 3. Because the ROC shows performance for all possible specifications of FAR, as can be seen in the FIG. 3, it is an excellent tool for comparing performance of competing systems, and we use it throughout our analyses.

[0050] Fusion and Decision Methods: Because different applications have different requirements for error rates, there is an interest in having a fusion scheme that has the flexibility to allow for the specification of a Type-I error yet have a theoretical basis for providing the most powerful test, as defined in Definition 2. Furthermore, it would be beneficial if the fusion scheme could handle the general problem, so there are no restrictions on the underlying statistics. That is to say that:

• • [0051] Biometrics may or may not be statistically independent of each other.
  • [0052] The class conditional pdf can be multi-modal (have local maximums) or have no modes.
  • [0053] The underlying pdf is assumed to be nonparametric (no set of parameters define its shape, such as the mean and standard deviation for a Gaussian distribution).

[0054] A number of combination strategies were examined, all of which are listed by Jain [4] on page 243. Most of the schemes, such as “Demptster-Shafer” and “fuzzy integrals” involve training. These were rejected mostly because of the difficulty in analyzing and controlling their performance mechanisms in order to obtain optimal performance. Additionally, there was an interest in not basing the fusion logic on empirical results. Strategies such as SUM, MEAN, MEDIAN, PRODUCT, MIN, MAX were likewise rejected because they assume independence between biometric features.

[0055] When combining two biometrics using a Boolean “AND” or “OR” it is easily shown to be suboptimal when the decision to accept or reject H₁ is based on fixed score thresholds. However, the “AND” and the “OR” are the basic building blocks of verification (OR) and identification (AND). Since we are focused on fixed score thresholds, making an optimal decision to accept or reject H₁ depends on the structure of: R_Au⊂S and R_Im⊂S, and simple thresholds almost always form suboptimal partitions.

[0056] If one accepts that accuracy dominates the decision making process and cost dominates the combination strategy, certain conclusions may be drawn. Consider a two-biometric verification system for which a fixed FAR has been specified. In that system, a person's identity is authenticated if H₁ is accepted by the first biometric OR if accepted by the second biometric. If H₁ is rejected by the first biometric AND the second biometric, then manual intervention is required. If a cost is associated with each stage of the verification process, a cost function can be formulated. It is a reasonable assumption to assume that the fewer people that filter down from the first biometric sensor to the second to the manual check, the cheaper the system.

[0057] Suppose a fixed threshold is used as the decision making process to accept or reject H₁ at each sensor and solve for thresholds that minimize the cost function. It will obtain settings that minimize cost, but it will not necessarily be the cheapest cost. Given a more accurate way to make decisions, the cost will drop. And if the decision method guarantees the most powerful test at each decision point it will have the optimal cost.

[0058] It is clear to us that the decision making process is crucial. A survey of methods based on statistical decision theory reveals many powerful tests such as the Maximum-Likelihood Test and Bayes' Test. Each requires the class conditional probability density function, as given by Equation 1. Some, such as the Bayes' Test, also require the a priori probabilities P(H₀) and P(H₁), which are the frequencies at which we would expect an impostor and authentic match attempt. Generally, these tests do not allow the flexibility of specifying a FAR—they minimize making a classification error.

[0059] In their seminal paper of 1933 [3], Neyman and Pearson presented a lemma that guarantees the most powerful test for a fixed FAR requiring only the joint class conditional pdf for H₀ and H₁. This test may be used as the centerpiece of the biometric fusion logic employed in the invention. The Neyman-Pearson Lemma guarantees the validity of the test. The proof of the Lemma is slightly different than those found in other sources, but the reason for presenting it is because it is immediately amenable to proving the Corollary to the Neyman Pearson Lemma. The corollary states that fusing two biometric scores with Neyman-Pearson, always provides a more powerful test than either of the component biometrics by themselves. The corollary is extended to state that fusing N biometric scores is better than fusing N−1 scores

[0060] Deriving the Neyman-Pearson Test: Let FAR=α be fixed. It is desired to find R_Au⊂S such that

[0061] $\alpha ={\int }_{R_{\mathrm{Au}}}f\left(x|\mathrm{Im}\right)dx$
and

[0062] ${\int }_{R_{\mathrm{Au}}}f\left(x|\mathrm{Au}\right)dx$
is most powerful. To do this the objective set function is formed that is analogous to Lagrange's Method:

[0063] $u\left(R_{\mathrm{Au}}\right)={\int }_{R_{\mathrm{Au}}}f\left(x|\mathrm{Au}\right)dx-\lambda \left[{\int }_{R_{\mathrm{Au}}}f\left(x|\mathrm{Im}\right)dx-\alpha \right]$
where λ≧0 is an undetermined Lagrange multiplier and the external value of u is subject to the constraint

[0064] $\alpha ={\int }_{R_{\mathrm{Au}}}f\left(x|\mathrm{Im}\right)dx.$
Rewriting the above equation as

[0065] $u\left(R_{\mathrm{Au}}\right)={\int }_{R_{\mathrm{Au}}}\left[f\left(x|\mathrm{Au}\right)-f\left(x|\mathrm{Im}\right)\right]dx+\lambda \alpha$
which ensures that the integrand in the above equation is positive for all x∈R_Au. Let λ≧0, and, recalling that the class conditional pdf is positive on S is assumed, define

[0066] $R_{\mathrm{Au}}=\left\{x\text{:}\left[f\left(x|\mathrm{Au}\right)-\lambda f\left(x|\mathrm{Im}\right)\right]>0\right\}=\left\{x\text{:}\frac{f\left(x|\mathrm{Au}\right)}{f\left(x|\mathrm{Im}\right)}>\lambda \right\}$
Then u is a maximum if λ is chosen such that

[0067] $\alpha ={\int }_{R_{\mathrm{Au}}}f\left(x\text{❘}\mathrm{Im}\right)dx$
is satisfied.

[0068] The Neyman-Pearson Lemma: Proofs of the Neyman-Pearson Lemma can be found in their paper [4] or in many texts [1] [2]. The proof presented here is somewhat different. An “in-common” region, R_IC is established between two partitions that have the same FAR. It is possible this region is empty. Having R_IC makes it easier to prove the corollary presented.

[0069] Neyman-Pearson Lemma: Given the joint class conditional probability density functions for a system of order N in making a decision with a specified FAR=α, let λ be a positive real number, and let

[0070] $\begin{array}{cc}{R}_{\mathrm{Au}}=\left\{x\in S\text{❘}\frac{f\left(x\text{❘}\mathrm{Au}\right)}{f\left(x\text{❘}\mathrm{Im}\right)}>\lambda \right\}& \left(5\right)\\ R_{\mathrm{Au}}^C=R_{\mathrm{Im}}=\left\{x\in S\text{❘}\frac{f\left(x\text{❘}\mathrm{Au}\right)}{f\left(x\text{❘}\mathrm{Im}\right)}\le \lambda \right\}& \left(6\right)\end{array}$
such that

[0071] $\begin{array}{cc}\alpha ={\int }_{R_{\mathrm{Au}}}f\left(x\text{❘}\mathrm{Im}\right)dx.& \left(7\right)\end{array}$
then R_Au is the best critical region for declaring a match—it gives the largest correct-acceptance-rate (CAR), hence R_λ^C gives the smallest FRR.

