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Quantifying the weight of ﬁngerprint evidence through the spatial relationship, directions and types of minutiae observed on ﬁngermarks Cedric Neumann a,b,c,*, Christophe Champod d, Mina Yoo b, Thibault Genessay d, Glenn Langenburg e a

The South Dakota State University, Department of Mathematics and Statistics, Harding Hall, Brookings, SD 57007, United States The Pennsylvania State University, Department of Statistics, University Park, PA 16802, United States Two N’s Forensics Inc., Brookings, SD 57006, United States d School of Criminal Justice, Forensic Science Institute, University of Lausanne, Batochime, quartier Sorge, CH-1015 Lausanne-Dorigny, Switzerland e Elite Forensic Services, LLC, Saint-Paul, MN 55117, United States b c

A R T I C L E I N F O

A B S T R A C T

Article history: Received 19 May 2014 Received in revised form 30 December 2014 Accepted 7 January 2015 Available online 16 January 2015

This paper presents a statistical model for the quantiﬁcation of the weight of ﬁngerprint evidence. Contrarily to previous models (generative and score-based models), our model proposes to estimate the probability distributions of spatial relationships, directions and types of minutiae observed on ﬁngerprints for any given ﬁngermark. Our model is relying on an AFIS algorithm provided by 3M Cogent and on a dataset of more than 4,000,000 ﬁngerprints to represent a sample from a relevant population of potential sources. The performance of our model was tested using several hundreds of minutiae conﬁgurations observed on a set of 565 ﬁngermarks. In particular, the effects of various sub-populations of ﬁngers (i.e., ﬁnger number, ﬁnger general pattern) on the expected evidential value of our test conﬁgurations were investigated. The performance of our model indicates that the spatial relationship between minutiae carries more evidential weight than their type or direction. Our results also indicate that the AFIS component of our model directly enables us to assign weight to ﬁngerprint evidence without the need for the additional layer of complex statistical modeling involved by the estimation of the probability distributions of ﬁngerprint features. In fact, it seems that the AFIS component is more sensitive to the sub-population effects than the other components of the model. Overall, the data generated during this research project contributes to support the idea that ﬁngerprint evidence is a valuable forensic tool for the identiﬁcation of individuals. ß 2015 Elsevier Ireland Ltd. All rights reserved.

Keywords: Fingerprint evidence Strength of evidence Sub-population effect Spatial relationship Statistical model

1. Introduction The skin of the digits (ﬁngers and toes), palms and soles of human beings is formed of papillary ridges, also known as friction ridges. Fingerprints have been used with considerable success over the past century to determine or verify the identity of individuals using ﬁnger impressions taken under controlled conditions, or from friction ridge impressions left inadvertently on crime scenes. In particular, ﬁngerprint examiners are concerned with the

* Corresponding author at: The South Dakota State University, Department of Mathematics and Statistics, Harding Hall, Brookings, SD 57007, United States. Tel.: +1 415 272 67 52. E-mail address: [email protected] (C. Neumann). http://dx.doi.org/10.1016/j.forsciint.2015.01.007 0379-0738/ß 2015 Elsevier Ireland Ltd. All rights reserved.

determination of the identity of criminals through the examination of partial, potentially distorted and degraded friction ridge impressions recovered on crime scenes. In line with the European terminology, we will refer to crime-scene impressions using the term ﬁngermarks, to control impressions from a known individual of interest using the term ﬁngerprints, and to impressions from individuals in a given population using the term reference prints. Currently, the ﬁngerprint examiner community uses one general protocol to guide ﬁngerprint examination. This protocol consists of 4 main stages, summarized by the acronym ACE-V (for Analysis, Comparison, Evaluation and Veriﬁcation). Albeit this acronym is not always mentioned, this protocol is commonly described by the different professional bodies [1,2], in the relevant literature [3–5], and in US courts when examiners report ﬁngerprint evidence [6–10].

C. Neumann et al. / Forensic Science International 248 (2015) 154–171

The practical implementation of this protocol may vary between agencies. However, ﬁngerprint professionals, and scientiﬁc and legal scholars, generally accept that it aims at minimizing the risk of errors and provides a measure of quality assurance. That said, this protocol requires examiners to take a series of decisions after each stage, and the same scholars have stressed the need to develop quantiﬁable measures to support these decisions: from the initial decision that a ﬁngermark is worth examining, through its level of (dis)agreement with a ﬁngerprint, to the ﬁnal quantiﬁcation of its evidential value [11] Several authors [see 4,5,12 for reviews] have argued that these decisions should be supported by a probabilistic framework, and possibly by the use of a statistical model enabling the quantiﬁcation of ﬁngerprint information, in a similar fashion as for DNA data. The aim of this research is to develop and test a model to help with the assignment of the weight of ﬁngerprint evidence. While several models have already been proposed, none of them has reached the appropriate level of statistical rigor, and has been tested extensively enough, to be used in casework. This paper proposes a model that is developed using a different concept than the two main approaches used so far, and presents some measures of its performances on different datasets. The paper is structured as follows: it starts with a brief presentation of the past modeling efforts to clarify the novel nature of the model that is proposed in this research. Then the developed model is presented, and is studied using various datasets. Finally, its performance is discussed. 2. Models applied to ﬁngerprints: a short review Several models have been proposed during the past century to quantify the weight of ﬁngerprint evidence and provide support for the conclusions reached at the end of a ﬁngerprint examination. Models pre-dating 2001 have been reviewed by Stoney [12]. More recent models were reviewed in [5,13,14]. These models can be classiﬁed in two groups: 1) so-called score-based models; 2) so-called generative models. 2.1. Score-based models: Contrary to DNA, friction ridge skin cannot be characterized by an easily deﬁnable and quantiﬁable set of features. Indeed, while DNA can be summarized, for forensic purposes, using alleles at given loci, which are easily measurable, friction ridge skin contains patterns with many different levels of details that cannot be readily summarized by discrete variables. In addition, impressions from these patterns are affected by numerous factors (such as distortion, substrate, detection technique), which lower the reproducibility of their characteristics and increase the complexity of their modeling. Several research projects have attempted to lower the complexity of representing the multi-dimensionality and heterogeneity in friction ridge patterns by measuring the similarity between pairs of impressions. Such similarity can be typically summarized by a univariate random variable (a similarity metric or score). Score-based models attempt to provide a measure of the weight of the ﬁngerprint evidence, represented by the level of similarity between the ﬁngermark and the ﬁngerprint from a considered individual, by assigning the likelihood of that level of similarity under two mutually exclusive sets of circumstances. Score-based statistical models have intrinsic limitations:

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2) By design (the evidence score has to be computed between the ﬁngermark and the ﬁngerprint from the considered individual), such models cannot be used to support decisions made at the end of the analysis phase (since the examiner is not supposed to have had access to the ﬁngerprint during that phase). 3) Adding new features to an existing model requires the redevelopment and re-optimization of the scoring algorithm.

2.2. Feature-based models Other researchers have attempted to model the underlying distributions of some of the features that can be observed on friction ridge skin impressions. After having investigated the structure of these distributions, and estimated their parameters, it is theoretically possible to randomly sample ﬁngermarks and ﬁngerprints from these distributions, and to assign the probability of observing any constellation of features detected on a ﬁngermark. In general, these models were developed on datasets that were too limited in size to account properly for the dependencies between the multiple highly dimensional variables used to describe friction ridge pattern, and to account for the variability between impressions from different ﬁngers. It appears that these models do not ﬁt the data well, most particularly the spatial relationships between neighboring minutiae [14]. Furthermore, currently these models do not allow for conditioning the underlying distributions on the observations made on the ﬁngerprint; thus they do not account for the level of similarity between the ﬁngermark and the ﬁngerprint. Most of these models rely on some heuristic to determine whether these two impressions are sufﬁciently similar or not. This limits the support that those models can provide during the comparison and evaluation phases of the examination process. A third type of model emerged [16] recently, where similarity measures are used to reduce the dimensionality of the problem by mapping all impressions from their original multi-dimensional space onto a single-dimensional space. This new type of models quantiﬁes the evidence represented by the one-dimensional projection of the ﬁngermark and the ﬁngerprint in the new space, and not the evidence represented by the measure of similarity between them. This new type of model can be used to support the decisions made by examiners at the end of each of the ﬁrst 3 stages of ACE-V [5]. However, the model presented in [16] uses an ad hoc ‘‘weighting function’’ for the Monte-Carlo estimation of the value of the probability density of the observations made on the ﬁngermark in both numerator and denominator distributions. It has been rightfully argued [17,18] that this model is only a very ad hoc approximation of a likelihood ratio. This paper proposes a novel approach for the quantiﬁcation of the weight of ﬁngerprint ﬁndings. In this approach, we attempt to reduce the dimensionality of the sets of variables used to describe minutiae conﬁgurations by using shape variables as proposed in [19]. The use of shape variables allows for (a) reducing the complexity of the problem, while accounting for the dependencies between ﬁngerprint features, as in score-based models, and (b) providing a measure of the speciﬁcity of the crime scene mark without involving the ﬁngerprint, as in generative models and in the model proposed in [16]. Overall, the direct modeling of ﬁngerprint features through the use of shape variables enables a more statistically appropriate construction of the model when compared to [16] and other score-based models. 3. Development of the model

1) The integration of scores in the full statistical framework for quantifying the weight of forensic ﬁngerprint evidence is not well understood, and still subject to debate [5,15].

