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ICALAB for image processing and each package

contains a variety of algorithms.

2. FastICA (http://www.cis.hut.ﬁ/projects/ica/fastica/):

It is the FastICA Matlab package that implements

fast ﬁxed-point algor ithms for non-Gaussianity

maximization [3]. It was developed in Helsinki

University of Technology, Finland and other envir-

onments (R, C++, Physon) are also available.

3. Infomax ICA (http://www.cnl.salk.edu/tewon/

ica_cnl.html): Matlab and C codes for Bell and

Sejnowski’s Infomax algorithm [8] and extended

infomax [2] where, a parametric density model is

incorporated into Infomax to handle both super-

Gaussian and sub-Gaussian sources.

4. EEGLAB (http://sccn.ucsd.edu/eeglab/): EEGLAB

is an interactive Matlab toolbox for processing

continuous and event-related EEG, MEG and

other electrophysiological data using ICA, time/

frequency analysis, artifact rejection, and several

modes of data visualization.

5. ICA: DTU Toolbox (http://isp.imm.dtu.dk/toolbox/

ica/): ‘ICA: DTU Toolbox’ is a collection of ICA

algorithms that includes: (1) ‘icaML’ which is an

efﬁcient implementation of Infomax; (2) ‘icaMF’

which is an iterative algorithm that offers a variety

of possible source priors and mixing matrix con-

straints (e.g., positivity) and can also handle over

and under-complete mixing; (3) ‘icaMS’ which is

an ‘one shot’ fast algorithm that requires time cor-

relation between samples.

Further Issues

1. Overcomplete representation:Overcompleterepresen-

tation enforces the latent space dimension n to be

greater than the data dimension m in the linear model

(1). Sparseness constraints on latent variables are

necessary to learn fruitful representation [20].

2. Bayesian ICA: In contrast to the simple ICA model

(2), Bayesian IC A incorporates uncertainty and

prior distributions of latent variables into the

model (1). Independent factor analysis [21]isa

pioneering work along this direction. EM algo-

rithm for ICA was developed in [22] and full Bayes-

ian learning was adopted [23].

3. Kernel ICA: Kernel methods were introduced to

consider statistical independence in reproducing

kernel Hilbert space [24], developing kernel ICA.

4. Nonnegative ICA: Nonnegativity constraints were

imposed on latent variables, yielding nonnegative

ICA [25]. Rectiﬁed Gaussian prior can also be used

in Bayesian ICA to handle nonnegative latent vari-

ables. It was successfully applied in medical imag-

ing [26].

5. Nonstationarity: Nonstationary characteristics was

used in developing second-order source separation

methods [27, 28].

6. Sparseness: Sparse component analysis is one of hot

issues [29].

7. Beyond ICA: Independent subspace analysis [30]

and tree-dependent component analysis [31] gen-

eralizes ICA, allowing intra-dependence structure

in feature subspaces or clusters.

Summary

ICA has been successfully applied to various applications

of pattern recognition. We have presented a brief over-

view of ICA, starting from the fundamental idea on

learning a linear latent variable model for parsimonious

representation. Natural gradient ICA algorithms were

derived in the framework of maximum likelihood esti-

mation or Kullback matching. Various softwares for ICA

were introduced, so that one could easily apply ICAto his

or her own applications. Further issues were also brieﬂy

mentioned so that readers could follow the status ICA.

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Independent Factor Analysis

▶ Independent Component Analysis

Indexing

▶ Biometric Algorithms

Individuality of Biometric Traits

▶ Individuality of Fingerprints

Individuality of Fingerprints

SARAT C. DASS

,S.PANKANTI

,S.PRABHAKAR

Y. Z

Michigan State University, East Lansing, MI, USA

IBM T.J. Watson Research Center, Hawthorne,

NY, USA

DigitalPersona, Redwood City, CA, USA

Discover Financial Services, Riverwoods, IL, USA

Synonyms

Fingerprint individuality; Fingerprint, Forensic Evi-

dence of; Individuality of biometric traits

Individuality of Fingerprints

741

Definition

Fingerprint individuality is the study of the extent of

uniqueness of ﬁngerprints. It is the most important

measure to be ascertained when ﬁngerprint evidence is

presented in court by experts. A measure of ﬁngerprint

individuality reﬂects the amount of uncertainty asso-

ciated with the experts’ decision, which arises primarily

due to the variability of feature characteristics in a pair

of ﬁngerprints. This inherent variability can cause ran-

dom matching between the pair of ﬁngerprints even

if they are not from the same person. Fingerprint indi-

viduality aims to characterize this randomness in match-

ing them quantitatively in terms of statistical models.

