**Others**

[24] Goethals, B., Laur, S., Lipmaa, H., & Mielikainen, T. (2004). On Private Scalar Product

[25] Erkin, Z., Franz, M., Guajardo, J., Katzenbeisser, S., Lagendijk, I., & Toft, T. (2009). Privacy-Preserving Face Recognition. *Privacy Enhancing Technologies*, Springer Ber‐

[26] Osadchy, M., Pinkas, B., Jarrous, A., & Moskovich, B. (2010). SCiFI- A System for Se‐

[27] Kumara Krishnan, Arun P., & Sy, Bon K. (2012). SIPPA-2.0- Secure Information Proc‐ essing with Privacy Assurance (version 2.0). *Proceeding of the PST 2012*, Paris, France.

[30] Shannon, C. (1949). Communication Theory of Secrecy Systems. *Bell System Technical*

[31] "Wikipedia," [Online]. Available: http://en.wikipedia.org/wiki/General\_linear\_group.

[32] "Wikipedia," [Online]. Available: http://en.wikipedia.org/wiki/Secure\_channel. [Ac‐

[33] Overbey, J., Traves, W., & Wojdylo, J. (2005). On the Keyspace of the Hill Cipher.

[34] "Wikipedia PP Complexity," Wikipedia, [Online]. Available: http://en.wikipedia.org/

[35] Damgard, M. Jurik, & Nielsen, J. (2010). A generalization of Paillier's public-key sys‐ tem with applications to electronic voting. *International Journal of Information Security*

[36] Fiat & Shamir, A. (1986). How to Prove Yourself: Practical Solutions to Identification

[37] Katz, J., & Lindell, Y. (2007). Introduction to Modern Cryptography, Chapman &

[39] Yang, J. C. (2011). Non-minutiae based fingerprint descriptor. book chapter, *Biomet‐*

[40] Yang, J. C., & Park, D. S. (2008). A Fingerprint Verification Algorithm Using Tessel‐

.

[38] http://biometrics.idealtest.org/downloadDB.do?id=7 [Accessed 30 3 2012].

lated Invariant Moment Features. *Neurocomputing*, 71(10-12), 1939-1946.

*rics*, Intech, Vienna, Austria, June, 978-9-53307-618-8.

.

.

,

Computation for Privacy-Preserving Data Mining. *ICISC*

cure Face Identification. *IEEE Symposium on Security Privacy*

[28] http://www.futronic-tech.com/product\_fs88.html.

[29] http://www.neurotechnology.com/verifinger.html.

wiki/PP\_(complexity). [Accessed 20 3 2012].

and Signature Problems. *CRYPTO*

lin / Heidelberg, 235-253.

218 New Trends and Developments in Biometrics

*Journal*, 28(4), 656-715.

[Accessed 16 3 2012].

*Cryptologia*, 29(1), 59-72.

cessed 16 3 2012].

9(6), 1615-5262.

Hall.

**Chapter 10**

**Provisional chapter**

**An AFIS Candidate List Centric Fingerprint Likelihood**

**Likelihood Ratio Model Based on Morphometric and**

The use of fingerprints for identification purposes boasts worldwide adoption for a large variety of applications, from governance centric applications such as border control to personalised uses such as electronic device authentication. In addition to being an inexpensive and widely used form of biometric for authentication systems, fingerprints are also recognised as an invaluable biometric for forensic identification purposes such as law enforcement and disaster victim identification. Since the very first forensic applications, fingerprints have been utilised as one of the most commonly used form of forensic evidence

Applications of fingerprint identification are founded on the intrinsic characteristics of the friction ridge arrangement present at the fingertips, which can be generally classified at different levels or resolutions of detail (Figure 1). Generally speaking, fingerprint patterns can be described as numerous curved lines alternated as ridges and valleys that are largely regular in terms orientation and flow, with relatively few key locations being of exception (singularities). A closer examination reveals a more detail rich feature set allowing for greater discriminatory analysis. In addition, analysis of local textural detail such as ridge shape, orientation, and frequency, have been used successfully in fingerprint matching algorithms

as primary features [1] [2] or in conjunction with other landmark-based features [3].

Both biometric and forensic fingerprint identification applications rely on premises that such fingerprint characteristics are highly discriminatory and immutable amongst the general population. However, the collectability of such fingerprint characteristics from biometric scanners, ink rolled impressions, and especially, latent marks, are susceptible to adverse factors such as partiality of contact, variation in detail location and appearance due to skin elasticity (specifically for level 2 and 3 features) and applied force, environmental noises such

> ©2012 Abraham et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Abraham et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Abraham et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Ratio Model Based on Morphometric and Spatial**

**An AFIS Candidate List Centric Fingerprint**

Joshua Abraham, Paul Kwan, Christophe Champod,

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Joshua Abraham, Paul Kwan, Christophe Champod,

**Analyses (MSA)**

Chris Lennard and Claude Roux

Chris Lennard and Claude Roux

**Spatial Analyses (MSA)**

http://dx.doi.org/10.5772/51184

**1. Introduction**

worldwide.

**Provisional chapter**

## **An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA) An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)**

Joshua Abraham, Paul Kwan, Christophe Champod, Chris Lennard and Claude Roux Joshua Abraham, Paul Kwan, Christophe Champod, Chris Lennard and Claude Roux

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51184

## **1. Introduction**

The use of fingerprints for identification purposes boasts worldwide adoption for a large variety of applications, from governance centric applications such as border control to personalised uses such as electronic device authentication. In addition to being an inexpensive and widely used form of biometric for authentication systems, fingerprints are also recognised as an invaluable biometric for forensic identification purposes such as law enforcement and disaster victim identification. Since the very first forensic applications, fingerprints have been utilised as one of the most commonly used form of forensic evidence worldwide.

Applications of fingerprint identification are founded on the intrinsic characteristics of the friction ridge arrangement present at the fingertips, which can be generally classified at different levels or resolutions of detail (Figure 1). Generally speaking, fingerprint patterns can be described as numerous curved lines alternated as ridges and valleys that are largely regular in terms orientation and flow, with relatively few key locations being of exception (singularities). A closer examination reveals a more detail rich feature set allowing for greater discriminatory analysis. In addition, analysis of local textural detail such as ridge shape, orientation, and frequency, have been used successfully in fingerprint matching algorithms as primary features [1] [2] or in conjunction with other landmark-based features [3].

Both biometric and forensic fingerprint identification applications rely on premises that such fingerprint characteristics are highly discriminatory and immutable amongst the general population. However, the collectability of such fingerprint characteristics from biometric scanners, ink rolled impressions, and especially, latent marks, are susceptible to adverse factors such as partiality of contact, variation in detail location and appearance due to skin elasticity (specifically for level 2 and 3 features) and applied force, environmental noises such

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Abraham et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Abraham et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Abraham et al., licensee InTech. This is an open access chapter distributed under the terms of the

a) b) c) d) e) f)

**Figure 2.** Examples of different fingerprint impressions, including an ink rolled print (a), latent mark (b), scanned fingerprint flats of ideal quality (c), dry skin (d), slippage (e), and over saturation (f). Fingerprints are sourced from the NIST 27 [48],

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

http://dx.doi.org/10.5772/51184

223

Historically, the forensic identification of fingerprints has had near unanimous acceptance as a gold standard of forensic evidence, where the scientific foundations of such testimonies were rarely challenged in court proceedings. In addition, fingerprint experts have generally been regarded as expert witnesses with adequate training, scientific knowledge, relevant experience, and following a methodical process for identification, ultimately giving

Fingerprint experts largely follow a friction ridge identification process called ACE-V (Analysis, Comparison, Evaluation, and Verification) [5] to compare an unknown fingermark with known fingerprint exemplars. The ACE-V acronym also details the ordering of the identification process (Figure 3). In the analysis stage, all visible ridge characteristics (level 1, 2, and 3) are noted and assessed for reliability, while taking into account variations caused by pressure, distortion, contact medium, and development techniques used in the laboratory. The comparison stage involves comparing features between the latent mark and either the top *n* fingerprint exemplars return from an AFIS search, or specific pre-selected exemplars. If a positive identification is declared, all corresponding features are charted, along with any differences considered to be caused by environmental influence. The Evaluation stage consists of an expert making an inferential decision based on the comparison stage

• exclusion: a discrepancy of features are discovered so it precludes the possibility of a

• identification: a significant correspondence of features are discovered that is considered

The Verification stage consists of a peer review of the prior stages. Any discrepancies in

Identification evaluation conclusions [7] made by fingerprint experts have historical influence

• inconclusive: not enough evidence is found for either an exclusion or identification.

from Edmond Locard's *tripartite rule* [8]. The tripartite rule is defined as follows:

to be sufficient in itself to conclude to a common source, and

evaluations are handled by a conflict resolution procedure.

FVC2004 [51], and our own databases.

**1.1. Forensic fingerprint identification**

credibility to their expert witness testimonies.

observations. The possible outcomes [6] are:

common source,

**Figure 1.** Level 1 features include features such as pattern class (a), singularity points and ridge frequency (b). Level 2 features (c) include minutiae with primitive types ridge endings and bifurcations. Level 3 features (d) include pores (open/closed) and ridge shape. These fingerprints were sourced from the FVC2002 [47], NIST4 [46], and NIST24 [48] databases

as moisture, dirt, slippage, and skin conditions such as dryness, scarring, warts, creases, and general ageing. Such influences generally act as a hindrance for identification, reducing both the quality and confidence of assessing matching features between impressions (Figure 2).

In this chapter, we will firstly discuss the current state of forensic fingerprint identification and how models play an important role for the future, followed by a brief introduction and review into relevant statistical models. Next, we will introduce a Likelihood Ratio (LR) model based on Support Vector Machines (SVMs) trained with features discovered via the morphometric and other spatial analyses of matching minutiae for both genuine and close imposter (or match and close non-match) populations typically recovered from Automated Fingerprint Identification System (AFIS) candidate lists. Lastly, experimentation performed on a set of over 60,000 publicly available fingerprint images (mostly sourced from NIST and FVC databases) and a distortion set of 6,000 images will be presented, illustrating that the proposed LR model is reliably guiding towards the right proposition in the identification assessment for both genuine and high ranking imposter populations, based on the discovered distortion characteristic differences of each population.

<sup>222</sup> New Trends and Developments in Biometrics An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA) 3 An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA) http://dx.doi.org/10.5772/51184 223

**Figure 2.** Examples of different fingerprint impressions, including an ink rolled print (a), latent mark (b), scanned fingerprint flats of ideal quality (c), dry skin (d), slippage (e), and over saturation (f). Fingerprints are sourced from the NIST 27 [48], FVC2004 [51], and our own databases.

## **1.1. Forensic fingerprint identification**

2 New Trends and Developments in Biometrics

Arch Tented Arch

Right Loop

Ridge Ending

**Level 2**

other

c) d)

distortion characteristic differences of each population.

a) b)

Core

**Pattern Classification Pattern Singularities**

Left Loop

Whorl

**Level 1**

**Minutiae Pores/Ridge Shape**

Open Pores

Bifurcation

**Figure 1.** Level 1 features include features such as pattern class (a), singularity points and ridge frequency (b). Level 2 features (c) include minutiae with primitive types ridge endings and bifurcations. Level 3 features (d) include pores (open/closed) and

as moisture, dirt, slippage, and skin conditions such as dryness, scarring, warts, creases, and general ageing. Such influences generally act as a hindrance for identification, reducing both the quality and confidence of assessing matching features between impressions (Figure 2). In this chapter, we will firstly discuss the current state of forensic fingerprint identification and how models play an important role for the future, followed by a brief introduction and review into relevant statistical models. Next, we will introduce a Likelihood Ratio (LR) model based on Support Vector Machines (SVMs) trained with features discovered via the morphometric and other spatial analyses of matching minutiae for both genuine and close imposter (or match and close non-match) populations typically recovered from Automated Fingerprint Identification System (AFIS) candidate lists. Lastly, experimentation performed on a set of over 60,000 publicly available fingerprint images (mostly sourced from NIST and FVC databases) and a distortion set of 6,000 images will be presented, illustrating that the proposed LR model is reliably guiding towards the right proposition in the identification assessment for both genuine and high ranking imposter populations, based on the discovered

ridge shape. These fingerprints were sourced from the FVC2002 [47], NIST4 [46], and NIST24 [48] databases

Delta

**Level 3**

Closed Pores

**Frequency/Orientation Maps**

Historically, the forensic identification of fingerprints has had near unanimous acceptance as a gold standard of forensic evidence, where the scientific foundations of such testimonies were rarely challenged in court proceedings. In addition, fingerprint experts have generally been regarded as expert witnesses with adequate training, scientific knowledge, relevant experience, and following a methodical process for identification, ultimately giving credibility to their expert witness testimonies.

Fingerprint experts largely follow a friction ridge identification process called ACE-V (Analysis, Comparison, Evaluation, and Verification) [5] to compare an unknown fingermark with known fingerprint exemplars. The ACE-V acronym also details the ordering of the identification process (Figure 3). In the analysis stage, all visible ridge characteristics (level 1, 2, and 3) are noted and assessed for reliability, while taking into account variations caused by pressure, distortion, contact medium, and development techniques used in the laboratory. The comparison stage involves comparing features between the latent mark and either the top *n* fingerprint exemplars return from an AFIS search, or specific pre-selected exemplars. If a positive identification is declared, all corresponding features are charted, along with any differences considered to be caused by environmental influence. The Evaluation stage consists of an expert making an inferential decision based on the comparison stage observations. The possible outcomes [6] are:


The Verification stage consists of a peer review of the prior stages. Any discrepancies in evaluations are handled by a conflict resolution procedure.

Identification evaluation conclusions [7] made by fingerprint experts have historical influence from Edmond Locard's *tripartite rule* [8]. The tripartite rule is defined as follows:

**1.2. Daubert and criticisms**

commentators were:

community.

discriminability, and

non-match, or inconclusive).

scientific method given were as follows:

• must be based on testable and falsifiable theories/techniques,

• must have standards and controls concerning its applications, and • must be generally accepted by a relevant scientific community.

• must be subjected to peer-review and publication,

• must have known or predictable error rates,

Recently, there has been a number of voiced criticisms on the scientific validity of forensic fingerprint identification [10] [11] [12] [13] [14] [15]. Questions with regards to the scientific validity of forensic fingerprint identification began shortly after the *Daubert* case [17]. In the 1993 case of Daubert v. Merrell Dow Pharmaceuticals [18] the US Supreme Court outlined criteria concerning the admissibility of scientific expert testimony. The criteria for a valid

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

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225

The objections which followed [13] [14] [15] from a number of academic and legal

• the contextual bias of experts for decisions made within the ACE-V (Analysis, Comparison, Evaluation, and Verification) framework used in fingerprint identification • the unfounded and unfalsifiable theoretical foundations of fingerprint feature

• the 'unscientific' absolute conclusions of identification in testimonies (i.e., either match,

There have been a number of studies [16] over the last 5 years concerning contextual bias and the associated error rates of ACE-V evaluations in practice. The experiments reported by [19] led to conclusions that experts appear more susceptible to bias assessments of 'inconclusive' and 'exclusion', while false positive rates are reasonably low within simulation of the ACE-V framework. It has also been suggested from results in [20] and [21] that not all stages of ACE-V are equally vulnerable to contextual bias, with primary effects occurring in the analysis stage, with proposals on how to mediate such variability found in [22]. While contextual bias is solely concerned with the influence of the expert, the remaining criticisms can be summarised as the non-existence of a scientifically sound probabilistic framework for fingerprint evidential assessment, that has the consensual approval from the forensic science

The theoretical foundations of fingerprint identification primarily rest on rudimentary observational science, where a high discriminability of feature characteristics exists. However, there is a lack of consensus regarding quantifiable error rates for a given pair of 'corresponding' feature configurations [23]. Some critics have invoked a more traditional interpretation for discriminability [24] [25], claiming that an assumption of 'uniqueness' is used. This clearly violates the falsifiable requirement of Daubert. However, it has been argued that modern day experts do not necessarily associate discriminability with uniqueness [26]. Nevertheless, a consensus framework for calculating accurate error rates

for corresponding fingerprint features needs to be established.

**Figure 3.** Flowchart of modern ACE-V process used in conjunction with AFIS. The iterative comparison of each exemplar fingerprint in the AFIS candidate list is performed until identification occurs or no more exemplars are left. The red flow lines indicate the process for the verification stage analysis. The purple flow line from the agreement of features test shows the ACE process that skips the evaluation stage.


Holistically, the tripartite rule can be viewed as a probabilistic framework, where the successful applications of the first and second rules are analogous to a statement with 100% certainty that the mark and the print share the same source, whereas the third rule covers the probability range between 0% to 100%. While some jurisdictions only apply the first rule to set a numerical standard within the ACE-V framework, other jurisdictions (such as Australia, UK, and USA [9]) adopt a holistic approach, where no strict numerical standard or feature combination is prescribed. Nevertheless, current fingerprint expert testimony is largely restricted to conclusions that convey a statement of certainty, ignoring the third rule's probabilistic outcome.

#### **1.2. Daubert and criticisms**

4 New Trends and Developments in Biometrics

**crime**

**crime scene processing**

**lab processing**

**Analysis**

**not found**

**consistent evaluation ?**

**Yes**

**No**

process that skips the evaluation stage.

quality fingermarks.

at least 2 experts.

probabilistic outcome.

minutiae.