The general framework of the model, and the notation, are similar to the ones described in Neumann et al. [16] and Neumann

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et al. [20]. We denote the entire collection of observations made on the ﬁngermark by the multi-dimensional quantity Y. We denote the observations made on corresponding properties on a set of 10 ﬁngerprints from a given individual by X. The model uses Y and X to address the following propositions: Hp: the ﬁngermark and the set of ﬁngerprints have both been left by Mr. X. Hd: the set of 10 ﬁngerprints has been provided by Mr. X., but the ﬁngermark comes from another individual within a considered population. Following Lindley [21], the objective is to evaluate the weight of the ﬁngerprint evidence, using a likelihood ratio (LR), which we write here, after some simpliﬁcations [16], as: LR ¼

pY j X ðX; Y H p pY j X ðX; Y jHd Þ

(1)

In general, we note that it is not possible to calculate the LR described in Eq. (1) since we do not know the structure and the parameters of the numerator and denominator likelihood function. Therefore, we resort to approximate this likelihood ratio by making some assumptions and simpliﬁcations. While we will use the term likelihood ratio and its corresponding acronym LR in the development presented below, it should remain clear to the reader that we mean approximate likelihood ratio. In [16], it is explained that the number of minutiae k recorded on the ﬁngermark deﬁnes the dimensionality of the problem. Following [16], we denote the vector of observations made on the ﬁngermark by y(k) and on the ﬁngerprint by x(k). When comparing a ﬁngermark with a set of 10 ﬁngerprints from a given individual (such as, Mr. X), an examiner will attempt to select the subset of features x(k) of X that corresponds best to the observations y(k) made on the ﬁngermark: 1) The examiner ﬁrst selects the ﬁngerprint with the ridge ﬂow that best corresponds to the ﬁngermark, taking into account the

LR ¼

rewritten as follows: LR ¼

pY jX min ðyðkÞ H p pY ðyðkÞ jHd Þ

At this point in the development of the model, it is critical to realize that the k minutiae on the ﬁngermark, and the corresponding k minutiae on the selected ﬁngerprint are assumed to be paired by virtue of the comparison process by the examiner: the ith minutia on the ﬁngermark is associated to one and only one minutia on the selected ﬁngerprint from Mr. X. This information is implied in the model. The numerator of the model in Eq. (2) involves estimating the probability density of the k minutiae conﬁguration observed on the ﬁngermark using a probability density function describing the variability of the closest k minutiae on the selected ﬁnger of Mr. X. The denominator of the model in Eq. (2) involves estimating the probability density of the k minutiae conﬁguration observed on the ﬁngermark using a hierarchical model of the distributions of (a) the individuals in the relevant population and (b) the set of observations made on the closest k minutiae observed on the set of 10-ﬁngers of each individual in the population deﬁned in (a). We will come back later on the selection of the closest k minutiae on ﬁngers from individuals in a relevant reference population. In order to simplify the model, we consider the possibility that, in practice, the friction ridge skin of a given individual may not present a k minutiae conﬁguration that would be considered sufﬁciently similar to the conﬁguration observed on the ﬁngermark. While we realize that we have yet to deﬁne what constitutes sufﬁciently similar (this will be done later), and that we could theoretically always select the most similar k minutiae conﬁguration for any individual (even if it is only remotely similar), we chose to introduce V as an indicator variable that takes value 1 if a considered impression from a given individual has a k minutiae conﬁguration that is sufﬁciently similar to the one observed on the ﬁngermark, and 0 otherwise. We include the additional information provided by V in Eq. (2) as follows:

pY jX min ;V ðyðkÞ H p ; v ¼ 1 pV ðv ¼ 1H p þ pY jX min ;V ðyðkÞ H p ; v ¼ 0 pV ðv ¼ 0H p pY jV ðyðkÞ jHd ; v ¼ 1Þ pV ðv ¼ 1jHd Þ þ pY jV ðyðkÞ jHd ; v ¼ 0Þ pV ðv ¼ 0jHd Þ

tolerances allowed by the mark due to ﬁnger pad distortion and other adverse factors. 2) Secondly, the examiner focuses on the general location within the ridge ﬂow (i.e., core, delta, periphery) of the selected ﬁngerprint, where the minutiae were observed on the ﬁngermark (if known). 3) Thirdly, the examiner determines whether a set of features x(k) on the selected ﬁngerprint could correspond to the set y(k) observed on the ﬁngermark at the corresponding location within the ridge ﬂow. 4) Finally, the examiner compares the details of the features between both impressions. of a Mathematically, this process corresponds to theselection P10 ni single k minutiae conﬁguration, out of the possible i¼1 k conﬁgurations on Mr. X’s 10-ﬁngerprint set (in this paper, we are not considering palms and other friction ridge areas, but the concepts underlying this model could be extended to these areas), such that its location, shape and other features are as similar as possible to the ones of the k minutiae conﬁguration observed on ðkÞ the ﬁngermark. We denote this conﬁguration by xmin . Eq. (1) can be

(2)

(3)

Eq. (3) can be simpliﬁed by making the following assumptions: 1) pY jX ;V ðyðkÞ H p ; v ¼ 0 and pV ðv ¼ 1H p bothtend tozero when min all Mr. X.’s ﬁngers show unexplainable differences with the k conﬁgurations observed on the ﬁngermark. This typically happens when Mr. X. is not the donor of the ﬁngermark and the observations made on his ﬁngers are not compatible with the ones made on the ﬁngermark. This may also happen when the mark shows extreme distortion or degradation. At this point, we have LR = 0; 2) pV ðv ¼ 1H p tends to 1 when Hp is true (i.e., Mr. X. is truly the source) or when Mr. X, while not strictly being the true source, displays a sufﬁciently similar k minutiae conﬁguration on one of his ﬁngers. This assumption is not a very strong one, and in practice it may be possible to assign a probability to pV ðv ¼ 1H p for any considered Mr. X; 3) pY jV ðyðkÞ jHd ; v ¼ 0Þ tends to zero for individuals in the reference populations whose ﬁngerprints have unexplainable differences with the k minutiae observed on the ﬁngermark. In the proposed model, the terms pY jX ;V ðyðkÞ H p ; v ¼ 1 and min ðkÞ pY jV ðy jHd ; v ¼ 1Þ are estimated by characterizing k minutiae

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conﬁgurations using three different variables: shape of conﬁguration S, minutiae direction D and minutiae type T. Rewriting Eq. (3), we obtain:

LR ¼

ðkÞ ðkÞ ðkÞ pY jX min ;V ðyS ; yD ; yT H p ; v ¼ 1 ðkÞ

ðkÞ

ðkÞ

pY jV ðyS ; yD ; yT jHd ; v ¼ 1Þ

1 pV ðv ¼ 1jHd Þ

(4)

In Eq. (4), we consider that the shapes of minutiae conﬁgurations, and the types and directions of the minutiae are inﬂuenced by the general pattern of the prints and by the location of the conﬁgurations on the ridge ﬂow. This dependency is captured by the variable V. Therefore, we make the assumption that within a particular location (e.g., core, delta or periphery) of a given pattern (e.g., whorl, loop, arch), conﬁguration shapes, minutiae types and minutiae directions are independent of each other. Using this assumption, we obtain:

LR ¼

ðkÞ ðkÞ pY jX min ;V ðyS H p ; v ¼ 1 pY jX min ;V ðyD H p ; v ¼ 1 ðkÞ

ðkÞ

pY jV ðyS jHd ; v ¼ 1Þ pY jV ðyD jHd ; v ¼ 1Þ ðkÞ pY jX min ;V ðyT H p ; v ¼ 1 1 ðkÞ pV ðv ¼ 1jHd Þ p ðy jHd ; v ¼ 1Þ Y jV

(5)

T

Given V, our model has four conditionally independent components. The ﬁrst component focuses on the shape of the ﬁngermark conﬁguration, the second component focuses on the directions of the minutiae in the ﬁngermark conﬁguration, the third component focuses on their types, and the last component includes information on the general pattern of the ridge ﬂow. Note that the design of the model enables the consideration of additional ﬁngerprint features, conditioned on V, without the need for changing the existing elements of the model. Thus, it is possible to consider other elements commonly used by latent print examiners, such as the presence of differences between the features observed on the trace and control prints, the presence/ absence of scares, warts and creases, as well as the presence/ absence of impressions from sweat pores on the prints, or the shape of the ridges.