Introduction

The two fundamental premises on which ﬁngerprint

identiﬁcation is based are: (1) ﬁngerprint details are

permanent, and (2) ﬁngerprints of an individual are

unique. The validit y of the ﬁrst premise has been

established by empirical observations as well as based

on the anatomy and morphogenesis of friction ridge

skin. It is the second premise that is being challenged in

recent court cases. The notion of

▶ ﬁngerprint indi-

viduality has been widely accepted based on a manual

inspection (by experts) of millions of ﬁngerprints .

Based on this notion, expert testimony is delivered in

a courtroom by comparing salient features of a latent

print lifted from a crime scene with those taken from

the defendant. A reasonably high degree of match

between the salie nt features leads the experts to testify

irrefutably that the owner of the latent print and the

defendant are one and the same person. For decades,

the testimony of forensic ﬁngerprint experts was

almost never excluded from these cases, and on cross-

examination, the foundations and basis of this testi-

mony were rarely questioned. Central to establishing

an identity based on ﬁngerprint evidence is the as-

sumption of discernible uniqueness; salient features

of ﬁngerprints of different individuals are observably

different, and therefore, when two prints share many

common features, the experts conclude that the own-

ers of the two different prints are one and the same

person. The assumption of discernible uniqueness,

although lacking sound theoretical and empirical fou-

ndations, allows forensic experts to offer an unques-

tionable proof toward the defendant’s guilt.

A signiﬁcant event that questioned this trend oc-

curred in 1993 in the case of Daubert v. Merrell Dow

Pharmaceuticals [1], where the U.S. Supreme Court

ruled that in order for an expert forensic testimony to

be allowed in courts, it had to be subject to ﬁve main

criteria of scientiﬁc validation, that is, whether (1) the

particular technique or methodology has been sub-

jected to statistical hypothesis testing, (2) its error

rates has been established, (3) standards controlling

the technique’s operation exist and have been main-

tained, (4) it has been peer reviewed, and (5) it has a

general widespread acceptance [2]. Forensic evidence

based on ﬁngerprints was ﬁrst challenged in the 1999

case of USA v. Byron Mitchell [3] under Daubert’s

ruling, stating that the fundamental premise for assert-

ing the uniqueness of ﬁngerprints had not been objec-

tively tested and its potential matching error rates

were unknown. After USA versus Byron Mitchell,

ﬁngerprint-based identiﬁcation has been challenged

in more than 20 court cases in the U.S.

The main issue with the admissibility of ﬁnger-

print evidence is that the underlying scientiﬁc basis

of ﬁngerprint individuality has not been rigorously

studied or tested. In particular, the central question

is: What is the uncertainty associated with the ex-

perts’ judgment? How likely can an erroneous deci-

sion be made for the given latent print? In March

2000, the U.S. Department of Justice admitted that no

such testing has been done and acknowledged the

need for such a study [4]. In response to this, the

National Institute of Justice issued a formal solic itation

for ‘‘Forensic Friction Ridge (Fingerprint) Examina-

tion Validation Studies,’’ whose goal is to conduct

‘‘basic research to determine the scientiﬁc validity of

individuality in friction ridge examination based on

measurement of features, quantiﬁcation, and statistical

analysis’’ [4]. The two main topics of basic research

under this solicitation include: (1) the amount of detail

in a single ﬁngerprint that is available for comparison,

and (2) the amount of detail in correspondence be-

tween two ﬁngerprints.

This article gives an overview of the problem of

ﬁngerprint individuality, the challenges faced and the

models and methods that have been developed to

study the extent of uniqueness of a ﬁnger. Interest

in the ﬁngerprint individuality problem is twofold.

