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WO2009032020A1 - Système logique de décision de fusion multimodale utilisant le modèle de copules - Google Patents

Système logique de décision de fusion multimodale utilisant le modèle de copules Download PDF

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Publication number
WO2009032020A1
WO2009032020A1 PCT/US2007/082130 US2007082130W WO2009032020A1 WO 2009032020 A1 WO2009032020 A1 WO 2009032020A1 US 2007082130 W US2007082130 W US 2007082130W WO 2009032020 A1 WO2009032020 A1 WO 2009032020A1
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indices
probability
far
pfm
biometric
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Fred W. Kiefer
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Ultra Scan Corp
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Ultra Scan Corp
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N20/00—Machine learning
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N5/00—Computing arrangements using knowledge-based models
    • G06N5/02—Knowledge representation; Symbolic representation

Definitions

  • 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.
  • 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”.
  • 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.
  • biometric identification systems 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.
  • NIST National Institute of Standards and Technology
  • 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.
  • False Acceptance Rate (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. [0009] Systematic research in the area of combining biometric modalities is nascent and sparse.
  • the present invention includes a method of deciding whether a data set is acceptable for making a decision.
  • 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.
  • 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(ij) 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(ij)" 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(ij) may be similar to a Neyman-Pearson Lemma probability partition array.
  • the Pm(i,j) and/or the Pfhi(i,j) may be Copula models.
  • a method according to the invention may identify a no-match zone.
  • 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(ij).
  • a second index set may be identified, the indices of being the (i,j) indices in set A where both Pfm(i,j) is larger than zero and Pm(ij) is equal to zero.
  • ⁇ 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 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.
  • FAR false- acceptance-rate
  • the method may further include identifying a third index set ZM ⁇ , the indices of being the (i,j) indices in plus those indices where both Pfm(ij) and Pm(ij) 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 .
  • the indices of C may be
  • 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: R .
  • the FRR may be determined, where > an d compared to a desired false-rejection-rate.
  • the data set may be rejected, even though FAR zoo is less than or equal to the desired false-acceptance-rate. Otherwise, the data set may be accepted.
  • the false-rejection-rate calculations and comparisons may be executed before the false-acceptance-rate calculations and comparisons.
  • 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 may be identified, the indices of Z ⁇ being the (i,j) indices of A where Pm(ij) is equal to zero.
  • a third index set (“C”) may be identified, the indices of C being the (i,j) indices that
  • the indices of C may be arranged such that 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: .
  • the FRR may be determined, where > an d 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 ⁇ 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.
  • the invention may also be embodied as a computer readable memory device for executing any of the methods described above.
  • Figure 1 which represents example PDFs for three biometrics data sets
  • Figure 2 which represents a joint PDF for two systems
  • FIG 3 which is a plot of the receiver operating curves (ROC) for three biometrics
  • Figure 4 which is a plot of the two-dimensional fusions of the three biometrics taken two at a time versus the single systems
  • Figure 5 which is a plot the fusion of all three biometric systems
  • Figure 6 which is a plot of the scores versus fingerprint densities
  • Figure 7 which is a plot of the scores versus signature densities
  • Figure 8 which is a plot of the scores versus facial recognition densities
  • Figure 9 which is the ROC for the three individual biometric systems
  • Figure 10 which is the two-dimensional ROC - Histogram interpolations for the three biometrics singly and taken two at a time
  • Figure 11 which is the two-dimensional ROC similar to Figure 10 but using the Parzen Window method
  • Figure 13 which is the three-dimensional ROC similar to Figure 12 but using the Parzen Window method
  • Figure 14 which is a cost function for determining the minimum cost given a desired FAR and FRR
  • Figure 15 which depicts a method according to the invention
  • FIG 16 which depicts another method according to the invention
  • Figure 17 which depicts a memory device according to the invention
  • Figure 18 which is an exemplary embodiment flow diagram demonstrating a security system typical of and in accordance with the methods presented by this invention.
  • Figure 19 which shows probability density functions (pdfs) of mismatch
  • Figure 20 which depicts empirical distribution of genuine scores of CUBS data generated on DBl of FVC 2004 fingerprint images.
  • Figure 21 which depicts a histogram of the CUBS genuine scores
  • Figure 25 which depicts the distribution of estimates F(t 0 ).
  • Figure 26 which depicts the bootstrap estimates ordered in ascending order.