[0072] Proof: The lemma is trivially true if R_Au is the only region for which Equation (7) holds. Suppose R_φ≠R_Au with m(R_φ∩R_Au^C)>0 (this excludes sets that are the same as R_Au except on a set of measure zero that contribute nothing to the integration), is any other region such that

[0073] $\begin{array}{cc}\alpha ={\int }_{R_{\varphi }}f\left(x\text{❘}\mathrm{Im}\right)dx.& \left(8\right)\end{array}$
Let R_IC=R_φ∩R_Au, which is the “in common” region of the two sets and may be empty. The following is observed:

[0074] $\begin{array}{cc}{\int }_{R_{\mathrm{Au}}}f\left(x\text{❘}\theta \right)dx={\int }_{R_{\mathrm{Au}}-R_{\mathrm{IC}}}f\left(x\text{❘}\theta \right)dx+{\int }_{R_{\mathrm{IC}}}f\left(x\text{❘}\theta \right)dx& \left(9\right)\\ {\int }_{R_{\varphi }}f\left(x\text{❘}\theta \right)dx={\int }_{R_{\varphi }-R_{\mathrm{IC}}}f\left(x\text{❘}\theta \right)dx+{\int }_{R_{\mathrm{IC}}}f\left(x\text{❘}\theta \right)dx& \left(10\right)\end{array}$
If R_Au has a better CAR than R_φthen it is sufficient to prove that

[0075] $\begin{array}{cc}{\int }_{R_{\mathrm{Au}}}f\left(x\text{❘}\mathrm{Au}\right)dx-{\int }_{R_{\varphi }}f\left(x\text{❘}\mathrm{Au}\right)dx>0& \left(11\right)\end{array}$
From Equations (7), (8), (9), and (10) it is seen that (11) holds if

[0076] $\begin{array}{cc}{\int }_{R_{\mathrm{Au}}-R_{\mathrm{IC}}}f\left(x\text{❘}\mathrm{Au}\right)dx-{\int }_{R_{\varphi }-R_{\mathrm{IC}}}f\left(x\text{❘}\mathrm{Au}\right)dx>0& \left(12\right)\end{array}$
Equations (7), (8), (9), and (10) also gives

[0077] $\begin{array}{cc}{\alpha }_{R_{\varphi }-R_{\mathrm{IC}}}={\int }_{R_{\mathrm{Au}}-R_{\mathrm{IC}}}f\left(x\text{❘}\mathrm{Im}\right)dx={\int }_{R_{\varphi }-R_{\mathrm{IC}}}f\left(x\text{❘}\mathrm{Im}\right)dx& \left(13\right)\end{array}$
When x∈R_Au−R_IC⊂R_Au it is observed from (5) that

[0078] $\begin{array}{cc}{\int }_{R_{\mathrm{Au}}-R_{\mathrm{IC}}}f\left(x\text{❘}\mathrm{Au}\right)dx>{\int }_{R_{\mathrm{Au}}-R_{\mathrm{IC}}}\lambda f\left(x\text{❘}\mathrm{Im}\right)dx={\mathrm{\lambda \alpha }}_{R_{\varphi }-R_{\mathrm{IC}}}& \left(14\right)\end{array}$
and when x∈R_φ−R_IC⊂R_Au^C it is observed from (6) that

[0079] $\begin{array}{cc}{\int }_{R_{\varphi }-R_{\mathrm{IC}}}f\left(x\text{❘}\mathrm{Au}\right)dx\le {\int }_{R_{\varphi }-R_{\mathrm{IC}}}\lambda f\left(x\text{❘}\mathrm{Im}\right)dx={\mathrm{\lambda \alpha }}_{R_{\varphi }-R_{\mathrm{IC}}}& \left(15\right)\end{array}$
Equations (14) and (15) give

[0080] $\begin{array}{cc}{\int }_{R_{\mathrm{Au}}-R_{\mathrm{IC}}}f\left(x\text{❘}\mathrm{Au}\right)dx>\lambda {\alpha }_{R\varphi -R_{\mathrm{IC}}}\ge {\int }_{R_{\varphi }-R_{\mathrm{IC}}}f\left(x\text{❘}\mathrm{Au}\right)dx& \left(16\right)\end{array}$
This establishes (12) and hence (11), which proves the lemma. // In this disclosure, the end of a proof is indicated by double slashes “//.”

[0081] Corollary: Neyman-Pearson Test Accuracy Improves with Additional Biometrics: The fact that accuracy improves with additional biometrics is an extremely important result of the Neyman-Pearson Lemma. Under the assumed class conditional densities for each biometric, the Neyman-Pearson Test provides the most powerful test over any other test that considers less than all the biometrics available. Even if a biometric has relatively poor performance and is highly correlated to one or more of the other biometrics, the fused CAR is optimal. The corollary for N-biometrics versus a single component biometric follows.

[0082] Corollarn to the Neyman Pearson Lemma: Given the joint class conditional probability density functions for an N-biometric system, choose α=FAR and use the Neyamn-Pearson Test to find the critical region R_Au that gives the most powerful test for the N-biometric system

[0083] $\begin{array}{cc}{\mathrm{CAR}}_{R_{\mathrm{Au}}}={\int }_{R_{\mathrm{Au}}}f\left(x\text{❘}\mathrm{Au}\right)dx.& \left(17\right)\end{array}$
Consider the i^th biometric. For the same α=FAR, use the Neyman-Pearson Test to find the critical collection of disjoint intervals I_Au⊂R¹ that gives the most powerful test for the single biometric, that is,

[0084] $\begin{array}{cc}{\mathrm{CAR}}_{I_{\mathrm{Au}}}={\int }_{\mathrm{IAu}}f\left(x_i\text{❘}\mathrm{Au}\right)dx& \left(18\right)\end{array}$
then
CAR_RAu≧CAR_IAu.  (19)
Proof: Let R_i=I_Au×γ₁×γ₂×. . . ×λ_N⊂R^N, where the Cartesian products are taken over all the γ_k except for k=i. From (4) the marginal pdf can be recast in terms of the joint pdf

[0085] $\begin{array}{cc}{\int }_{R_i}f\left(x\text{❘}\theta \right)dx={\int }_{I_{Au}}f\left(x_i\text{❘}\theta \right)dx& \left(20\right)\end{array}$
First, it will be shown that equality holds in (19) if and only if R_Au=R_i. Given that R_Au=R_i except on a set of measure zero, i.e., m(R_Au^C∩R_i)=0, then clearly CAR_RAu=CAR_1Au. On the other hand, assume CAR_RAu=CAR_IAu and m(R_Au^C∩R_i)>0, that is, the two sets are measurably different. But this is exactly the same condition previously set forth in the proof of the Neyman-Pearson Lemma from which from which it was concluded that CAR_RAu>CAR_IAu, which is a contradiction. Hence equality holds if and only if R_Au=R_i. Examining the inequality situation in equation (19), given that R_Au≠R_i such that m(R_Au^{C ∩R}_i)>0, then it is shown again that the exact conditions as in the proof of the Neyman-Pearson Lemma have been obtained from which we conclude that CAR_RAu>CAR_IAu, which proves the corollary.