First, a scientiﬁc basis (a reliable statistical estimate

of the matching error) for ﬁngerprint comparison

can determine the admissibility of ﬁngerprint

742

Individuality of Fingerprints

identiﬁcation in the courts of law as an ev idence of

identity. Secondly, it can establish an upper bound on

the performance of automatic ﬁngerprint veriﬁcation

systems.

The main challenge in assessing ﬁngerprint indi-

viduality is to elicit models that can capture the varia-

bility of ﬁngerprint features in a population of

individuals. Fingerprints are represented by a large

number of features, including the overall ridge ﬂow

pattern, ridge frequency, location and position of sin-

gular points (core(s) and delta(s)), type, direction, and

location of minutiae points, ridge counts between pairs

of minutia e, and location of pores. These features are

also used by forensic experts to establish an identity,

and therefore, contribute to the assessment of ﬁnger-

print individuality. Developing statistical models on

complex feature spaces is difﬁcult albeit necessary.

In this chapter, minutiae have been used as the ﬁnger-

print feature of choice to keep the problem tractable

and as a ﬁrst step. There are several reasons for

this choice: Minutiae is utilized by forensic experts,

it has been demonstrated to be relatively stable, and

it has been adopted by most of the commonly available

automatic ﬁngerprint matching systems. In principal,

the assessment of ﬁngerprint indi viduality can be

carried out for any particular matching mode, such

as by human exper ts or by automatic systems, as long

as appropriate statistical models are developed on

the relevant feature space used in the matching. Thus,

the framework also extends to the case where matching

is performed based on an automatic system.

Even for the simpler ﬁngerprint feature, nam ely

minutiae, capturing its variability in a population of

ﬁngerprints is challenging. For example, it is known

that ﬁngerprint minutiae tend to form clusters [5, 6],

minutiae information tend to be missed in poor

quality images and minutiae location and direction

information tend to be highly dependent on one

another. All these characteristics of minutiae variabil-

ity, in turn, affect the chance that two arbitrary ﬁnger-

prints will match. For example, if the ﬁngerprint pair

have minutiae that are clustered in the same region of

space, there is a high chance that minutiae in the

clustered region will randomly match one another. In

this case, the matches are spurious, or false, and statis-

tical models for ﬁngerprint individualit y should be

able to quantify the likelihood of spurious matc hes.

To summarize, candidate models for assessing ﬁnger-

print individuality must meet two important

requirements: (1) ﬂexibility, that is, the model can

represent the observed distributions of the minutiae

features in ﬁngerprint images over different databases,

and (2) associated measures of ﬁngerprint individuali-

ty can be easily obtained from these models.

Several works have been reported in the literature

on ﬁngerprint individuality. The reader is referred to

the overview by Pankanti et al. [2] on this subject. This

article focuses on two recent works of ﬁngerprint indi-

viduality where statistical models have been developed

for minutiae to address the questi on of ﬁngerprint

individuality. These two works are (1) Pankanti et al.

[2], and (2) Zhu et al. [7].

The Statistical Test of Biometric

Recognition

Fingerprint based recognition, and more generally bio-

metric recognition, can be described in terms of a test

of statistical hypotheses. Suppose a query image, Q,

corresponding to an unknown identity, I

, is acquired,

ﬁngerprint experts claim that Q belongs to individual

, say. This is done by retrieving information of a

template image T of I

and matching T with Q. The

two competing expert decision can be stated in terms

of two competing hypotheses: The null hypothesis, H

states that I

is not the owner of the ﬁngerprint Q (i.e.,

Q is an impostor impression of I

), and the alternative

hypothesis, H

, states that I

is the owner of Q (i.e., Q is

a genuine impression of I

). The hypotheses testing

scenario is

: I

6¼ I

versus H

: I

¼ I

: ð1Þ

Forensic experts match Q and T based on their

degree of similarity (see Fig . 1). For the present sce-

nario, it will be assumed that the degree of similarity

is given by the number of matched minutiae pairs,

S(Q,T), between Q and T. Large (respectively, small)

values of S (Q,T) indicate that T and Q are simila r to

(respectively, dissimilar to) each other. If S(Q,T )is

lower (respectively, higher) than a prespeciﬁed thresh-

old l, it leads to rejection (respectively, acceptance) of

. Since noise factors distort information in the

prints, two types of errors can be made: False match,

which is also called the Type I error in statistics (since

is rejected when it is true.) (FM) and false non-

match, also known as the Type II error in statistics

(since H

is accepted when H

is false.) False match

Individuality of Fingerprints

743

occurs when an expert incorrectly accepts an impostor

print as a match whereas false nonmatch occurs when

the expert incorrectly rejects a genuine ﬁngerprint as a

nonmatch. The false match and nonmatch rates (FMR

and FNMR, respectively), are the probability of FM

and FNM. The formulae for FMR and FNMR are:

FMRðlÞ¼PðSðQ; TÞ > l jI

6¼ I

Þ;

FNMRðlÞ¼PðSðQ; TÞl jI

¼ I

Þ:

ð2Þ

In case there is no external noise factors that affect the

acquisition of Q and T, it can decided without error

whether Q belongs to I

or not based on the premise of

the uniqueness of ﬁngerprints. However, the process

of ﬁngerprint acquisition is prone to many sources of

external noise factors that di stort the true information

present in Q (as well as T). For example, there can be

variability due to the placement of the ﬁnger on the

sensing plane, smudges and partial prints in the latent

that is lifted from the crime scene, nonlinear distortion

due to the ﬁnger skin elasticit y, poor quality image due

to dryness of the skin, and many other factors. These

noise factors cause information in Q to be distorted,

for example, true minutiae points may be missed and

spurious minutiae points can be generated, which in

turn affects the uncertainty associated with rejecting or

accepting H

The different noise factors can be grouped into

two major sources of variability: (1) inter- and (2) intra-

class ﬁngerprint variability. Intraclass variability

refers to the fact that ﬁngerprints from the same

ﬁnger look different from one another. As mentioned

earlier, sources for this variability includes nonlinear

deformation due to skin elasticity, partial print, non-

uniform ﬁngertip pressure, poor ﬁnger-condition

(e.g., dry ﬁnger), noisy environment, etc. Interclass

variability refers to the fact that ﬁngerprints from

different individuals look very similar. Unlike intra-

class variability, the cause of interclass variability is

intrinsic to the target population. Panels (b) of Fig. 1

show an example of interclass variability for two dif-

ferent ﬁngerprint images. Both intra- and interclass

variability need to be accounted for, when determining

whether Q and T match or not. It is easy to see that

ﬁngerprint experts will be able to make more reli-

able decisions if the inter-class ﬁngerprint variability

is large and the intra-class ﬁngerprint variability is

small. However, less reliable decisions will be made

if the reverse happens, that is, when intraclass varia-

bility is large and interclass variability is small. In

other words, the study of ﬁngerprint individuality

is the study of quantiﬁcation of inter- and intraclass

variability in Q and T, as well as to what extent these

sources of variability affect the ﬁngerprint expert’s

decision.

Statistical Models for Fingerprint

Individuality

The study and quantiﬁcation of inter- and intraclass

variability can be done by eliciting appropriate sto-

chastic (or, statistical) models on ﬁngerprint minutiae.

Figure 2 shows two examples of minutiae (ending and

bifurcation) and the corresponding location and direc-

tion information. Two such approaches are described

in this section, namely, the work done by Pankanti

et al. [2] and the subsequent model that was proposed

by Zhu et al. [7]. Both works focus on modeling the

Individuality of Fingerprints. Figure 1 Illustrating genuine and impostor minutiae matching (taken from [2]). (a) Two

impressions of the same ﬁnger are matched; 39 minutiae were detected in input (left), 42 in template (right), and 36 ‘‘true’’

correspondences were found. (b) Two different ﬁngers are matched; 64 minutiae were detected in input (left), 65 in

template (right), and 25 ‘‘false’’ correspondences were found. ß 2002 IEEE.

744

Individuality of Fingerprints

interclass ﬁngerprint variability, that is, the variability

inherent in ﬁngerprint minutiae of different ﬁngers in

a population.

Pankanti’s Fingerprint Individuality

Model

The set up of Pankanti et al. [2] is as follows: Suppose

the query ﬁngerprint Q has n minutiae and the tem-

plate T has m minutiae denoted by the sets

ffS

; D

g; fS

; D

g; ::::; fS

; D

gg ð3Þ

ffS

; D

g; fS

; D

g; ::::; fS

; D

gg; ð4Þ

where in (3) and (4), S and D refer to a generic

minutiae locatio n and direction pair. To assess a

measure of ﬁngerprint individuality, it is ﬁrst necessary

to deﬁne a minutiae correspondence between Q and T.