**No**

**consistent evaluation ?**

**Yes**

**suitable?**

**Yes**

**No**

**expert agreement ?**

Additional expert(s) analysis

**Yes**

**Evaluation Verification**

**No**

**of features?**

**fetch exemplar from AFIS**

**Yes**

**Comparison**

**conclusive? agreement**

**Figure 3.** Flowchart of modern ACE-V process used in conjunction with AFIS. The iterative comparison of each exemplar fingerprint in the AFIS candidate list is performed until identification occurs or no more exemplars are left. The red flow lines indicate the process for the verification stage analysis. The purple flow line from the agreement of features test shows the ACE

• Positive identifications are possible when there are more than 12 minutiae within sharp

• If 8 to 12 minutiae are involved, then the case is borderline. Certainty of identity will depend on additional information such as finger mark quality, rarity of pattern, presence of the core, delta(s), and pores, and ridge shape characteristics, along with agreement by

• If a limited number of minutiae are present, the fingermarks cannot provide certainty for an identification, but only a presumption of strength proportional to the number of

Holistically, the tripartite rule can be viewed as a probabilistic framework, where the successful applications of the first and second rules are analogous to a statement with 100% certainty that the mark and the print share the same source, whereas the third rule covers the probability range between 0% to 100%. While some jurisdictions only apply the first rule to set a numerical standard within the ACE-V framework, other jurisdictions (such as Australia, UK, and USA [9]) adopt a holistic approach, where no strict numerical standard or feature combination is prescribed. Nevertheless, current fingerprint expert testimony is largely restricted to conclusions that convey a statement of certainty, ignoring the third rule's

**No Yes**

**unexamined candidate ?**

**No**

**conflict resolution**

1 2 *n*

**AFIS matching score rank**

**consistent evaluation ?**

**No**

**Yes**

**candidate list**

**...**

**identification**

Recently, there has been a number of voiced criticisms on the scientific validity of forensic fingerprint identification [10] [11] [12] [13] [14] [15]. Questions with regards to the scientific validity of forensic fingerprint identification began shortly after the *Daubert* case [17]. In the 1993 case of Daubert v. Merrell Dow Pharmaceuticals [18] the US Supreme Court outlined criteria concerning the admissibility of scientific expert testimony. The criteria for a valid scientific method given were as follows:


The objections which followed [13] [14] [15] from a number of academic and legal commentators were:


There have been a number of studies [16] over the last 5 years concerning contextual bias and the associated error rates of ACE-V evaluations in practice. The experiments reported by [19] led to conclusions that experts appear more susceptible to bias assessments of 'inconclusive' and 'exclusion', while false positive rates are reasonably low within simulation of the ACE-V framework. It has also been suggested from results in [20] and [21] that not all stages of ACE-V are equally vulnerable to contextual bias, with primary effects occurring in the analysis stage, with proposals on how to mediate such variability found in [22]. While contextual bias is solely concerned with the influence of the expert, the remaining criticisms can be summarised as the non-existence of a scientifically sound probabilistic framework for fingerprint evidential assessment, that has the consensual approval from the forensic science community.

The theoretical foundations of fingerprint identification primarily rest on rudimentary observational science, where a high discriminability of feature characteristics exists. However, there is a lack of consensus regarding quantifiable error rates for a given pair of 'corresponding' feature configurations [23]. Some critics have invoked a more traditional interpretation for discriminability [24] [25], claiming that an assumption of 'uniqueness' is used. This clearly violates the falsifiable requirement of Daubert. However, it has been argued that modern day experts do not necessarily associate discriminability with uniqueness [26]. Nevertheless, a consensus framework for calculating accurate error rates for corresponding fingerprint features needs to be established.

#### **1.3. Role of statistical models**

While a probabilistic framework for fingerprint comparisons has not been historically popular and was even previously banned by professional bodies [8], a more favourable treatment within the forensic community is given in recent times. For example, the IAI have recently rescinded their ban on reporting possible, probable, or likely conclusions [27] and support the future use of valid statistical models (provided that they are accepted as valid by the scientific community) to aid the practitioner in identification assessments. It has also been suggested in [28] that a probabilistic framework is based on strong scientific principles unlike the traditional numerical standards.

• *LR* = 1: the evidence has equal support from both hypotheses, and

or latent mark) with *m* marked features (denoted as *x*(*m*)), the LR is defined as

*LRfinger* <sup>=</sup> *<sup>P</sup>*(*y*(*m*′

agree given that the marks were produced by the same finger, while *P*(*y*(*m*′

landmark based feature configurations required to derive values for *P*(*y*(*m*′

using the *closest q* corresponding features between *x*(*m*) and *y*(*m*′

• *H*0: *x* and *y* were produced by the same finger, and • *HA*: *x* and *y* were produced by different fingers.

**2.1. AFIS score based LR models**

*2.1.1. Parametric Based Models*

The general LR form of equation (1) can be restated specifically for fingerprint identification evaluations. Given an unknown query impression, *y*, (e.g., unknown latent mark) with *m*′

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

the probability that *x* and *y* agree given that the marks were *not* produced by the same finger,

hypotheses used to calculate the LR numerator and denominator probabilities are defined

The addendum crime scene information, *Ics*, may include detail of surrounding fingermarks, surficial characteristics of the contacted medium, or a latent mark quality/confidence assessment. In order to measure the within-finger and between-finger variability of

(used as a proxy for direct assessment) derived from either the analysis of spatial properties [33] [34] [35], or analysis of similarity score distributions produced by the AFIS [36] [37] [38].

AFIS score based LR models use estimates of the genuine and imposter similarity score distributions from fingerprint matching algorithm(s) within AFIS, in order to derive a LR measure. In a practical application, a given mark and exemplar may have an AFIS similarity score of *s*, from which the conditional probability of the score can be calculated (Figure 4) to

> *LR* <sup>=</sup> *<sup>P</sup>*(*s*|*H*0) *P*(*s*|*HA*)

In order to estimate the score distributions used in equation (3), the authors of [36] proposed using the Weibull *W*(*λ*, *β*) and Log-Normal ln N (*µ*, *σ*2) distributions with scale/shape parameters tuned to estimate the genuine and imposter AFIS score distributions, respectively.

Given query and template fingermarks with an AFIS similarity score, *s*, the LR is

)|*x*(*m*), *HA*, *Ics*), models either use statistical distributions of dissimilarity metrics

)), and a known impression, *x*, (e.g., known AFIS candidate

*<sup>P</sup>*(*y*(*m*′)|*x*(*m*), *HA*, *Ics*) (2)

) with *q* ≤ min(*m*, *m*′

. (3)

)|*x*(*m*), *HA*, *Ics*) is

http://dx.doi.org/10.5772/51184

227

)|*x*(*m*), *H*0, *Ics*)

). Thus,

)|*x*(*m*), *H*0, *Ics*)

)|*x*(*m*), *H*0, *Ics*) represents the probability that impressions *x* and *y*

• *LR* > 1: the evidence has more support for hypothesis *H*0.

marked features (denoted as *y*(*m*′

where the value *P*(*y*(*m*′

as:

and *P*(*y*(*m*′

give an LR of

Statistical models for fingerprint identification provide a probabilistic framework that can be applied to forensic fingerprint identification to create a framework for evaluations, that do not account for the inherent uncertainties of fingerprint evidence. Moreover, the use of such statistical models as an identification framework helps answer the criticisms of scientific reliability and error rate knowledge raised by some commentators. For instance, statistical models can be used to describe the discriminatory power of a given fingerprint feature configuration, which in hand can be used to predict and estimate error rates associated with the identification of specific fingerprint features found in any given latent mark.

Statistical models could potentially act as a tool for fingerprint practitioners with evaluations made within the ACE-V framework, specifically when the confidence in identification or exclusion is not overtly clear. However, such applications require statistical models to be accurate and robust to real work scenarios.

## **2. Likelihood Ratio models**

A *likelihood ratio* (LR) is a simple yet powerful statistic when applied to a variety of forensic science applications, including inference of identity of source for evidences such as DNA [29], ear-prints [30], glass fragments [31], and fingerprints [32] [33] [34] [35]. An LR is defined as the ratio of two likelihoods of a specific *event* occurring, each of which follow a different prior hypothesis, and thus, empirical distribution. In the forensic identification context, an event, *E*, may represent the recovered evidence in question, while the prior hypotheses considered for calculating the two likelihoods of *E* occurring are:


Noting any additional relevant prior information collected from the crime scene as *Ics*, the LR can be expressed as

$$LR = \frac{P(E|H\_{0\prime}I\_{\rm cs})}{P(E|H\_{A\prime}I\_{\rm cs})} \tag{1}$$

where *P*(*E*|*H*0, *Ics*) is the likelihood of the observations on the mark given that the mark was produced by the same finger as the print *P*, while *P*(*E*|*HA*, *Ics*) is the likelihood of the observations on the mark given that the mark was *not* produced by the same finger as *P*. The LR value can be interpreted as follows:

• *LR* < 1: the evidence has more support for hypothesis *HA*,


**1.3. Role of statistical models**

principles unlike the traditional numerical standards.

accurate and robust to real work scenarios.

for calculating the two likelihoods of *E* occurring are:

• *H*0: *E* comes from a specific known source, *P*, and

• *LR* < 1: the evidence has more support for hypothesis *HA*,

• *HA*: *E* has an alternative origin to *P*.

LR value can be interpreted as follows:

LR can be expressed as

**2. Likelihood Ratio models**

While a probabilistic framework for fingerprint comparisons has not been historically popular and was even previously banned by professional bodies [8], a more favourable treatment within the forensic community is given in recent times. For example, the IAI have recently rescinded their ban on reporting possible, probable, or likely conclusions [27] and support the future use of valid statistical models (provided that they are accepted as valid by the scientific community) to aid the practitioner in identification assessments. It has also been suggested in [28] that a probabilistic framework is based on strong scientific

Statistical models for fingerprint identification provide a probabilistic framework that can be applied to forensic fingerprint identification to create a framework for evaluations, that do not account for the inherent uncertainties of fingerprint evidence. Moreover, the use of such statistical models as an identification framework helps answer the criticisms of scientific reliability and error rate knowledge raised by some commentators. For instance, statistical models can be used to describe the discriminatory power of a given fingerprint feature configuration, which in hand can be used to predict and estimate error rates associated with the identification of specific fingerprint features found in any given latent mark.

Statistical models could potentially act as a tool for fingerprint practitioners with evaluations made within the ACE-V framework, specifically when the confidence in identification or exclusion is not overtly clear. However, such applications require statistical models to be

A *likelihood ratio* (LR) is a simple yet powerful statistic when applied to a variety of forensic science applications, including inference of identity of source for evidences such as DNA [29], ear-prints [30], glass fragments [31], and fingerprints [32] [33] [34] [35]. An LR is defined as the ratio of two likelihoods of a specific *event* occurring, each of which follow a different prior hypothesis, and thus, empirical distribution. In the forensic identification context, an event, *E*, may represent the recovered evidence in question, while the prior hypotheses considered

Noting any additional relevant prior information collected from the crime scene as *Ics*, the

*LR* <sup>=</sup> *<sup>P</sup>*(*E*|*H*0, *Ics*)

where *P*(*E*|*H*0, *Ics*) is the likelihood of the observations on the mark given that the mark was produced by the same finger as the print *P*, while *P*(*E*|*HA*, *Ics*) is the likelihood of the observations on the mark given that the mark was *not* produced by the same finger as *P*. The

*<sup>P</sup>*(*E*|*HA*, *Ics*) (1)

The general LR form of equation (1) can be restated specifically for fingerprint identification evaluations. Given an unknown query impression, *y*, (e.g., unknown latent mark) with *m*′ marked features (denoted as *y*(*m*′ )), and a known impression, *x*, (e.g., known AFIS candidate or latent mark) with *m* marked features (denoted as *x*(*m*)), the LR is defined as

$$LR\_{finger} = \frac{P(y^{(m')} | \mathbf{x}^{(m)}, H\_{0}, I\_{\rm cs})}{P(y^{(m')} | \mathbf{x}^{(m)}, H\_{\rm A}, I\_{\rm cs})} \tag{2}$$

where the value *P*(*y*(*m*′ )|*x*(*m*), *H*0, *Ics*) represents the probability that impressions *x* and *y* agree given that the marks were produced by the same finger, while *P*(*y*(*m*′ )|*x*(*m*), *HA*, *Ics*) is the probability that *x* and *y* agree given that the marks were *not* produced by the same finger, using the *closest q* corresponding features between *x*(*m*) and *y*(*m*′ ) with *q* ≤ min(*m*, *m*′ ). Thus, hypotheses used to calculate the LR numerator and denominator probabilities are defined as:


The addendum crime scene information, *Ics*, may include detail of surrounding fingermarks, surficial characteristics of the contacted medium, or a latent mark quality/confidence assessment. In order to measure the within-finger and between-finger variability of landmark based feature configurations required to derive values for *P*(*y*(*m*′ )|*x*(*m*), *H*0, *Ics*) and *P*(*y*(*m*′ )|*x*(*m*), *HA*, *Ics*), models either use statistical distributions of dissimilarity metrics (used as a proxy for direct assessment) derived from either the analysis of spatial properties [33] [34] [35], or analysis of similarity score distributions produced by the AFIS [36] [37] [38].

#### **2.1. AFIS score based LR models**

AFIS score based LR models use estimates of the genuine and imposter similarity score distributions from fingerprint matching algorithm(s) within AFIS, in order to derive a LR measure. In a practical application, a given mark and exemplar may have an AFIS similarity score of *s*, from which the conditional probability of the score can be calculated (Figure 4) to give an LR of

$$LR = \frac{P(s|H\_0)}{P(s|H\_A)}.\tag{3}$$

#### *2.1.1. Parametric Based Models*

In order to estimate the score distributions used in equation (3), the authors of [36] proposed using the Weibull *W*(*λ*, *β*) and Log-Normal ln N (*µ*, *σ*2) distributions with scale/shape parameters tuned to estimate the genuine and imposter AFIS score distributions, respectively. Given query and template fingermarks with an AFIS similarity score, *s*, the LR is

**Figure 4.** Typical AFIS imposter and genuine score distributions. The LR can be directly calculated for a given similarity score using the densities from these distributions.

$$LR = \frac{f\_W(s|\lambda\_\prime \beta)}{f\_{\ln \mathcal{N}}(s|\mu\_\prime \sigma^2)}\tag{4}$$

*P*(*s*|*H*0) either by normalised bin (method (i)) or probability density (methods (ii) and (iii)) values for respective distributions. Experimentation revealed that the parametric method was biased. In addition, the authors suggest that the kernel density method is the most ideal, as it does not suffer from bias while it can be used to extrapolate NMP scores where

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

AFIS score based LR models provide a framework that is both practically based and simple to implement in conjunction with the AFIS architecture. However, model performance is dependent on the matching algorithm of the AFIS. In fact, LR models presented will usually reflect the exact information contained in a candidate list of an AFIS query. A more complex construction, for instance, multiple AFIS matching algorithms with a mixture-of-experts statistical model would be more ideal and avoid LR values that are strictly algorithm

The scores produced from matching algorithms in AFIS detail pairwise similarity between two impressions (i.e., mark and exemplar). However, the methods used in [36] [38], which generalise the distributions for all minutiae configurations, do not allow evidential aspects such as the rarity of a given configuration to be considered. A more sound approach would be to base LR calculations on methods that do not have primary focus on only pairwise similarities, but consider statistical characteristics of features within a given population. For instance, the LR for a rare minutiae configuration should be weighted to reflect its significance. This is achieved in the method described in [37] by focusing distribution

Feature Vector (FV) based LR models are based on FVs constructed from landmark (i.e., minutiae) feature analyses. A dissimilarity metric is defined that is based on the resulting FV. The distributions of such vector dissimilarity metrics are then analysed for both genuine

The first FV based LR model proposed in the literature can be found in [33]. FVs are derived from Delaunay triangulation (Figure 5 **left**) for different regions of the fingerprint. Each FV

where *GPx* is the pattern of the mark, *Rx* is the region of the fingerprint, *Ntx* is the number of minutiae that are ridge endings in the triangle (with *Ntx* ∈ {0, 1, 2, 3}), *Aix* is the angle of

mod 3)*th* minutiae, for a given query fingerprint. Likewise, these structures are created for

*GPy*, *Ry*, *Nty*, {*A*1*y*, *L*1*y*−2*y*}, {*A*2*y*, *L*2*y*−3*y*}, {*A*3*y*, *L*3*y*−1*y*}

*th* minutia, and *Lix*−((*i*+1) mod 3)*<sup>x</sup>* is the length in pixels between the *<sup>i</sup>*

*x* = [*GPx*, *Rx*, *Ntx*, {*A*1*x*, *L*1*x*−2*x*}, {*A*2*x*, *L*2*x*−3*x*}, {*A*3*x*, *L*3*x*−1*x*}] (6)

*th* and the ((*i* + 1)

http://dx.doi.org/10.5772/51184

229

. (7)

no match has been observed, unlike the histogram based representation.

*2.1.3. Practicality of AFIS based LR Models*

estimates of scores for each minutiae configuration.

and imposter comparisons, from which an LR is derived.

**2.2. Feature Vector based LR models**

*2.2.1. Delauney Triangulation FV Model*

was constructed as follows:

candidate fingerprint(s):

*y* =

the *i*

dependent.

using the proposed probability density functions of the estimated AFIS genuine and imposter score distributions.

An updated variant can be found in [37], where imposter and genuine score distributions are modelled per minutiae configuration. This allows the rarity of the configuration to be accounted for.

#### *2.1.2. Non-Match Probability Based Model*

The authors of [38] proposed a model based on AFIS score distributions, using LR and Non-Match Probability (NMP) calculations. The NMP can be written mathematically as

$$NMP = P(H\_A|s) = \frac{P(s|H\_A)P(H\_A)}{P(s|H\_A)P(H\_A) + P(s|H\_0)P(H\_0)},\tag{5}$$

which is simply the complement of the probability that the null hypothesis (i.e., *x* and *y* come from the same known source) is true, given prior conditions *x*, *y*, and *Ics* (i.e., background information).