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To ease the description of the model, the three ﬁrst components of the model are described in separated sections below. However, we ﬁrst describe two algorithms for: 1) Extracting and quantifying the features observed on k minutiae conﬁguration present on friction ridge impressions. 2) Finding the most similar (within tolerances) k minutiae conﬁgurations on any ﬁngerprint or reference print based on the k minutiae conﬁguration observed on a ﬁngermark.

3.1. Feature extraction The process of extracting features from friction ridge impressions is image dependent: minutiae locations and directions are relatively measured to a coordinate system deﬁned by the ﬁngerprint image. Fig. 1 displays a set of 7 features on a ﬁngermark and the corresponding features on a ﬁngerprint. Fig. 1 also shows that the locations and directions of corresponding minutiae are different in the two images, and that it is not possible to build a statistical model relying directly on these measurements. Following Neumann et al. [16], we propose to describe conﬁgurations of k minutiae as a set of k triangles, whose vertices are deﬁned by pairs of consecutive minutiae and the virtual centroid of the k conﬁguration. This design enables the capture of the spatial relationships between minutiae, provides some robustness to the distortion affecting impressions when ﬁnger pads are pressed against a surface, and allows for measuring variables with respect to the triangles, thus breaking their dependency to the images. Fig. 2 illustrates how the considered variables are extracted from a given conﬁguration. At ﬁrst, the minutiae are annotated on the ﬁnger impression using markers indicating their locations, types and directions. This image dependent information is used to organize the minutiae around a virtual centroid, deﬁned by the arithmetic mean of the spatial coordinates of the minutiae. This process creates a series of triangles, whose vertices are deﬁned by pairs of consecutive minutiae and the centroid. The triangulation is

Fig. 1. Raw information extracted from minutiae location and direction, with indication of the image deﬁned axes.

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Fig. 2. Extraction of the variables considered by the model from the raw information available on the image of a ﬁnger impression. From left to right: (a) annotation of the minutiae on the ﬁngerprint image distinguishing ridge endings (round) and bifurcations (square); (b) deﬁnition of the centroid and organization of the minutiae with respect to the centroid; (c) creation of the triangles; (d) extraction of shape variables for one triangle and (e) extraction of the type and direction variables of the minutiae for one triangle (the variables for all triangles are similarly extracted).

rotationally independent: the minutiae will be organized in the same order, irrespective of the angle between the impression and the axes of the image. The triangulation also provides the capability to measure the considered variables according to the triangles, and thus to break their dependency to the images. In this research project, we decided to characterize each conﬁguration by the following variables: S—The shape of each triangle in the conﬁguration is described by two quantitative measurements: (a) the ratio between its area and perimeter (form factor), and (b) the ratio between the diameters of its circumcircle and incircle (aspect ratio). The shape of a ﬁngermark conﬁguration can be formally represented by YS = [YS,1, ..., YS,k]; D—The direction of each minutia in the conﬁguration is described by the angle between the direction of the minutia and an axis deﬁned by the centroid and the minutiae location (Fig. 2). The angle is measured counterclockwise from the axis to the minutiae. The directions of the minutiae in a latent print conﬁguration can be formally represented by YD = [YD,1, ..., YD,k]; T—The type of each minutia in the conﬁguration is described by a nominal variable, which can take the following values: RE for ridge ending minutiae; BI for bifurcation minutiae; UK for minutiae which type is unknown. The types of the minutiae in a latent print conﬁguration can be formally represented by YT = [YT,1, ..., YT,k].

3.2. Finding the most similar k conﬁguration and estimating pV ðv ¼ 1jHd Þ The numerator of our model assumes (in Eq. (2)) that a trained ﬁngerprint examiner has selected the single most similar k minutiae conﬁguration from all conﬁgurations observed on Mr. X’s 10 ﬁngerprints, following their comparison with the ﬁngermark. In Eq. (4), we have also assumed that if Mr. X.’s conﬁguration does not signiﬁcant discrepancies with the show H p would tend to 1. We note that a value ﬁngermark, pV ðv ¼ 1 for pV ðv ¼ 1H p could be computed to remove this assumption.1 By symmetry, the denominator needs to consider the distributions of the observation of the same variables on 1 Current technical limitations due to the set-up of our AFIS prevented us from doing it.

ﬁngerprints from individuals in the relevant reference population. Contrary to DNA proﬁling, which relies on population genetics and well-deﬁned variables (selected alleles), it is currently not possible to determine analytically the structure and parameters of the distribution of friction ridge features in a relevant population. The distributions of the various friction ridge features need to be estimated using a sample of ﬁngerprints from the individuals in this population. Since it is unrealistic to require a human examiner to determine the most similar k minutiae conﬁguration on each set of reference prints in our sample, we used the ﬁngermarkﬁngerprint matching algorithm of an Automatic Fingerprint Identiﬁcation System (AFIS) provided by 3M Cogent as a proxy for the human-based comparison process described above. The matching algorithm is used to search a large dataset of reference prints from a sample of individuals, and select, for each person, the set of k minutiae that is most similar to y(k) in terms of general pattern, location on the ridge ﬂow, and general appearance (i.e., shape). In practice, not all of the individuals in the reference dataset were found to have k minutiae that were sufﬁciently similar (as deﬁned by the factory settings of the 3M Cogent’s matching algorithm2) to the ones observed on the ﬁngermark. This process enables us to easily estimate pV ðv ¼ 1jHd Þ by simply counting the number of individuals retrieved by the system in relation to the number of prints in the database. The reader will have realized that the process described above has a major implication: the approximation of the denominator in Eq. (5) is performed based on a single impression of each k minutiae conﬁguration in our reference dataset. The denominator of our model does not account for the within variability (due for example to the distortion of ﬁnger pads) of the selected k minutiae for each individual in the reference dataset. This allows for considerable simpliﬁcations in the development of our model at the cost of some of its coherence. We believe that, for a large reference dataset, this process still allows for a reasonable approximation of the speciﬁcity of the k minutiae conﬁguration observed on the ﬁngermark.

2 These settings are proprietary and not known to the authors. They inﬂuence the model as they conditioned which conﬁgurations from the reference sets are subsequently used by the model. However, this algorithm has been extensively tested by the National Institute of Standards and Technology (NIST) and appears to be very efﬁcient at retrieving conﬁgurations that are similar in shape and location [22].

C. Neumann et al. / Forensic Science International 248 (2015) 154–171

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Table 1 Spearman rank correlation coefﬁcients between the form factors measured on triangles (T1 to T12) from approximately 100,000 12 minutiae reference conﬁgurations paired with a single ﬁngermark with 12 minutiae. Shape