[0086] Examples have been built such that CAR_RAu=CAR_1Au but it is hard to do and it is unlikely that such densities would be seen in a real world application. Thus, it is safe to assume that CAR_RAu>CAR_1Au is almost always true.

[0087] The corollary can be extended to the general case. The fusion of N biometrics using Neyman-Pearson theory always results in a test that is as powerful as or more powerful than a test that uses any combination of M<N biometrics. Without any loss to generality, arrange the labels so that the first M biometrics are the ones used in the M<N fusion. Choose α=FAR and use the Neyman-Pearson Test to find the critical region R_M⊂R^M that gives the most powerful test for the M-biometric system. Let R_N=R_M×γ_M+1×γ_{M +2}×. . . ×γ_N⊂R^N, where the Cartesian products are taken over all the γ-intervals not used in the M biometrics combination. Then writing

[0088] $\begin{array}{cc}{\int }_{R_N}f\left(x\text{❘}\theta \right)dx={\int }_{R_M}f\left(x_1,x_2\dots ,x_M\text{❘}\theta \right)dx& \left(21\right)\end{array}$
gives the same construction as in (20) and the proof flows as it did for the corollary.

[0089] We now state and prove the following five mathematical propositions. The first four propositions are necessary to proving the fifth proposition, which will be cited in the section that details the invention. For each of the propositions we will use the following: Let r={r₁,r₂, . . . ,r_n} be a sequence of real valued ratios such that r₁≧r₂≧. . . ≧r_n. For each r_i we know the numerator and denominator of the ratio, so that

[0090] $r_i=\frac{n_i}{d_i},n_i,d_i>0\forall i.$

[0091] Proposition #1: For r_i,r_i+1∈r and

[0092] $r_i=\frac{n_i}{d_i},r_{i+1}=\frac{n_{i+1}}{d_{i+1}},$
then

[0093] $\frac{n_i+n_{i+1}}{d_i+d_{i+1}}\le \frac{n_i}{d_i}=r_i.$
Proof: Because r_i,r_i+1∈r we have

[0094] $\frac{n_{i+1}}{d_{i+1}}\le \frac{n_i}{d_i}\Rightarrow \frac{d_i{n}_{i+1}}{n_i{d}_{i+1}}\le 1\Rightarrow \frac{d_i}{n_i}\left(n_i+n_{i+1}\right)\le d_i+d_{i+1}\Rightarrow \frac{n_i+n_{i+1}}{d_i+d_{i+1}}\le \frac{n_i}{d_i}=r_i$

[0095] Proposition #2: For r_i,r_i+1∈r and

[0096] $r_i=\frac{n_i}{d_i},r_{i+1}=\frac{n_{i+1}}{d_{i+1}},$
then

[0097] $r_{i+1}=\frac{n_{i+1}}{d_{i+1}}\le \frac{n_i+n_{i+1}}{d_i+d_{i+1}}.$
Proof: The proof is the same as for proposition #1 except the order is reversed.

[0098] Proposition #3: For

[0099] $r_i=\frac{n_i}{d_i},i=\left(1,\dots ,n\right),$
then

[0100] $\frac{\sum _{i=m}^M{n}_i}{\sum _{i=m}^M{d}_i}\le \frac{n_m}{d_m},1\le m\le M\le n.$
Proof: The proof is by induction. We know from proposition #1 that the result holds for N=1, that is

[0101] $\frac{n_i+n_{i+1}}{d_i+d_{i+1}}\le \frac{n_i}{d_i}.$
Now assume it holds for any N>1, we need to show that it holds for N+1. Let m=2 and M=N+1 (note that this is the N^th case and not the N+1 case), then we assume

[0102] $\frac{\sum _{i=2}^{N+1}{n}_i}{\sum _{i=2}^{N+1}{d}_i}\le \frac{n_2}{d_2}\le \frac{n_1}{d_1}\Rightarrow \frac{d_i}{n_1}\frac{\sum _{i=2}^{N+1}{n}_i}{\sum _{i=2}^{N+1}{d}_i}\le 1\Rightarrow \frac{d_1}{n_1}\sum _{i=2}^{N+1}{n}_i\le \sum _{i=2}^{N+1}{d}_i\Rightarrow \frac{d_i}{n_i}\sum _{i=2}^{N+1}{n}_i+d_i\le \sum _{i=2}^{N+1}{d}_i+d_1=\sum _{i=1}^{N+1}{d}_i\Rightarrow d_1\left(1+\frac{\sum _{i=2}^{N+1}{n}_i}{n_1}\right)\le \sum _{i=1}^{N+1}{d}_i\Rightarrow d_1\left(\frac{n_1+\sum _{i=2}^{N+1}{n}_i}{n_1}\right)=\frac{d_1}{n_1}\sum _{i=1}^{N+1}{n}_i\le \sum _{i=1}^{N+1}{d}_i\Rightarrow \frac{\sum _{i=1}^{N+1}{n}_i}{\sum _{i=1}^{N+1}{d}_i}\le \frac{n_1}{d_1}$
which is the N+1 case as was to be shown.

[0103] Proposition #4: For

[0104] $r_i=\frac{n_i}{d_i},i=\left(1,\dots ,n\right),$
then

[0105] $\frac{n_M}{d_M}\le \frac{\sum _{i=m}^M{n}_i}{\sum _{i=m}^M{d}_i},1\le m\le M\le n.$
Proof: As with proposition #3, the proof is by induction. We know from proposition #2 that the result holds for N=1, that is

[0106] $\frac{n_{i+1}}{d_{i+1}}\le \frac{n_i+n_{i+1}}{d_i+d_{i+1}}.$
The rest of the proof follows the same format as for proposition #3 with the order reversed.

[0107] Proposition #5: Recall that the r is an ordered sequence of decreasing ratios with known numerators and denominators. We sum the first N numerators to get S_n and sum the first N denominators to get S_d. We will show that for the value S_n, there is no other collection of ratios in r that gives the same S_n and a smaller S_d. For

[0108] $r_i=\frac{n_i}{d_i},i=\left(1,\dots ,n\right),$
let S be the sequence of the first N terms of r, with the sum of numerators given by

[0109] $S_n=\sum _{i=1}^N{n}_i,$
and the sum of denominators by

[0110] $S_d=\sum _{i=1}^N{d}_i,1\le N\le n.$
Let S′ be any other sequence of ratios in r, with numerator sum S′_n=Σn_i and denominator sum S′_d=Σd_i such that S′_n=S_n, then we have S_d≦S′_d. Proof: The proof is by contradiction. Suppose the proposition is not true, that is, assume another sequence, S′, exists such that S′_d<S_d. For this sequence, define the indexing sets A={indices that can be between 1 and N inclusive} and B={indices that can be between N+1 and n inclusive}. We also define the indexing set A^C={indices between 1 and N inclusive and not in A}, which means A∩A^C=Ø and A∪A^C={all indices between 1 and N inclusive}. Then our assumption states:
S′_n=Σ_i∈An_i+Σ_i∈Bn_i=S_n
and
S′_d=Σ_i∈Ad_i+Σ_i∈Bd_i<S_d.
This implies that