Three main methods for modelling the AFIS score distributions where tested, being (i) histogram based, (ii) Gaussian kernel density based, and (iii) parametric density based estimation using the proposed distributions found in [36]. Given an AFIS score, *s*, the NMP and LR were calculated by setting *P*(*HA*) = *P*(*H*0), while estimating both *P*(*s*|*HA*) and *P*(*s*|*H*0) either by normalised bin (method (i)) or probability density (methods (ii) and (iii)) values for respective distributions. Experimentation revealed that the parametric method was biased. In addition, the authors suggest that the kernel density method is the most ideal, as it does not suffer from bias while it can be used to extrapolate NMP scores where no match has been observed, unlike the histogram based representation.

## *2.1.3. Practicality of AFIS based LR Models*

8 New Trends and Developments in Biometrics

Density

using the densities from these distributions.

*2.1.2. Non-Match Probability Based Model*

score distributions.

accounted for.

information).

AFIS Score (**s**)

**Figure 4.** Typical AFIS imposter and genuine score distributions. The LR can be directly calculated for a given similarity score

*LR* <sup>=</sup> *fW*(*s*|*λ*, *<sup>β</sup>*)

using the proposed probability density functions of the estimated AFIS genuine and imposter

An updated variant can be found in [37], where imposter and genuine score distributions are modelled per minutiae configuration. This allows the rarity of the configuration to be

The authors of [38] proposed a model based on AFIS score distributions, using LR and Non-Match Probability (NMP) calculations. The NMP can be written mathematically as

which is simply the complement of the probability that the null hypothesis (i.e., *x* and *y* come from the same known source) is true, given prior conditions *x*, *y*, and *Ics* (i.e., background

Three main methods for modelling the AFIS score distributions where tested, being (i) histogram based, (ii) Gaussian kernel density based, and (iii) parametric density based estimation using the proposed distributions found in [36]. Given an AFIS score, *s*, the NMP and LR were calculated by setting *P*(*HA*) = *P*(*H*0), while estimating both *P*(*s*|*HA*) and

*P*(*s*|*HA*)*P*(*HA*) + *P*(*s*|*H*0)*P*(*H*0)

*NMP* <sup>=</sup> *<sup>P</sup>*(*HA*|*s*) = *<sup>P</sup>*(*s*|*HA*)*P*(*HA*)

P(**s**=66|G)

Genuine Impostor

*<sup>f</sup>*ln <sup>N</sup> (*s*|*µ*, *<sup>σ</sup>*2) (4)

, (5)

P(**s**=23|G)

P(**s**=66|I)

P(**s**=23|I)

AFIS score based LR models provide a framework that is both practically based and simple to implement in conjunction with the AFIS architecture. However, model performance is dependent on the matching algorithm of the AFIS. In fact, LR models presented will usually reflect the exact information contained in a candidate list of an AFIS query. A more complex construction, for instance, multiple AFIS matching algorithms with a mixture-of-experts statistical model would be more ideal and avoid LR values that are strictly algorithm dependent.

The scores produced from matching algorithms in AFIS detail pairwise similarity between two impressions (i.e., mark and exemplar). However, the methods used in [36] [38], which generalise the distributions for all minutiae configurations, do not allow evidential aspects such as the rarity of a given configuration to be considered. A more sound approach would be to base LR calculations on methods that do not have primary focus on only pairwise similarities, but consider statistical characteristics of features within a given population. For instance, the LR for a rare minutiae configuration should be weighted to reflect its significance. This is achieved in the method described in [37] by focusing distribution estimates of scores for each minutiae configuration.

#### **2.2. Feature Vector based LR models**

Feature Vector (FV) based LR models are based on FVs constructed from landmark (i.e., minutiae) feature analyses. A dissimilarity metric is defined that is based on the resulting FV. The distributions of such vector dissimilarity metrics are then analysed for both genuine and imposter comparisons, from which an LR is derived.

#### *2.2.1. Delauney Triangulation FV Model*

The first FV based LR model proposed in the literature can be found in [33]. FVs are derived from Delaunay triangulation (Figure 5 **left**) for different regions of the fingerprint. Each FV was constructed as follows:

$$\mathbf{x} = \begin{bmatrix} G P\_{\mathbf{x}\prime} R\_{\mathbf{x}\prime} \operatorname{Nt}\_{\mathbf{x}\prime} \begin{Bmatrix} A\_{1\mathbf{x}\prime} L\_{1\mathbf{x}-2\mathbf{x}\prime} \end{Bmatrix} \begin{Bmatrix} A\_{2\mathbf{x}\prime} L\_{2\mathbf{x}-3\mathbf{x}\prime} \end{Bmatrix} \begin{Bmatrix} A\_{3\mathbf{x}\prime} L\_{3\mathbf{x}-1\mathbf{x}\prime} \end{Bmatrix} \tag{6}$$

where *GPx* is the pattern of the mark, *Rx* is the region of the fingerprint, *Ntx* is the number of minutiae that are ridge endings in the triangle (with *Ntx* ∈ {0, 1, 2, 3}), *Aix* is the angle of the *i th* minutia, and *Lix*−((*i*+1) mod 3)*<sup>x</sup>* is the length in pixels between the *<sup>i</sup> th* and the ((*i* + 1) mod 3)*th* minutiae, for a given query fingerprint. Likewise, these structures are created for candidate fingerprint(s):

$$y = \begin{bmatrix} \mathbf{G} \mathbf{P}\_{y\prime} \mathbf{R}\_{y\prime} \mathbf{N} \mathbf{t}\_{y\prime} \begin{Bmatrix} A\_{1y\prime} \ L\_{1y-2y} \end{Bmatrix} , \begin{Bmatrix} A\_{2y\prime} \ L\_{2y-3y} \end{Bmatrix} , \begin{Bmatrix} A\_{3y\prime} \ L\_{3y-1y} \end{Bmatrix} \end{bmatrix} . \tag{7}$$

**Finger/Region LR True** < 1 **LR False** > 1 Index/All 2.94 % 1.99 % Middle/All 1.99 % 1.84 % Thumbs/All 3.27 % 3.24 % Index/Core 4.19 % 1.36 % Middle/Core 3.65 % 1.37 % Thumbs/Core 3.74 % 2.43 % Index/Delta 1.95 % 2.62 % Middle/Delta 2.96 % 2.58 % Thumbs/Delta 2.39 % 5.20 %

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

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231

Density functions of both *P*(*d*(*xc*, *yc*)|*xd*, *yd*, *H*0, *Ics*) and *P*(*d*(*xc*, *yc*)|*xd*, *yd*, *HA*, *Ics*) were estimated using a kernel smoothing method. All LR numerator and denominator likelihood

Two experiments were configured in order to evaluate within-finger (i.e., genuine) and between-finger (i.e., imposter) LRs. Ideally, LRs for within-finger comparisons should be larger than all between-finger ratios. The within-finger experiment used 216 fingerprints from 4 different fingers under various different distortion levels. The between-finger datasets included the same 818 fingerprints used in the minutia-type probability calculations. Delaunay triangulation had to be manually adjusted in some cases due to different triangulation results occurring under high distortion levels. Error rates for LRs greater than 1 for false comparisons (i.e., between-finger) and LRs less than 1 for true comparisons (i.e., within-finger) for index, middle, and thumbs, are given in Table 1. These errors rates indicate the power that 3 minutiae (in each triangle) have in creating an LR value dichotomy between

Although the triangular structures of [33] performed reasonably well in producing higher LRs for within-finger comparisons against between-finger comparisons, there are issues with the proposed FV structure's robustness towards distortion. In addition, LRs could potentially have increased dichotomy between imposter and genuine comparisons by including more minutiae in the FV structures, rather than restricting each FV to only have three minutiae. The authors of [34] defined *radial triangulation* FVs based on *n* minutiae *x* = [*GPx*, *xs*] with:

*<sup>x</sup>*(*n*) = [{*Tx*,1, *RAx*,1, *Rx*,1, *Lx*,1,2, *Sx*,1}, {*Tx*,2, *RAx*,2, *Rx*,2, *Lx*,2,3, *Sx*,2},

(and similarly for *<sup>y</sup>* and *<sup>y</sup>*(*n*)), where *GP* denotes the general pattern, *Tk* is the minutia type, *RAk* is the direction of minutia *k* relative to the image, *Rk* is the radius from the *kth* minutia to the centroid (Figure 5 **right**), *Lk*,*k*+<sup>1</sup> is the length of the polygon side from minutia *k* to *k* + 1, and *Sk* is the area of the triangle defined by minutia *k*, (*k* + 1) mod *n*, and the centroid.

..., {*Tx*,*n*, *RAx*,*n*, *Rx*,*n*, *Lx*,*<sup>n</sup>*,1, *Sx*,*n*}], (11)

**Table 1.** Some likelihood ratio error rate results for different finger/region combinations.

calculations were derived from these distribution estimates.

within and between finger comparisons.

*2.2.2. Radial Triangulation FV Model: I*

**Figure 5.** Delaunay triangulation (**left**) and radial triangulation (**right**) differences for a configuration of 7 minutiae. The blue point for the radial triangulation illustration represents the centroid (i.e., arithmetic mean of minutiae x-y coordinates).

The FVs can be decomposed into *continuous* and *discrete* components, representing the measurement based and count/categorical features, respectively. Thus, the likelihood ratio is rewritten as:

$$LR = \underbrace{\frac{P(\mathbf{x}\_{\mathcal{L}}, y\_{\mathcal{c}} | \mathbf{x}\_{d}, y\_{d}, H\_{0\prime}, I\_{\mathrm{cs}})}{P(\mathbf{x}\_{\mathcal{L}}, y\_{\mathcal{c}} | \mathbf{x}\_{d}, y\_{d}, H\_{A\prime}, I\_{\mathrm{cs}})}}\_{LR\_{\mathrm{c}|d}}.\underbrace{\frac{P(\mathbf{x}\_{d}, y\_{d} | H\_{0\prime}, I\_{\mathrm{cs}})}{P(\mathbf{x}\_{d}, y\_{d} | H\_{A\prime}, I\_{\mathrm{cs}})}}\_{LR\_{d}} = LR\_{\mathrm{c}|d}.LR\_{d}\tag{8}$$

where *LRd* is formed as a prior likelihood ratio with discrete FVs *xd* = [*GPx*, *Rx*, *Ntx*] and *yd* = *GPy*, *Ry*, *Nty* , while continuous FVs *xc* and *yc* contain then remaining features in *x* and *y*, respectively. The discrete likelihood numerator takes the value of 1, while the denominator was calculated using frequencies for general patterns multiplied by region and minutia-type combination probabilities observed from large datasets.

A dissimilarity metric, *d*(*xc*, *yc*), was created for comparing the continuous FV defined as:

$$d(\mathbf{x}\_{\mathcal{C}}, y\_{\mathcal{C}}) = \Delta^2 A\_1 + \Delta^2 L\_{1-2} + \Delta^2 A\_2 + \Delta^2 L\_{2-3} + \Delta^2 A\_3 + \Delta^2 L\_{3-1} \tag{9}$$

with ∆<sup>2</sup> as the squared difference of corresponding variables from *xc* and *yc*. This was used to calculate the continuous likelihood value, with:

$$LR\_{c|d} = \frac{P(d(\mathbf{x\_{c}}, y\_{c}) | \mathbf{x\_{d}}, y\_{d'} H\_{0'} I\_{\rm cs})}{P(d(\mathbf{x\_{c}}, y\_{c}) | \mathbf{x\_{d'}} y\_{d'} H\_{A'} I\_{\rm cs})}.\tag{10}$$

<sup>230</sup> New Trends and Developments in Biometrics An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA) 11 An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA) http://dx.doi.org/10.5772/51184 231


**Table 1.** Some likelihood ratio error rate results for different finger/region combinations.

Density functions of both *P*(*d*(*xc*, *yc*)|*xd*, *yd*, *H*0, *Ics*) and *P*(*d*(*xc*, *yc*)|*xd*, *yd*, *HA*, *Ics*) were estimated using a kernel smoothing method. All LR numerator and denominator likelihood calculations were derived from these distribution estimates.

Two experiments were configured in order to evaluate within-finger (i.e., genuine) and between-finger (i.e., imposter) LRs. Ideally, LRs for within-finger comparisons should be larger than all between-finger ratios. The within-finger experiment used 216 fingerprints from 4 different fingers under various different distortion levels. The between-finger datasets included the same 818 fingerprints used in the minutia-type probability calculations. Delaunay triangulation had to be manually adjusted in some cases due to different triangulation results occurring under high distortion levels. Error rates for LRs greater than 1 for false comparisons (i.e., between-finger) and LRs less than 1 for true comparisons (i.e., within-finger) for index, middle, and thumbs, are given in Table 1. These errors rates indicate the power that 3 minutiae (in each triangle) have in creating an LR value dichotomy between within and between finger comparisons.

#### *2.2.2. Radial Triangulation FV Model: I*

10 New Trends and Developments in Biometrics

is rewritten as:

*yd* =

*GPy*, *Ry*, *Nty*

**Delaunay Triangulation Radial Triangulation**

**Figure 5.** Delaunay triangulation (**left**) and radial triangulation (**right**) differences for a configuration of 7 minutiae. The blue point for the radial triangulation illustration represents the centroid (i.e., arithmetic mean of minutiae x-y coordinates).

The FVs can be decomposed into *continuous* and *discrete* components, representing the measurement based and count/categorical features, respectively. Thus, the likelihood ratio

where *LRd* is formed as a prior likelihood ratio with discrete FVs *xd* = [*GPx*, *Rx*, *Ntx*] and

*y*, respectively. The discrete likelihood numerator takes the value of 1, while the denominator was calculated using frequencies for general patterns multiplied by region and minutia-type

A dissimilarity metric, *d*(*xc*, *yc*), was created for comparing the continuous FV defined as:

with ∆<sup>2</sup> as the squared difference of corresponding variables from *xc* and *yc*. This was used

*LRc*<sup>|</sup>*<sup>d</sup>* <sup>=</sup> *<sup>P</sup>*(*d*(*xc*, *yc*)|*xd*, *yd*, *<sup>H</sup>*0, *Ics*)

*P*(*d*(*xc*, *yc*)|*xd*, *yd*, *HA*, *Ics*)

. *<sup>P</sup>*(*xd*, *yd*|*H*0, *Ics*) *P*(*xd*, *yd*|*HA*, *Ics*) *LRd*

, while continuous FVs *xc* and *yc* contain then remaining features in *x* and

*d*(*xc*, *yc*) = ∆2*A*<sup>1</sup> + ∆2*L*1−<sup>2</sup> + ∆2*A*<sup>2</sup> + ∆2*L*2−<sup>3</sup> + ∆2*A*<sup>3</sup> + ∆2*L*3−<sup>1</sup> (9)

= *LRc*<sup>|</sup>*d*.*LRd* (8)

. (10)

*LR* <sup>=</sup> *<sup>P</sup>*(*xc*, *yc*|*xd*, *yd*, *<sup>H</sup>*0, *Ics*) *P*(*xc*, *yc*|*xd*, *yd*, *HA*, *Ics*) *LRc*<sup>|</sup>*<sup>d</sup>*

combination probabilities observed from large datasets.

to calculate the continuous likelihood value, with:

Although the triangular structures of [33] performed reasonably well in producing higher LRs for within-finger comparisons against between-finger comparisons, there are issues with the proposed FV structure's robustness towards distortion. In addition, LRs could potentially have increased dichotomy between imposter and genuine comparisons by including more minutiae in the FV structures, rather than restricting each FV to only have three minutiae.