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11

T12

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12

1 0.211 0.021 0.040 0.042 0.113 0.028 0.009 0.021 0.009 0.007 0.180

1 0.278 0.055 0.008 0.001 0.074 0.047 0.001 0.100 0.014 0.031

1 0.188 0.008 0.027 0.004 0.041 0.035 0.059 0.052 0.009

1 0.308 0.037 0.020 0.072 0.041 0.119 0.051 0.046

1 0.156 0.033 0.004 0.023 0.018 0.019 0.022

1 0.326 0.008 0.000 0.029 0.033 0.043

1 0.185 0.054 0.000 0.020 0.005

1 0.194 0.066 0.067 0.051

1 0.221 0.016 0.069

1 0.119 0.012

1 0.325

1

3.3. Shape component of the model From Eq. (5) and Section 3.1, we rewrite the shape element of the model as: ðkÞ pY S jX min ;V ðyS H p ; v ¼ 1 LRS ¼ ðkÞ pY S jV ðyS jHd ; v ¼ 1Þ ðkÞ ðkÞ pY S jX min ;V ðyS;1 ; :::; yS;k H p ; v ¼ 1 (6) ¼ ðkÞ ðkÞ pY S jV ðyS;1 ; :::; yS;k jHd ; v ¼ 1Þ ðkÞ

where YS;i represents the shape measurements performed on the ith triangle in the k minutiae conﬁguration. In order to simplify the modeling of the joint distributions in Eq. (6), we assume that the shape of triangle i is mostly inﬂuenced by its immediate neighbors. This assumption is reasonable as adjacent triangles share one side with each other, while non-adjacent triangles share only one vertex. Table 1 presents the Spearman rank correlation coefﬁcients between the form factors measured on triangles from more than 100,000 12 minutiae conﬁgurations that have been paired with a single ﬁngermark conﬁguration. The coefﬁcients show weak correlation between immediately adjacent triangles, and no correlation between non-adjacent triangles on the same conﬁguration. Removing dependencies between non-adjacent triangles forces us to select a ﬁrst triangle in each conﬁguration, and to assign a marginal probability to this triangle, rather than a joint probability. ðkÞ To set the ﬁrst triangle, we remember that each YS;i is a bidimensional variable containing the form factor and the aspect ratio of triangle i. The form factor and the aspect ratio are functionally independent and may capture the shape of a triangle in complimentary ways. To select the ﬁrst triangle, we decided to use the aspect ratio information of the k triangles in the ðkÞ ðkÞ conﬁguration. We set YS;1 ¼ min1ik YS;i based on the aspect ratio variable of each triangle, and then register the remaining

k 1 triangles counterclockwise. Since the k minutiae in the ﬁngermark are paired with the k minutiae in the ﬁngerprint under consideration (by the human examiner) and the reference prints (by the AFIS matching algorithm), all triangles in these prints can ðkÞ be reordered according to YS . Once organized counterclockwise starting from the triangle with the smallest aspect ratio, the aspect ðkÞ ðkÞ ratio component from YS;i is dropped and YS;i is considered to be univariate and to only include the form factor of the triangle. In ðkÞ other words, YS;i is a variable capturing the form factor of each ðkÞ triangle, and the representations yS;i of this variable are organized according to the aspect ratio of the triangles. Based on this development, Eq. (6) can be rewritten as:

LRS ¼

ðkÞ pY S jX min ;V ðyS;1 H p ; v ¼ 1 ðkÞ

pY S jV ðyS;1 jHd ; v ¼ 1Þ ðkÞ ðkÞ ðkÞ ðkÞ pY S jX min ;V ðyS;2 yS;1 ; H p ; v ¼ 1 pY S jX min ;V ðyS;k yS;k1 ; H p ; v ¼ 1 ðkÞ ðkÞ ðkÞ ðkÞ pY S jV ðyS;2 yS;1 ; Hd ; v ¼ 1 pY S jV ðyS;k yS;k1 ; Hd ; v ¼ 1 (7)

The numerator of an LRS describes the probability of observing the conﬁguration of k minutiae on the ﬁngermark if it originates from the same ﬁnger as the selected ﬁngerprint. Ideally, assigning this probability would require the source of the ﬁngerprint to generate multiple pseudo-marks in various conditions of distortion and pressure, and to model the distribution of the shape of the considered k minutiae conﬁguration across these pseudo-marks. In practice, this is unrealistic and we used the distortion model from Neumann et al. [16], based on Bookstein [23], to generate pseudomarks from the ﬁngerprint. The denominator relates to the probability of observing the conﬁgurations of k minutiae on the ﬁngermark in a relevant population deﬁned by Hd. The dataset necessary to approximate the denominator is obtained as

Table 2 Spearman rank correlation coefﬁcients between the directions of neighboring minutiae in 100,000 reference prints paired with a single ﬁngermark with 12 minutiae. Direction

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11

T12

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12

1 0.022 0.056 0.003 0.015 0.020 0.098 0.020 0.042 0.012 0.044 0.083

1 0.022 0.004 0.003 0.049 0.164 0.128 0.077 0.053 0.046 0.000

1 0.006 0.026 0.048 0.039 0.064 0.056 0.015 0.005 0.016

1 0.062 0.167 0.084 0.003 0.059 0.002 0.027 0.069

1 0.101 0.168 0.015 0.169 0.127 0.152 0.018

1 0.093 0.015 0.069 0.016 0.037 0.063

1 0.030 0.068 0.038 0.019 0.070

1 0.059 0.065 0.020 0.082

1 0.122 0.142 0.047

1 0.118 0.085

1 0.049

1

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described above, using the sorting power of our AFIS algorithm to ﬁnd the appropriate conﬁgurations to consider under Hd. In general, the histogram estimates for the distributions of the form factors of the triangles extracted from multiple impressions of a single k minutiae conﬁguration (i.e., under Hp) are reasonably symmetrical and unimodal. However, the histogram estimates for the distributions of the form factors of triangles extracted from k minutiae conﬁgurations originated from different donors (i.e., under Hd) are moderately to highly skewed on their right or left tails. Therefore, we decided to model the numerator distributions using uni- and bivariate normal densities; and we choose not to impose any parametric assumption on the structure of the densities the denominator distributions by learning the density functions from the data using kernel density estimation. 3.4. Direction element of the model From Eq. (5) and Section 3.1, we rewrite the shape element of the model as: ðkÞ pY D jX min ;V ðyD H p ; v ¼ 1 LRD ¼ ðkÞ pY D jV ðyD jHd ; v ¼ 1Þ ðkÞ ðkÞ pY D jX min ;V ðyD;1 ; :::; yD;k H p ; v ¼ 1 (8) ¼ ðkÞ ðkÞ pY D jV ðyD;1 ; :::; yD;k jHd ; v ¼ 1Þ As explained previously, the direction of each minutia in a conﬁguration is described with respect to an axis deﬁned by the centroid of the conﬁguration and the minutia itself. This allows for obtaining directional information on the minutiae in the conﬁguration regardless of the rotation and location of the impression on the image. This transformation also enables the reduction of the dependency between the directions measured for neighboring minutiae. Table 2 shows the Spearman rank correlation coefﬁcients between the minutiae directions of the same 12 minutiae conﬁgurations as in Table 1. It conﬁrms that few neighboring minutiae have weakly correlated directions (as measured in our study) and that most show low to no correlation. By taking advantage of the low correlation between directions of neighboring minutiae, we make the assumption of independence between the minutiae direction (as speciﬁcally measured in our project) and obtain the following ratio: ðkÞ k p Y Y D jX min ;V ðyD;i H p ; v ¼ 1 (9) LRD ¼ ðkÞ pY D jV ðyD;i jHd ; v ¼ 1Þ i¼1 As explained previously, we used a distortion model to generate pseudo-marks for the estimation of the density function under Hp and our large reference dataset for the estimation of the density function under Hd. The histogram estimates for the density function of minutiae directions in the ﬁngerprint show that they tend to be skewed to the right, while the estimates for the reference prints show multiple modes. We decided to approximate both distributions using non-parametric distributions based on von Mises kernels [24]. 3.5. Type element of the model From Eq. (5) and Section 3.1, the type element of the model can be rewritten as: ðkÞ pY T jX min ;V ðyT H p ; v ¼ 1 LRT ¼ ðkÞ pY T jV ðyT jHd ; v ¼ 1Þ ðkÞ ðkÞ pY T jX min ;V ðyT;1 ; :::; yT;k H p ; v ¼ 1 (10) ¼ ðkÞ ðkÞ pY T jV ðyT;1 ; :::; yT;k jHd ; v ¼ 1Þ

In order to simplify the dimensionality of the probabilities in Eq. (10), we assume that minutiae types are inﬂuenced by the location of the minutiae within the pattern of the ridge ﬂow (accounted for in V) but not by each other. Thus, given V, we can make the following simpliﬁcation: ðkÞ k p Y Y T jX min ;V ðyT;i H p ; v ¼ 1 (11) LRT ¼ ðkÞ pY T jV ðyT;i jHd ; v ¼ 1Þ i¼1 We have deﬁned previously minutiae type as nominal variable such that any i minutia can take one of the following values ðkÞ YT;i ¼ fRE; BI; UKg. That said, the observation of the type of a minutia on a potentially distorted and degraded ﬁngermark is not only conditioned by the true type of that minutia, but also by the ability of the examiner to correctly interpret the ridge ﬂow. Therefore, we have for the numerator: X fRE;BI;UKg ðkÞ ðkÞ pY T jX min ;V ðyT;i H p ;v ¼1 ¼ pY T jX min ;V ðyT;i ¼ jH p ;v ¼1 j