[0111] $\begin{array}{cc}\begin{array}{c}\sum _{i\in A}{d}_i+\sum _{i\in B}{d}_i<\sum _{i=1}^N{d}_i=\sum _{i\in A}{d}_i+\sum _{i\in A^c}{d}_i.\\ \Rightarrow \sum _{i\in B}{d}_i<\sum _{i\in A^c}{d}_i\\ \Rightarrow \frac{1}{\sum _{i\in A^c}{d}_i}<\frac{1}{\sum _{i\in B}{d}_i}\end{array}& \left(22\right)\end{array}$
Because the numerators of both sequences are equal, we can write

[0112] $\begin{array}{cc}\sum _{i\in A}{n}_i+\sum _{i\in B}{n}_i=\sum _{i\in A\bigcup A^c}{n}_i\Rightarrow \sum _{i\in B}{n}_i=\sum _{i\in A\bigcup A^c}{n}_i-\sum _{i\in A}{n}_i=\sum _{i\in A^c}{n}_i& \left(23\right)\end{array}$
Combining (22) and (23), and from propositions #3 and #4, we have

[0113] $r_N=\frac{n_N}{d_N}\le \frac{\sum _{i\in A^c}{n}_i}{\sum _{i\in A^c}{d}_i}<\frac{\sum _{i\in B}{n}_i}{\sum _{i\in B}{d}_i}\le \frac{n_{N+1}}{d_{N+1}}=r_{N+1},$
which contradicts the fact that r_N>r_N+, hence the validity of the proposition follows.

[0114] Cost Functions for Optimal Verification and Identification: In the discussion above, it is assumed that all N biometric scores are simultaneously available for fusion. The Neyman-Pearson Lemma guarantees that this provides the most powerful test for a fixed FAR. In practice, however, this could be an expensive way of doing business. If it is assumed that all aspects of using a biometric system, time, risk, etc., can be equated to a cost, then a cost function can be constructed. The inventive process described below constructs the cost function and shows how it can be minimized using the Neyman-Pearson test. Hence, the Neyman-Pearson theory is not limited to “all-at-once” fusion; it can be used for serial, parallel, and hierarchal systems.

[0115] Having laid a basis for the invention, a description of two cost functions is now provided as a mechanism for illustrating an embodiment of the invention. A first cost function for a verification system and a second cost function for an identification system are described. For each system, an algorithm is presented that uses the Neyman-Pearson test to minimize the cost function for a second order biometric system, that is a biometric system that has two modalities. The cost functions are presented for the general case of N-biometrics. Because minimizing the cost function is recursive, the computational load grows exponentially with added dimensions. Hence, an efficient algorithm is needed to handle the general case.

[0116] First Cost Function—Verification System. A cost function for a 2-stage biometric verification (one-to-one) system will be described, and then an algorithm for minimizing the cost function will be provided. In a 2-stage verification system, a subject may attempt to be verified by a first biometric device. If the subject's identity cannot be authenticated, the subject may attempt to be authenticated by a second biometric. If that fails, the subject may resort to manual authentication. For example, manual authentication may be carried out by interviewing the subject and determining their identity from other means.

[0117] The cost for attempting to be authenticated using a first biometric, a second biometric, and a manual check are c₁, c₂, and c₃, respectively. The specified FAR_sys is a system false-acceptance-rate, i.e. the rate of falsely authenticating an impostor includes the case of it happening at the first biometric or the second biometric. This implies that the first-biometric station test cannot have a false-acceptance-rate, FAR₁, that exceeds FAR_sys. Given a test with a specified FAR₁, there is an associated false-rejection-rate, FRR₁, which is the fraction of subjects that, on average, are required to move on to the second biometric station. The FAR required at the second station is FAR₂=FAR_sys=FAR₁. It is known that P(A∪B)=P(A)+P(B)−P(A)P(B), so the computation of FAR₂ appears imperfect. However, if FAR_sys=P(A∪B) and FAR₁=P(A), in the construction of the decision space, it is intended that FAR₂=P(B)−P(A)P(B).

[0118] Note that FAR₂ is a function of the specified FAR_sys and the freely chosen FAR₁; FAR₂ is not a free parameter. Given a biometric test with the computed FAR₂, there is an associated false-rejection-rate, FRR₂ by the second biometric test, which is the fraction of subjects that are required to move on to a manual check. This is all captured by the following cost function:
Cost=c₁+FRR₁(c₂+c₃FRR₂)  (30)

[0119] There is a cost for every choice of FAR₁≦FAR_sys, so the Cost in Equation 30 is a function of FAR₁. For a given test method, there exists a value of FAR₁ that yields the smallest cost and we present an algorithm to find that value. In a novel approach to the minimization of Equation 30, a modified version of the Neyman-Pearson decision test has been developed so that the smallest cost is optimally small.

[0120] An algorithm is outlined below. The algorithm seeks to optimally minimize Equation 30. To do so, we (a) set the initial cost estimate to infinity and, (b) for a specified FAR_sys, loop over all possible values of FAR₁≦FAR_sys. In practice, the algorithm may use a uniformly spaced finite sample of the infinite possible values. The algorithm may proceed as follows: (c) set FAR₁=FAR_sys, (d) set Cost=∞, and (e) loop over possible FAR₁ values. For the first biometric at the current FAR₁ value, the algorithm may proceed to (f) find the optimal Match-Zzone, R₁, and (g) compute the correct-acceptance-rate over R₁ by:

[0121] $\begin{array}{cc}{\mathrm{CAR}}_1={\int }_{R_1}f\left(x|\mathrm{Au}\right)dx& \left(31\right)\end{array}$
and (h) determine FRR₁ using FRR₁=1−CAR₁.

[0122] Next, the algorithm may test against the second biometric. Note that the region R₁ of the score space is no longer available since the first biometric test used it up. The Neyman-Pearson test may be applied to the reduced decision space, which is the compliment of R₁. So, at this time, (h) the algorithm may compute FAR₂=FAR_sys−FAR₁, and FAR₂ may be (i) used in the Neyman-Pearson test to determine the most powerful test, CAR₂, for the second biometric fused with the first biometric over the reduced decision space R₁^c. The critical region for CAR₂ is R₂, which is disjoint from R₁ by our construction. Score pairs that result in the failure to be authenticated at either biometric station must fall within the region R₃=(R₁∪R₂)^C, from which it is shown that

[0123] ${\mathrm{FRR}}_2=\frac{{\int }_{R_3}f\left(x|\mathrm{Au}\right)dx}{{\mathrm{FRR}}_1}$

[0124] The final steps in the algorithm are (0) to compute the cost using Equation 30 at the current setting of FAR₁ using FRR₁ and FRR₂, and (k) to reset the minimum cost if cheaper, and keep track of the FAR₁ responsible for the minimum cost.

[0125] To illustrate the algorithm, an example is provided. Problems arising from practical applications are not to be confused with the validity of the Neyman-Pearson theory. Jain states in [5]: “In case of a larger number of classifiers and relatively small training data, a classifier my actually degrade the performance when combined with other classifiers . . . ” This would seem to contradict the corollary and its extension. However, the addition of classifiers does not degrade performance because the underlying statistics are always true and the corollary assumes the underlying statistics. Instead, degradation is a result of inexact estimates of sampled densities. In practice, a user may be forced to construct the decision test from the estimates, and it is errors in the estimates that cause a mismatch between predicted performance and actual performance.