The authors of [34] defined *radial triangulation* FVs based on *n* minutiae *x* = [*GPx*, *xs*] with:

$$\begin{aligned} \mathbf{x}^{(\mathsf{n})} &= [\{T\_{\mathbf{x},\mathsf{l}}, \mathsf{R}A\_{\mathbf{x},\mathsf{l}}, \mathsf{R}\_{\mathbf{x},\mathsf{l}}, L\_{\mathbf{x},\mathsf{l},\mathsf{2}}, \mathsf{S}\_{\mathbf{x},\mathsf{l}}\}\_{\mathsf{l}}\{T\_{\mathbf{x},\mathsf{2}}, \mathsf{R}A\_{\mathbf{x},\mathsf{2}}, R\_{\mathbf{x},\mathsf{2}}, L\_{\mathbf{x},\mathsf{2},\mathsf{3}}, \mathsf{S}\_{\mathbf{x},\mathsf{2}}\}\_{\mathsf{l}} \\ &\dots , \{T\_{\mathbf{x},\mathsf{n}}, \mathsf{R}A\_{\mathbf{x},\mathsf{n}}, R\_{\mathbf{x},\mathsf{n}}L\_{\mathbf{x},\mathsf{n},\mathsf{l}}, S\_{\mathbf{x},\mathsf{n}}\} \}\_{\mathsf{l}} \end{aligned} \tag{11}$$

(and similarly for *<sup>y</sup>* and *<sup>y</sup>*(*n*)), where *GP* denotes the general pattern, *Tk* is the minutia type, *RAk* is the direction of minutia *k* relative to the image, *Rk* is the radius from the *kth* minutia to the centroid (Figure 5 **right**), *Lk*,*k*+<sup>1</sup> is the length of the polygon side from minutia *k* to *k* + 1, and *Sk* is the area of the triangle defined by minutia *k*, (*k* + 1) mod *n*, and the centroid. The LR was then calculated as

$$LR = \underbrace{\frac{P(\mathbf{x}^{(n)}, \mathbf{y}^{(n)} | GP\_{\mathbf{x}}, GP\_{\mathbf{y}}, H\_{\mathbf{0}}, I\_{\mathbf{G}})}{P(\mathbf{x}^{(n)}, \mathbf{y}^{(n)} | GP\_{\mathbf{x}}, GP\_{\mathbf{y}}, H\_{A\mathbf{A}}, I\_{\mathbf{G}})}}\_{LR\_{\mathbf{x}|\mathbf{g}}} \cdot \underbrace{\frac{P(GP\_{\mathbf{x}}, GP\_{\mathbf{y}} | H\_{\mathbf{0}}, I\_{\mathbf{G}})}{P(GP\_{\mathbf{x}}, GP\_{\mathbf{y}} | H\_{A\mathbf{A}}, I\_{\mathbf{G}})}}\_{LR\_{\mathbf{g}}} = LR\_{\mathbf{n}|\mathbf{g}}.\tag{12}$$

*y* (*n*)

distance between the *j*

constituted by the *j*

where *x*(*n*)(*δj*) (and *y*

where

with

*z* (*k*)

and *τ<sup>j</sup>* is the type of the *j*

for a configuration *y*(*n*) starting from the *i*

*<sup>i</sup>* = ({*δj*, *σj*, *θj*, *αj*, *τj*}, *i* = *j*,(*j* + 1) mod *n*,...,(*j* − 1) mod *n*),

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

minutia and the next contiguous minutia (in a clockwise direction), *θ<sup>j</sup>* is the angle between the direction of a minutia and the line from the centroid point, *α<sup>j</sup>* is the area of the triangle

The distance between configurations *x*(*n*) and *y*(*n*), each representing *n* minutiae, is

) = min *i*=1,...,*n*

*d*(*x*(*n*)

<sup>∆</sup>*<sup>j</sup>* <sup>=</sup> *<sup>q</sup>δ*.(*x*(*n*)

(*n*)

*qτ*) are tuned via a heuristic based procedure. The proposed LR calculation makes use of:

labelling minutiae in fingerprint images.

Let *y*(*k*) be the configuration of a fingermark, *x*

*<sup>i</sup>*,min the closest configuration for the *i*

focused on finger variability,

<sup>+</sup>*q<sup>θ</sup>* .*d<sup>θ</sup>* (*x*(*n*)

, *y*(*n*)

*dc*(*x*(*n*) , *y* (*n*) *<sup>i</sup>* ) =

(*n*)

(*δj*) − *y*

(*θj*), *y* (*n*)

*th* minutia (ridge ending, bifurcation, unknown).

*th* minutia and the centroid point, *σ<sup>j</sup>* is the distance between the *j*

*th* minutia, the next contiguous minutia and the centre of the polygon,

*dc*(*x*(*n*) , *y* (*n*)

*n* ∑ *j*=1

+*qτ*.*dT*(*x*(*n*)

(*σj*) − *y*

(*αj*) − *y*

(*τj*), *y* (*n*) *<sup>i</sup>* (*τj*))<sup>2</sup>

(*n*) *<sup>i</sup>* (*σj*))<sup>2</sup>

(*n*) *<sup>i</sup>* (*αj*))<sup>2</sup>

min the closest *k* configuration found, and

*th* member of a reference database containing *N*

*<sup>i</sup>* (*δj*))<sup>2</sup> <sup>+</sup> *<sup>q</sup>σ*.(*x*(*n*)

*<sup>i</sup>* (*θj*))<sup>2</sup> <sup>+</sup> *<sup>q</sup>α*.(*x*(*n*)

*<sup>i</sup>* (*δj*)) is the normalised value for *δ* for the *j*

for all other normalised vector components *σ*, *θ*, *α*, and *τ*, while *d<sup>θ</sup>* is the angular difference and *dT* is the defined minutiae type difference metric. The multipliers (i.e., *qδ*, *qσ*, *q<sup>θ</sup>* , *qα*, and

• distortion model: based on the Thin Plate Spline (TPS) bending energy matrices representing the non-affine differences of minutiae spatial detail trained from a dataset

• examiner influence model: created to represent the variability of examiners when

impressions. Synthetic FVs can be generated from minute modifications to minutiae locations

(*k*)

*th* minutia, for *i* = 1, 2, . . . , *n*, where *δ<sup>j</sup>* is the

*<sup>i</sup>* ) (15)

http://dx.doi.org/10.5772/51184

<sup>∆</sup>*<sup>j</sup>* (16)

*th* minutiae, and likewise

*th*

233

(17)

The component *LRg* is formed as a prior likelihood with *P*(*GPx*, *GPy*|*H*0, *Ics*) = 1 and *P*(*GPx*, *GPy*|*HA*, *Ics*) equal to the FBI pattern frequency data. Noting that the centroid FVs can be arranged in *n* different ways (accounting for clockwise rotation):

$$\begin{aligned} y\_j^{(n)} &= (\{T\_{y,k\prime} R A\_{y,k\prime} R\_{y,k\prime} L\_{y,k,(k+1)\bmod n\prime} S\_{y,k}\}, \\ k &= j\_\prime(j+1) \bmod n\_\prime \dots (j-1) \bmod n\}, \end{aligned}$$

for *j* = 1, 2, . . . , *n*, *LRn*<sup>|</sup>*<sup>g</sup>* was defined as

$$LR\_{n|g} = \frac{P(d(\mathbf{x}^{(n)}, \mathbf{y}^{(n)}) | GP\_{\mathbf{x}\prime} GP\_{\mathbf{y}\prime} H\_{0\prime} I\_{\text{cs}})}{P(d(\mathbf{x}^{(n)}, \mathbf{y}^{(n)}) | GP\_{\mathbf{x}\prime} GP\_{\mathbf{y}\prime} H\_{A\prime} I\_{\text{cs}})} \tag{13}$$

where the dissimilarity metric is

$$d(\mathfrak{x}^{(n)}, \mathfrak{y}^{(n)}) = \min\_{i=1,\ldots,n} d(\mathfrak{x}^{(n)}, \mathfrak{y}\_i^{(n)}).\tag{14}$$

The calculation of each of the *d*(*x*(*n*), *y* (*n*) *<sup>i</sup>* ) is the Euclidean distance of respective FVs which are normalised to take a similar range of values. The two conditional probability density functions of *P*(*d*(*x*(*n*), *y*(*n*))|*GPx*, *GPy*, *H*0, *Ics*) and *P*(*d*(*x*(*n*), *y*(*n*))|*GPx*, *GPy*, *HA*, *Ics*) were estimated using mixture models of normal distributions with a mixture of three and four distributions, respectfully, using the EM algorithm to estimate distributions for each finger and number of minutiae used.

This method modelled within and between finger variability more accurately in comparison to the earlier related work in [33], due to the flexibility of the centroid structures containing more than three minutiae. For example, the addition of one extra minutia halved the LR error rate for some fingerprint patterns. In addition, the prior likelihood is more flexible in real life applications as it is not dependent on identifying the specific fingerprint region (which is more robust for real life fingermark-to-exemplar comparisons).

#### *2.2.3. Radial Triangulation FV Model: II*

The authors of [35] proposed a FV based LR model using radial triangulation structures. In addition, they tuned the model using distortion and examination influence models. The radial triangulation FVs used were based on the structures defined in [34], where five features are stored per minutia, giving

$$y\_j^{(n)} = (\{\delta\_{j'} \sigma\_{j'} \theta\_{j'} \mathfrak{a}\_{j'} \tau\_j\}, i = j, (j+1) \bmod n, \dots, (j-1) \bmod n), j$$

for a configuration *y*(*n*) starting from the *i th* minutia, for *i* = 1, 2, . . . , *n*, where *δ<sup>j</sup>* is the distance between the *j th* minutia and the centroid point, *σ<sup>j</sup>* is the distance between the *j th* minutia and the next contiguous minutia (in a clockwise direction), *θ<sup>j</sup>* is the angle between the direction of a minutia and the line from the centroid point, *α<sup>j</sup>* is the area of the triangle constituted by the *j th* minutia, the next contiguous minutia and the centre of the polygon, and *τ<sup>j</sup>* is the type of the *j th* minutia (ridge ending, bifurcation, unknown).

The distance between configurations *x*(*n*) and *y*(*n*), each representing *n* minutiae, is

$$d(\mathbf{x}^{(n)}, y^{(n)}) = \min\_{i=1,\ldots,n} d\_c(\mathbf{x}^{(n)}, y\_i^{(n)}) \tag{15}$$

where

12 New Trends and Developments in Biometrics

The LR was then calculated as

*LR* <sup>=</sup> *<sup>P</sup>*(*x*(*n*), *<sup>y</sup>*(*n*)|*GPx*, *GPy*, *<sup>H</sup>*0, *Ics*) *P*(*x*(*n*), *y*(*n*)|*GPx*, *GPy*, *HA*, *Ics*) *LRn*<sup>|</sup>*<sup>g</sup>*

> *y* (*n*)

for *j* = 1, 2, . . . , *n*, *LRn*<sup>|</sup>*<sup>g</sup>* was defined as

where the dissimilarity metric is

The calculation of each of the *d*(*x*(*n*), *y*

finger and number of minutiae used.

*2.2.3. Radial Triangulation FV Model: II*

are stored per minutia, giving

can be arranged in *n* different ways (accounting for clockwise rotation):

*d*(*x*(*n*)

, *y*(*n*)

(which is more robust for real life fingermark-to-exemplar comparisons).

.

The component *LRg* is formed as a prior likelihood with *P*(*GPx*, *GPy*|*H*0, *Ics*) = 1 and *P*(*GPx*, *GPy*|*HA*, *Ics*) equal to the FBI pattern frequency data. Noting that the centroid FVs

*<sup>j</sup>* = ({*Ty*,*k*, *RAy*,*k*, *Ry*,*k*, *Ly*,*k*,(*k*+1) mod *<sup>n</sup>*, *Sy*,*k*},

*LRn*<sup>|</sup>*<sup>g</sup>* <sup>=</sup> *<sup>P</sup>*(*d*(*x*(*n*), *<sup>y</sup>*(*n*))|*GPx*, *GPy*, *<sup>H</sup>*0, *Ics*)

) = min *i*=1,...,*n*

which are normalised to take a similar range of values. The two conditional probability density functions of *P*(*d*(*x*(*n*), *y*(*n*))|*GPx*, *GPy*, *H*0, *Ics*) and *P*(*d*(*x*(*n*), *y*(*n*))|*GPx*, *GPy*, *HA*, *Ics*) were estimated using mixture models of normal distributions with a mixture of three and four distributions, respectfully, using the EM algorithm to estimate distributions for each

This method modelled within and between finger variability more accurately in comparison to the earlier related work in [33], due to the flexibility of the centroid structures containing more than three minutiae. For example, the addition of one extra minutia halved the LR error rate for some fingerprint patterns. In addition, the prior likelihood is more flexible in real life applications as it is not dependent on identifying the specific fingerprint region

The authors of [35] proposed a FV based LR model using radial triangulation structures. In addition, they tuned the model using distortion and examination influence models. The radial triangulation FVs used were based on the structures defined in [34], where five features

(*n*)

*d*(*x*(*n*) , *y* (*n*)

*k* = *j*,(*j* + 1) mod *n*,...,(*j* − 1) mod *n*),

*P*(*GPx*, *GPy*|*H*0, *Ics*) *P*(*GPx*, *GPy*|*HA*, *Ics*) *LRg*

*<sup>P</sup>*(*d*(*x*(*n*), *<sup>y</sup>*(*n*))|*GPx*, *GPy*, *HA*, *Ics*) (13)

*<sup>i</sup>* ) is the Euclidean distance of respective FVs

*<sup>i</sup>* ). (14)

= *LRn*<sup>|</sup>*g*.*LRg* (12)

$$d\_{\mathbb{C}}(\mathfrak{x}^{(n)}, \mathfrak{y}\_{i}^{(n)}) = \sum\_{j=1}^{n} \Delta\_{j} \tag{16}$$

with

$$\begin{split} \Delta\_{\hat{j}} = q\_{\delta^\*} (\mathbf{x}^{(n)}(\delta\_{\hat{j}}) - y\_i^{(n)}(\delta\_{\hat{j}}))^2 + q\_{\sigma^\*} (\mathbf{x}^{(n)}(\sigma\_{\hat{j}}) - y\_i^{(n)}(\sigma\_{\hat{j}}))^2 \\ + q\_{\theta^\*} d\_{\theta} (\mathbf{x}^{(n)}(\theta\_{\hat{j}}), y\_i^{(n)}(\theta\_{\hat{j}}))^2 + q\_{a^\*} (\mathbf{x}^{(n)}(\mathbf{a}\_{\hat{j}}) - y\_i^{(n)}(\mathbf{a}\_{\hat{j}}))^2 \\ + q\_{\tau^\*} d\_T (\mathbf{x}^{(n)}(\tau\_{\hat{j}}), y\_i^{(n)}(\tau\_{\hat{j}}))^2 \end{split} \tag{17}$$

where *x*(*n*)(*δj*) (and *y* (*n*) *<sup>i</sup>* (*δj*)) is the normalised value for *δ* for the *j th* minutiae, and likewise for all other normalised vector components *σ*, *θ*, *α*, and *τ*, while *d<sup>θ</sup>* is the angular difference and *dT* is the defined minutiae type difference metric. The multipliers (i.e., *qδ*, *qσ*, *q<sup>θ</sup>* , *qα*, and *qτ*) are tuned via a heuristic based procedure.

The proposed LR calculation makes use of:


Let *y*(*k*) be the configuration of a fingermark, *x* (*k*) min the closest *k* configuration found, and *z* (*k*) *<sup>i</sup>*,min the closest configuration for the *i th* member of a reference database containing *N* impressions. Synthetic FVs can be generated from minute modifications to minutiae locations represented by a given FV, via Monte-Carlo simulation of both distortion and examiner influence models. A set of *M* synthetic FVs are created for *x* (*k*) min ({*ζ* (*k*) <sup>1</sup> ,..., *ζ* (*k*) *<sup>M</sup>* }) and for each *z* (*k*) *<sup>i</sup>*,min ({*ζ* (*k*) *<sup>i</sup>*,1 ,..., *ζ* (*k*) *<sup>i</sup>*,*M*}), from which the LR is given as

$$LR = \frac{N\sum\_{i=1}^{M} \psi\left(d(y^{(k)}, \zeta\_i^{(k)})\right)}{\sum\_{i=1}^{N} \sum\_{j=1}^{M} \psi\left(d(y^{(k)}, \zeta\_{i,j}^{(k)})\right)}\tag{18}$$

The LR models proposed in [33] and [34] use dissimilarity measures of FVs (equations (9) and (14)) which are potentially erroneous as minutiae types can change, particularly in distorted impressions. While the method in [35] has clearly improved the dissimilarity function by introducing tuned multipliers, squared differences in angle, area, and distance based measures are ultimately not probabilistically based. A joint probabilistic based metric for each FV component using distributions for both imposter and genuine populations would

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

http://dx.doi.org/10.5772/51184

235

With regards to skin distortion, the radial triangulation FV structures of [34] [35] are robust, unlike the Delaunay triangulation structure of [33]. Furthermore, the model proposed in [35] models realistic skin distortion encountered on flat surfaces by measuring the bending energy matrix for a specialised distortion set. However, this only accounts for the non-affine variation. Affine transformations such as shear and uniform compression/dilation are not accounted for. Such information can be particularly significant for comparisons of small minutiae configurations encountered in latent marks. For instance, a direct downward application of force may have prominent shear and scale variations (in addition to non-affine differences) for minutiae configurations, in comparison to the corresponding configurations of another impression from the same finger having no notable downward force applied.

**3. Proposed method: Morphometric and Spatial Analyses (MSA) based**

In this section, we present a newly formulated FV based LR model that focuses on the important sub-population of close non-matches (i.e., highly similar imposters), with intended practicality for fingermark-to-exemplar identification scenarios where only sparse minutiae triplet information may be available for comparisons. First we discuss relevant background material concerning morphometric and spatial measures to be used in the FVs of the proposed model. The proposed model is presented, which is based on a novel machine learning framework, followed by a proposed LR calculation that focuses on the candidate list population of an AFIS match query (i.e., containing close non-match exemplars and/or a matching exemplar). Finally, an experimental framework centred around the simulation of fingermark-to-exemplar close non-match discovery is introduced, followed by experimental

The foundations of the morphometric and spatial analyses used in the proposed FV based LR model are presented. This includes a non-parametric multidimensional goodness-of-fit statistic, along with several other morphometrical measures that describe and contrast shape characteristics between two given configurations. In addition, a method for finding close

A general multidimensional Kolmogorov-Smirnov (KS) statistic for two empirical distributions has been proposed in [39] with properties of high efficiency, high statistical power, and distributional freeness. Like the classic one dimensional KS test, the multidimensional variant looks for the largest absolute difference between the empirical

be more consistent with the overall LR framework.

**Likelihood Ratio model**

**3.1. Morphometric and spatial metrics**

non-match minutiae configurations is presented.

*3.1.1. Multidimensional Kolmogorov-Smirnov Statistic for Landmarks*

results.

where *ψ* is defined as

$$\psi(d(y^{(k)},\bullet)) = \exp\left(\frac{-\lambda\_1 d(y^{(k)},\bullet)}{T^{(k)}}\right) + \frac{B(d(y^{(k)},\bullet),\lambda\_2 k)}{B(d\_0,\lambda\_2 k)}\tag{19}$$

which is a mixture of Exponential and Beta functions with tuned parameters *λ*<sup>1</sup> and *λ*2, while *d*<sup>0</sup> is the smallest value into which distances were binned, and *T*(*k*) is the 95th percentile of simulated scores from the examiner influence model applied on *y*(*k*). Experimental results from a large validation dataset showed that the proposed LR model can generally distinguish within and between finger comparisons with high accuracy, while an increased dichotomy arose from increasing the configuration size.