A minutiae in Q,(S

), is said to match (or, corre-

spond) to a minutiae in T,(S

), if for ﬁxed positive

numbers r

and d

, the following inequalities are valid:

 S

 r

and jD

 D

 d

; ð5Þ

where

 S



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðx

 x

þðy

 y

ð6Þ

is the Euclidean distance between the minutiae loca-

tions S

 (x

, y

) and S

 (x

, y

), and

 D

 minðjD

 D

j; 2p jD

 D

jÞ

ð7Þ

is the angular distance between the minutiae directions

and D

. The choice of parameters r

and d

deﬁnes

a tolerance region (see Fig. 3), which is critical in

determining a match according to (5). Large (respec-

tively, small) values of the pair (r

) will lead to

Individuality of Fingerprints. Figure 2 Minutiae features consisting of the location, s, and direction, y, for a typical

ﬁngerprint image (b): The top (respectively, bottom) panel in (a) shows s and y for a ridge bifurcation (respectively,

ending). The top (respectively, bottom) panel in (a) shows two subregions in which orientations of minutiae points that

are spatially close tend to be very similar. ß 2007 IEEE.

Individuality of Fingerprints. Figure 3 Identifying the

matching region for a query minutiae (image taken from

[2] and [7]). ß 2002 IEEE.

Individuality of Fingerprints

745

spurious (missed) minutiae matches. Thus, it is neces-

sary to selec t (r

) judiciously so that both kinds of

matching errors are minimized. A discussion on how

to select (r

) is given subsequently.

In [2], ﬁngerprint individuality was measured in

terms of the probability of random correspondence

(PRC). The PRC of w matches is the probability that

two arbitrary ﬁngerprints from a target population

have at least w pairs of minutiae correpondences be-

tween them. Recall the hypothesis testing scenario of

(1) for biometr ic authentication. When the similarity

measure S(Q,T ) is above the threshold l, the claimed

identity (I

) is accepted as true identity. Based on the

statistical hypothesis in (1), the PRC is actually the

false match rate, FMR, given by

PRCðwÞ¼PðSðQ; TÞw jI

6¼ I

Þð8Þ

evaluated at l ¼ w.

To estimate the PRC, the following assumptions

were made in [2]: (1) Only minutiae ending and bifur-

cation are considered as salient ﬁngerprint features for

matching. Other types of minutiae, such as islands,

spur, crossover, lake, etc., rarely appear and can be

thought of as combination of endings and bifurcations.

(2) Minutiae location and direction are uniformly

distributed and independent of each other. Further,

minutiae locations cannot occur very close to each

other. (3) Different minutiae correspondences between

Q and T are independent of each other, and any two

correspondences are equally important. (4) All minu-

tiae are assumed true, that is there are no missed or

spurious minutiae. (5) Ridge width is unchanged

across the whole ﬁngerprint. (6) Alignment between

Q and T exists, and can be uniquely determined.

Based on the above assumptions, Pankanti et al.

were able to come up with the uniform distribution as

the statistical model for ﬁngerprint individuality. The

probability of matching w minutiae in both position as

well as direction is given by

pðM; m; n; wÞ¼

minðm;nÞ

r¼w

M  m

n  r



lðÞ

1  lðÞ

rw

;

ð9Þ

where M ¼ A∕C with A and C deﬁned, respectively,

as the area of overlap between Q and T and C

¼ pr

the area of the circle with radius r

. Pankanti et al.

further improved their model based on several con-

siderations of the occurrence of minutiae. The ridges

occupy approximately

of the total area with the

other half occupied by the valleys. Assuming that the

number (or the area) of ridges across all ﬁngerprint

types is the same and that the minutiae can lie only on

ridges, i.e., along a curve of length

; where o is

the ridge period, the value of M in (9) is changed

from M

¼ A∕C to

M ¼

A=o

; ð10Þ

where 2r

is the length tolerance in minutiae location.