  • Figures 27A and 27B which show GPD fit to tails of imposter data and authentic data.
  • Figures 28A and 28B which depict the test CDFs (in black color) are matching or close to the estimates of training CDFs.
  • Figure 29 which depicts an ROC with an example of confidence box.
  • Figure 30 which shows a red curve that is the frequency histogram of raw data for 390,198 impostor match scores from a fingerprint identification system, and shows a blue curve that is the histogram for 3052 authentic match scores from the same system.
  • Figure 31 which depicts the QQ-plot for the data depicted in Figure 30.
  • Figure 32A and 32B which depict probability density functions showing the Parzen window fit and the GPD (Gaussian Probability Distribution) fit.
  • Figure 32A shows a subset fit
  • Figure 32B shows the fit for the full set of data.
  • Figure 33A and 33B which are enlarged views of the right-side tail sections of Figures 32A and 32B respectively.
  • 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.
  • test observation is the score sample space, is a vector function of a random variable X from which is observed a random sample (X 1 , X 2 , ... , X N ) .
  • the distribution of scores is provided by the joint class-conditional probability density function ("pdf '):
  • the decision logic is to accept Hi (declare a match) if a test observation, x, belongs to R Au or to accept Ho (declare no match and hence reject Hi) if x belongs toR Im .
  • Each hypothesis is associated with an error type: A Type I error occurs when H 0 is rejected (accept H 1 ) when H 0 is true; this is a false accept (FA); and a Type II error occurs when Hj is rejected (accept H 0 ) when Hi is true; this is a false reject (FR).
  • F false accept
  • FR false reject
  • 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.
  • the class-conditional pdf for each individual biometric 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, .
  • S the sample space
  • the marginal pdf which we will often reference, for any of the individual biometrics can be written in terms of the joint pdf:
  • 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 Figure 3. Because the ROC shows performance for all possible specifications of FAR, as can be seen in the Figure 3, it is an excellent tool for comparing performance of competing systems, and we use it throughout our analyses. [0023] 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:
  • Biometrics may or may not be statistically independent of each other.
  • the class conditional pdf can be multi-modal (have local maximums) or have no modes.
  • 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).
  • Neyman and Pearson presented a lemma that guarantees the most powerful test for a fixed FAR requiring only the joint class conditional pdf for H 0 and H 1 .
  • 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-I scores
  • Equations (7), (8), (9), and (10) also gives
  • Equations (14) and (15) give This establishes (12) and hence (11), which proves the lemma. // In this disclosure, the end of a proof is indicated by double slashes "//.”
  • 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.
  • 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 Ot FAR and use the Neyman-Pearson Test to find the critical region that gives the most powerful test for the M-biometric system. , where the Cartesian products are taken over all the ⁇ -intervals not used in the M biometrics combination. Then writing gives the same construction as in (20) and the proof flows as it did for the corollary.
  • Proposition #2 For , then Proof: The proof is the same as for proposition #1 except the order is reversed.
  • 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 Sa- For , let S be the sequence of the first N terms of r, with the sum of numerators given by ⁇ and the sum of denominators by S , ⁇ ⁇ N ⁇ n . Let S ' be any other sequence of ratios in r, with numerator sum and denominator sum such that , then we have .
  • a first cost function for a verification system and a second cost function for an identification system are described.
  • 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.
  • 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.
  • 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.
  • the cost for attempting to be authenticated using a first biometric, a second biometric, and a manual check arec, ,c 2 , andc 3 , 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, FAR, , that exceeds FAR sys .
  • FRR false-rejection-rate
  • FAR 2 is a function of the specified FAR sys and the freely chosen FAR, ; FAR 2 is not a free parameter.
  • FRR 2 false-rejection-rate
  • 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 OfFAR 1 ⁇ FAR sys .
  • the algorithm may test against the second biometric. Note that the region R 1 of the score space is no longer available since the first biometric test used it up.
  • the critical region for CAR 2 is R 2 , which is disjoint from R 1 by our construction. Score pairs that result in the failure to be authenticated at either biometric station must fall within the region , from which it is shown that
  • the final steps in the algorithm are (j) to compute the cost using Equation 30 at the current setting of FAR 1 using FRR 1 and FRR 2 , and (k) to reset the minimum cost if cheaper, and keep track of the FAR 1 responsible for the minimum cost.
  • 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 (p) 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 Figure 2, where the correlation between the authentic distributions is quite evident.
  • 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.