#### *2.2.4. Practicality of FV based LR Models*

Generally speaking, to implement robust FV based statistical models for forensic applications, the following must be considered:


The LR models proposed in [33] and [34] use dissimilarity measures of FVs (equations (9) and (14)) which are potentially erroneous as minutiae types can change, particularly in distorted impressions. While the method in [35] has clearly improved the dissimilarity function by introducing tuned multipliers, squared differences in angle, area, and distance based measures are ultimately not probabilistically based. A joint probabilistic based metric for each FV component using distributions for both imposter and genuine populations would be more consistent with the overall LR framework.

With regards to skin distortion, the radial triangulation FV structures of [34] [35] are robust, unlike the Delaunay triangulation structure of [33]. Furthermore, the model proposed in [35] models realistic skin distortion encountered on flat surfaces by measuring the bending energy matrix for a specialised distortion set. However, this only accounts for the non-affine variation. Affine transformations such as shear and uniform compression/dilation are not accounted for. Such information can be particularly significant for comparisons of small minutiae configurations encountered in latent marks. For instance, a direct downward application of force may have prominent shear and scale variations (in addition to non-affine differences) for minutiae configurations, in comparison to the corresponding configurations of another impression from the same finger having no notable downward force applied.

## **3. Proposed method: Morphometric and Spatial Analyses (MSA) based Likelihood Ratio model**

In this section, we present a newly formulated FV based LR model that focuses on the important sub-population of close non-matches (i.e., highly similar imposters), with intended practicality for fingermark-to-exemplar identification scenarios where only sparse minutiae triplet information may be available for comparisons. First we discuss relevant background material concerning morphometric and spatial measures to be used in the FVs of the proposed model. The proposed model is presented, which is based on a novel machine learning framework, followed by a proposed LR calculation that focuses on the candidate list population of an AFIS match query (i.e., containing close non-match exemplars and/or a matching exemplar). Finally, an experimental framework centred around the simulation of fingermark-to-exemplar close non-match discovery is introduced, followed by experimental results.

## **3.1. Morphometric and spatial metrics**

14 New Trends and Developments in Biometrics

(*k*) *<sup>i</sup>*,1 ,..., *ζ*

where *ψ* is defined as

(*k*)

*ψ*(*d*(*y*(*k*)

arose from increasing the configuration size.

applications, the following must be considered:

occur when different distortion exists.

*2.2.4. Practicality of FV based LR Models*

finger data is essential.

each *z* (*k*) *<sup>i</sup>*,min ({*ζ*

represented by a given FV, via Monte-Carlo simulation of both distortion and examiner

*<sup>i</sup>*,*M*}), from which the LR is given as

*<sup>i</sup>*=<sup>1</sup> *ψ d*(*y*(*k*), *ζ*

−*λ*1*d*(*y*(*k*), •) *T*(*k*)

which is a mixture of Exponential and Beta functions with tuned parameters *λ*<sup>1</sup> and *λ*2, while *d*<sup>0</sup> is the smallest value into which distances were binned, and *T*(*k*) is the 95th percentile of simulated scores from the examiner influence model applied on *y*(*k*). Experimental results from a large validation dataset showed that the proposed LR model can generally distinguish within and between finger comparisons with high accuracy, while an increased dichotomy

Generally speaking, to implement robust FV based statistical models for forensic

• Any quantitative measures used should be based on the data driven discovery of statistical relationships of features. Thus, a rich dataset for both within and between

• Effects of skin distortion must be considered in models. Latent marks can be highly distorted from skin elasticity and applied pressure. For instance, differences in both minutiae location (relative to other features) and type (also known as type transfer) can

• Features used in models must be robust to noisy environmental factors, whilst maintaining a high level of discriminatory power. For instance, level 1 features such as classification may not be available due to partiality. In addition, level 2 sub-features such as ridge count between minutiae, minutiae type, and level 3 features such as pores, may not be available in a latent mark due to the material properties of the contacted medium

• The model should be robust towards reasonable variations in feature markings from practitioners in the analysis phase of ACE-V. For instance, minutiae locations can vary

or other environmental noise that regularly exist in latent mark occurrences.

slightly depending on where a particular practitioner marks a given minutia.

 +

*LR* <sup>=</sup> *<sup>N</sup>* <sup>∑</sup>*<sup>M</sup>*

∑*N <sup>i</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>M</sup> <sup>j</sup>*=<sup>1</sup> *ψ d*(*y*(*k*), *ζ*

(*k*) min ({*ζ*

*B*(*d*(*y*(*k*), •), *λ*2*k*)

(*k*) *<sup>i</sup>* ) 

> (*k*) *<sup>i</sup>*,*<sup>j</sup>* )

(*k*) <sup>1</sup> ,..., *ζ*

(*k*)

(18)

*<sup>B</sup>*(*d*0, *<sup>λ</sup>*2*k*) (19)

*<sup>M</sup>* }) and for

influence models. A set of *M* synthetic FVs are created for *x*

, •)) = exp

The foundations of the morphometric and spatial analyses used in the proposed FV based LR model are presented. This includes a non-parametric multidimensional goodness-of-fit statistic, along with several other morphometrical measures that describe and contrast shape characteristics between two given configurations. In addition, a method for finding close non-match minutiae configurations is presented.

## *3.1.1. Multidimensional Kolmogorov-Smirnov Statistic for Landmarks*

A general multidimensional Kolmogorov-Smirnov (KS) statistic for two empirical distributions has been proposed in [39] with properties of high efficiency, high statistical power, and distributional freeness. Like the classic one dimensional KS test, the multidimensional variant looks for the largest absolute difference between the empirical

and cumulative distribution functions, as a measure of fit. Without losing generality, let two sets with *m* and *n* points in **R**<sup>3</sup> be denoted as *X* = {(*x*1, *y*1, *z*1),...,(*xm*, *ym*, *zm*)} and *Y* = {(*x*′ <sup>1</sup>, *<sup>y</sup>*′ 1, *z*′ 1),...,(*x*′ *<sup>n</sup>*, *y*′ *<sup>n</sup>*, *z*′ *<sup>n</sup>*)}, respectively. For each point (*xi*, *yi*, *zi*) ∈ *X* we can divide the plane into eight defined regions

$$\begin{aligned} q\_{i,1} &= \{ (\mathbf{x}, y\_\prime z) | \mathbf{x} < \mathbf{x}\_{i\prime} y < y\_{i\prime} z < z\_i \}, \\ q\_{i,2} &= \{ (\mathbf{x}, y\_\prime z) | \mathbf{x} < \mathbf{x}\_{i\prime} y < y\_{i\prime} z > z\_i \}, \\ &\vdots \\ q\_{i,8} &= \{ (\mathbf{x}, y\_\prime z) | \mathbf{x} \ge \mathbf{x}\_{i\prime} y \ge y\_{i\prime} z \ge z\_i \}, \end{aligned}$$

and similarly for each (*x*′ *j* , *y*′ *j* , *z*′ *j* ) ∈ *Y*,

$$\begin{aligned} q'\_{j,1} &= \{ (\mathbf{x}, y, z) | \mathbf{x} < \mathbf{x}'\_{i\prime} y < y'\_{i\prime} z < z'\_i \}, \\ q'\_{j,2} &= \{ (\mathbf{x}, y, z) | \mathbf{x} < \mathbf{x}'\_{i\prime} y < y'\_{i\prime} z > z'\_i \}, \\ &\vdots \\ q'\_{j,8} &= \{ (\mathbf{x}, y, z) | \mathbf{x} \ge \mathbf{x}'\_j y \ge y'\_j z \ge z'\_j \}. \end{aligned}$$

Further defining

$$D\_m = \max\_{\substack{i=1,\ldots,m\\s=1,\ldots,8}} |\left| X \cap q\_{i,s} \right| - \left| Y \cap q\_{i,s} \right| \; | \tag{20}$$

*<sup>z</sup>* <sup>=</sup> *<sup>z</sup>*(*θ*, *<sup>θ</sup>*0) = *<sup>π</sup>*

where *z* ∈ [−*<sup>π</sup>*

<sup>2</sup> , *<sup>π</sup>*

in the third dimension by *z* ≥ 0 and *z* < 0.

*3.1.2. Thin Plate Spline and Derived Measures*

to be invariant under both rotation and scale.

from an input image in **R**<sup>2</sup> and control points

� **p**′ <sup>1</sup> = (*x*′ <sup>1</sup>, *<sup>y</sup>*′ <sup>1</sup>), **<sup>p</sup>**′

Given *n* control points

**P** =

by the equation

 

1 *x*<sup>1</sup> *y*<sup>1</sup> 1 *x*<sup>2</sup> *y*<sup>2</sup> ... ... ... 1 *xn yn*

 , **<sup>V</sup>** <sup>=</sup> <sup>2</sup> <sup>−</sup> min(2*<sup>π</sup>* − |*<sup>θ</sup>* <sup>−</sup> *<sup>θ</sup>*0|, <sup>|</sup>*<sup>θ</sup>* <sup>−</sup> *<sup>θ</sup>*0|) (23)

http://dx.doi.org/10.5772/51184

237

<sup>2</sup> ]. Each minutia, (*x*, *y*, *θ*), is then transformed to (*x*, *y*, *z*(*θ*, *θ*0)) if the centred

minutia used to create the eight regions has a direction of *θ*0, while region borders are defined

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

The Thin Plate Spline (TPS) [40] is based on the algebraic expression of physical bending energy of an infinitely thin metal plate on point constraints after finding the optimal affine transformations for the accurate modelling of surfaces that undergo *natural warping* (i.e., where a diffeomorphism exists). Two sets of landmarks from each surface are paired in order to provide an interpolation map on **R**<sup>2</sup> → **R**2. TPS decomposes the interpolation into an affine transform that can be considered as the transformation that expresses the global geometric dependence of the point sets, and a non-affine transform that fine tunes the interpolation of the point sets. The inclusion of the affine transform component allows TPS

{**p**<sup>1</sup> = (*x*1, *y*1), **p**<sup>2</sup> = (*x*2, *y*2),..., **p***<sup>n</sup>* = (*xn*, *yn*)}

<sup>2</sup>),..., **<sup>p</sup>**′

0 *u*(*r*12) ... *u*(*r*1*n*) *u*(*r*21) 0 ... *u*(*r*2*n*) ... ... ... ... *u*(*rn*1) *u*(*rn*2) ... 0

where **K**, **P**, **V**, **Y**, **L** have dimensions *n* × *n*, 3 × *n*, 2 × *n*, (*n* + 3) × 2, and (*n* + 3) × (*n* + 3), respectively. The vector *W* = (*w*1, *w*2,..., *wn*) and the coefficients *a*1, *ax*, *ay*, can be calculated

*<sup>n</sup>* = (*x*′

 ,

**V 0**2×<sup>3</sup>

�*<sup>T</sup>* , **<sup>L</sup>** =

**<sup>L</sup>**−1**<sup>Y</sup>** = (**W**| *<sup>a</sup>*<sup>1</sup> *ax ay*)*T*. (24)

� **K P <sup>P</sup><sup>T</sup> <sup>0</sup>**3×<sup>3</sup> � ,

*<sup>n</sup>*, *<sup>y</sup>*′ *n*) �

<sup>2</sup> = (*x*′ 2, *<sup>y</sup>*′

from a target image **R**2, the following matrices are defined in TPS:

 

where *u*(*r*) = *r*<sup>2</sup> log *r*<sup>2</sup> with *r* as the Euclidean distance, *rij* = �*pi* − *pj*�,

� *x*′ 1 *x*′ <sup>2</sup> ... *<sup>x</sup>*′ *n*

*y*′ 1 *y*′ <sup>2</sup> ... *<sup>y</sup>*′ *n* � , **Y** = �

**K** =

which is the maximum pairwise difference of point tallies for *X* and *Y* within each of the eight defined regions centred and evaluated at each point in *X*, and likewise,

$$D\_n = \max\_{\substack{j=1,\ldots,n\\s=1,\ldots,8}} |\lfloor X \cap q\_{j,s}' \rfloor - |Y \cap q\_{j,s}'| \ | \tag{21}$$

which is the maximum pairwise difference of point tallies for the eight defined regions centred and evaluated at each point in *Y*, the three dimensional KS statistic is

$$Z\_{m,n,\Im D} = \sqrt{n.m / (n+m)} \cdot \left(\frac{D\_m + D\_n}{2}\right) \,. \tag{22}$$

The three dimensional KS statistic can be specific to the minutiae triplet space where each minutia spatial and directional detail is represented as a three dimensional point, (*x*, *y*, *θ*). Given *m* = *n* matching minutiae correspondences from two configurations *X* and *Y*, alignment is performed prior to calculating the statistic, in order to ensure that minutiae correspondences are close together both spatially and directionally. However, direction has a circular nature that must be handled differently from the spatial detail. Instead of raw angular values, we use the orientation difference defined as

<sup>236</sup> New Trends and Developments in Biometrics An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA) 17 An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA) http://dx.doi.org/10.5772/51184 237

$$z = z(\theta, \theta\_0) = \frac{\pi}{2} - \min(2\pi - |\theta - \theta\_0|, |\theta - \theta\_0|) \tag{23}$$

where *z* ∈ [−*<sup>π</sup>* <sup>2</sup> , *<sup>π</sup>* <sup>2</sup> ]. Each minutia, (*x*, *y*, *θ*), is then transformed to (*x*, *y*, *z*(*θ*, *θ*0)) if the centred minutia used to create the eight regions has a direction of *θ*0, while region borders are defined in the third dimension by *z* ≥ 0 and *z* < 0.

#### *3.1.2. Thin Plate Spline and Derived Measures*

The Thin Plate Spline (TPS) [40] is based on the algebraic expression of physical bending energy of an infinitely thin metal plate on point constraints after finding the optimal affine transformations for the accurate modelling of surfaces that undergo *natural warping* (i.e., where a diffeomorphism exists). Two sets of landmarks from each surface are paired in order to provide an interpolation map on **R**<sup>2</sup> → **R**2. TPS decomposes the interpolation into an affine transform that can be considered as the transformation that expresses the global geometric dependence of the point sets, and a non-affine transform that fine tunes the interpolation of the point sets. The inclusion of the affine transform component allows TPS to be invariant under both rotation and scale.

Given *n* control points

16 New Trends and Developments in Biometrics

1),...,(*x*′

the plane into eight defined regions

*<sup>n</sup>*, *y*′ *<sup>n</sup>*, *z*′

> *j* , *y*′ *j* , *z*′ *j* ) ∈ *Y*,

> > *q*′

*q*′

. . . *q*′

. . .

*Y* = {(*x*′

<sup>1</sup>, *<sup>y</sup>*′ 1, *z*′

and similarly for each (*x*′

Further defining

and cumulative distribution functions, as a measure of fit. Without losing generality, let two sets with *m* and *n* points in **R**<sup>3</sup> be denoted as *X* = {(*x*1, *y*1, *z*1),...,(*xm*, *ym*, *zm*)} and

> *qi*,1 = {(*x*, *y*, *z*)|*x* < *xi*, *y* < *yi*, *z* < *zi*}, *qi*,2 = {(*x*, *y*, *z*)|*x* < *xi*, *y* < *yi*, *z* > *zi*},

> *qi*,8 = {(*x*, *y*, *z*)|*x* ≥ *xi*, *y* ≥ *yi*, *z* ≥ *zi*},

*<sup>i</sup>*, *y* < *y*′ *i* , *z* < *z*′ *i*},

*<sup>i</sup>*, *y* < *y*′ *i* , *z* > *z*′ *i*},

*j* , *y* ≥ *y*′ *j* , *z* ≥ *z*′ *j* }.

which is the maximum pairwise difference of point tallies for *X* and *Y* within each of the


which is the maximum pairwise difference of point tallies for the eight defined regions

*n*.*m*/(*n* + *m*).