Parameter s (r

) determine the minutiae match-

ing region. In the ideal situation, a genuine pair of

matching minutiae in the query and template will

correspond exactly, which leads to the choice of

, d

) as (0,0). However, intraclass variability factors

such as skin elasticity and nonuniform ﬁngertip pres-

sure can cause the minutiae pair that is supposed to

perfectly match, to slightly deviate from one another.

To avoid rejecting such pairs as nonmatches, nonzero

values of r

and d

need to be speciﬁed for matching

pairs of genuine minutiae. The value of r

is deter-

mined based on the distribution of the Euclidean dis-

tance between every pair of matched minutiae in the

genuine case. To ﬁnd the corresponding pairs of min-

utiae, pairs of genuine ﬁngerprints were aligned, and

Euclidean distance between each of the genuine min-

utiae pairs was then calculated. The value of r

was

selected so that only the upper 5% of the genuine

matching distances (corresponding to large values of r)

were rejected. In a similar fashion, the value of d

was

determined to be the 95th percentile of this distribu-

tion (i.e., the upper 5% of the

▶ genuine matching

angular distances were rejected).

To ﬁnd the actual r

and d

, Pankanti et al. used a

database of 450 mated ﬁngerprint pairs from IBM

ground truth database (see [2] for details). The true

minutiae locations in this database and the minutiae

correspondences between each pair of genuine ﬁnger-

prints in the databas e were determined by a ﬁngerprint

expert. Using the ground truth correspondences, r

and d

were estimated to be 15 and 22.5, respectively.

These values will be used to estimate the PRC in the

experiments presented in this paper.

Pankanti et al. [2] were the ﬁrst to attempt in

quantifying a measure of ﬁngerprint individuality

746

Individuality of Fingerprints

based on statistical models. However, the proposed

uniform model does have some drawbacks. Compari-

son between model prediction and empirical obser va-

tions showed that the corrected uniform model grossly

underestimated the matching probabilities. The inher-

ent drawbacks of the uniform model motivated the

research by Zhu et al. [7] to propose statistical distri-

butions that can better represent minutiae variability

in ﬁngerprints.

Mixture Models for Fingerprint

Features

Zhu et al. [ 7] proposed a mixture model to model the

minutiae variability of a ﬁnger by improving Assump-

tion (2) of [2]. A joint distribution model for the

k pairs of minutiae features {(S

, D

), j ¼ 1, 2, ... k}

is proposed to account for (1) clustering tendencies

(i.e., nonuniformity) of minutiae, and (2) dependence

between minutiae location (S

) and direction (D

)in

different regions of the ﬁngerprint. The mixture model

on (S,D) is given by

f ðs;y jY

Þ¼

g¼1

ðs jm

; S

Þf

ðyjn

; k

Þ; ð11Þ

where G is the total number of mixture components,

() is the bivariate Gaussian density with mean m

and covariance matrix S

, and

ðyjn

; p

Þ¼

vðyÞ if 0 y < p

ð1 p

Þvðy pÞ if p y < 2p;

(

ð12Þ

where v(y) is the Von-Mises distribution for the min-

utiae direction given by

vðyÞvðyjn

; k

Þ¼

ðk

expfk

cos2ðy n

Þg ð13Þ

with I

) deﬁned as

ðk

Þ¼

expfk

cosðy n

Þgdy: ð14Þ

In (13), n

and k

represent the mean angle and the

precision (inverse of the variance) of the Von-Mises

distribution, respectively (see [7] for details). The dis-

tribution f

in (12) can be interpreted in the following

way: The ridge ﬂow orientation, o, is assumed to follow

the Von-Mises distribution in (13) with mean n

and

precision k

. Subsequently, minutiae arising from the

gth component have directions that are either o or

oþp with probabilities p

and 1p

, respectively.

The model described by (11) has three distinct

advantages over the uniform model: (1) it allows for

different clustering tendencies in minutiae locations

and directions via G different clusters, (2) it incorpo-

rates dependence between minutiae location and di-

rection since, if S is known to come from the gth

component, the direction D also comes from the

gth component, and (3) it is ﬂexible in that it can ﬁt

a variety of observed minutiae distributions adequ-

ately. The estimation of the unknown parameters in

(11) has been described in details in [7].