  • 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.
  • the costs for using the first biometric, the second biometric, and the manual check are c x ,c 2 , and c 3 , respectively, and FAR sys is specified.
  • the first-station test cannot have a false-acceptance- rate, FAR 1 , that exceeds FAR sys .
  • FAR 1 false-acceptance- rate
  • FRR 1 false-rejection-rate
  • FAR 2 FAR sys - FAR 1 .
  • Step 1 set the initial cost estimate to infinity and, for a specified FAR sys , loop over all possible values of FAR 1 ⁇ FAR sys .
  • Step 4 compute the cost at the current setting of FAR 1 using FRR 1 and FRR 2 .
  • Step 5 reset the minimum cost if cheaper, and keep track of the FAR 1 responsible for the minimum cost.
  • a typical cost function is shown in Figure 14.
  • Equation 35 there is 1 degree of freedom, namely FAR 1 .
  • Equation 35 has N-I degrees of freedom.
  • FAR ⁇ FAR sys are set, then FAR 2 ⁇ FAR, can bet set, then FAR 3 ⁇ FAR 2 , and so on - and to minimize thus, N-I levels of recursion.
  • Identification System Identification (one-to-many) systems are discussed.
  • 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 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.
  • the system administrator must specify a system false-rejection-rate, FRR sys instead of a FAR sys .
  • equation 37 may be written as:
  • Equations 34 and 35 are a mathematical dual of Equations 34 and 35 are thus minimized using the logic of the algorithm that minimizes the verification system.
  • 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 iy ⁇ 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.
  • cdf marginal cumulative distribution functions
  • the joint pdf for both the authentic and impostor cases may be built by mapping simulated score pairs to the appropriate 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 Pg n 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, , resulting in a match is . Likewise, the probability of a score pair resulting in a match when it should be a match is . The PPA for a false reject (does not match when it should) is .
  • Step 1 Assume a required FAR has been given.
  • Step 2. Allocate the array P 1 ⁇ Z to have the same dimensions as P m and P fm . This is the match zone array. Initialize all of its elements as belonging to the match zone.
  • Step 6. Identify all indices in B such that
  • 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.
  • Step 7. Tag all the sub-squares in P mz indexed by Z 00 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: Furthermore, if then we are done and can exit the algorithm. Step 8. Otherwise, we construct a new index set
  • Step 9 Let C N be the index set that contains the first N indices in C. We determine N so that:
  • 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 and furthermore this FRR is optimal.
  • 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.
  • Figure 15 illustrates one method according to the invention in which Pm(ij) is provided 10 and Pfm(i,j) 13 is provided.
  • 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(ij).
  • a second index set may be identified, the indices of Z ⁇ being the (i,j) indices in set A where both Pfm(ij) is larger than zero and Pm(i,j) is equal to zero.
  • FAR z ⁇ 19 where • It should be noted that the indices of Z ⁇ may be the indices in set A where Pm(ij) is equal to zero, since the indices where both
  • [0085] may be compared 22 to a desired false-acceptance-rate ("FAR"), and if 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 is greater than the desired false- acceptance-rate, then the data set may be rejected 25.
  • FAR false-acceptance-rate
  • indices in a match zone may be selected, ordered, and some of the indices may be selected for further calculations 28.
  • the method may further include identifying a third index set which may be thought of as a modified Z ⁇ , that is to say a modified no-match zone.
  • Z ⁇ may be thought of as a modified Z ⁇ , that is to say a modified no-match zone.
  • 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 maybe the (i,j) indices in Z ⁇ plus those indices where both Pfm(ij) and Pm(i,j) are equal to zero.
  • the indices of C may be the (i,j) indices that are in A but not ZM ⁇ .
  • the indices of C may be arranged
  • 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: .
  • the FRR may be determined 31 , where , 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.
  • Figure 16 illustrates another method according to the invention in which the
  • FRR may be calculated first.
  • a first index set (“A") may be identified, the indices in A being the (i,j) indices that have values in both Pfm(ij) and Pm(i,j).
  • a second index set which is the no match zone, may be identified 16.
  • the indices of Z ⁇ may be the (i,j) indices of A where Pm(ij) 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
  • 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: .
  • the FRR may be determined, where , and compared to a desired false-rejection-rate.