The three dimensional KS statistic can be specific to the minutiae triplet space where each minutia spatial and directional detail is represented as a three dimensional point, (*x*, *y*, *θ*). Given *m* = *n* matching minutiae correspondences from two configurations *X* and *Y*, alignment is performed prior to calculating the statistic, in order to ensure that minutiae correspondences are close together both spatially and directionally. However, direction has a circular nature that must be handled differently from the spatial detail. Instead of raw

*<sup>j</sup>*,*s*|−|*<sup>Y</sup>* <sup>∩</sup> *<sup>q</sup>*′

 *Dm* + *Dn* 2

*<sup>j</sup>*,1 <sup>=</sup> {(*x*, *<sup>y</sup>*, *<sup>z</sup>*)|*<sup>x</sup>* <sup>&</sup>lt; *<sup>x</sup>*′

*<sup>j</sup>*,2 <sup>=</sup> {(*x*, *<sup>y</sup>*, *<sup>z</sup>*)|*<sup>x</sup>* <sup>&</sup>lt; *<sup>x</sup>*′

*<sup>j</sup>*,8 <sup>=</sup> {(*x*, *<sup>y</sup>*, *<sup>z</sup>*)|*<sup>x</sup>* <sup>≥</sup> *<sup>x</sup>*′

eight defined regions centred and evaluated at each point in *X*, and likewise,

centred and evaluated at each point in *Y*, the three dimensional KS statistic is

*Dm* = max *<sup>i</sup>*=1,...,*<sup>m</sup> <sup>s</sup>*=1,...,8

*Dn* = max *j*=1,...,*n s*=1,...,8

*Zm*,*n*,3*<sup>D</sup>* =

angular values, we use the orientation difference defined as

*<sup>n</sup>*)}, respectively. For each point (*xi*, *yi*, *zi*) ∈ *X* we can divide


*<sup>j</sup>*,*s*| | (21)

. (22)

$$\{\mathbf{p}\_1 = (\mathbf{x}\_{1\prime} y\_1)\_{\prime} \mathbf{p}\_2 = (\mathbf{x}\_{2\prime} y\_2)\_{\prime} \dots \mathbf{p}\_n = (\mathbf{x}\_{n\prime} y\_n)\}$$

from an input image in **R**<sup>2</sup> and control points

$$\{\mathbf{p}'\_1 = (\mathbf{x}'\_{1'}, y'\_1), \mathbf{p}'\_2 = (\mathbf{x}'\_{2'} y'\_2), \dots, \mathbf{p}'\_n = (\mathbf{x}'\_{n'} y'\_n)\}$$

from a target image **R**2, the following matrices are defined in TPS:

$$\mathbf{K} = \begin{bmatrix} 0 & \boldsymbol{\mu}(r\_{12}) & \dots \boldsymbol{\mu}(r\_{1n}) \\ \boldsymbol{\mu}(r\_{21}) & 0 & \dots \boldsymbol{\mu}(r\_{2n}) \\ \dots & \dots & \dots & \dots \\ \boldsymbol{\mu}(r\_{n1}) \ \boldsymbol{\mu}(r\_{n2}) & \dots & 0 \end{bmatrix}^{\prime}$$

where *u*(*r*) = *r*<sup>2</sup> log *r*<sup>2</sup> with *r* as the Euclidean distance, *rij* = �*pi* − *pj*�,

$$\mathbf{P} = \begin{bmatrix} 1 & \boldsymbol{\chi}\_1 \ \boldsymbol{y}\_1 \\ 1 & \boldsymbol{\chi}\_2 \ \boldsymbol{y}\_2 \\ \vdots & \cdots & \cdots \\ 1 & \boldsymbol{\chi}\_n \ \boldsymbol{y}\_n \end{bmatrix}, \mathbf{V} = \begin{bmatrix} \boldsymbol{\chi}\_1' \ \boldsymbol{x}\_2' \ \dots \ \boldsymbol{x}\_n' \\ \boldsymbol{y}\_1' \ \boldsymbol{y}\_2' \ \dots \ \boldsymbol{y}\_n' \end{bmatrix}, \mathbf{Y} = \begin{bmatrix} \mathbf{V} \ \mathbf{0}\_{2 \times 3} \end{bmatrix}^T, \mathbf{L} = \begin{bmatrix} \mathbf{K} & \mathbf{P} \\ \mathbf{P} \ \mathbf{0}\_{3 \times 3} \end{bmatrix}.$$

where **K**, **P**, **V**, **Y**, **L** have dimensions *n* × *n*, 3 × *n*, 2 × *n*, (*n* + 3) × 2, and (*n* + 3) × (*n* + 3), respectively. The vector *W* = (*w*1, *w*2,..., *wn*) and the coefficients *a*1, *ax*, *ay*, can be calculated by the equation

$$\mathbf{L}^{-1}\mathbf{Y} = (\mathbf{W}|\,\boldsymbol{a}\_1\,\,\boldsymbol{a}\_\mathbf{x}\,\,\boldsymbol{a}\_\mathbf{y})^T.\tag{24}$$

The elements of **L**−1**Y** are used to define the TPS interpolation function

$$f(\mathbf{x}, y) = \left[ f\_{\mathbf{x}}(\mathbf{x}, y), f\_{\mathbf{y}}(\mathbf{x}, y) \right], \tag{25}$$

with *θ* = |(arctan(**V**1,2, **V**1,1) − arctan(**U**1,2, **U**1,1)|, a shear cost

*dscale* <sup>=</sup> log

*3.1.3. Shape Size and Difference Measures*

define the shape size difference as

and rotation affine operators:

*OSSp*(**X***c*, **<sup>Y</sup>***c*) = trace

with *ρ* (**X***c*, **Y***c*) as the Procrustes distance defined as

max

*S*(**X**) =

min Γ,*γ*

*<sup>k</sup>* , the ordinary partial Procrustes sum of squares is

**X***T <sup>c</sup>* **<sup>X</sup>***<sup>c</sup>* 

where *λ*1,..., *λ<sup>m</sup>* are the square roots of the eigenvalues of *Z<sup>T</sup>*

 *k* ∑ *i*=1

**<sup>S</sup>**1,1, **<sup>S</sup>**2,2, <sup>1</sup>

Shape size measures are useful metrics for comparing general shape characteristics. Given a matrix **X** of dimensions *k* × *m*, representing a set of *k m*-dimensional points, the *centroid size*

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

point). Given a second landmark configuration **Y** also with *k m*-dimensional points, we

Another useful shape metric is derived from the partial Procrustes method [41], which finds the optimal superimposition of one set of landmarks, **X**, onto another, **Y**, using translation

where 1*<sup>k</sup>* is a (*<sup>k</sup>* × <sup>1</sup>) vector of ones, <sup>Γ</sup> is a *<sup>m</sup>* × *<sup>m</sup>* rotation matrix and *<sup>γ</sup>* is the (*<sup>m</sup>* × <sup>1</sup>) translation offset vector. Using centred landmarks, **X***<sup>c</sup>* = *C***X** and **Y***<sup>c</sup>* = *C***Y** where *C* =

+ trace

*<sup>ρ</sup>* (**X***c*, **<sup>Y</sup>***c*) <sup>=</sup> arccos *<sup>m</sup>*

*H***X**/�*H***X**� and *ZY* = *H***Y**/�*H***Y**� for the Helmert sub-matrix, *H*, with dimension *k* × *k*.

**Y***T <sup>c</sup>* **<sup>Y</sup>***<sup>c</sup>* 

> ∑ *i*=1 *λi*

**S**1,1 , 1

*th* row of **X** and **X**¯ is the arithmetic mean of the points in **X** (i.e., centroid

and a scale cost

[41] is defined as

where (**X**)*<sup>i</sup>* is the *i*

**I***<sup>k</sup>* − <sup>1</sup>

*<sup>k</sup>* <sup>1</sup>*k*1*<sup>T</sup>*

*dshear* = log(**S**1,1/**S**2,2), (33)

**<sup>S</sup>**2,2 . (34)

http://dx.doi.org/10.5772/51184

239

�(**X**)*<sup>i</sup>* − **X**¯ �2, (35)

*dS* = |*S*(**X**) − *S*(**Y**)|. (36)

�**<sup>Y</sup>** − **<sup>X</sup>**<sup>Γ</sup> − <sup>1</sup>*kγT*�<sup>2</sup> (37)

− 2�**X***c*��**Y***c*� cos *ρ* (**X***c*, **Y***c*) (38)

*XZYZ<sup>T</sup>*

(39)

*<sup>Y</sup>ZX* with *ZX* =

with the coordinates compiled from the first column of **L**−1**Y** giving

$$f\_{\mathbf{x}}(\mathbf{x}, \mathbf{y}) = a\_{\mathbf{1}, \mathbf{x}} + a\_{\mathbf{x}, \mathbf{x}} \mathbf{x} + a\_{\mathbf{y}, \mathbf{x}} \mathbf{y} + \sum\_{i=1}^{n} w\_{i, \mathbf{x}} \mathcal{U}(\|\mathbf{p}\_{i} - (\mathbf{x}, \mathbf{y})\|) \tag{26}$$

where *a*1,*<sup>x</sup> ax*,*<sup>x</sup> ay*,*<sup>x</sup> <sup>T</sup>* is the affine transform component for *x*, and likewise for the second column, where

$$f\_{\mathbf{y}}(\mathbf{x}, \mathbf{y}) = a\_{1, \mathbf{y}} + a\_{\mathbf{x}, \mathbf{y}} \mathbf{x} + a\_{\mathbf{y}, \mathbf{y}} \mathbf{y} + \sum\_{i=1}^{n} w\_{i, \mathbf{y}} \mathcal{U}(\|\mathbf{p}\_i - (\mathbf{x}, \mathbf{y})\|) \tag{27}$$

with *a*1,*<sup>y</sup> ax*,*<sup>y</sup> ay*,*<sup>y</sup> <sup>T</sup>* as the affine component for *y*. Each point (or minutia location in our application) can now be updated as

$$(\mathbf{x}\_{new}, y\_{new}) = (f\_{\mathbf{x}}(\mathbf{x}, y), f\_{\mathbf{y}}(\mathbf{x}, y)). \tag{28}$$

It can be shown that the function *f*(*x*, *y*) is the interpolation that minimises

$$I\_f \propto \mathbf{W} \mathbf{K} \mathbf{W}^T = \mathbf{V} (\mathbf{L\_n^{-1} K L\_n^{-1}}) \mathbf{V}^T,\tag{29}$$

where *If* is the *bending energy* measure

$$I\_f = \int \int\_{\mathbb{R}^2} \left(\frac{\partial^2 z}{\partial x^2}\right)^2 + 2\left(\frac{\partial^2 z}{\partial x \partial y}\right)^2 + \left(\frac{\partial^2 z}{\partial y^2}\right)^2 dx dy \tag{30}$$

and **Ln** is the *n* × *n* sub-matrix of **L**. Affine transform based metrics relating to shear, rotation, and scale (i.e., compression and dilation) can be calculated straight from Singular Value Decomposition (SVD) of the affine matrix

$$\mathbf{U}\mathbf{S}\mathbf{V}^T = \text{SVD}\left(\begin{bmatrix} a\_{\mathbf{x},\mathbf{x}} \ a\_{\mathbf{x},\mathbf{y}}\\ a\_{\mathbf{y},\mathbf{x}} \ a\_{\mathbf{y},\mathbf{y}} \end{bmatrix}\right). \tag{31}$$

From this decomposition, we define an angle cost

$$d\_{\theta} = \min(\theta, 2\pi - \theta) \tag{32}$$

with *θ* = |(arctan(**V**1,2, **V**1,1) − arctan(**U**1,2, **U**1,1)|, a shear cost

$$d\_{shear} = \log(\mathbf{S}\_{1,1}/\mathbf{S}\_{2,2})\_\prime \tag{33}$$

and a scale cost

18 New Trends and Developments in Biometrics

where

with

column, where

*a*1,*<sup>x</sup> ax*,*<sup>x</sup> ay*,*<sup>x</sup>*

*a*1,*<sup>y</sup> ax*,*<sup>y</sup> ay*,*<sup>y</sup>*

application) can now be updated as

where *If* is the *bending energy* measure

*If* =

Decomposition (SVD) of the affine matrix

From this decomposition, we define an angle cost

 **R**2

The elements of **L**−1**Y** are used to define the TPS interpolation function

with the coordinates compiled from the first column of **L**−1**Y** giving

*fx*(*x*, *y*) = *a*1,*<sup>x</sup>* + *ax*,*<sup>x</sup> x* + *ay*,*xy* +

*fy*(*x*, *y*) = *a*1,*<sup>y</sup>* + *ax*,*yx* + *ay*,*yy* +

It can be shown that the function *f*(*x*, *y*) is the interpolation that minimises

 *∂*2*z ∂x*<sup>2</sup>

*If* <sup>∝</sup> **WKW***<sup>T</sup>* <sup>=</sup> **<sup>V</sup>**(**L**−**<sup>1</sup>**

2 + 2

**USV***<sup>T</sup>* = *SVD*

*f*(*x*, *y*) =

*fx*(*x*, *y*), *fy*(*x*, *y*)

*n* ∑ *i*=1

*n* ∑ *i*=1

*<sup>T</sup>* is the affine transform component for *x*, and likewise for the second

*<sup>T</sup>* as the affine component for *y*. Each point (or minutia location in our

**<sup>n</sup> KL**−**<sup>1</sup>**

2 + *∂*2*z ∂y*<sup>2</sup>

 *ax*,*<sup>x</sup> ax*,*<sup>y</sup> ay*,*x ay*,*y*

 *∂*2*z ∂x∂y*

and **Ln** is the *n* × *n* sub-matrix of **L**. Affine transform based metrics relating to shear, rotation, and scale (i.e., compression and dilation) can be calculated straight from Singular Value

**<sup>n</sup>** )**V***<sup>T</sup>*

2

*d<sup>θ</sup>* = min(*θ*, 2*π* − *θ*) (32)

(*xnew*, *ynew*)=(*fx*(*x*, *y*), *fy*(*x*, *y*)). (28)

, (25)

*wi*,*xU*(�**p***<sup>i</sup>* − (*x*, *y*)�) (26)

*wi*,*yU*(�**p***<sup>i</sup>* − (*x*, *y*)�) (27)

, (29)

*dxdy* (30)

. (31)

$$d\_{scale} = \log\left(\max\left(\mathbf{S}\_{1,1}, \mathbf{S}\_{2,2}, \frac{1}{\mathbf{S}\_{1,1}}, \frac{1}{\mathbf{S}\_{2,2}}\right)\right). \tag{34}$$

#### *3.1.3. Shape Size and Difference Measures*

Shape size measures are useful metrics for comparing general shape characteristics. Given a matrix **X** of dimensions *k* × *m*, representing a set of *k m*-dimensional points, the *centroid size* [41] is defined as

$$S(\mathbf{X}) = \sqrt{\sum\_{i=1}^{k} \| (\mathbf{X})\_i - \bar{\mathbf{X}} \|^2} \,\tag{35}$$

where (**X**)*<sup>i</sup>* is the *i th* row of **X** and **X**¯ is the arithmetic mean of the points in **X** (i.e., centroid point). Given a second landmark configuration **Y** also with *k m*-dimensional points, we define the shape size difference as

$$d\_{\mathbb{S}} = |\mathcal{S}(\mathbb{X}) - \mathcal{S}(\mathbb{Y})|.\tag{36}$$

Another useful shape metric is derived from the partial Procrustes method [41], which finds the optimal superimposition of one set of landmarks, **X**, onto another, **Y**, using translation and rotation affine operators:

$$\min\_{\Gamma,\gamma} \|\mathbf{Y} - \mathbf{X}\Gamma - \mathbf{1}\_k \gamma^T\|^2 \tag{37}$$

where 1*<sup>k</sup>* is a (*<sup>k</sup>* × <sup>1</sup>) vector of ones, <sup>Γ</sup> is a *<sup>m</sup>* × *<sup>m</sup>* rotation matrix and *<sup>γ</sup>* is the (*<sup>m</sup>* × <sup>1</sup>) translation offset vector. Using centred landmarks, **X***<sup>c</sup>* = *C***X** and **Y***<sup>c</sup>* = *C***Y** where *C* = **I***<sup>k</sup>* − <sup>1</sup> *<sup>k</sup>* <sup>1</sup>*k*1*<sup>T</sup> <sup>k</sup>* , the ordinary partial Procrustes sum of squares is

$$\text{OSS}\_p(\mathbf{X}\_{\mathcal{C}}, \mathbf{Y}\_{\mathcal{C}}) = \text{trace}\left(\mathbf{X}\_{\mathcal{C}}^T \mathbf{X}\_{\mathcal{C}}\right) + \text{trace}\left(\mathbf{Y}\_{\mathcal{C}}^T \mathbf{Y}\_{\mathcal{C}}\right) - 2\|\mathbf{X}\_{\mathcal{C}}\| \|\mathbf{Y}\_{\mathcal{C}}\| \cos \rho\left(\mathbf{X}\_{\mathcal{C}}, \mathbf{Y}\_{\mathcal{C}}\right) \tag{38}$$

with *ρ* (**X***c*, **Y***c*) as the Procrustes distance defined as

$$\rho\left(\mathbf{X}\_{\mathcal{C}}\,\mathbf{Y}\_{\mathcal{C}}\right) = \arccos\left(\sum\_{i=1}^{m} \lambda\_{i}\right) \tag{39}$$

where *λ*1,..., *λ<sup>m</sup>* are the square roots of the eigenvalues of *Z<sup>T</sup> XZYZ<sup>T</sup> <sup>Y</sup>ZX* with *ZX* = *H***X**/�*H***X**� and *ZY* = *H***Y**/�*H***Y**� for the Helmert sub-matrix, *H*, with dimension *k* × *k*.

