1. INTRODUCTION
Biometrics-based
personal identification attempts to answer the questions "Who are
you?" and "Are you who you claim to be?",
Weicheng and Tieniu (1999).
Personal identification is very vital in our daily lives. For instance, we are
required to prove our identity to gain access to a bank account, to enter a
protected site, to draw cash from an ATM, to login to a computer, to claim
welfare benefits, to cross national borders, and so on. Conventionally, we
identify ourselves and gain access by physically carrying passports, keys,
badges, tokens, and access cards or by remembering passwords, secret codes, and
personal identification numbers (PINs). Unfortunately, these means of
identification can be lost, duplicated, stolen, or forgotten. Such loop-holes
have caused major problems to all concerned. Biometrics is personal physical or
biological measurements about an individual. To uniquely identify an individual
based on biometrics data, the following characteristics of biometrics data are
needed: highly unique to each individual, easily obtainable, time-invariant,
easily transmittable, able to be acquired as non-intrusive as possible and
distinguishable by humans without much special training. Human beings are so uniquely made that there are many
prints that cannot be duplicated or seen by two individuals. These are
fingerprints, palm-prints, foot-prints, toes-prints and other unique features such
as iris, etc. These are biometric markers that can be used to identify human
beings since they are known to be unique for each individual. Using
fingerprints to identify or verify an individual's identity requires special
fingerprint comparison skill. For a pair of untrained eyes, it may be difficult
to distinguish one set of fingerprints from another.
The
criminal investigation department in the police force studies the patterns of
the person’s fingerprints. These patterns are little ridges on the end of an
individual’s fingers and thumb that are arranged in a pattern of spirals and
loops. Nature has made the patterns in order to help human beings to grip and
hold onto things otherwise, human hands would be frictionless, slippery and
could not hold things without sliding. These fingerprints are so unique that no
two persons have exactly the same pattern of fingerprints. These patterns are
formed during the gestation period and remain throughout a person’s life.
Fingerprint is a unique method of individual identification. Fingerprints could
be described as having three basic patterns namely: arches, loops and whorls.
These shapes and contours were later sub-divided into eight basic patterns used
to identify an individual, they are;
Arches:
these occur in about 5% of the
encountered fingerprints. The ridges of the finger run continuously from one
side of the finger to the other and make no backward turn. Normally, there is
no delta in an arch pattern but if it exists, there must be no re-curving ridge
that intervenes between the core and delta points. Loops: these can be seen in almost 60 to 70% of the
fingerprints. The ridges make a backward turn in loops but they do not twist.
This backward turn or loop is distinguished by how the loop flows on the hand and
not by how the loop flows on the card where the imprint is taken. This imprint
on the fingerprint is similar to the reverse image that we see when we look at
ourselves in the mirror. A loop pattern has only one delta.
Whorls:
these can be found in about 25 to
35% of the fingerprints. Some of the ridges in a whorl make a turn through at
least one circuit. Therefore any pattern that contains two or more deltas will
be a whorl. From the three basic types of fingerprints, we obtain eight
sub-classifications according to the combinations of the basic three, they are
as follows;
(1) Plain Arch: it starts on one
side of the finger and the ridge then slightly cascades upward. (2) Tented Arch: it lies in the ridges in the
centre and is not continuous like the plane
arch. (3) Ulnar Loop: they are not
very common and most of the times would be found on the index fingers. (4)
Radial Loop: the flow of this pattern runs from the thumb towards the little
finger on the hand. (5) Double Loop: it has two distinct and separate shoulders
for each core, two deltas and one or more ridges that make a complete circuit.
(6) Plain Whorl: the ridges in these whorls makes a turn of one complete
circuit with two deltas and are therefore circular or spiral in shape. (7) Central
Pocket Loop Whorl: these whorl ridges make one complete circuit and may be
oval, circular, spiral or any variant of a circle. (8) Accidental Whorl: whorls
containing ridges that match the characteristics of a particular whorl
sub-grouping are referred to as accidental whorls, Laufer (1917) and Tyler
(1981).