  • step 1 provide a first probability partition array ("Pm(i,j)"), the Pm(ij) being comprised of probability values for information pieces in the data set, each probability value in the Pm(ij) corresponding to the probability of an authentic match;
  • Pm(i,j) a first probability partition array
  • step 2 provide a second probability partition array ("Pfm(ij)"), 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(ij) a second probability partition array
  • step 3 identify a first index set ("A"), the indices in set A being the (i,j) indices that have values in both Pfm(ij) and Pm(i,j);
  • step 4 execute at least one of the following:
  • 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;
  • indices of C such that 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: and determine FRR, where , 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.
  • the data set may be accepted.
  • 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 Figure 17.
  • 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 Figure 18.
  • each dimension can require a partition of 1000 bins or more, an "n" of four or more could easily swamp storage capacity.
  • copulas provides an accurate estimate of the joint pdf by weighting a product of the marginal densities with a n- dimensional normal distribution that has the covariance of the correlated marginals.
  • This Gaussian copula model is derived in [1 : Nelsen, Roger B., An Introduction to Copulas, 2 nd Edition. Springer 2006, ISBN 10-387-28659-4].
  • Copulas are a relatively new area of statistics; as is stated in [1], the word “copula” did not appear in the Encyclopedia of Statistical Sciences until 1997 and [1] is essentially the first text on the subject. The theory allows us to defeat the "curse” by storing the one-dimensional marginal densities and an n- by-n covariance matrix, which transforms storage requirements from products of dimensions to sums of dimensions - or in other words, from billions to thousands.
  • Equation 74
  • ⁇ is the pdf of the standard normal distribution and is the joint pdf of and is n-dimensional normal distribution with mean zero and covariance matrix R.
  • the Gaussian copula function can be written in terms of the exponential function:
  • Equation 74 Sklar's Theorem guarantees the existence of a copula function that constructs multidimensional joint distributions by coupling them with the one-dimensional marginals [1],[2].
  • Equation 77 is the copula model that we use to compute the densities for the
  • J 11n are the sample means of S Au and S Im respectively.
  • the diagonal entries of the associated R matrices are unity because the copula components are standard normal distributions, and the off diagonal entries are the correlation coefficients in the C matrices just computed. Recall that the correlation coefficient between the i"' and j" 1
  • X 1 F 1 (S 1 ) by numerically computing the integral: .
  • X 1 F 1 (S 1 ) is the probability of obtaining a score less than S 1 , hence .
  • the value is the random deviate of the standard normal distribution for which there is probability X 1 of drawing a value less than Z 1 , hence .
  • erf ⁇ x (x) is the inverse error function.
  • Statistical modeling of biometric systems at score level may be extremely important. It can be the foundation for biometric systems performance assessment, including determination of confidence intervals, test sample size of a simulation and prediction of real world system performances.
  • Statistical modeling used in multimodal biometric systems integrates information from multiple biometric sources to compensate for the limitations in performance of each individual biometric system.
  • a general non-parametric method is used for fitting the center body part of a distribution curve.
  • a parametric model the general Pareto distribution, is used for fitting the tail parts of the curve, which is theoretically supported by extreme value theory.
  • a biometric fusion scheme using Gaussian copula may be adapted to incorporate a set of marginal distributions estimated by the proposed procedures.
  • Biometric systems authenticate or identify subjects based on their physiological or behavioral characteristics such as fingerprints, face, voice and signature. In an authentication system, one is interested in confirming whether the presented biometric is in some sense the same as or close to the enrolled biometrics of the same person.
  • biometric systems compute and store a compact representation (template) of a biometric sample rather than the sample of a biometric (a biometric signal). Taking the biometric samples or their representations (templates) as patterns, an authentication of a person with a biometric sample is then an exercise in pattern recognition.
  • FAR False Accept Rate
  • FRR False Reject Rate
  • Biometric applications can be divided into two types: verification tasks and identification tasks.
  • verification tasks a single matching score is produced, and an application accepts or rejects a matching attempt by thresholding the matching score. Based on the threshold value, performance characteristics of FAR and FRR can be estimated.
  • identification tasks a set of N matching scores ⁇ _?, , s 2 ,...,s N ⁇ , S 1 > s 2 > ... > s N is produced for N enrolled persons. The person can be identified as an enrollee corresponding to the best ranked score S 1 , or the identification attempt can be rejected.
  • FAR and FRR can be estimated.
  • the usual decision algorithm for verification and identification involves setting some threshold t 0 and accepting a matching score s if this score is greater than threshold: s > t 0 .