## *3.1.4. Close Non-Match Discovery and Alignment*

In order to reproduce the process of an examiner querying a minutiae configuration marked on fingermark with an AFIS, a method for finding close configurations was developed. To find close non-matches for a particular minutiae configuration, we employed a simple search algorithm based solely on minutiae triplet features, in order to maintain robustness towards such fingermark-to-exemplar match scenarios. The minutiae triplet features are extracted in a fully automated manner using the NIST mindtct tool [49] without particular attention to spurious results, besides minimum quality requirements as rated by the mindtct algorithm.

potential match due to the likely *unnatural* distortion encountered. Finally, a candidate list

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

http://dx.doi.org/10.5772/51184

241

We now propose an LR model based on what is found in [4], developed specifically to aid AFIS candidate list assessments, using the intrinsic differences of morphometric and spatial analyses (which we label as MSA) between match and close non-match comparisons, learnt

Given two matching configurations *X* and *Y* (discovered from the procedure described in Section 3.1.4) a FV based on the previously discussed morphometric and spatial analyses is

where *Zm*,*n*,3*<sup>D</sup>* is the three dimensional KS statistic of equation (22) using the transformed triplet points, *If* , *d<sup>θ</sup>* , *dshear*, and *dscale* are the defined measures of equations (29) and (32-34) resulting from registering **X** onto **Y** via TPS, *S*(**X**) and *dS* are the shape size and difference metric of equations (35-36), *OSSp*(**X***c*, **Y***c*) is the ordinary partial Procrustes sum of squares of equation (38), and *dmc* is the difference of the number of interior minutiae within the convex hulls of **X** and **Y**. The *dmc* measure is an optional component to the FV dependent on the clarity of a fingermark's detail within the given minutiae configuration. For the experiments

The compulsory measures used in the proposed feature vector rely solely on features that are robust to the adverse environmental conditions of latent marks, all of which are based on minutiae triplet detail. The FV structures are categorised by genuine/imposter (or match/close non-match) classes, number of minutiae in the matching configurations, and

Using the categories prescribed for the defined FVs, a probabilistic machine learning framework is applied for finding the probabilities for match and close non-match classes. The probabilistic framework employed [42] is based on Support Vector Machines (SVMs)

*<sup>h</sup>*(**x**) = ∑

*i*

*f*(**x**) = *h*(**x**) + *b* (41)

*yiαik*(**x***i*, **x**) (42)

**x***<sup>i</sup>* = {*Zm*,*n*,3*D*, *If* , *d<sup>θ</sup>* , *dshear*, *dscale*, *S*(**X**), *dS*,*OSSp*(**X***c*, **Y***c*), *dmc*} (40)

with all close minutiae configurations is produced for analysis.

from a two-class probabilistic machine learning framework.

presented later in this chapter, we will exclude this measure.

configuration area (categorised as small, medium, and large).

*3.2.2. Machine Learning of Feature Vectors*

with unthresholded output, defined as

**3.2. Proposed model**

*3.2.1. Feature Vector Definition*

defined as:

with

```
Algorithm 1 f indCloseTripletCon f igs: Find all close triplet configurations to X
```

```
Require: A minutiae triplet set X and a dataset of exemplars D.
  candidateList = null
  for all minutiae configurations Y ∈ D with |X| = |Y| do
    for all minutiae (xY, yY, θY) ∈ Y do
       f ound ← false
       for all minutiae (xX, yX, θX) ∈ X do
          Y′ ← Y
          rotate Y′ by (θX − θY) {This includes rotating minutiae angles.}
          translate Y′ by offset (xX − xY, yX − yY)
          if Y′ is close to X then
             f ound = true
            goto finished:
          end if
       end for
    end for
    finished:
    if f ound = true then
       Y′ ← PartialProcrustes(X, Y′
                                     ) {Translate/Rotate Y′ using partial Procrustes}
       TPS(X, Y′
                 ) {non-affine registration by TPS}
       if If < Imax then
          add Y′ to candidateList {Add if bending energy < limit (equation (29))}
       end if
    end if
  end for
  return candidateList
```
Once feature extraction is complete, the close match search algorithm (Algorithm 1) finds all equally sized close minutiae configurations in a given dataset of exemplars to a specified minutiae set configuration (i.e., potentially marked from a latent) in an iterative manner by assessing all possible minutiae triplet pairs via a crude affine transform based alignment on configuration structures. Recorded close minutiae configurations are then re-aligned using the partial Procrustes method using the discovered minutiae pairings. Unlike the Procrustes method, the partial Procrustes method does not alter scale of either landmarks. For the application of fingerprint alignment, ignoring scale provides a more accurate comparison of landmarks since all minutiae structures are already normalised by the resolution and dimensions of the digital image. The TPS registration is then applied for a non-affine transformation. If the bending energy is higher than a defined threshold, we ignore the potential match due to the likely *unnatural* distortion encountered. Finally, a candidate list with all close minutiae configurations is produced for analysis.

## **3.2. Proposed model**

20 New Trends and Developments in Biometrics

*candidateList* = *null*

*f ound* ← false

**Y**′ ← **Y**

**end if end for end for** *finished*:

**if** *f ound* = true **then**

**if** *If* < *Imax* **then**

*TPS*(**X**, **Y**′

**return** *candidateList*

**end if end if end for**

*3.1.4. Close Non-Match Discovery and Alignment*

**for all** minutiae (*xY*, *yY*, *θY*) ∈ **Y do**

**if Y**′ is *close* to **X then** *f ound* = true **goto** *finished*:

**Y**′ ← *PartialProcrustes*(**X**, **Y**′

**for all** minutiae (*xX*, *yX*, *θX*) ∈ **X do**

In order to reproduce the process of an examiner querying a minutiae configuration marked on fingermark with an AFIS, a method for finding close configurations was developed. To find close non-matches for a particular minutiae configuration, we employed a simple search algorithm based solely on minutiae triplet features, in order to maintain robustness towards such fingermark-to-exemplar match scenarios. The minutiae triplet features are extracted in a fully automated manner using the NIST mindtct tool [49] without particular attention to spurious results, besides minimum quality requirements as rated by the mindtct algorithm.

**Algorithm 1** *f indCloseTripletCon f igs*: Find all close triplet configurations to **X**

rotate **Y**′ by (*θ<sup>X</sup>* − *θY*) {This includes rotating minutiae angles.}

add **Y**′ to *candidateList* {Add if bending energy < limit (equation (29))}

Once feature extraction is complete, the close match search algorithm (Algorithm 1) finds all equally sized close minutiae configurations in a given dataset of exemplars to a specified minutiae set configuration (i.e., potentially marked from a latent) in an iterative manner by assessing all possible minutiae triplet pairs via a crude affine transform based alignment on configuration structures. Recorded close minutiae configurations are then re-aligned using the partial Procrustes method using the discovered minutiae pairings. Unlike the Procrustes method, the partial Procrustes method does not alter scale of either landmarks. For the application of fingerprint alignment, ignoring scale provides a more accurate comparison of landmarks since all minutiae structures are already normalised by the resolution and dimensions of the digital image. The TPS registration is then applied for a non-affine transformation. If the bending energy is higher than a defined threshold, we ignore the

) {Translate/Rotate **Y**′ using partial Procrustes}

**Require:** A minutiae triplet set **X** and a dataset of exemplars *D*.

**for all** minutiae configurations **Y** ∈ *D* with |**X**| = |**Y**| **do**

translate **Y**′ by offset (*xX* − *xY*, *yX* − *yY*)

) {non-affine registration by TPS}

We now propose an LR model based on what is found in [4], developed specifically to aid AFIS candidate list assessments, using the intrinsic differences of morphometric and spatial analyses (which we label as MSA) between match and close non-match comparisons, learnt from a two-class probabilistic machine learning framework.

#### *3.2.1. Feature Vector Definition*

Given two matching configurations *X* and *Y* (discovered from the procedure described in Section 3.1.4) a FV based on the previously discussed morphometric and spatial analyses is defined as:

$$\mathbf{x}\_{i} = \{Z\_{\mathrm{m},\mathrm{n},\mathrm{3D},\prime}I\_{f}, d\_{\theta}, d\_{\mathrm{shear}}, d\_{\mathrm{scale},\prime} \text{S}(\mathbf{X}), d\_{\mathrm{S},\prime} \text{OSS}\_{p}(\mathbf{X}\_{\mathrm{c}}, \mathbf{Y}\_{\mathrm{c}}), d\_{\mathrm{mc}}\} \tag{40}$$

where *Zm*,*n*,3*<sup>D</sup>* is the three dimensional KS statistic of equation (22) using the transformed triplet points, *If* , *d<sup>θ</sup>* , *dshear*, and *dscale* are the defined measures of equations (29) and (32-34) resulting from registering **X** onto **Y** via TPS, *S*(**X**) and *dS* are the shape size and difference metric of equations (35-36), *OSSp*(**X***c*, **Y***c*) is the ordinary partial Procrustes sum of squares of equation (38), and *dmc* is the difference of the number of interior minutiae within the convex hulls of **X** and **Y**. The *dmc* measure is an optional component to the FV dependent on the clarity of a fingermark's detail within the given minutiae configuration. For the experiments presented later in this chapter, we will exclude this measure.

The compulsory measures used in the proposed feature vector rely solely on features that are robust to the adverse environmental conditions of latent marks, all of which are based on minutiae triplet detail. The FV structures are categorised by genuine/imposter (or match/close non-match) classes, number of minutiae in the matching configurations, and configuration area (categorised as small, medium, and large).

#### *3.2.2. Machine Learning of Feature Vectors*

Using the categories prescribed for the defined FVs, a probabilistic machine learning framework is applied for finding the probabilities for match and close non-match classes. The probabilistic framework employed [42] is based on Support Vector Machines (SVMs) with unthresholded output, defined as

$$f(\mathbf{x}) = h(\mathbf{x}) + b \tag{41}$$

with

$$h(\mathbf{x}) = \sum\_{i} y\_i \alpha\_i k(\mathbf{x}\_{i\prime} \mathbf{x}) \tag{42}$$

where *k*(•, •) is the kernel function, and the target output *yi* ∈ {−1, 1} represents the two classes (i.e., 'close non-match' and 'match', respectively). We use the radial basis function

$$k(\mathbf{x}\_{i\prime}\mathbf{x}) = \exp(-\gamma \|\mathbf{x}\_i - \mathbf{x}\|^2) \tag{43}$$

where *a* reflects the proportion of close minutiae configuration comparisons that are ground

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

For future consideration, the probabilities *P*(**x***<sup>i</sup>* is a match) and *P*(**x***<sup>i</sup>* is a close non-match) can be adaptively based on Cumulative Match Characteristic (CMC) curve [44] statistics of a

As already noted, the LR formulas are based on different distributions specified per FV categories of minutiae count and the area of the given configuration. This allows the LR models to capture any spatial and morphometric relational differences between such defined categories. Unlike previous LR methods that are based on the distributions of a dissimilarity metric, the proposed method is based on class predictions based on a number of measures, some of which do not implicitly or explicitly rate or score a configuration's dissimilarity (e.g. centroid size, *S*(**X***i*)). Instead, statistical relationships of the FV measures and classes are

In its current proposed form, the LR of equation (49) is not an evidential weight for the entire

Without access to large scale AFISs, a sparse number of fingermark-to-exemplar datasets exists in the public domain (i.e., NIST27 is the only known dataset with only 258 sets). Thus,

We follow a methodology similar to that of [35] where live scanned fingerprints have eleven directions applied, eight of which are linear directions, two torsional, and central application of force. Using a readily available live scan device (Suprema Inc. Realscan-D: 500ppi with rolls, single and dual finger flats), we follow a similar methodology, described as follows:

• finally, numerous impressions with emphasis on partiality and high distortion are obtained by recording fifteen frames per second, while each finger manoeuvres about

This gave a minimum total of 968 impressions per finger. A total of 6,000 impressions from six different fingers (from five individuals) were obtained for our within-finger dataset, most of which are partial impressions from the freestyle methodology. For the between-finger

population, but rather, an evidential weight specifically for a given candidate list.

. *<sup>P</sup>*(**x***<sup>i</sup>* is a match<sup>|</sup> *<sup>f</sup>*(**x***i*))

*<sup>P</sup>*(**x***<sup>i</sup>* is a close non-match<sup>|</sup> *<sup>f</sup>*(**x***i*)). (49)

http://dx.doi.org/10.5772/51184

243

truth matches. Thus, the LR is equivalent to the posterior ratio (PR)

given AFIS system or any other relevant background information.

learnt by SVMs in a supervised manner, only for class predictions.

to study the within-finger characteristics, a distortion set was built.

• all directions described above have at least three levels of force applied,

the scan area in a freestyle manner for a minimum of sixty seconds.

1 − *a a*

*LR* =

**3.3. Experimentation**

*3.3.1. Experimental Databases*

• sixteen different linear directions of force,

• at least five rolled acquisitions are collected,

• four torsion directions of force,

• central direction of force,

1 − *a a*

 .*PR* =

due to the observed non-linear relationships of the proposed FV. Training the SVM minimises the error function

$$\mathbb{C}\sum\_{i}(1-y\_{i}f(\mathbf{x}\_{i}))\_{+}+\frac{1}{2}||h||\_{\mathcal{F}}\tag{44}$$

where *C* is the soft margin parameter (i.e., regularisation term which provides a way to control overfitting) and F is the Reproducing Kernel Hilbert Space (RKHS) induced by the kernel *k*. Thus, the norm of *h* is penalised in addition to the approximate training misclassification rate. By transforming the target values with

$$t\_i = \frac{y\_i + 1}{2} \,\prime \tag{45}$$

the posterior probabilities *P*(*yi* = 1| *f*(**x***i*)) and *P*(*yi* = −1| *f*(**x***i*)) which represents the probabilities that **x***<sup>i</sup>* is of classes 'match' and 'close non-match', respectively, can now be estimated by fitting a sigmoid function after the SVM output with

$$P(\mathbf{x}\_i \text{ is a match} | f(\mathbf{x}\_i)) = P(y\_i = 1 | f(\mathbf{x}\_i)) = \frac{1}{1 + \exp(A f(\mathbf{x}\_i) + B)} \tag{46}$$

and

$$P(\mathbf{x}\_i \text{ is a close non-match} | f(\mathbf{x}\_i)) = P(y\_i = -1 | f(\mathbf{x}\_i)) = 1 - \frac{1}{1 + \exp(A f(\mathbf{x}\_i) + B)}.\tag{47}$$

The parameters *A* and *B* are found by minimising the negative log-likelihood of the training data:

$$\arg\min\_{A,B} \left[ -\left( \sum\_{i} t\_i \log \left( \frac{1}{1 + \exp(A f(\mathbf{x}\_i) + \mathcal{B})} \right) + (1 - t\_i) \log \left( 1 - \frac{1}{1 + \exp(A f(\mathbf{x}\_i) + \mathcal{B})} \right) \right) \right] \tag{48}$$

using any optimisation algorithm, such as the Levenberg-Marquardt algorithm [43].

#### *3.2.3. Likelihood Ratio Calculation*

The probability distributions of equations (47-48) are posterior probabilities. Nevertheless, for simplicity of the initial application, we assume uniform distributions for *P*(*f*(**x***i*)) = *z* for some constant, *z*, whereas *P*(**x***<sup>i</sup>* is a match) = *a* and *P*(**x***<sup>i</sup>* is a close non-match) = 1 − *a*

where *a* reflects the proportion of close minutiae configuration comparisons that are ground truth matches. Thus, the LR is equivalent to the posterior ratio (PR)

$$LR = \left(\frac{1-a}{a}\right).PR = \left(\frac{1-a}{a}\right).\frac{P(\mathbf{x}\_i \text{ is a match} | f(\mathbf{x}\_i))}{P(\mathbf{x}\_i \text{ is a close non-match} | f(\mathbf{x}\_i))}.\tag{49}$$

For future consideration, the probabilities *P*(**x***<sup>i</sup>* is a match) and *P*(**x***<sup>i</sup>* is a close non-match) can be adaptively based on Cumulative Match Characteristic (CMC) curve [44] statistics of a given AFIS system or any other relevant background information.

As already noted, the LR formulas are based on different distributions specified per FV categories of minutiae count and the area of the given configuration. This allows the LR models to capture any spatial and morphometric relational differences between such defined categories. Unlike previous LR methods that are based on the distributions of a dissimilarity metric, the proposed method is based on class predictions based on a number of measures, some of which do not implicitly or explicitly rate or score a configuration's dissimilarity (e.g. centroid size, *S*(**X***i*)). Instead, statistical relationships of the FV measures and classes are learnt by SVMs in a supervised manner, only for class predictions.

In its current proposed form, the LR of equation (49) is not an evidential weight for the entire population, but rather, an evidential weight specifically for a given candidate list.

## **3.3. Experimentation**

22 New Trends and Developments in Biometrics

the error function

and

data:

arg min*A*,*<sup>B</sup>*

 − ∑*<sup>i</sup> ti* log

*3.2.3. Likelihood Ratio Calculation*

where *k*(•, •) is the kernel function, and the target output *yi* ∈ {−1, 1} represents the two classes (i.e., 'close non-match' and 'match', respectively). We use the radial basis function

due to the observed non-linear relationships of the proposed FV. Training the SVM minimises

1 2

(1 − *yi f*(**x***i*))+ +

where *C* is the soft margin parameter (i.e., regularisation term which provides a way to control overfitting) and F is the Reproducing Kernel Hilbert Space (RKHS) induced by the kernel *k*. Thus, the norm of *h* is penalised in addition to the approximate training

*ti* <sup>=</sup> *yi* <sup>+</sup> <sup>1</sup>

the posterior probabilities *P*(*yi* = 1| *f*(**x***i*)) and *P*(*yi* = −1| *f*(**x***i*)) which represents the probabilities that **x***<sup>i</sup>* is of classes 'match' and 'close non-match', respectively, can now be

The parameters *A* and *B* are found by minimising the negative log-likelihood of the training

The probability distributions of equations (47-48) are posterior probabilities. Nevertheless, for simplicity of the initial application, we assume uniform distributions for *P*(*f*(**x***i*)) = *z* for some constant, *z*, whereas *P*(**x***<sup>i</sup>* is a match) = *a* and *P*(**x***<sup>i</sup>* is a close non-match) = 1 − *a*

using any optimisation algorithm, such as the Levenberg-Marquardt algorithm [43].