¼

fRE;BIg X fRE;BI;UKg X l

j

ðkÞ ðkÞ pY T jX min ;V ðyT;i ¼ jxT;i ¼l;H p ;v¼ 1

ðkÞ

Ideally, the pX T jV ðxT;i

ðkÞ pX T jV ðxT;i ¼ lH p ;v¼ 1

(12) ¼ lH p ; v ¼ 1 terms should be assigned by

having the examiner annotate the type of the ith minutia on series of pseudo-marks generated by the source of the ﬁngerprint. ðkÞ Indeed, pX T jV ðxT;i ¼ lH p ; v ¼ 1 could be developed in a similar ðkÞ fashion as pY jX ;V ðyT;i ¼ jH p ; v ¼ 1 by conditioning on the true T

min

type of the minutiae observed on the friction ridge skin of the ﬁnger pad. However, for all intents and purposes of this project, we consider that there is no uncertainty affecting the determination of the type of a given minutia when observed on ﬁngerprints or ðkÞ pseudo-traces. Thus, in our model, p ðx ¼ lH p ; v ¼ 1 takes X T jV

T;i

values {0,1} depending on whether the lth type is observed by the examiner on the ith minutia of the k conﬁguration present on the ﬁngerprint. ðkÞ ðkÞ The pY jX ;V ðyT;i ¼ jxT;i ¼ l; H p ; v ¼ 1 terms take different T min values depending on the type observed on the ﬁngermark for the ith minutia and for the corresponding type observed on the ﬁngerprint for the corresponding minutiae. A survey of a series of 82 minutiae [25], each annotated by more than 200 latent prints examiners on 12 pairs of ﬁngermarks and ﬁngerprints, reveals that (for any i): When the ith minutia on the ﬁngermark is deemed to be a ridge ending: ðkÞ ðkÞ pY T jX min ;V ðyT;i ¼ RExT;i ¼ RE; H p ; v ¼ 1 ¼ 0:76; ðkÞ ðkÞ pY T jX min ;V ðyT;i ¼ RExT;i ¼ BI; H p ; v ¼ 1 ¼ 0:16; all other terms equal 0; When the ith minutia on the latent print is deemed to be a bifurcation: ðkÞ ðkÞ pY T jX min ;V ðyT;i ¼ BIxT;i ¼ RE; H p ; v ¼ 1 ¼ 0:16; ðkÞ ðkÞ pY T jX min ;V ðyT;i ¼ BIxT;i ¼ BI; H p ; v ¼ 1 ¼ 0:75; all other terms equal 0;

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161

Table 3 Number of reference prints by general pattern and ﬁnger number (‘‘Other’’ includes amputated and missing ﬁngers). Finger #

Pattern

1

2

3

4

5

6

7

8

9

10

Arch Right loop Left loop Whorl Other

6,643 1,53,400 714 1,99,017 45,661

24,552 1,02,885 33,382 1,23,994 1,20,622

13,691 2,40,039 1,625 71,248 78,832

3,539 1,38,992 1,667 1,74,406 86,831

1,577 2,56,983 504 54,671 91,700

11,262 614 1,80,496 1,56,064 56,999

26,451 29,363 1,13,208 1,13,069 1,23,344

17,992 1,349 2,19,008 71,032 96,054

5,018 994 1,57,587 1,35,197 1,06,639

2,454 311 2,55,813 39,323 1,07,534

Total

4,05,435

4,05,435

4,05,435

4,05,435

4,05,435

4,05,435

4,05,435

4,05,435

4,05,435

4,05,435

When the type of the ith minutia on the latent print is unknown: ðkÞ ðkÞ pY T jX min ;V ðyT;i ¼ UK xT;i ¼ RE; H p ; v ¼ 1 ¼ 0:08; ðkÞ ðkÞ pY jX ;V ðyT;i ¼ UK xT;i ¼ BI; H p ; v ¼ 1 ¼ 0:09;all T min equal 0.

other

terms ðkÞ

We note that under Hp only one of the pY jX ;V ðyT;i ¼ T min ðkÞ ðkÞ terms is non-null jxT;i ¼ l; H p ; v ¼ 1 pX T jV ðxT;i ¼ lH p ; v ¼ 1 depending on the types observed on the ith pair of minutiae on the ﬁngermark and ﬁngerprint. Similarly for the denominator, we have: ðkÞ

pY T jV ðyT;i jHd ; v ¼ 1Þ ¼

fRE;BI;UKg X

ðkÞ

pY T jV ðyT;i ¼ jjHd ; v ¼ 1Þ

j

¼

fRE;BIg X X fRE;BI;UKg l

(13)

ðkÞ

pY T jV ðyT;i ¼ jjHd ; v ¼ 1Þ pX T jV ðljHd ; v ¼ 1Þ

j

elements of the model were obtained by generating 2500 pseudomarks from each ﬁngerprint, using the distortion model from [16]. The dataset used for studying the histogram estimates of the denominators of the different elements of the model contained approximately 12,000 reference prints as in [16]. The minutiae on all ﬁngermarks, ﬁngerprints and reference prints were annotated or veriﬁed manually as described in [16]. 4.2. Reference dataset A reference dataset of approximately 4,000,000 reference prints from 405,435 anonymous donors was used in conjunction with our AFIS algorithm to support the assignment of the probability pV ðv ¼ 1jHd Þ. Each reference print is characterized by an identiﬁer number, which is unique to the donor of the print, by its ﬁnger number (1–10) and by its general pattern (arch, whorl, loop, other— classiﬁcation made automatically by the AFIS system) (Table 3). The minutiae on the 4,000,000 prints were extracted automatically using the minutiae detection algorithm of the 3M Cogent AFIS. 4.3. Test datasets

The pX T jV ðljHd ; v ¼ 1Þ terms can be assigned by using the distribution of the type of the ith minutia in all reference prints’ k minutiae conﬁgurations retrieved by the matching algorithm as described in the previous sections. ðkÞ

The pY T jV ðyT;i ¼ jjHd ; v ¼ 1Þ terms are assigned using the same values as for the numerator depending on whether a ridge ending, bifurcation or unknown type was observed on the ith minutia of the ﬁngermark. 4. Datasets The development of the model and the measures of its performance described in the next sections were conducted using multiple datasets that are described below. 4.1. Development dataset The model was developed using 48, 45 and 33 conﬁgurations of, respectively, 4, 8 and 12 minutiae sampled from ﬁngermarks and corresponding ﬁngerprints obtained from archived casework [16]. The histogram estimates of the numerators of the different Table 4 Conﬁgurations used to test the model, presented by number of minutiae and region. Note that ‘‘All regions’’ means conﬁgurations spanning across multiple regions, while the ‘‘core’’, ‘‘delta’’ or ‘‘periphery’’ implies that all minutiae were sampled from a given region. # Minutiae

3

4

5

6

7

8

9

10

11

12

All region Core Delta Periphery

96 151 61 159

99 170 70 180

98 159 66 125

97 144 57 142

97 125 61 101

100 97 29 76

100 72 25 53

96 60 24 39

93 47 17 27

89 33 14 16

The performance of the model was tested using 565 ﬁngermarks and their corresponding ﬁngerprints: the ﬁrst 364 ﬁngermarks originate from casework and correspond to the data used to test the model in [16]; an additional 201 ﬁngermarks, developed in casework-like conditions, and their corresponding prints were added to complete the test datasets. Different trained analysts manually annotated the minutiae on the 565 ﬁngermarks and their corresponding ﬁngerprints, in different batches, using PiAnoS4 [26]. Each minutia was paired between the latent and control prints using PiAnoS4’s pairing feature. The following numbers of conﬁgurations of 3v12 minutiae (Table 4) were sampled from different regions of the 565 ﬁngermarks and used to test the model3: Two test datasets were constructed using the conﬁgurations listed in Table 4: 1) A dataset aimed at measuring the performance of the model under Hp. This dataset includes the ﬁngermark conﬁgurations listed in Table 4 and the corresponding conﬁgurations taken from the paired ﬁngerprints. 2) A dataset aimed at measuring the performance of the model under Hd. To test the model under the most difﬁcult conditions, each ﬁngermark conﬁguration listed in Table 4 was searched against our reference dataset, using our AFIS matching algorithm, in order to retrieve the most similar k conﬁguration 3 Note that more conﬁgurations than those listed in Table 4 were sampled; however, when searched in the reference dataset, some conﬁgurations were not associated with a sufﬁcient number of reference conﬁgurations to estimate the density functions.