The effectiveness of the mixture models can also be

shown by simulating from the ﬁtted models and check-

ing to see if a similar pattern of minutiae is obtained as

observed. Figure 4a shows a ﬁngerprint whose

Individuality of Fingerprints. Figure 4 All (S,D) realizations from the ﬁtted mixture model (b), and from the uniform

distribution (c) for the original image in (a). The true minutiae locations and directions are marked in (a). Images are taken

from [7]. ß 2007 IEEE.

Individuality of Fingerprints

747

minutiae features were ﬁtted with the mixture distri-

bution in (11). Figure 4c shows a simulated realization

when each S and D is assumed to be uniformly

distributed independent of each other. Note that

there is a good agreement, in the distributional sense,

between the observed (Fig. 4a) and simulated minutiae

locations and directions from the ﬁtted mixture model

(Fig. 4b), but no such agreement exists for the uniform

model.

Zhu et al. [7] obtains a closed form expression for

the PRC corresponding to w matches under similar

assumptions of Pankanti et al. [ 2] (barring Assu-

mption (2)). The probability of obtaining exactly

w matches given there are m and n minu tiae in Q

and T, respectively, is given by the expression



ðw ; Q; TÞ¼

lðQ;TÞ

lðQ; TÞ

ð15Þ

for large m and n;(15) corresponds to the Poisson

probability mass function with mean l(Q, T) given by

lðQ; TÞ¼mnpðQ; TÞ; ð16Þ

where

pðQ; TÞ¼PðjS

 S

 r

and jD

 D

 d

ð17Þ

denotes the probability of a match when (S

, D

) and

) are random minutiae from the mixture dis-

tributions ﬁtted to Q and T, respectively. The mean

parameter l(Q, T) can be interpreted as the expected

number of matches from the total number of mn

possible pairings between m minutiae in Q and n

minutiae points in T with the probability of each

match being p(Q, T ).

Incorporating Interclass Variability Via

Clustering

The above PRC was obtained for a single query and

template ﬁngerprint pair. An important difference be-

tween the proposed methodolog y and previous work is

that mixture models are ﬁtted to each ﬁnger, whereas

previous studies assumed a common distribution for

all ﬁngers/impressions. Assuming a common minutiae

distribution for all ﬁngerprint impressions has a serious

drawback, namely, that the true distribution of minu-

tiae may not be mode led well and important cluster

information may be smoothed out. Zhu et al. [7] adopt

an agglomerative hierarchical clustering procedure on

the space of all ﬁtted mixture models to obtain a

reliable representation of the minutiae in all ﬁnger-

prints in a population as well as to reduce computa-

tional time to obtain the PRC estimates. In this

framework, the probability of obtaining exactly u

matches corresponding to clusters C

and C

is given by



ðu ; C

; C

Þ¼e

lðC

lðC

; C

; ð18Þ

where l(C

) is interpreted as the mean number of

matches between any Q arising from C

and T arising

from C

, and is calculated based on the mean (average)

mixture den sities in clusters C

and C

. The overall

probability of exactly u matches is



ðuÞ¼

ij

jjC



ðu ; C

; C

ij

jjC

ð19Þ

where jC

j is the total number of mixtures in cluster

. The overall PRC corresponding to w matches is

given by

PRC ¼

u¼w



ðuÞ: ð20Þ

To remove the effect of very high or very low PRCs,

the 100(1a)% trimmed mean is used instead of

the ordinary mean as in (19). The lower and upper

100a∕2th percentiles of {p

∗

(u; C

), 1  i, j  N

∗

}

are denoted by p

∗

(u;a∕2) and p

∗

(u;1a∕2). Also,

deﬁne the set of all trimmed p

∗

(u; C

) probabili-

ties as Tfði; jÞ : p



ðu; a=2Þp



ðu ; C

; C

Þ



ðu; 1  a=2Þg. Then, the 100(1a)% trimmed

mean PRC is

PRC

u¼w



ðuÞ; ð21Þ

where



ðuÞ¼

ði ;jÞ2T

jjC



ðu ; C

; C

ði;jÞ2T

jjC

ð22Þ

In the next section, the trimmed mean is with a¼0.05.