Another frequently used biometric is the
face. Face pictures are often used as a means for verification, as evidenced by
various picture ID cards because a person's face is an easily accessible and verifiable
biometric to human eyes. Nevertheless, faces are known to be ambiguous for
identification/verification purposes, as different individuals can have similar
facial features. Voice is another frequently used biometric, which is used for
differentiating one from another. On the negative side, using voice as
biometric data to identify or verify identity also lacks the uniqueness that
fingerprint-based identification/verification techniques can provide. The human
iris recently has attracted the attention of biometrics-based
identification/verification research and development community. It has been
demonstrated that with a feature vector of relative small size, the human iris
exhibited high discrimination power for different individuals.
Biometrics is simple and the results offer a
solid business case for deployment, James and Rick (2006). Biometrics offer
organizations solutions to many of the security problems that occur on a daily
and repetitive basis. Deployment of biometrics has saved commercial businesses
reasonable payroll costs per annum by eliminating fraudulent practices by both
business and government. The use of biometrics better guarantees a specific
individual's access, whether that access is a door or an attendance verifier.
The most common technologies for biometric authentication range from hand scans
to retina scans, and from voice to facial patterns. But all biometric
technologies share certain main principles for recognizing an individual and
determining whether a person will gain access. Biometrics clearly enables
organizations to quickly upgrade security postures. Different biometric
technologies fit different applications; in other words, one technology may not
be the best product for all of the possible applications at a given facility.
Out of the numerous biometric identifications
parameters discussed so far, our interest is on the fingerprint. We want to
develop a mathematical model for fingerprints and fit a probability
distribution to it. As we have observed, there are escalation of crimes in
Nigeria because people easily get away with crimes undetected. The fingerprint
model is meant to identify an individual based on their prior and posterior
probabilities. If the developed system sees the prior information as the
posterior information represented by their various probabilities, the
individual will be verified as being authenticated and as the person he is
claimed to be. Though fingerprints have different security application, but we
are focusing on crime control. We observed from the literatures and to the best
of our knowledge that there was no mathematical model developed to quantify the
fingerprint-based biometric parameters. Hence, in this study, we have developed
a fingerprint-based biometric identification model for crime prevention and
control.
The major aim of this study is to develop a
fingerprint-based biometric identification method for crime identification and
control in Nigeria and the specific objective includes:
1.
To
develop a Poisson Distribution model for fingerprint (thumbprint)
identification.
2.
To
fit the developed model.
3.
To
apply a Bayesian Statistical model to relate prior information to posterior
information of an individual’s fingerprints.
4.
To
determine prior and posterior probabilities of individual’s fingerprints.
5.
To
make a valid conclusion based on the result from the study.
2. LITERATURE
REVIEW
Sir
William J. Herschel was the one who conceive the notion for fingerprints and
elaborated the system which was subsequently developed and placed on a truly
scientific basis by Sir Francis Galton, see Laufer
(1917). Ever since then, the development of finger prints has metamorphosed to
other Biometric variable identifications. Fingerprints has advanced with the
application of modern technologies such as scanners, computerization etc to identify an individual. In the words of Tyler (1981),
a fingerprint is an impression from the ridge crests of the friction-skin of
the ventral surface of a digit. As the terminal phalange is the only one
constantly exhibiting a pattern configuration, it is the one utilized for
making identification records, although identification may as accurately be
made from an impression of either remaining phalange, or the palm of the hand
or the sole of the foot. An impression may be naturally or artificially made.
By naturally, it means the absence of any transfer medium other than nature's
perspiration residuum. A natural print is also sometimes called a latent print
because it is often invisible, and, to make a permanent record available for
inspection and comparison, it must be visualized and fixed by one of the usual
methods. An artificial impression may be intentionally or unintentionally made,
but the medium for recording it is externally applied and is not nature's skin
excretions; for example, paint on the hand of a painter; grease on a mechanic's
hand; blood on the hand of a butcher; ink on a printer's hand; or the surface
is purposely inked as in commercial and institutional spheres. Artificial
impressions made with oil, cold cream or other invisible medium must be
visualized and fixed as are latent prints.
The bloody print of a finger
tip that is found near the scene of crime and leads to the apprehension
of the criminal; and today no detective who carries "coke" in his
left arm is necessary in a case where so substantial a clue is found as an
impress of the lineation of the guilty person's thumb. Moreover, Bertillon has
taught the police the reliability of finger prints as a part of a system of
identification far superior to the ordinary photograph, Smith (1917). Also a
case was recently tried at the Highgate police court
in London which brought out the infallibility of the finger-print test as a
means of identifying criminals. There was also a Conviction on Finger-Print
Evidence in Norway. The accused denied the commission of the offense, but the
jury, after a half-hour's deliberation, returned a unanimous verdict of guilty.