+ (1 − *ti*)log

1 − <sup>1</sup>

*<sup>P</sup>*(**x***<sup>i</sup>* is a match<sup>|</sup> *<sup>f</sup>*(**x***i*)) = *<sup>P</sup>*(*yi* <sup>=</sup> <sup>1</sup><sup>|</sup> *<sup>f</sup>*(**x***i*)) = <sup>1</sup>

*<sup>P</sup>*(**x***<sup>i</sup>* is a close non-match<sup>|</sup> *<sup>f</sup>*(**x***i*)) = *<sup>P</sup>*(*yi* <sup>=</sup> <sup>−</sup>1<sup>|</sup> *<sup>f</sup>*(**x***i*)) = <sup>1</sup> <sup>−</sup> <sup>1</sup>

*<sup>C</sup>*∑ *i*

misclassification rate. By transforming the target values with

estimated by fitting a sigmoid function after the SVM output with

 1 1+exp(*A f*(**x***i*)+*B*)

*<sup>k</sup>*(**x***i*, **<sup>x</sup>**) = exp(−*γ*�**x***<sup>i</sup>* − **<sup>x</sup>**�2) (43)

�*h*�F (44)

<sup>2</sup> , (45)

<sup>1</sup> <sup>+</sup> exp(*A f*(**x***i*) + *<sup>B</sup>*) (46)

1 + exp(*A f*(**x***i*) + *B*)

1+exp(*A f*(**x***i*)+*B*)

. (47)

(48)

#### *3.3.1. Experimental Databases*

Without access to large scale AFISs, a sparse number of fingermark-to-exemplar datasets exists in the public domain (i.e., NIST27 is the only known dataset with only 258 sets). Thus, to study the within-finger characteristics, a distortion set was built.

We follow a methodology similar to that of [35] where live scanned fingerprints have eleven directions applied, eight of which are linear directions, two torsional, and central application of force. Using a readily available live scan device (Suprema Inc. Realscan-D: 500ppi with rolls, single and dual finger flats), we follow a similar methodology, described as follows:


This gave a minimum total of 968 impressions per finger. A total of 6,000 impressions from six different fingers (from five individuals) were obtained for our within-finger dataset, most of which are partial impressions from the freestyle methodology. For the between-finger comparisons, we use the within-finger set in addition to the public databases of NIST 14 [45] (27000 × 2 impressions), NIST 4 [46] (2000 × 2 impressions), FVC 2002 [47] (3 × 110 × 8 flat scan/swipe impressions), and the NIST 27 database [48] (258 exemplars + 258 latents), providing over 60,000 additional impressions.

## *3.3.2. SVM Training Procedure*

A simple training/evaluation methodology was used in the experiments. After finding all FVs for similar configurations, a random selection of 50% of the FVs were used to train each respective SVM by the previously defined categories (i.e., minutiae configuration count and area). The remaining 50% of FVs were used to evaluate the LR model accuracy. The process was then repeated by swapping the training and test sets (i.e., two-fold cross-validation). Due to the large size of the within-finger database, a substantially larger number of within-finger candidates are returned. To alleviate this, we randomly sampled the within-finger candidates to be of equal number to the between-finger counterparts (i.e., *a* = 0.5 in equation (49)). All individual features within each FV were scaled to have a range of [0, 1], using pre-defined maximum and minimum values specific to each feature component.

−15 −10 −5 0 5 10

−15 −10 −5 0 5 10

−15 −10 −5 0 5 10

**Tippett Plot for 6 minutiae 'large' configurations**

http://dx.doi.org/10.5772/51184

245

Log\_Likelihood

**Tippett Plot for 7 minutiae 'large' configurations**

−15 −10 −5 0 5 10

Log\_Likelihood

**Tippett Plot for 8 minutiae 'large' configurations**

−15 −10 −5 0 5 10

Log\_Likelihood

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

**Tippett Plot for 6 minutiae 'medium' configurations**

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

Log\_Likelihood

**Tippett Plot for 7 minutiae 'medium' configurations**

−15 −10 −5 0 5 10

Log\_Likelihood

**Tippett Plot for 8 minutiae 'medium' configurations**

−15 −10 −5 0 5 10

Log\_Likelihood

**Figure 6.** Tippett plots for minutiae configurations of 6 (top row), 7 (middle row), and 8 (bottom row) minutiae with small, medium, and large area categories (left to right, respectively), calculated from *P*(**x***<sup>i</sup>* is a match| *f*(**x***i*)) and *P*(**x***<sup>i</sup>* is a close non-match| *f*(**x***i*)) distributions. The *x*-axes represents the logarithm (base 2) of the LR values in equation (49) for match (blue line) and close non-match (red line) populations, while the *y*-axes represents proportion of such values being

greater than *x*. The green vertical dotted line at *x* = 0 signifies a marker for *LR* = 1 (i.e., *x* = *log*21 = 0).

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

**Tippett Plot for 6 minutiae 'small' configurations**

Log\_Likelihood

**Tippett Plot for 7 minutiae 'small' configurations**

−15 −10 −5 0 5 10

Log\_Likelihood

**Tippett Plot for 8 minutiae 'small' configurations**

−15 −10 −5 0 5 10

Log\_Likelihood

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

A naive approach was used to find the parameters for the SVMs. The radial basis kernel parameter, *γ*, and the soft learning parameter, *C*, of equations (43) and (44), respectively, were selected using a grid based search, using the cross-validation framework to measure the test accuracy for each parameter combination, (*γ*, *C*). The parameter combination with the highest test accuracy was selected for each constructed SVM.

### *3.3.3. Experimental Results*

Experiments were conducted for minutiae configurations of sizes of 6, 7, and 8 (Figure 6) from the within-finger dataset, using configurations marked manually by an iterative circular growth around a first minutiae until the desired configuration sizes were met. From the configuration sizes, a total of 12144, 4500, and 1492 candidates were used, respectively, from both the within (50%) and between (50%) finger datasets. The focus on these configuration settings were due to three reasons: firstly, the high computational overhead involved in the candidate list retrieval for the prescribed datasets, secondly, configurations of such sizes perform poorly in modern day AFIS systems [50], and finally, such configuration sizes are traditionally contentious in terms of Locard's tripartite rule, where a probabilistic approach is prescribed to be used.

The area sizes used for categorising the minutiae configurations were calculated by adding up the individual areas of triangular regions created using Delaunay triangulation. Small, medium, and large configuration area categories were defined as 0 < *A* < 4.2*mm*2, 4.2*mm*<sup>2</sup> ≤ *A* < 6.25*mm*2, and *A* ≥ 6.25*mm*2, respectively.

The results clearly indicate a stronger dichotomy of match and close non-match populations when the number of minutiae was increased. In addition, the dichotomy was marginally stronger for larger configuration areas with six minutiae. Overall, the majority of FV's of class 'match' derive significantly large LR values.

<sup>244</sup> New Trends and Developments in Biometrics An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA) 25 An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA) http://dx.doi.org/10.5772/51184 245

*3.3.2. SVM Training Procedure*

*3.3.3. Experimental Results*

is prescribed to be used.

*A* < 6.25*mm*2, and *A* ≥ 6.25*mm*2, respectively.

class 'match' derive significantly large LR values.

providing over 60,000 additional impressions.

comparisons, we use the within-finger set in addition to the public databases of NIST 14 [45] (27000 × 2 impressions), NIST 4 [46] (2000 × 2 impressions), FVC 2002 [47] (3 × 110 × 8 flat scan/swipe impressions), and the NIST 27 database [48] (258 exemplars + 258 latents),

A simple training/evaluation methodology was used in the experiments. After finding all FVs for similar configurations, a random selection of 50% of the FVs were used to train each respective SVM by the previously defined categories (i.e., minutiae configuration count and area). The remaining 50% of FVs were used to evaluate the LR model accuracy. The process was then repeated by swapping the training and test sets (i.e., two-fold cross-validation). Due to the large size of the within-finger database, a substantially larger number of within-finger candidates are returned. To alleviate this, we randomly sampled the within-finger candidates to be of equal number to the between-finger counterparts (i.e., *a* = 0.5 in equation (49)). All individual features within each FV were scaled to have a range of [0, 1], using pre-defined

A naive approach was used to find the parameters for the SVMs. The radial basis kernel parameter, *γ*, and the soft learning parameter, *C*, of equations (43) and (44), respectively, were selected using a grid based search, using the cross-validation framework to measure the test accuracy for each parameter combination, (*γ*, *C*). The parameter combination with

Experiments were conducted for minutiae configurations of sizes of 6, 7, and 8 (Figure 6) from the within-finger dataset, using configurations marked manually by an iterative circular growth around a first minutiae until the desired configuration sizes were met. From the configuration sizes, a total of 12144, 4500, and 1492 candidates were used, respectively, from both the within (50%) and between (50%) finger datasets. The focus on these configuration settings were due to three reasons: firstly, the high computational overhead involved in the candidate list retrieval for the prescribed datasets, secondly, configurations of such sizes perform poorly in modern day AFIS systems [50], and finally, such configuration sizes are traditionally contentious in terms of Locard's tripartite rule, where a probabilistic approach

The area sizes used for categorising the minutiae configurations were calculated by adding up the individual areas of triangular regions created using Delaunay triangulation. Small, medium, and large configuration area categories were defined as 0 < *A* < 4.2*mm*2, 4.2*mm*<sup>2</sup> ≤

The results clearly indicate a stronger dichotomy of match and close non-match populations when the number of minutiae was increased. In addition, the dichotomy was marginally stronger for larger configuration areas with six minutiae. Overall, the majority of FV's of

maximum and minimum values specific to each feature component.

the highest test accuracy was selected for each constructed SVM.

**Figure 6.** Tippett plots for minutiae configurations of 6 (top row), 7 (middle row), and 8 (bottom row) minutiae with small, medium, and large area categories (left to right, respectively), calculated from *P*(**x***<sup>i</sup>* is a match| *f*(**x***i*)) and *P*(**x***<sup>i</sup>* is a close non-match| *f*(**x***i*)) distributions. The *x*-axes represents the logarithm (base 2) of the LR values in equation (49) for match (blue line) and close non-match (red line) populations, while the *y*-axes represents proportion of such values being greater than *x*. The green vertical dotted line at *x* = 0 signifies a marker for *LR* = 1 (i.e., *x* = *log*21 = 0).

## **4. Summary**

A new FV based LR model using morphometric and spatial analysis (MSA) with SVMs, while focusing on candidate list results of AFIS, has been proposed. This is the first LR model known to the authors that use machine learning as a core component to learn spatial feature relationships of close non-match and match populations. For robust applications for fingermark-to-exemplar comparisons, only minutiae triplet information were used to train the SVMs. Experimental results illustrate the effectiveness of the proposed method in distinguishing match and close non-match configurations.

[7] C. Champod (2009). Friction Ridge Examination (Fingerprints): Interpretation Of, *in Wiley Encyclopedia of Forensic Science (Vol. 3), A. Moenssens and A. Jamieson (Eds.),*

An AFIS Candidate List Centric Fingerprint Likelihood Ratio Model Based on Morphometric and Spatial Analyses (MSA)

http://dx.doi.org/10.5772/51184

247

[8] C. Champod (1995). Edmond Locard–numerical standards and "probable"

[9] J. Polski (2011), R. Smith, R. Garrett, et.al. The Report of the International Association for Identification, Standardization II Committee,*Grant no. 2006-DN-BX-K249 awarded by*

[10] M. Saks (2010). Forensic identification: From a faith-based "Science" to a scientific

[11] S. A. Cole (2008). The 'Opinionization' of Fingerprint Evidence, *BioSocieties, Vol. 3*,

[12] J. J. Koehler (2010), M. J. Saks. Individualization Claims in Forensic Science: Still

[13] L. Haber (2008), R. N. Haber. Scientific validation of fingerprint evidence under

[14] S. A. Cole (2007). Toward Evidence-Based Evidence: Supporting Forensic Knowledge

[15] S. A. Cole (2009). Forensics without Uniqueness, Conclusions without Individualization: The New Epistemology of Forensic Identification, *Law Probability*

[16] I. E. Dror (2010), S. A. Cole. The Vision in 'Blind' Justice: Expert Perception, Judgement, and Visual Cognition in Forensic Pattern Recognition, *Psychonomic Bulletin & Review,*

[17] M. Page (2011), J. Taylor, M. Blenkin. Forensic Identification Science Evidence Since Daubert: Part I-A Quantitative Analysis of the Exclusion of Forensic Identification

[19] G. Langenburg (2009), C. Champod, P. Wertheim. Testing for Potential Contextual Bias Effects During the Verification Stage of the ACE-V Methodology when Conducting Fingerprint Comparisons, *Journal of Forensic Sciences, Vol. 54, No. 3*, pages 571-582.

[20] L. J. Hall (2008), E. Player. Will the introduction of an emotional context affect fingerprint analysis and decision-making?, *Forensic Science International, Vol. 181*, pages

Science Evidence, *Journal of Forensic Sciences, Vol. 56, No. 5*, pages 1180-1184.

[18] Daubert v. Merrel Dow Pharmaceuticals (1993), *113 S. Ct. 2786*.

Claims in the Post-Daubert Era, *Tulsa Law Review, Vol 43*, pages 263-283.

*Chichester, UK: John Wiley & Sons*, pages 1277-1282.

identifications, *J. Forensic Ident., Vol. 45*, pages 136-163.

*the U.S. Department of Justice, Washington, DC, March 2011*.

science, *Forensic Science International Vol. 201*, pages 14-17.

Daubert Law, *Probability and Risk Vol. 7, No. 2*, pages 87-109.

Unwarranted, *75 Brook. L. Rev. 1187-1208*.

*and Risk, Vol. 8*, pages 233-255.

*Vol. 17*, pages 161-167.

36-39.

pages 105-113.

The proposed model is a preliminary proposal and is not focused on evidential value for judicial purposes. However, minor modifications can potentially allow the model to also be used for evidential assessments. For future research, we hope to evaluate the model with commercial AFIS environments containing a large set of exemplars.

## **Author details**

Joshua Abraham1,⋆, Paul Kwan2, Christophe Champod3, Chris Lennard4 and Claude Roux1

<sup>⋆</sup> Address all correspondence to: joshua.abraham@uts.edu.au


## **References**


distinguishing match and close non-match configurations.

Joshua Abraham1,⋆, Paul Kwan2, Christophe Champod3,

<sup>⋆</sup> Address all correspondence to: joshua.abraham@uts.edu.au

*(Ed.), ISBN: 978-953-307-618-8, InTech*, pages 79-98.

*(Eds.), ISBN: 978-953-307-489-4, InTech*, pages 25-56.

Ridge Skin Impressions, *CRC Press.* 2004.

*Forensic Science International*, DOI: 10.1016/j.forsciint.2012.10.034

1 Centre for Forensic Science, University of Technology, Sydney, Australia 2 School of Science and Technology, University of New England, Australia 3 Institute of Forensic Science, University of Lausanne, Switzerland 4 National Centre for Forensic Studies, University of Canberra, Australia

commercial AFIS environments containing a large set of exemplars.

A new FV based LR model using morphometric and spatial analysis (MSA) with SVMs, while focusing on candidate list results of AFIS, has been proposed. This is the first LR model known to the authors that use machine learning as a core component to learn spatial feature relationships of close non-match and match populations. For robust applications for fingermark-to-exemplar comparisons, only minutiae triplet information were used to train the SVMs. Experimental results illustrate the effectiveness of the proposed method in

The proposed model is a preliminary proposal and is not focused on evidential value for judicial purposes. However, minor modifications can potentially allow the model to also be used for evidential assessments. For future research, we hope to evaluate the model with

[1] J.C. Yang (2008), D.S. Park. A Fingerprint Verification Algorithm Using Tessellated Invariant Moment Features, *Neurocomputing, Vol. 71, No. 10-12*, pages 1939-1946.

[2] J.C. Yang (2011). Non-minutiae based fingerprint descriptor, in *Biometrics, Jucheng Yang*

[3] J. Abraham (2011), P. Kwan, J. Gao. Fingerprint Matching using a Hybrid Shape and Orientation Descriptor, in *State of the art in Biometrics, Jucheng Yang and Loris Nanni*

[4] J. Abraham (2012), C. Champod, C. Lennard, C. Roux. Spatial Analysis of Corresponding Fingerprint Features from Match and Close Non-Match Populations,

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**4. Summary**

**Author details**

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**Chapter 11**

**Physiological Signal Based Biometrics for Securing**

Nowadays, the constraints in the healthcare of developing countries, including high pop‐ ulation growth, a high burden of disease prevalence, low health care workforce, large numbers of rural inhabitants, and limited financial resources to support healthcare infra‐ structure and health information systems, accompanied with the improvement of poten‐ tial of lowering information and transaction costs in healthcare delivery due to the explosively access of mobile phones to all segments of a country, has motivated the de‐ velopment of mobile health or m-health field. M-health is known as the practice of medi‐ cal and public health supported by mobile devices such as mobile phones and PDAs for delivering medical and healthcare services. Thus, the popularity of m-health can be sub‐ jected to the development of wearable medical devices and wireless communication tech‐ nology. In order to fully utilize wireless technology between the wearable medical devices, the concept of body sensor network (BSN), which is a kind of wireless sensor

BSN, which has great potential in being the main front-end platform of telemedicine and mobile health systems, is currently being heavily developed to keep pace with the continu‐ ously rising demand for personalized healthcare. Comprised of sensors attached to the hu‐ man body for collecting and transmitting vital signs, BSN is able to facilitate the joint processing of spatially and temporally collected medical data from different parts of the body for resource optimization and systematic health monitoring. In a typical BSN, each sensor node collects various physiological signals in order to monitor the patient's health status no matter their location and then instantly transmit all information in real time to the

> © 2012 Miao et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 Miao et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

**Body Sensor Network**

Fen Miao, Shu-Di Bao and Ye Li

http://dx.doi.org/10.5772/51856

**1. Introduction**

**1.1. Body sensor network**

Additional information is available at the end of the chapter

network around human body, was proposed in 2002.

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