  • PDFs Probability Density Functions
  • Performance prediction a) Statistical prediction of operational performance from laboratory testing. b) Prediction of large database identification performance from small- scale performance.
  • Data quality evaluation a) Quality evaluation of biometric data, relating biometric sampling quality to performance improvement. b) Standardization of data collection to ensure quality; determine meaningful metrics for quality.
  • System performance a) Modeling the matching process as a parameterized system accounting for all the variables (joint probabilities). b) Determine the best payoff for system performance improvement.
  • the behavior of distributions can be modeled by straightforward parametric methods, for example, a Gaussian distribution can be assumed and a priori training data can be used to estimate parameters, such as the mean and the variance of the model. Such a model is then taken as a normal base condition for signaling error or being used in classification, such as a Bayesian based classifiers. More sophisticated approaches use Gaussian mixture models [2] or kernel density estimates [3].
  • the main limitation of all of these methods is that they make an unwarranted assumption about the nature of the distribution. For instance, the statistical modeling based on these methods weigh on central statistics such as the mean and the variance, and the analysis is largely insensitive to the tails of the distribution. Therefore, using these models to verify biometrics can be problematic in the tail part of the distribution. An incorrect estimation of a distribution especially at tails or making an unwarranted assumption about the nature of the PDFs can lead to a false positive or a false negative matching.
  • PDFs probability density functions
  • a simple naive estimator that is also called empirical distribution function is used:
  • the kernel- based methods are generally considered to be a superior non-parametric model than the naive formula in (2).
  • V* is a random variable with distribution function F .
  • the linear interpolation H(O of H(O) is then an approximation of the random variable.
  • F(O is
  • the distribution function (d.f.) of the truncated scores can be defined as in
  • Figure 20 depicts the empirical distribution of genuine scores of CUBS data generated on DBl of FVC2004 fingerprint images. The vertical line indicates the 95% level.
  • Figure 20 shows the empirical distribution function of the CUBS genuine fingerprint matching data evaluated at each of the data points.
  • the empirical d.f. for a sample of size n is defined by
  • the CUBS data comprise 2800 genuine matching scores that is generated using DBl of FVC 2004 fingerprint images (database #1 of a set of images available from the National Bureau Of Standards, fingerprint verification collection).
  • the limit distribution must be an extreme value distribution for some value of the parameters ⁇ , ⁇ ,
  • Distributions in the maximum domain of attraction of the Gumbel include the normal, exponential, gamma and lognormal distributions.
  • the lognormal distribution has a moderately heavy tail and has historically been a popular model for loss severity distributions; however it is not as heavy-tailed as the distributions in MDA(H ⁇ ) for ⁇ ⁇ 0 .
  • Distributions in the domain of attraction of the Weibull are short tailed distributions such as the uniform and beta distributions.
  • the GPD is defined to be .
  • the CUBS Fingerprint Matching Data consist of 2800 genuine matching scores and 4950 impostor matching scores that are generated by CUBS fingerprint verification/identification system using DBl of FVC 2004 fingerprint images. Results described in this section are from using the genuine matching scores for clarity, although the same results can be generated using the impostor scores.
  • EVS EVS
  • the histogram of the scores gives us an intuitive view of the distribution.
  • the CUBS data values are re-scaled into range of [0,1]. See Figure 21.
  • A. Quantile-Quantile Plot The quantile-quantile plot (QQ-plot) is a graphical technique for determining if two data sets come from populations with a common distribution.
  • a QQ-plot is a plot of the quantiles of the first data set against the quantiles of the second data set. By a quantile, we mean the fraction (or percent) of points below the given value.
  • a 45-degree reference line is also plotted. If the two sets come from a population with the same distribution, the points should fall approximately along this reference line. The greater the departure from this reference line, the greater the evidence for the conclusion that the two data sets have come from populations with different distributions.
  • the QQ-plot against the exponential distribution is a very useful guide.
  • the points should lie approximately along a straight line if the data are an i.i.d. sample from an exponential distribution.
  • the first method is a useful graphical tool which is the plot of the sample mean excess function of the distribution. IfX has a GPD with parameters ⁇ and ⁇ , the excess over the threshold u has also a GPD and its mean excess function over this threshold is given by
  • the sample mean plot is the plot of the points: where X 1 n and X 11n are the first and nth order statistics and e n (u) is the sample mean excess
  • the mean excess function describes the expected overshoot of a threshold given that exceedance occurs.