It is said this is the first conviction in Norway solely on finger-print
evidence. In another report, the Boston police authorities, says the
Transcript, have recently established the most efficient and up-to-date finger
print system in the country. Every country in Europe, it adds, is now using the
finger print system in connection with the Bertillon method of measurement. All
that is needed for finger prints is a pad of paper and some ink, John (1910).
Kim (2000) observed that the Army-led
biometrics program is working to create a product list of commercially
available systems that comply with emerging industry and government standards
for the Defense Department to use when acquiring fingerprint scanners, voice
authentication devices and other biometrics equipment, according to documents
and officials. Biometrics involves safeguarding access to computers, weapons
systems and facilities using technologies that can read a person's physical
characteristics. Technologies of interest include fingerprint verification,
hand geometry, retinal scanning, iris scanning, signature verification, facial
recognition, voice authentication and gait authentication. Biometric
technologies are poised to play a key role in the military’s transformation to
a 21st-century fighting force, Hampton (2001). Biometric technology is capital
intensive; Fawzia (2009) observed that the Pentagon
plans to boost funding for a biometrics science and technology office that
developed a high-speed iris capture system, a rugged field-portable fingerprint
workstation and a face-recognition system, according to fiscal year 2010 budget
documents. The department has budgeted $10.5 million for the office in FY-09
and wants to raise the investment to $11 million in FY-10, which would help to
refine a S&T roadmap.
Fawzia (2010) observed that
a new NATO working group assessing how to create a long-term capability to use
biometrics information in operations ranging from the Afghanistan war to
humanitarian missions will develop technology standards and concepts of
operation in an effort that could last years. There is an ongoing effort to
create a biometric database for coalition forces. Hampton (2002) observed that
Biometrics, a technology area of interest to the Defense Department for years,
is set to move out of the realm of research and development and into mainstream
application, but Anil and Arun (2015) were of the
opinion that biometric recognition, or simply biometrics, refers of individuals
based on their biological and behavioral. Biometrics is a field that grew out
of fingerprinting to encompass emerging areas like gait and heartbeat
recognition and it is extensively used in China, Mara (2012). Sebastian (2007)
observed that senior defense department officials have tentatively approved a
proposal to consolidate the military’s biometrics programs and fund them
through the Pentagon’s base budgets, breaking with the practice of funding this
key growth area for the military through supplemental spending. Kim (2000) was
of the opinion that the Automated Biometrics Identification System,
and its role in the U.S.-Iraqi cooperation is in question. The database holds
biometric information from 2.5 million individuals, primarily collected in Iraq
and Afghanistan. One issue that has sparked some concern is the idea that the
data, if it falls into the wrong hands, could underpin an effective “enemies”
list that could be used in crackdowns.
From the literatures, we observed that no
researcher quantifies biometric-based identification, rather, they were
theoretically presenting their findings, for this reason, we model
biometric-based identification, fingerprints; determine its probability
distribution, fit the biometric markers into the distribution and establish a
relationship between the prior information and the posterior information to
track criminality using the fingerprints (thumbprints).
3. MATERIALS AND
METHODS
3.1. Method Data Collection
It was difficult to collect biometric data in an already made form from agencies
and data centers because of the sensitive nature of such data. The biometric
data in the data base of banks and National Identity management (the fingerprint
and facial pictures) was impossible to collect from the organizations. Finally,
we lack the capacity to collect such data and make meaning out of it because of
the instruments needed; however, we resort to simulation to produce a near-real
life data that imitate the real data obtainable from fingerprints measurements.
3.2. Nature of the Problem
Haven studied the characteristics of fingerprints (thumbprints) such as
uniqueness, invariants properties, random occurrence, non-over lapping,
discreteness, counting system, etc we observed that the
fingerprints follow Poisson random variable. We
shall model a fingerprint of individuals with the aim of detecting and
controlling crimes. In this type of data, there is no identity theft because the
biometric data is peculiar to individuals. Studies have shown that no two
persons can share the same biometric information. Our interest is on thumb
prints even though there are so many other prints peculiar to individuals. We
shall model the fingerprints and fit a probability distribution to it, determine
the prior and posterior probability of the fingerprint in order to identify
individuals and detect criminality.