[0126] Given the true underlying class conditional pdf for, H₀ and H₁, the corollary is true. This is demonstrated with a challenging example using up to three biometric sensors. The marginal densities are assumed to be Gaussian distributed. This allows a closed form expression for the densities that easily incorporates correlation. The general n^th order form is well known and is given by

[0127] $\begin{array}{cc}f\left(x|\theta \right)=\frac{1}{{\left(2\pi \right)}^{\frac{n}{2}}{C}^{1/2}}\exp \left(-\frac{1}{2}\left[{\left(x-\mu \right)}^T{C}^{-1}\left(x-\mu \right)\right]\right)& \left(32\right)\end{array}$
where μ is the mean and C is the covariance matrix. The mean (μ) and the standard deviation (σ) for the marginal densities are given in Table 1. Plots of the three impostor and three authentic densities are shown in FIG. 1.

[0128]

TABLE 1
Biometric	Impostors		Authentics
Number	μ	Σ	μ	σ
#1	0.3	0.06	0.65	0.065
#2	0.32	0.065	0.58	0.07
#3	0.45	0.075	0.7	0.08

[0129]

TABLE 2
Correlation Coefficient	Impostors	Authentics
ρ12	0.03	0.83
ρ13	0.02	0.80
ρ23	0.025	0.82

[0130] To stress the system, the authentic distributions for the three biometric systems may be forced to be highly correlated and the impostor distributions to be lightly correlated. The correlation coefficients (ρ) are shown in Table 2. The subscripts denote the connection. A plot of the joint pdf for the fusion of system #1 with system #2 is shown in FIG. 2, where the correlation between the authentic distributions is quite evident.

[0131] The single biometric ROC curves are shown in FIG. 3. As could be predicted from the pdf curves plotted in FIG. 1, System #1 performs much better than the other two systems, with System #3 having the worst performance.

[0132] Fusing 2 systems at a time; there are three possibilities: #1+#2, #1+#3, and #2+#3. The resulting ROC curves are shown in FIG. 4. As predicted by the corollary, each 2-system pair outperforms their individual components. Although the fusion of system #2 with system #3 has worse performance than system #1 alone, it is still better than the single system performance of either system #2 or system #3.

[0133] Finally, FIG. 5 depicts the results when fusing all three systems and comparing its performance with the performance of the 2-system pairs. The addition of the third biometric system gives substantial improvement over the best performing pair of biometrics.

[0134] Tests were conducted on individual and fused biometric systems in order to determine whether the theory presented above accurately predicts what will happen in a real-world situation. The performance of three biometric systems were considered. The numbers of score samples available are listed in Table 3. The scores for each modality were collected independently from essentially disjoint subsets of the general population.

[0135]

TABLE 3
		Number of Authentic
Biometric	Number of Impostor Scores	Scores
Fingerprint	8,500,000	21,563
Signature	325,710	990
Facial Recognition	4,441,659	1,347

[0136] To simulate the situation of an individual obtaining a score from each biometric, the initial thought was to build a “virtual” match from the data of Table 3. Assuming independence between the biometrics, a 3-tuple set of data was constructed. The 3-tuple set was an ordered set of three score values, by arbitrarily assigning a fingerprint score and a facial recognition score to each signature score for a total of 990 authentic score 3-tuples and 325,710 impostor score 3-tuples.

[0137] By assuming independence, it is well known that the joint pdf is the product of the marginal density functions, hence the joint class conditional pdf for the three biometric systems, ƒ(x|θ)=ƒ(χ₁,χ₂,χ₃|θ) can be written as
ƒ(X|θ)=ƒ_fingerprint(χ|θ)ƒ_signature(χ|θ)ƒ_facial(χ|θ)  (33)

[0138] So it is not necessary to dilute the available data. It is sufficient to approximate the appropriate marginal density functions for each modality using all the data available, and compute the joint pdf using Equation 33.

[0139] The class condition density functions for each of the three modalities, ƒ_fingerprint(χ|θ), ƒ_signature(χ|θ) and ƒ_facial(χ|θ), were estimated using the available sampled data. The authentic score densities were approximated using two methods: (1) the histogram-interpolation technique and (2) the Parzen-window method. The impostor densities were approximated using the histogram-interpolation method. Although each of these estimation methods are guaranteed to converge in probability to the true underlying density as the number of samples goes to infinity, they are still approximations and can introduce error into the decision making process, as will be seen in the next section. The densities for fingerprint, signature, and facial recognition are shown in FIG. 6, FIG. 7, and FIG. 8 respectively. Note that the authentic pdf for facial recognition is bimodal. The Neyman-Pearson test was used to determine the optimal ROC for each of the modalities. The ROC curves for fingerprint, signature, and facial recognition are plotted in FIG. 9.

[0140] There are 3 possible unique pairings of the three biometric systems: (1) fingerprints with signature, (2) fingerprints with facial recognition, and (3) signatures with facial recognition. Using the marginal densities (above) to create the required 2-D joint class conditional density functions, two sets of 2-D joint density functions were computed - one in which the authentic marginal densities were approximated using the histogram method, and one in which the densities were approximated using the Parzen window method. Using the Neyman-Pearson test, an optimal ROC was computed for each fused pairing and each approximation method. The ROC curves for the histogram method are shown in FIG. 10 and the ROC curves for the Parzen window method are shown in FIG. 11.

[0141] As predicted by the corollary, the fused performance is better than the individual performance for each pair under each approximation method. But, as we cautioned in the example, error due to small sample sizes can cause pdf distortion. This is apparent when fusing fingerprints with signature data (see FIG. 10 and FIG. 11). Notice that the Parzen-window ROC curve (FIG. 11) crosses over the curve for fingerprint-facial-recognition fusion, but does not cross over when using the histogram interpolation method (FIG. 10). Small differences between the two sets of marginal densities are magnified when using their product to compute the 2-dimensional joint densities, which is reflected in the ROC.

[0142] As a final step, all three modalities were fused. The resulting ROC using histogram interpolating is shown in FIG. 12, and the ROC using the Parzen window is shown FIG. 13. As might be expected, the Parzen window pdf distortion with the 2-dimensional fingerprint-signature case has carried through to the 3-dimensional case. The overall performance, however, is dramatically better than any of the 2-dimensional configurations as predicted by the corollary.

[0143] In the material presented above, it was assumed that all N biometric scores would be available for fusion. Indeed, the Neyman-Pearson Lemma guarantees that this provides the most powerful test for a fixed FAR. In practice, however, this could be an expensive way of doing business. If it is assumed that all aspects of using a biometric system, time, risk, etc., can be equated to a cost, then a cost function can be constructed. Below, we construct the cost function and show how it can be minimized using the Neyman-Pearson test. Hence, the Neyman-Pearson theory is not limited to “all-at-once” fusion—it can be used for serial, parallel, and hierarchal systems.

[0144] In the following section, a review of the development of the cost function for a verification system is provided, and then a cost function for an identification system is developed. For each system, an algorithm is presented that uses the Neyman-Pearson test to minimize the cost function for a second order modality biometric system. The cost function is presented for the general case of N-biometrics. Because minimizing the cost function is recursive, the computational load grows exponentially with added dimensions. Hence, an efficient algorithm is needed to handle the general case.