  • Figure 24 shows the sample mean excess plot of the CUBS data.
  • a confidence interval is defined as a range within which a distribution F(t 0 ) is reliably estimated by F(t Q ) .
  • F(t Q ) a distribution of the probability that the true distribution F at ⁇ 0 can be guaranteed as
  • ZIM is distributed according to a norm distribution, i.e., Z I M ⁇ N(F(t o ), ⁇ (t o )) .
  • a confidence interval can be determined. For example, a 90% interval of confidence is Figure 25 depicts F(t 0 ).
  • the bootstrap uses the available data many times to estimate confidence intervals or to perform hypothesis testing [10].
  • a parametric approach again, makes assumptions about the shape of the error in the probability distribution estimates.
  • a non-parametric approach makes no such assumption about the errors in
  • H F (z) Pr ob(F(t 0 ) ⁇ z) , where H F (z) is the probability distribution of the estimate F(V 0 )
  • the sample population X has empirical distribution F(t 0 ) defined in (2) or
  • the bootstrap principle prescribes sampling, with replacement, the set X a large number (B) of times, resulting in the bootstrap sets . Using these sets,
  • Determining confidence intervals now essentially amounts to a counting exercise.
  • the bootstrap estimates are ordered in ascending order and the middle 90% (the samples between
  • a quantile estimator x ⁇ p) is defined by substituting estimated parameters ⁇ and ⁇ for the parameters in (17).
  • the variance of x(p) is given asymptotically by [12]
  • Figures 27 A and 27B show GPD fit to tails of imposter data and authentic data. From the figure we can see that the tail estimates using GPD represented by red lines are well fitted with the real data, which are inside the empirical estimate of the score data in blue color. The data are also enclosed safely inside a confidence interval. The thick red bars are examples of confidence intervals.
  • EVT Error Transcription theory
  • Methods based around assumptions of normal distributions are likely to underestimate the tails.
  • Methods based on simulation can only provide very imprecise estimates of tail distributions.
  • EVT is the most scientific approach to an inherently difficult problem - predicting the probability of a rare event.
  • copulas provides an accurate estimate of the joint pdf by weighting a product of the marginal densities with an n- dimensional normal distribution that has the co variance of the correlated marginals.
  • This Gaussian copula model is derived in [1 : Nelsen, Roger B., An Introduction to Copulas, 2 nd Edition. Springer 2006, ISBN 10-387-28659-4].
  • Copulas are a relatively new area of statistics; as is stated in [1], the word “copula” did not appear in the Encyclopedia of Statistical Sciences until 1997 and [1] is essentially the first text on the subject. The theory allows us to defeat the "curse” by storing the one-dimensional marginal densities and an n- by-n co variance matrix, which transforms storage requirements from products of dimensions to sums of dimensions, from billions to thousands.
  • a biometric system with "n” recognition systems that produce “n” scores for each identification event has both an authentic and an impostor cdf
  • Equation 65 is the cdf of the standard normal distribution and ⁇ "1 is its inverse, and is an n-dimensional Gaussian cdf whose arguments are the random deviates of standard normal distributions, that is they have a mean of zero and a standard deviation of unity.
  • ⁇ ⁇ is assumed to have covariance given by R and a mean of zero.
  • the diagonal entries of R are unity, and, in general, the off diagonal entries are non-zero and measure the correlation between components.
  • Equation 68 is the copula model that we use to compute the densities for the
  • the diagonal entries of the associated R matrices are unity because the copula components are standard normal distributions, and the off diagonal entries are the correlation coefficients in the C matrices just computed. Recall that the correlation coefficient between the i"' and j"' element of C,
  • X 1 F 1 [S 1 ) by numerically computing the integral: .
  • We numerically solve is the inverse error function.
  • Our invention uses a two-prong approach to modeling a pdf from a given set of representative sample data.
  • EDT Extreme Value Theory
  • We use results from Extreme Value Theory (EVT) see [1 : Statistical Modeling of Biometric Systems Using Extreme Value Theory, A USC STTR Project Report to the U .S. Army] and [2: An Introduction to Statistical Modeling of Extreme Values, Stuart Coles, Springer, ISBN 1852334592]), which provides techniques for predicting rare events; that is, events not included in the observed data.
  • EVT uses rigorous statistical theory in the analysis of residuals to derive a family of extreme value distributions, which is known collectively as the generalized extreme value (GEV) distribution.