3.3. 1 Method of Data Analysis
We
shall fit the Poisson probability distribution model to the data. From the
probability model, we shall determine the prior probability of the
fingerprints; adapt the Bayesian statistical model to determine the posterior
probabilities of the individual. Bayesian statistical model has memory and
relates the prior to posterior probabilities. This type of information needs to
be stored by a system with memory and can easily be recall to relate the
current information with the previously stored information and produced results
based on the authentication. If the probability of the priors matches that of
posterior according to the system’s classification, we classify the data as
coming from the same individual and classify as not coming from the same
individual if otherwise. The system should see the prior probability as the
posterior probability using the unique classification property inherent in
fingerprint. The number of thumbprints is assumed to follow a Poisson
distribution according to the properties of Poisson random variables.
For
us to model this problem, we should have in mind that each thumbprint of an
individual, Ai :i = 1, 2, . .
. , n, is an independent random variable that occur naturally and do not vary
with time; therefore, it has a stationery increment in addition to the other
properties of Poisson random variables listed above. Hence, the random
variables follows a homogenous Poisson process with parameter
; that is Ai ~HPP().
3.3.2 Notation and Assumptions
A
counting process {N(t), t ≥ 0} is called a
non-homogeneous Poisson Process with rate function {l
(t), t ≥ 0} if
(i).
The number of events at time zero is equal to zero or N(o)
= 0
(ii)
The number of events in non-overlapping time interval are independent or {N(t), t ≥ 0} has independent increment
(iii) o(h) - some function of smaller order than h which
satisfy the condition 
(iv) The probability that exactly one event occur in
a small interval of length t + h approximately equal to l(t).
h or P{N(t + h)
= K + 1/N(t) = K} = l(t).h + 0 (h)
(v).
The probability that no event occur in the time
interval t + h is
P{N(t + h) = K/N(t)
= K} = 1 - l(t).h + o(h)
(vi)
The probability that more than one event will occur in a small interval of
length t + h is negligible or P{N(t + h) = K + j/N(t)
= K} = 0(h); j ≥ 2.
(vi) The events must occur at random
We
write {N(t), t ≥ 0) ~
NHPP (l(.)) to denote that {N(t), t ≥ 0) is a
non-homogeneous Poisson process with
rate function l(.); When l(t) = l
for all t ≥ 0, then; NHPP becomes a HPP. Thus, NHPP is a generalization
of HPP. In both HPP and NHPP, events take place one at a time, Amuji et al (2019).
Still buttressing the point that thumbprints
follow Poisson distribution, we have the following: Poisson distribution
provides a realistic model for many random phenomena. Since the value of
Poisson random variable are the non-negative integers, any random phenomenon
for which a count of some sort is of interest is a candidate for modelling by assuming a Poisson distribution. Such a count
might be the number of fatal traffic accident per week in a given state, the
number of radioactive particle emissions per unit of time, the number of
telephone calls per hour coming into the switchboard of a large business, the
number of meteorites that collide with a test satellite during a single orbit, etc; if certain assumptions regarding the phenomenon under
observation are satisfied, the Poisson model is the correct model, Mood et al
(1974). The random variables (thumbprints) is independent of time (i.e., does
not change with time) and hence has a stationery increment and therefore a
homogenous Poisson process with parameter (HPP()). From the above, we observed that thumbprint
satisfies the assumptions above.
The information from individuals is collected
together and stored in the memory for future retrieval; hence, the joint
density (likelihood) function does that. The distribution function that has the
ability to remember (store and recall) the past history of Ai and
relate it with the current information about Ai is Bayesian
Statistic model. We need the posterior probability or the current chances of
getting the information about the individual given that the information had
been collected and stored. Bayesian statistic has a memory and can link the
prior information of Ai to the posterior information of Ai
and store them. The biometric expert
access this stored information and link it with the current information about Ai;
if it corresponds, then Ai can be identified as the owner of the
thumbprints and either granted access or arrested. Hence, posterior probability
of Ai links the prior probability of Ai to produce a
result.
3.3.3 Formulation of the Model
We
formulate the model as follows:
Let 
;
i = 1, 2, . . . , n
Where
Ai = xi are random variables (thumbprints)
The
probability that the random variable Ai assumes a value ai is equal to the probability that the random
variable Xi assumes value xi.