[0145] From the prior discussion of the cost function for a 2-station system, the costs for using the first biometric, the second biometric, and the manual check are c₁, c₂, and c₃, respectively, and FAR_sys is specified. The first-station test cannot have a false-acceptance-rate, FAR₁, that exceeds FAR_sys. Given a test with a specified FAR₁, there is an associated false-rejection-rate, FRR₁, which is the fraction of subjects that, on average, are required to move on to the second station. The FAR required at the second station is FAR₂=FAR_sys−FAR₁. It is known that P(A∪B)=P(A)+P(B)−P(A)P(B), so the computation of FAR₂ appears imperfect. However, if FAR_sys=P(A∪B) and FAR₁=P(A), in the construction of the decision space, it is intended that FAR₂=P(B)−P(A)P(B).

[0146] Given a test with the computed FAR₂, there is an associated false-rejection-rate, FRR₂ by the second station, which is the fraction of subjects that are required to move on to a manual checkout station. This is all captured by the following cost function
Cost=c₁+FRR₁(c₂+c₃FRR₂)  (30)

[0147] There is a cost for every choice of FAR₁≦FAR_sys, so the Cost in Equation 30 is a function of FAR₁. For a given test method, there exists a value of FAR₁ that yields the smallest cost and we develop an algorithm to find that value.

[0148] Using a modified Neyman-Pearson test to optimally minimize Equation 30, an algorithm can be derived. Step 1: set the initial cost estimate to infinity and, for a specified FAR_sys, loop over all possible values of FAR₁≦FAR_sys. To be practical, in the algorithm a finite sample of the infinite possible values may be used. The first step in the loop is to use the Neyman-Pearson test at FAR₁ to determine the most powerful test, CAR₁, for the first biometric, and FRR₁=1−CAR₁ is computed. Since it is one dimensional, the critical region is a collection of disjoint intervals I_Au. As in the proof to the corollary, the 1-dimensional I_Au is recast as a 2-dimensional region, R₁=I_Au×γ₂⊂R², so that

[0149] $\begin{array}{cc}{\mathrm{CAR}}_1={\int }_{R_1}f\left(x|\mathrm{Au}\right)dx={\int }_{I_{\mathrm{Au}}}f\left(x_1|\mathrm{Au}\right)dx& \left(34\right)\end{array}$

[0150] When it is necessary to test against the second biometric, the region R₁ of the decision space is no longer available since the first test used it up. The Neyman-Pearson test can be applied to the reduced decision space, which is the compliment of R₁. Step 2: FAR₂=FAR_sys−FAR₁ is computed. Step 3: FAR₂ is used in the Neyman-Pearson test to determine the most powerful test, CAR₂, for the second biometric fused with the first biometric over the reduced decision space R₁^C. The critical region for CAR₂ is R ₂, which is disjoint from R₁ by the construction. Score pairs that result in the failure to be authenticated at either biometric station must fall within the region R₃=(R₁∪R₂)^C, from which it is shown that

[0151] ${\mathrm{FRR}}_2=\frac{{\int }_{R_3}f\left(x|\mathrm{Au}\right)dx}{{\mathrm{FRR}}_1}$

[0152] Step 4: compute the cost at the current setting of FAR₁ using FRR₁ and FRR₂. Step 5: reset the minimum cost if cheaper, and keep track of the FAR₁ responsible for the minimum cost. A typical cost function is shown in FIG. 14.

[0153] Two special cases should be noted. In the first case, if c₁>0, c₂>0, and c₃=0, the algorithm shows that the minimum cost is to use only the first biometric—that is, FAR₁=FAR_sys. This makes sense because there is no cost penalty for authentication failures to bypass the second station and go directly to the manual check.

[0154] In the second case, c₁=c₂=0 and c₃>0, the algorithm shows that scores should be collected from both stations and fused all at once; that is, FAR₁=1.0. Again, this makes sense because there is no cost penalty for collecting scores at both stations, and because the Neyman-Pearson test gives the most powerful CAR (smallest FRR) when it can fuse both scores at once.

[0155] To extend the cost function to higher dimensions, the logic discussed above is simply repeated to arrive at

[0156] $\begin{array}{cc}\mathrm{Cost}=c_1+c_2{\mathrm{FRR}}_1+c_3\prod _{i=1}^2{\mathrm{FRR}}_i+\dots +c_{N+1}\prod _{i=1}^N{\mathrm{FRR}}_i& \left(35\right)\end{array}$
In minimizing Equation 35, there is 1 degree of freedom, namely FAR₁. Equation 35 has N-1 degrees of freedom. When FAR₁≦FAR_sys are set, then FAR₂≦FAR₁ can bet set, then FAR₃≦FAR₂, and so on—and to minimize thus, N-1 levels of recursion.

[0157] Identification System: Identification (one-to-many) systems are discussed. With this construction, each station attempts to discard impostors. Candidates that cannot be discarded are passed on to the next station. Candidates that cannot be discarded by the biometric systems arrive at the manual checkout. The goal, of course, is to prune the number of impostor templates thus limiting the number that move on to subsequent steps. This is a logical AND process—for an authentic match to be accepted, a candidate must pass the test at station 1 and station 2 and station 3 and so forth. In contrast to a verification system, the system administrator must specify a system false-rejection-rate, FRR_sys instead of a FAR_sys. But just like the verification system problem the sum of the component FRR values at each decision point cannot exceed FRR_sys. If FRR with FAR are replaced in Equations 34 or 35 the following cost equations are arrived at for 2-biometric and an N-biometric identification system
Cost=c₁+FAR₁(c₂+c₃FAR₂)  (36)

[0158] $\begin{array}{cc}\mathrm{Cost}=c_1+c_2{\mathrm{FAR}}_1+c_3\prod _{i=1}^2{\mathrm{FAR}}_i+\dots +c_{N+1}\prod _{i=1}^N{\mathrm{FAR}}_i& \left(37\right)\end{array}$
Alternately, equation 37 may be written as:

[0159] $\begin{array}{cc}\mathrm{Cost}=\sum _{j=0}^N\left(c_{j+1}\prod _{i=1}^j{\mathrm{FAR}}_i\right),& \left(37A\right)\end{array}$
These equations are a mathematical dual of Equations 34 and 35 are thus minimized using the logic of the algorithm that minimizes the verification system.

[0160] Algorithm: Generating Matching and Non-Matching PDF Surfaces. The optimal fusion algorithm uses the probability of an authentic match and the probability of a false match for each p_ij∈P. These probabilities may be arrived at by numerical integration of the sampled surface of the joint pdf. A sufficient number of samples may be generated to get a “smooth” surface by simulation. Given a sequence of matching score pairs and non-matching score pairs, it is possible to construct a numerical model of the marginal cumulative distribution functions (cdf). The distribution functions may be used to generate pseudo random score pairs. If the marginal densities are independent, then it is straightforward to generate sample score pairs independently from each cdf. If the densities are correlated, we generate the covariance matrix and then use Cholesky factorization to obtain a matrix that transforms independent random deviates into samples that are appropriately correlated.