Experimental Results

The results in this section are taken from Zhu et al. [7];

the interested reader is referred to more details

748

Individuality of Fingerprints

discussed in the chapter. The methodology for asses-

sing the individuality of ﬁngerprints are validated on

three target populations, namely, the NIST Special

Database 4 [8], FVC2002 DB1 and FVC2002 DB2 [9]

ﬁngerprint databases. The NIST ﬁngerprint database

[8] is publicly available and contains 2,000 8-bit gray

scale ﬁngerprint image pairs of size 512  512 pix els.

Because of the relative large size of the images in the

NIST database, the ﬁrst image of each pair is used

for statistical modeling. Minutiae could not be auto-

matically extracted from two images of the NIST data-

base due to poor quality . Thus, the total number of

NIST ﬁngerprints used in the authors’ experiments is

F ¼ 1,998.

For the FVC2002 database, also available in the pub-

lic domain, two of its subsets DB1 and DB2 is used. The

DB1 impressions (images size ¼388 374) are acquired

using the optical sensor ‘‘TouchView II’’ by Identix,

while the DB2 impressions (image size ¼ 296  560)

are acquired using the optical sensor ‘‘FX2000’’ by

Biometrika. Each database consists of F ¼ 100 different

ﬁngers with eight impressions (L ¼ 8) per ﬁnger.

Because of the small size of the DB1 and DB2 data-

bases, a minutiae consolidation procedure was adopted

to obtain a master (see [7] for the details). The mixture

models were subsequently ﬁtted to each master.

Zhu et al. developed a measure of goodness of ﬁt of

hypothesized distributions to the observed minutiae

based on a chi-square type criteria. Two tests were

considered, namely, the Freeman–Tukey and Chi-

square tests. The results for the goodness of ﬁt for

two hypothesized distributions, namely, mixture and

uniform models are reported in Table 1. For all the

three databases, the number of ﬁngerprint images

with p-values above (corresponding to accepting the

hypothesized distribution) and below the thres-

hold 0.01 (corresponding to rejecting the hypothesized

distribution) were obtained. Note that the entries

in Table 1 imply that the mixture model is generally

a better ﬁt to the observed minutiae compared to

the uniform; for example, the mixture model is a good

ﬁt to 1,666 images from the NIST database (corres-

ponding to p-values above 0.01) based on the Freeman–

Tukey test. For the Chi-square test, this number is

1,784. In comparison, the uniform model is a good

ﬁt to only 905 and 762 images, respectively.

Zhu et al. compared the PRC obtained by [7] with

those of Pankanti et al. [2]. The query and template

ﬁngerprints in the NIST and FVC databases are ﬁrst

aligned using the matcher described in [10], and an

overlapping area between the two ﬁngerprints are

determined. To compute the PRCs, the mixture models

are restricted onto overlapping area (see [7] for more

details). Table 2 gives the mean m, mean n, mean

overlapping area and M (see equation (10)) for the

NIST and Fvc databses, whereas Table 3 gives the

corresponding PRCs. The empirical PRC is computed

as the proportion of impostor pairs with 12 or greater

matches among all pairs with m

and n values within  5

of the mean in the overlapping area. The empirical

probabilities of at least w matches are obtained by

counting the number of ﬁngerprint pairs with 12 or

more matches divided by the total number of pairs.

Thus, one should note that the empirical probability is

Individuality of Fingerprints. Table 1 Results from the Freeman–Tukey and Chi-square tests for testing the goodness of

fit of the mixture and uniform models

Mixture model

Freeman–Tukey Chi-square

p-value NIST (1,998) DB1 (100) DB2 (100) NIST (1,998) DB1 (100) DB2 (100)

p-value>0.01 (mixture accepted) 1,864 71 67 1,569 65 52

p-value0.01 (mixture rejected) 134 29 33 429 35 48

Uniform model

Freeman–Tukey Chi-square

p-value NIST (1,998) DB1 (100) DB2 (100) NIST (1,998) DB1 (100) DB2 (100)

p-value>0.01 (uniform accepted) 550 1 0 309 1 0

p-value0.01 (uniform rejected) 1,448 99 100 1,689 99 100

Entries correspond to the number of fingerprints in each database with p-values above and below 0.01. The total number of ﬁngerprints

in each database is indicated in parenthesis. Table entries are taken from [7]

Individuality of Fingerprints

749