But
That
is, each of the random variables above the same probability distribution function
(1)
See
Mood et al. (1974)
Since
the random variables follows Poisson distribution, the probability mass
function (pmf) is as given in equation (1). Equation
(1) is written in equation (2) for one of the random variable.
(2)
For
many of the random variables (many individuals thumb prints), we write equation
(2) as shown in equation (3) below.
(3)
But
since each of the random variables is the same as the data it supplied which is
invariant and has stationary increment, that is, 
is not
associated with time, , then the likelihood (joint distribution) function of
the distribution in equation (3) is given in equation (4) as
(4)
Equation
(4) can be written as equation (5) below,
(5)
Let
(6)
Since
n! is large, we approximate the value of n! by the Sterling’s formula
(7)
Taking
the natural log of both sides, we have
(8)
(9)
Therefore
substituting equation (9) into equation (5), we have equation (10) as,
(10)
Equation
(10) can now be written as equation (11) which is the probability model for the
biometric identification.
; (11)
n
= the sample size
Equation
(11) is the probability of unique number of thumbprint per individual at any
period x. The probability model is used to determine the prior probability of
individuals.
To determine the posterior probability, we
apply the Bayesian Statistics model given in equation (12), see Arua et al (2000).
Relating
the above equation to the fingerprint, we have equation (12) below as,
(12)
where 
is the
probability of obtaining biometric information about individual thumbprints, Ai
, given that we previously had the thumbprint history about the individual, Ai.
3.4. Implementation
To
implement the developed model of equations (11) and (12), we simulated the
individual thumbprint data using Minitab 16.0 software and used MS Excel for
the computations. According to the literatures on thumbprints, the average
number of lines present in a thumbprint is 64. We made use of this information
in simulating forty (40) sample data points for the study as presented in Table
4.1.
4. Data Presentation
and Analysis
4.1. Data Presentation
4.3 Interpretation of Results
The prior probabilities
are represented by the column “prior pr” and the
posterior probabilities are represented by the column “post. prob” in table 4.2. The
fingerprint biometric reader (system) matches these two probabilities as coming
from one individual. Though the value may differ from human eyes but using a
unique identification and invariant property of biometric identification and classification,
the system sees both probabilities as coming from one individual and hence
classifies them as the owner of the finger print. For example, the first person
has a prior probability of 1.54E-34 and posterior probability of 1.75E-14, etc.
the system should see these two probabilities as coming from one persons finger prints. We made use of equations (11) and
(12) in the computation of priors and posterior probabilities.
5. DISCUSSIONS AND
CONCLUSION
5.1. Discussion
In
this study, we have developed thumbprint-based biometric identification model
for biometric authentication to determine the identity of individuals in other
to prevent crime. The authentication is based on prior and posterior
information represented by their respective probabilities. These biometric data
(information) is unique to the individual and eliminates the problem of
identity theft which is common to cryptographic authentication such as PIN
(personal identification number) etc that can be
stolen by other individuals to commit fraud with it. In biometric
authentication as a preferred means of detecting crime, the biometric data is
unique to the individual and cannot be stolen. What is actually done is to
pre-collect biometric data (represented by prior probabilities) from the
individual and store it in a common pool (data base), then, in the case where
an individual is being suspected of crime, a fresh data is collected from him
and a match and comparison test is done automatically by the system to detect
if the current biometric data matches the previous data stored on the same
individual (represented by posterior probabilities). If it does, the individual
automatically become a culprit. The authenticating system must have a memory
that stored the prior probabilities, recognizes and matches it with the
corresponding posterior probabilities. In this study, we developed a
probability model and applied Bayesian statistical model as a biometric
authentication model. We match the prior probability of the biometric data with
those of the posterior probabilities. With this link, we can identify
individuals, determine who has a criminal record, arrest the situation on time,
prevent and track crimes.
5.2. Conclusion
In
this study, we modeled thumbprints of individuals as a biometric based
identification and authentication for effective crime controlling and
prevention. Two probability models were developed and applied to achieve the
purpose, namely; the homogenous Poisson process model and Bayesian statistical
models. These two models are collectively called the Biometric authentication
and identification models. Since this kind of data is rear and difficult to
collect, we resort to simulation as an alternative to data
collection using Minitab 16.0 as a simulation tool.
Competing Interests
There
is no competing interest
Authors Contributions
Each
author contributed in the preparation of the manuscript, Literature review
analysis and presentation of the work.
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