[0161] Assume a given partition P. The joint pdf for both the authentic and impostor cases may be built by mapping simulated score pairs to the appropriate P_ij∈P and incrementing a counter for that sub-square. That is, it is possible to build a 2-dimensional histogram, which is stored in a 2-dimensional array of appropriate dimensions for the partition. If we divide each array element by the total number of samples, we have an approximation to the probability of a score pair falling within the associated sub-square. We call this type of an array the probability partition array (PPA). Let P_fm be the PPA for the joint false match distribution and let P_m be the PPA for the authentic match distribution. Then, the probability of an impostor's score pair, (s₁, s₂)∈p_ij, resulting in a match is P_fm(i,j). Likewise, the probability of a score pair resulting in a match when it should be a match is P_m(i,j). The PPA for a false reject (does not match when it should) is P_fr=1−P_m.

[0162] Consider the partition arrays, P_fm and P_m, defined in Section #5. Consider the ratio

[0163] $\frac{P_{\mathrm{fm}}\left(i,j\right)}{P_m\left(i,j\right)},P_{\mathrm{fr}}>0.$
The larger this ratio the more the sub-square indexed by (i,j) favors a false match over a false reject. Based on the propositions presented in Section 4, it is optimal to tag the sub-square with the largest ratio as part of the no-match zone, and then tag the next largest, and so on.

[0164] Therefore, an algorithm that is in keeping with the above may proceed as:

• • [0165] Step 1. Assume a required FAR has been given.
  • [0166] Step 2. Allocate the array P_mz to have the same dimensions as P_m and
    • [0167] P_fm. This is the match zone array. Initialize all of its elements as belonging to the match zone.
  • [0168] Step 3. Let the index set A={all the (i,j) indices for the probability partition arrays}.
  • [0169] Step 4. Identify all indices in A such that P_m(i,j)=P_fm(i,j)=0 and store those indices in the indexing set Z={(i,j):P_m(i,j)=P_fm(i,j)=0 }. Tag each element in P_mz indexed by Z as part of the no-match zone.
  • [0170] Step 5. Let B=A−Z={(i,j): P_m(i,j)≠0 and P_fm(i,j)≠0}. We process only those indices in B. This does not affect optimality because there is most likely zero probability that either a false match score pair or a false reject score pair falls into any sub-square indexed by elements of Z.
  • [0171] Step 6. Identify all indices in B such that
    • [0172] P_fm(i,j)>0 and P_m(i,j)=0 and store those indices in Z∞={(i,j)_k:
    • [0173] P_fm(i,j)>0, P_m(i,j)=0}. Simply put, this index set includes the indexes to all the sub-squares that have zero probability of a match but non-zero probability of false match.
  • [0174] Step 7. Tag all the sub-squares in P_mz indexed by Z∞ as belonging to the no-match zone. At this point, the probability of a matching score pair falling into the no-match zone is zero. The probability of a non-matching score pair falling into the match zone is:
    FAR_Z∞=1−Σ_(i,j)∈Z∞P_fm(i,j)
  • [0175] Furthermore, if FAR_Z∞<=FAR then we are done and can exit the algorithm.
  • [0176] Step 8. Otherwise, we construct a new index set

[0177] $C=A-Z-Z_{\infty }=\left\{{\left(i,j\right)}_k:{P_{\mathrm{fm}}\left(i,j\right)}_k>0,{P_m\left(i,j\right)}_k>0,\frac{{P_{\mathrm{fm}}\left(i,j\right)}_k}{{P_m\left(i,j\right)}_k}>=\frac{{P_{\mathrm{fm}}\left(i,j\right)}_{k+1}}{{P_m\left(i,j\right)}_{k+1}}\right\}.$

• • [0178] We see that C holds the indices of the non-zero probabilities in a sorted order—the ratios of false-match to match probabilities occur in descending order.
  • [0179] Step 9. Let C_N be the index set that contains the first N indices in C. We determine N so that:
    FAR_z∞∪CN=1−Σ_(i,j)∈Z∞P_fm(i,j)−Σ_(i,j)∈CNP_fm(i,j)≦FAR
  • [0180] Step 10. Label elements of P_mz indexed by members of C_N as belonging to the no-match zone. This results in a FRR given by
    FRR=Σ_(i,j)∈CNP_m(i,j),
  • [0181]  and furthermore this FRR is optimal.

[0182] It will be recognized that other variations of these steps may be made, and still be within the scope of the invention. For clarity, the notation “(i,j)” is used to identify arrays that have at least two modalities. Therefore, the notation “(i,j)” includes more than two modalities, for example (i,j,k), (i,j,k,l), (i,j,k,l,m), etc.

[0183] For example, FIG. 15 illustrates one method according to the invention in which Pm(i,j) is provided 10 and Pfm(i,j) 13 is provided. As part of identifying 16 indices (i,j) corresponding to a no-match zone, a first index set (“A”) may be identified. The indices in set A may be the (i,j) indices that have values in both Pfm(i,j) and Pm(i,j). A second index set (“Z∞”) may be identified, the indices of Z∞ being the (i,j) indices in set A where both Pfm(i,j) is larger than zero and Pm(i,j) is equal to zero. Then determine FAR_Z∞ 19, where FAR_Z∞=1−Σ_(i,j)∈Z∞P_fm(i,j). It should be noted that the indices of Z∞ may be the indices in set A where Pm(i,j) is equal to zero, since the indices where both Pfm(i,j)=Pm(i,j)=0 will not affect FAR_Z∞, and since there will be no negative values in the probability partition arrays. However, since defining the indices of Z∞ to be the (i,j) indices in set A where both Pfm(i,j) is larger than zero and Pm(i,j) is equal to zero yields the smallest number of indices for Z∞, we will use that definition for illustration purposes since it is the least that must be done for the mathematics of FAR_Z∞ to work correctly. It will be understood that larger no-match zones may be defined, but they will include Z∞, so we illustrate the method using the smaller Z∞ definition believing that definitions of no-match zones that include Z∞ will fall within the scope of the method described.

[0184] FAR_Z∞ may be compared 22 to a desired false-acceptance-rate (“FAR”), and if FAR_Z∞ is less than or equal to the desired false-acceptance-rate, then the data set may be accepted, if false-rejection-rate is not important. If FAR_Z∞ is greater than the desired false-acceptance-rate, then the data set may be rejected 25.