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out of 4,000,000 impressions. Thus, this dataset includes the ﬁngermark conﬁgurations listed in Table 4 and their most similar (according to the speciﬁcations of the matching algorithm) counterparts sampled from the reference dataset.

5. Results and discussion on model performance Neumann et al. [16] provided data on the expected value of the weight of ﬁngerprint evidence for conﬁgurations of 3–12 minutiae under Hp and Hd. This data was used to support the admissibility of ﬁngerprint evidence in U.S. courts [27,28]. However, no data was generated to study the expected value of the weight of ﬁngerprint evidence when the ﬁngermark is thought to have originated from a particular ﬁnger, a ﬁnger with a particular ridge ﬂow pattern, or from a particular region of the ﬁngerprint. Therefore, the experiments reported below were designed to not only measure the global performance of the model under Hp and Hd as in [16], but also to study the differences between the expected weights of ﬁngerprint evidence under different conditions. Four types of experiments were conducted: 1) Experiment #1: Measure of the global performance of the model under Hp and Hd—this experiment was designed to study the global performance of the model by assigning weights to samesource ﬁngermark/ﬁngerprint comparisons, and to different (but very similar)-sources comparisons, using minutiae conﬁgurations indistinctively spanning across all regions of the test ﬁngermarks (1st row of Table 4). During this experiment, we used the entire reference dataset. 2) Experiment #2: Expected weight of the evidence when the ﬁngermark conﬁguration is originating from a speciﬁc region— this experiment was designed to study the differences in weight of evidence between conﬁgurations with the same number of minutiae, but sampled from different regions of the test ﬁngermarks (2nd–4th rows of Table 4). During this experiment, we used the entire reference dataset. 3) Experiment #3: Expected weight of the evidence when the ﬁngermark conﬁguration is thought to have come from a ﬁnger with a speciﬁc general pattern—this experiment was designed to study the differences between the weight of evidence of a set of given ﬁngermark conﬁgurations when there is information (typically from the ﬁngermark itself) leading the examiner to believe that they have been left by ﬁngers with a speciﬁc general pattern. During this experiment, a set of conﬁgurations indistinctively spanning across all regions of the test ﬁngermarks (1st row of Table 4) were used, and our reference dataset was conditioned on three different types of patterns (loop, arch, whorl). 4) Experiment #4: Expected weight of the evidence when the ﬁngermark conﬁguration is thought to have come from a ﬁnger with a speciﬁc ﬁnger number—this experiment was designed to study the differences between the weight of evidence of a set of given ﬁngermark conﬁgurations when there is information (typically from the location of the mark) leading the examiner to believe that they originate from a speciﬁc ﬁnger number (thumb, fore ﬁnger, . . .). During this experiment, a set of conﬁgurations indistinctively spanning across all regions of the test ﬁngermarks (1st row of Table 4) were used, and our reference dataset was conditioned based on ﬁngerprints located on thumbs vs. the other 8 ﬁngers. We were unable to study the behavior of the model when the test conﬁgurations are assumed to come from a speciﬁc region of the print, since it is currently not possible to constrain our AFIS matching algorithm to only search among cores or deltas of the

reference prints. Nevertheless, since our 3 M Cogent AFIS algorithm is using image information in addition to the minutiae information to perform its searches, it is reasonable to assume that test conﬁgurations containing cores or deltas (as in experiment #2) were primarily searched in areas including cores and deltas on the reference prints. For each type of experiments, we differentiated the behavior of the modeled part (i.e., the shape (S), direction (D) and type (T) components) from the AFIS part (i.e., pV ðv ¼ 1jHd Þ) of the model. Experiment #1. Overall performance of the model (shape, direction, type components)—all regions Fig. 3a–d depicts the estimates for the denominator of the three components (S, D and T) of the model, separately (a–c), and jointly (d). We observe that the contribution of the shape variable to the overall weight of evidence is much larger than the other ones. We also observe the lack of contribution of the direction component of the model. Overall, Fig. 3a–d shows that the model has the ability to assess the speciﬁcity of minutiae conﬁgurations; that this speciﬁcity increases with the number of minutiae; but that it also varies between conﬁgurations with a given number of minutiae, as these conﬁgurations have different shapes and combinations of minutiae types. These observations are similar to [16], but also to most of the previous studies listed in [5,14]. Fig. 4a–d presents estimates for the numerators of the three components (S, D and T) of the model, separately (a–c), and jointly (d). The ﬁgures on the left-hand-side present the data obtained for ﬁngermark and ﬁngerprint conﬁgurations originating from the same source, while the ﬁgures on the right-hand-side present data obtained for the same ﬁngermark conﬁgurations when compared with ﬁngerprints provided by different sources (as explained above). Fig. 4a–d shows that the expected probability of observing the features on a ﬁngermark, based on potentially corresponding features observed on the ﬁngerprint, decreases with the number of minutiae. This is not surprising as the increase in the number of minutiae induces increasing variability between multiple impressions of the same set of k minutiae due to pressure, distortion and other factors. When comparing the left (same source) to the right (different sources) columns of Fig. 4a–d, we realize that the expected numerator probability decreases faster when ﬁngermarks are compared to ﬁngerprints originating from different sources. This effect is the result of the added discrimination introduced by the increasing number of features. Although it seems that there are not many differences between numerators calculated for same/ different sources situations, we remind the reader that the ﬁngerprint conﬁgurations used when Hd is assumed to be true were selected to be very similar to the ﬁngermark conﬁgurations: we anticipate that the expected numerator probability would decrease even faster when the model is not tested in the most difﬁcult conditions. The somewhat large ranges of values calculated for the numerator of the model can be explained by two elements: (1) the distortion model used in this project is not providing enough variability in the set of pseudo-marks generated from the ﬁngerprints, and thus cannot compensate for medium to large distortion effects of ﬁngermarks; (2) the model is affected by a lack of accuracy from the users annotating the test marks and prints in PiAnoS4. Fig. 5a–d presents the estimates obtained for the overall LR of the three components (S, D and T) of the model, separately (a–c), and jointly (d). The ﬁgures on the left-hand-side column present the estimates obtained for ﬁngermarks compared with ﬁngerprints provided by the true source, while the ﬁgures on the

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163

Fig. 3. (a) Estimates for the denominators of the shape component of the model—all regions. (b) Estimates for the denominators of the direction component of the model—all regions. (c) Estimates for the denominators of the type component of the model—all regions. (d) Estimates for the denominators of the model (S, D and T)—all regions.

right-hand-side column present data obtained for the same ﬁngermarks when compared with ﬁngerprints provided by different sources. Overall, we observe that the LRs calculated for pairs of ﬁngermarks and ﬁngerprints originating from the same source increases with the number of minutiae, while it remains centered around LR = 1 for pairs of ﬁngermarks and ﬁngerprints originating from different sources. These results are similar to the ones obtained by Neumann et al. [16] and for most models reported in [5,14]: (1) the expected value of the LR increases with the number of minutiae, (2) a range of LR values is observed for each number of minutiae, indicating that each conﬁguration of minutiae needs to be considered on its own merits, (3) there is not clear cut-off point that would entertain the idea of a scientiﬁc basis for a numerical standard. We observe a signiﬁcant number of LR calculated for pairs of ﬁngermarks/ﬁngerprints from different sources that are above 1 (Fig. 5d—right column), and therefore would misleadingly provide

support for the hypothesis that the ﬁngermark originates from the same ﬁnger as the ﬁngerprint (even though it is not true). Furthermore, we observe that a minority of these LRs have extremely high values, thus they are not only misleading, but they carry a signiﬁcant weight in favor of the wrong hypothesis. Overall, this observation is not concerning. This result can be explained by the method used to select the putative sources’ ﬁngerprints when Hd is assumed to be true: by design, we searched our ﬁngermarks in our reference dataset of 4,000,000 prints, and we considered the most similar reference ﬁngerprints as being from the same putative sources in order to provide a ‘‘worst case scenario’’ snapshot of the performance of the model. The selection was based on the 3M Cogent AFIS algorithm, and thus on the similarity between the shapes of ﬁngermark and reference prints conﬁgurations. We can observe, by comparing Fig. 5a–d (right column) that the overall LRs calculated when the ﬁngermarks and ﬁngerprints are not coming from the same source (Fig. 5d—right column) is mainly driven by the shape component of the LR, while

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164

a

b

c

d

Fig. 4. (a) Estimates for the numerators of the shape component of the model—all regions. Left: Same source ﬁngerprints; Right: different source ﬁngerprints. (b) Estimates for the numerators of the direction component of the models—all regions. Left: Same source ﬁngerprints; right: different source ﬁngerprints. (c) Estimates for the numerators of the type component of the model—all regions. Left: Same source ﬁngerprints; right: different source ﬁngerprints. (d) Estimates for the numerators of the model (S, D and T)— All regions. Left: Same source ﬁngerprints; right: different source ﬁngerprints.