  • GEV generalized extreme value
  • the GEV distribution is the limiting distribution for sample extrema and is analogous to the normal distribution as the limiting distribution of sample means as explained by the central limit theorem.
  • Equation 70 Given a sample of matching scores for a given biometric, we employ the algorithmic steps explained below to determine the values for ⁇ and ⁇ to be used in Equation 70, which will be used to model the tail that represents the rare event side of the pdf. In order to more clearly see what it is we are going to model and how we are going to model it, consider the densities shown in Figure 30.
  • the red curve is the frequency histogram of the raw data for 390,198 impostor match scores from a fingerprint identification system.
  • the blue curve is the histogram for 3052 authentic match scores from the same system.
  • Step 1 Sort the sample scores in ascending order and store in the ordered set
  • Step 2 Let S 1 be an ordered sequence of threshold values that partition S into body and tail. Let this sequence of threshold be uniformly spaced between . The values of are chosen by visual inspection of the histogram of S with set at the score value in S above which there is approximately 7% of the tail data. The value may be chosen so that the tail has at least 100 elements. For each we construct the set of tail values [00204] Step 3: In this step, we estimate the values of ⁇ ; and ⁇ ( . the parameters of the
  • Step 4 Having estimated the values of ⁇ ,. and ⁇ . , we may now compute a linear correlation coefficient that tells us how good the fit is to the values in T 1 . To do this we may use the concept of a QQ plot.
  • a QQ plot is the plot of the quantiles of one probability distribution versus the quantiles of another distribution. If the two distributions are the same, then the plot will be a straight line. In our case, we would plot the set of sorted scores in T f , which we call , versus the quantiles of the GPD with ⁇ ; and ⁇ . , which is the set
  • Step 5 The with the maximum r t is chosen as the model for the tail of the pdf.
  • Step 6 The complete modeling of the pdf uses the fit from a non-parametric technique for scores up to the associated , and for scores above is used.
  • Steps 1 through 4 above we have the best fit with , .
  • the QQ plot is shown in Figure 31 revealing an excellent linear fit.
  • Figure 32A and 32B it appears that the Parzen window and GPD models provide a good fit to both the subset of scores (left-hand plot) and the complete set of scores (right-hand plot). However, a closer look at the tails tells the story.
  • Figure 33A and 33B we have zoomed in on the tail section and show the GPD tail-fit in the black line, the Parzen window fit in red, and the normalized histogram of the raw subset data in black dots. As can be seen, both the Parzen fit and the GPD fit look good but they have remarkably different shapes.
  • United States provisional patent application number 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.

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Abstract

La présente invention concerne un procédé permettant de déterminer si un ensemble de données est acceptable pour une prise de décision. Un premier système de partition de probabilités et un second système de partition de probabilité peuvent être prévus. Un ou les deux réseaux de partition de probabilités peut/peuvent être un modèles de copules. Une zone de non appariement peut être établie et utilisée pour calculer un taux de fausses acceptations (FAR) et/ou un taux de faux rejets (FRR) pour l'ensemble de données. Le taux de fausses acceptations et/ou le taux de faux rejets peuvent être comparés à des taux souhaités. En fonction de la comparaison, l'ensemble de données peut être soit accepté soit rejeté. L'invention peut se présenter également sous la forme d'un dispositif de mémoire lisible par ordinateur pour exécuter les procédés.
PCT/US2007/082130 2007-09-07 2007-10-22 Système logique de décision de fusion multimodale utilisant le modèle de copules Ceased WO2009032020A1 (fr)

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US8190540B2 (en) 2005-01-14 2012-05-29 Ultra-Scan Corporation Multimodal fusion decision logic system for determining whether to accept a specimen
WO2013112275A1 (fr) * 2012-01-27 2013-08-01 The State Of Oregon Acting By And Through The State Board Of Higher Education On Behalf Of The Portland State University Système à base de copules et procédé de gestion de spécifications de test de fabrication et de produit pendant le cycle de vie du produit pour des systèmes électroniques ou des circuits intégrés
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CN113468721B (zh) * 2021-06-07 2024-03-29 太原科技大学 一种对齿轮减速箱中齿轮和轴承剩余寿命的预测方法
CN119671289A (zh) * 2025-02-18 2025-03-21 湖南大学 天空地一体、通导感融合的电力系统典型灾害关键诱发因素辨识方法及相关装置

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