[0185] If false-rejection-rate is important to determining whether a data set is acceptable, then indices in a match zone may be selected, ordered, and some of the indices may be selected for further calculations 28. Toward that end, the following steps may be carried out. The method may further include identifying a third index set ZM∞, which may be thought of as a modified Z∞, that is to say a modified no-match zone. Here we modify Z∞ so that ZM∞ includes indices that would not affect the calculation for FAR_Z∞, but which might affect calculations related to the false-rejection-rate. The indices of ZM∞ may be the (i,j) indices in Z∞ plus those indices where both Pfm(i,j) and Pm(i,j) are equal to zero. The indices where Pfm(i,j)=Pm(i,j)=0 are added to the no-match zone because in the calculation of a fourth index set (“C”), these indices should be removed from consideration. The indices of C may be the (i,j) indices that are in A but not ZM∞. The indices of C may be arranged such that

[0186] $\frac{{P_{\mathrm{fm}}\left(i,j\right)}_k}{{P_m\left(i,j\right)}_k}>=\frac{{P_{\mathrm{fm}}\left(i,j\right)}_{k+1}}{{P_m\left(i,j\right)}_{k+1}}$
to provide an arranged C index. A fifth index set (“Cn”) may be identified. The indices of Cn may be the first N (i,j) indices of the arranged C index, where N is a number for which the following is true:

[0187] ${\mathrm{FAR}}_{Z_{\infty }\bigcup C_N}=1-\sum _{\left(i,j\right)\in Z_{\infty }}{P}_{\mathrm{fm}}\left(i,j\right)-\sum _{\left(i,j\right)\in C_N}{P}_{\mathrm{fm}}\left(i,j\right)\le \mathrm{FAR}.$

[0188] The FRR may be determined 31, where FRR=Σ_(i,j)∈CNP_m(i,j), and compared 34 to a desired false-rejection-rate. If FRR is greater than the desired false-rejection-rate, then the data set may be rejected 37, even though FAR_Z∞ is less than or equal to the desired false-acceptance-rate. Otherwise, the data set may be accepted.

[0189] FIG. 16 illustrates another method according to the invention in which the FRR may be calculated first. In FIG. 16, the reference numbers from FIG. 15 are used, but some of the steps in FIG. 16 may vary somewhat from those described above. In such a method, a first index set (“A”) may be identified, the indices in A being the (i,j) indices that have values in both Pfm(i,j) and Pm(i,j). A second index set (“Z∞”), which is the no match zone, may be identified 16. The indices of Z∞ may be the (i,j) indices of A where Pm(i,j) is equal to zero. Here we use the more inclusive definition for the no-match zone because the calculation of set C comes earlier in the process. A third index set (“C”) may be identified, the indices of C being the (i,j) indices that are in A but not Z∞. The indices of C may be arranged such that

[0190] $\frac{{P_{\mathrm{fm}}\left(i,j\right)}_k}{{P_m\left(i,j\right)}_k}>=\frac{{P_{\mathrm{fm}}\left(i,j\right)}_{k+1}}{{P_m\left(i,j\right)}_{k+1}}$
to provide an arranged C index, and a fourth index set (“Cn”) may be identified. The indices of Cn may be the first N (i,j) indices of the arranged C index, where N is a number for which the following is true: FAR_Z∞∪CN=1−Σ_(i,j)∈Z∞P_fm(i,j)−Σ_(i,j)∈CNP_fm(i,j)≦FAR . The FRR may be determined, where FRR=Σ_(i,j)∈CNP_m(i,j), and compared to a desired false-rejection-rate. If the FRR is greater than the desired false-rejection-rate, then the data set may be rejected. If the FRR is less than or equal to the desired false-rejection-rate, then the data set may be accepted, if false-acceptance-rate is not important. If false-acceptance-rate is important, the FAR_Z∞ may be determined, where FAR_Z∞=1−Σ_(i,j)∈Z∞P_fm(i,j). Since the more inclusive definition is used for Z∞ in this method, and that more inclusive definition does not affect the value of FAR_Z∞ we need not add back the indices where both Pfm(i,j)=Pm(i,j)=0. The FAR_Z∞ may be compared to a desired false-acceptance-rate, and if FAR_Z∞ is greater than the desired false-acceptance-rate, then the data set may be rejected. Otherwise, the data set may be accepted.

[0191] It may now be recognized that a simplified form of a method according to the invention may be executed as follows:

• • [0192] step 1: provide a first probability partition array (“Pm(i,j)”), the Pm(i,j) being comprised of probability values for information pieces in the data set, each probability value in the Pm(i,j) corresponding to the probability of an authentic match;
  • [0193] step 2: provide a second probability partition array (“Pfm(i,j)”), the Pfrn(i,j) being comprised of probability values for information pieces in the data set, each probability value in the Pfm(i,j) corresponding to the probability of a false match;
  • [0194] step 3: identify a first index set (“A”), the indices in set A being the (i,j) indices that have values in both Pfm(i,j) and Pm(i,j);
  • [0195] step 4: execute at least one of the following:
    • [0196] (a) identify a first no-match zone (“Z1∞”) that includes at least the indices of set A for which both Pfm(i,j) is larger than zero and Pm(i,j) is equal to zero, and use Z1∞ to determine FAR _Z∞, where FAR_Z∞=1−Σ_(i,j)∈Z∞P_fm(i,j), and compare FAR_Z∞ to a desired false-acceptance-rate, and if FAR_Z∞ is greater than the desired false-acceptance-rate, then reject the data set;
    • [0197] (b) identify a second no-match zone (“Z2∞”) that includes the indices of set A for which Pm(i,j) is equal to zero, and use Z2∞ to identify a second index set (“C”), the indices of C being the (i,j) indices that are in A but not Z2∞, and arrange the (i,j) indices of C such that

[0198] $\frac{{P_{\mathrm{fm}}\left(i,j\right)}_k}{{P_m\left(i,j\right)}_k}>=\frac{{P_{\mathrm{fm}}\left(i,j\right)}_{k+1}}{{P_m\left(i,j\right)}_{k+1}}$
to provide an arranged C index, and identify a third index set (“Cn”), the indices of Cn being the first N (i,j) indices of the arranged C index, where N is a number for which the following is true:
FAR_Z∞∪CN=1−Σ_(i,j)∈Z∞P_fm(i,j)−Σ_(i,j)∈CNP_fm(i,j)≦FAR

• • • [0199]  and determine FRR, where FRR=Σ_(i,j)∈CNP_m(i,j), and compare FRR to a desired false-rejection-rate, and if FRR is greater than the desired false-rejection-rate, then reject the data set.
      If FAR_Z∞ is less than or equal to the desired false-acceptance-rate, and FRR is less than or equal to the desired false-rejection-rate, then the data set may be accepted. Z1∞ may be expanded to include additional indices of set A by including in Z1∞ the indices of A for which Pm(i,j) is equal to zero. In this manner, a single no-match zone may be defined and used for the entire step 4.

[0200] The invention may also be embodied as a computer readable memory device for executing a method according to the invention, including those described above. See FIG. 17. For example, in a system 100 according to the invention, a memory device 103 may be a compact disc having computer readable instructions 106 for causing a computer processor 109 to execute steps of a method according to the invention. A processor 109 may be in communication with a data base 112 in which the data set may be stored, and the processor 109 may be able to cause a monitor 115 to display whether the data set is acceptable. Additional detail regarding a system according to the invention is provided in FIG. 18. For brevity, the steps of such a method will not be outlined here, but it should suffice to say that any of the methods outlined above may be translated into computer readable code which will cause a computer to determine and compare FAR_Z∞ and/or FRR as part of a system designed to determine whether a data set is acceptable for making a decision.

[0201] U.S. provisional patent application No. 60/643,853 discloses additional details about the invention and additional embodiments of the invention. The disclosure of that patent application is incorporated by this reference.

[0202] Although the present invention has been described with respect to one or more particular embodiments, it will be understood that other embodiments of the present invention may be made without departing from the spirit and scope of the present invention. Hence, the present invention is deemed limited only by the appended claims and the reasonable interpretation thereof.

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