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165

Fig. 5. (a) Estimates for the LRs of the shape component of the model—all regions. Left: Same source ﬁngerprints; right: different source ﬁngerprints. (b) Estimates for the LRs of the direction component of the model—all regions. Left: Same source ﬁngerprints; right: different source ﬁngerprints. (c) Estimates for the LRs of the type component of the model—all regions. Left: same source ﬁngerprints; right: different source ﬁngerprints. (d) Estimates for the LRs (S, D and T)—all regions. Left: Same source ﬁngerprints; right: different source ﬁngerprints.

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its direction and type components tend to correctly provide support for Hd. The results presented in Fig. 5 support the idea that our model has good performance when the putative source has not been generated as a result of a ﬁngerprint database search. By extension, our results conﬁrm that conclusions of ﬁngerprint examinations, when the suspect has been generated through a database search, need to rely on features that exhibit a greater degree of speciﬁcity (and of potentially higher quality) than when the suspect has been generated through investigative work. Experiment #2. Expected value of the weight of ﬁngerprint evidence (S, D and T components)—by region

In general, we observe that the component-wise behavior of the model in that experiment is similar to the one reported above. This observation is valid under both Hp and Hd. The overall model’s behavior when considering the conﬁgurations sampled in three

different regions of the test ﬁngermarks (core, delta and peripheral regions) under Hp is presented Fig. 6a–c. The similarity of the behavior of the model can be observed by comparing Fig. 6 to Figs. 3–5. Interestingly, it appears that the expected weight of the evidence, as calculated for the shape, direction and type components of the model, is not different between regions of the friction ridge skin. This seems counterintuitive: we would be expecting conﬁgurations in core and delta regions to be less discriminative than conﬁgurations in the periphery of the prints (at least shape-wise), and therefore, have lower weight of evidence. The explanation of this observation requires us to take a look at the results obtained for the AFIS part of the model, which we have not considered so far. Experiments #1&2 – Part II. Overall performance of the model and expected value of the weight of evidence by region (AFIS component)

Fig. 6. (a) Estimates for the numerators of the model—shape, direction and type components. Same source—Left: core region; middle: delta region; right: peripheral region. (b) Estimates for the denominators of the model—shape, direction and type components. Same source—Left: core region; middle: delta region; right: peripheral region. (c) Estimates for the LRs of the model—shape, direction and type components. Same source—Left: core region; middle: delta region; right: peripheral region.

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Fig. 7a–d presents the values for pV ðv ¼ 1jHd Þ estimated for the core (a), delta (b) and peripheral regions (c), and for all regions together (d). Fig. 7 shows that as the number of minutiae increases in the conﬁguration, the number of sufﬁciently similar reference conﬁgurations retrieved by the AFIS algorithm decreases. We also observe that this decrease in the number of retrieved conﬁgurations varies between the different patterns of the tested conﬁgurations sampled from our ﬁngermarks. In particular, it appears that the AFIS algorithm retrieves more reference conﬁgurations when a query conﬁguration originating from a delta region is searched, even if that conﬁguration includes a large number of minutiae. We would have expected more differences between the results obtained between peripheral and core regions, since it would seem that the variability of conﬁgurations in the periphery of the pattern should be larger than in the core. Nevertheless, we note that cores usually do not include many minutiae (especially in the automatic encoding of AFIS) and that it is difﬁcult to deﬁne conﬁgurations that are entirely in the core region of a pattern. Thus, there may have been a signiﬁcant overlap between peripheral and core conﬁgurations.

167

With respect to the observation that the behavior of the shape, direction and type components of the model do not show any difference between the different regions (in relation to Fig. 6), we can now see that most of the difference is already captured by the AFIS part of the model. By design, the shape, direction and type components of the model are only focusing on the reference conﬁgurations that are found to be similar to the selected conﬁguration on the ﬁngermark. These components do not account for the difﬁculty of ﬁnding these similar reference conﬁgurations. This is done by the AFIS component. These results are conﬁrmed in the next sections for ﬁnger numbers and ﬁnger friction ridge patterns (i.e., whorl, loop, arch). Critically, the results presented in Fig. 8 show that a suitably conﬁgured commercial AFIS can readily provide information on the weight of ﬁngerprint evidence. Experiments #3&4. Expected value of the weight of ﬁngerprint evidence when there is information on the origin (general pattern or ﬁnger number) of the ﬁngermark conﬁguration (S, D, T and AFIS components)

Fig. 7. (a) Estimates for the AFIS component of the model—core region. (b) Estimates for the AFIS component of the model—delta region. (c) Estimates for the AFIS component of the model—peripheral region. (d) Estimates for the AFIS component of the model—all regions.

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When information is available on the general pattern (arch, loop, whorl, . . .), and/or the ﬁnger number (thumb, fore ﬁnger, . . .) of the ﬁnger that has produced the ﬁngermark, the shape, type, direction components of the model behave in a very similar fashion as discussed in relation to Figs. 3–6. The use of the different reference datasets does not allow us to observe any difference between the expected weight of the evidence calculated by the shape, direction and type components of the model (neither under Hp, nor Hd); and most of the effect of using different reference datasets is captured by the AFIS part of the model. Fig. 8a–f presents the effect of using information on the source ﬁnger being a thumb vs. another ﬁnger, and being an arch, a whorl, or a loop on the AFIS component of the model. Overall, Fig. 8 seems to indicate that the probability of observing any given minutiae conﬁguration on thumbs is lower than on other ﬁngers; and that the probability of observing any given minutiae conﬁguration on a ﬁnger with an arch pattern is lower than on a ﬁnger with another pattern (i.e., loop and whorl). These observations seem to indicate that there is more variability between conﬁgurations located on thumbs and on arch patterns than between conﬁgurations located on other ﬁngers/patterns. This may seem counterintuitive to any latent print examiner, whose experience is that it is more difﬁcult to identify latent prints with arch patterns (due precisely to a much lower variability between conﬁgurations located on arch patterns). This seemingly counterintuitive result can partially be explained. The pV ðv ¼ 1jHd Þ in Fig. 8 were calculated as described in [20,29]. That is, only k minutiae conﬁgurations deemed similar to the ﬁngermarks by our AFIS algorithm on the speciﬁc ﬁngers/ general patterns were considered; however the number of these conﬁgurations was weighted by the total number of ﬁngerprints in the reference dataset to reﬂect the actual probability of observing the considered ﬁnger/pattern combination in the population. This is justiﬁed by the choice of propositions for Hp, and Hd, which are person propositions and not ﬁnger propositions [20]. This manner of estimating pV ðv ¼ 1jHd Þ explains partially why Fig. 8 shows that the weights of conﬁgurations on thumbs are shifted when compared to the weights of the same conﬁgurations on the other ﬁngers. For example, Table 3 shows that there are 17,905 thumbs and 95,274 other ﬁngers with arch patterns out of 4,054,350 ﬁngers, which corresponds to a ratio of 0.188 thumb for every other ﬁnger with arch pattern and to a base shift of 0.72 on the log10 scale in Fig. 8. Table 5 presents the log10 of the base shift for the various combinations of ﬁngers/patterns used in this study. Fig. 8 provides the actual weight carried by any given conﬁguration when it is thought to be on a particular ﬁnger or on a particular pattern: the weight presented in Fig. 8 are the weights that could be reported in casework if one was to present the weights of the test conﬁgurations in court. However, Table 5 is provided as a way to quantify how much of the weight of any particular conﬁguration is due simply to the ﬁnger number/ pattern combination, and how much of that speciﬁcity is due to the variability between minutiae conﬁgurations on that ﬁnger/ pattern. We note the inconsistency of some of the results for conﬁgurations of 12 minutiae thought to originate from thumbs and other ﬁngers on arch patterns: in these cases, a large number of test conﬁgurations were not associated with any reference conﬁgurations by the AFIS algorithm. Since it was not possible to approximate the pV ðv ¼ 1jHd Þ for these conﬁgurations, their results are not shown in Fig. 8a and b; thus the results shown for conﬁgurations of 11–12 minutiae in Fig. 8a and b are based a very small number of test ﬁngermark conﬁgurations, which decreases the accuracy of the estimation of the expected value of the weight of evidence for these numbers of minutiae.

6. General discussion and conclusion In this paper, we present a statistical model for the quantiﬁcation of the weight of forensic evidence. Our model is based on the likelihood ratio framework. According to this framework, the weight of a particular ﬁngerprint evidence is assigned by comparing (a) the likelihood of observing a given ﬁngermark considering that it originates from a particular person and (b) the likelihood of observing that ﬁngermark considering that it originates from a random individual in a relevant population. Contrary to a majority of models proposed in the past years (since [30]), the model is not based on similarity measures, but directly attempts to assign probability distributions to ﬁngerprint features. The model has 3 components, which focus respectively on the spatial relationship, the direction and the type of minutiae that can be observed in any given ﬁngermark conﬁguration. These components are all conditioned on a fourth component, which is designed to mimic the ﬁngerprint comparison process where an examiner will select a set of k features on the ﬁngermark, and select the k potentially corresponding features on a ﬁngerprint out of the n possible features that can be observed on that ﬁngerprint. In our model, an examiner does the selection of the k minutiae on the ﬁngerprint from the considered individual, while an AFIS matching algorithm provided by 3M Cogent, Inc. (Pasadena, CA, USA) performs this selection for a sample of individuals from the relevant population. In practice, any means of selecting the most similar k minutiae out of a set of n can be used. The set of k minutiae conﬁgurations selected by the ﬁngerprint examiners and by the AFIS matching algorithm are then used to support the estimation of the probability density of the spatial relationship, the direction and the type of the minutiae observed on the ﬁngermark. On the one hand, the development of the 3 ﬁrst components requires a series of assumptions and simpliﬁcations in order to maintain the statistical and computational complexity of the model to an acceptable level. On the other hand, the fourth component simply involves searching the features observed on a considered ﬁngermark into (a) the ﬁngerprint of a given individual and (b) a large reference set of ﬁngerprints. The performance of the model has been tested in different situations. In particular, the expected weight of evidence of a series of ﬁngermark conﬁgurations has been calculated using different assumptions on their origin in terms of ﬁnger number or pattern of the ridge ﬂow. The results presented above for all 4 components of the model shows that the model works reasonably well; and in particular that the spatial relationship between minutiae observed on ﬁngermarks carries more weight than the direction and the type of these minutiae. This effect is mainly due to the better robustness of distance measurements between minutiae when compared to direction and type assignments to the minutiae. Interestingly, the results show that the AFIS component of the model already captures a signiﬁcant amount of the speciﬁcity of ﬁngerprint features. In other words, our results show that, while the shape, direction and type components of the models enable the quantiﬁcation of some of the weight of the ﬁngerprint evidence, this weight is provided in addition to the weight already provided by the AFIS algorithm. In fact, it seems that the AFIS component of the model, by design, has a better sensitivity to the various subpopulations contained in the relevant population of alternative sources, and that it does not require as many assumptions and simpliﬁcations as the other components. Thus, this research rejoins Egli et al. [31] and conﬁrms that a model for the quantiﬁcation of the weight of ﬁngerprint evidence can be designed directly based on an AFIS algorithm, without the need for an added layer of complexity induced by the modeling of ﬁngerprint features. However, we need to reiterate the caveat made earlier in this paper: the use of AFIS scores in models aimed

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Fig. 8. (a) Estimates for the AFIS component of the model—arch pattern on thumbs. (b) Estimates for the AFIS component of the model—arch pattern on other ﬁngers. (c) Estimates for the AFIS component of the model—loop pattern on thumbs. (d) Estimates for the AFIS component of the model—loop pattern on other ﬁngers. (e) Estimates for the AFIS component of the model—whorl pattern on thumbs. (f) Estimates for the AFIS component of the model—whorl pattern on other ﬁngers.

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Table 5 log10 base shift for the various combinations of ﬁngers/patterns used in this study. This base shift represents the weight of evidence that is simply due to the ﬁnger/ pattern combination as opposed to the weight of evidence due to differences in the distributions of conﬁgurations on the different ﬁnger/pattern. Thumb

Thumb Arch Loop Whorl Other ﬁnger Arch Loop Whorl

Other Fingers

Arch

Loop

Whorl

Arch

Loop

Whorl

1.00

1.27 1.00

1.30 0.02 1.00

0.73 0.02 0.57

1.94 0.67 0.64

1.64 0.37 0.34

1.00

1.21 1.00

0.91 0.30 1.00

at quantifying the weight of ﬁngerprint evidence is not well understood. Thus, it may be that future models should rely on AFIS technology, but not directly on AFIS scores. Our results also show that the model seems to generate a signiﬁcant amount of misleading evidence when a ﬁngermark is compared to a ﬁngerprint from a different source, and that in some cases, the misleading evidence strongly supports the hypothesis that these impressions originate from the same source. These results root in our experimental design. Indeed, the results provided above concern comparisons between a set of ﬁngermarks and the most similar ﬁngerprints found in a dataset of more than 4,000,000 ﬁngerprints. This provides an idea of the performance of the model in a ‘‘worst case scenario’’ environment. Nevertheless, some improvements in the model are needed to keep the rate of misleading evidence to an acceptable level, even in the harshest conditions. Overall, this paper provides signiﬁcant data on the expected probative value of ﬁngermark conﬁgurations under different conditions and assumptions regarding sub-populations of potential donors. This globally provides information on the validity of the use of ﬁngerprint evidence to discriminate between individuals in a relatively restrained geographical area. In particular, our results conﬁrm the ones presented in [16]: 1) The expected value of the LR increases with the number of minutiae. 2) A range of LR values is observed for each number of minutiae, indicating that each conﬁguration of minutiae needs to be considered on its own merits. 3) There is not a clear cut-off point that would lend itself to the idea of a scientiﬁc basis for a numerical standard. In addition, our results stress that different magnitude of probative values are needed to reach a conclusion on the source of a mark, when the case is assessed in the light of the potential source being generated through non-ﬁngerprint evidence, compared to the situation where the potential source was generated through a search in a ﬁngerprint database. In the latter case, a greater degree of speciﬁcity and quality in the selected ﬁngerprint features will be necessary for the examiner to reach a conclusion. Our data can readily be used to answer the need for statistical data to support the admissibility of ﬁngerprint evidence in court as it was done for the data presented in [16] on behalf of the State of Minnesota [27,28]. The need for statistical data to support the general scientiﬁc foundations of ﬁngerprint evidence was expressed by many legal and scientiﬁc commentators and summarized in the recent report from the National Research Council of the American National Academy of Sciences [32]; however, we also believe that ﬁngerprint statistical models could

be used in any given case to quantify the evidential weight of any ﬁngerprint comparison. Naturally, such case-speciﬁc usage of statistical models implies that the models have gone through extensive scientiﬁc and operational validation as described by SWGFAST [33]. Ideally, the rates of misleading evidences [30] of the models should be provided based on a large sample, the various assumptions supporting the models should be tested and their general accuracy should be assessed. This latter point may be the most challenging. Notwithstanding the importance of the validation process, the use of statistical models in casework also needs to be framed within a larger context of operating procedures, workﬂow management and operational beneﬁts and limitations. Our model has not been validated in that sense. Overall, the main issue faced by scientists developing statistical models for ﬁngerprint evidence is related with the dimensionality and complexity of the variables used to capture friction ridge characteristics. Our attempt to build a model in the feature space required us to make a number of assumptions related to the independence of the features; attempts to build models based on AFIS scores are slowed down by the lack of understanding of the mapping of the feature space to the score space. We believe that the solution lives in building a model in the feature space, where the characteristics are summarized by some means, such as a score as in [16]. Acknowledgments This research was funded in part by Grant 2010-DN-BX-K267 from the National Institute of Justice of the U.S. Department of Justice awarded to The Pennsylvania State University. This research was supported by data and technology provided by 3M Cogent, Inc. to Two N’s Forensics, Inc. In particular, the authors wish to thank Ms. Teresa Wu, Dr. Xian Tang, Mr. Walt Seltz, Mr. Chris Kopcsak, Mr. Martin Kenner and the rest of the team at 3M Cogent for their help in setting up the research AFIS algorithm. Finally, the authors want to thank Ms. Haleigh Boswell, Ms. Chelsea Ellis and Mr. Jonathan Duffy for annotating the ﬁngerprint images used to test the model.

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