By Amuji, HO; Okoroji, LI; Osuji, WI; Mbachu, JC; Obasi, A (2023). Greener Journal of Science, Engineering and Technological Research, 12(1): 1-10.
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ISSN: 2276-7835
Vol. 12(1), pp. 1-10, 2023
Copyright ©2023, the copyright of this article is retained by the author(s)
DOI: https://doi.org/10.5281/zenodo.7765092
https://gjournals.org/GJSETR
1,5Department of Statistics, Federal University of Technology Owerri, Nigeria
2Department of Transport Management Technology, Federal University of Technology Owerri, Nigeria
3Department of Mathematics, Federal University of Technology Owerri, Nigeria
4Department of Maritime Management Technology, Federal University of Technology, Owerri Nigeria.
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Type: Research
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DOI: 10.5281/zenodo.7765092
In this paper, we developed the probability distribution for a fingerprint-based biometric identification model for crime prevention and control. We fitted Poisson distribution to the fingerprint data and applied Bayesian statistical model to determine and relate the prior to the posterior probabilities and determine the prior and posterior probabilities of the finger prints. The earlier information about the owner of the fingerprint is represented by the prior probabilities and the information from the individual on the second check of the fingerprint is represented by the posterior probabilities. Using the 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. The first person has a prior probability to be 1.54E-34 and the posterior probability to be 1.75E-14, etc.
Accepted: 02/03/2023
Published: 23/03/2023
Harrison O. Amuji
E-mail: harrison.amuji@ futo.edu.ng
Phone: +234 803 647 8180
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:
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
3.3.2 Notation and Assumptions
A counting process {N(t), t ≥ 0} is called a non-homogeneous Poisson Process with rate function {λ (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 λ(t). h or P{N(t + h) = K + 1/N(t) = K} = λ(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 – λ(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 (λ(.)) to denote that {N(t), t ≥ 0) is a non-homogeneous Poisson process with rate function λ(.); When λ(t) = λ 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 . 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:
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
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.
For many of the random variables (many individuals thumb prints), we write equation (2) as shown in equation (3) below.
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
Equation (4) can be written as equation (5) below,
Let
Since n! is large, we approximate the value of n! by the Sterling’s formula
Taking the natural log of both sides, we have
Therefore substituting equation (9) into equation (5), we have equation (10) as,
Equation (10) can now be written as equation (11) which is the probability model for the biometric identification.
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,
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
Table 4.1 Thumbprint Data
Simulated using Minitab 16.0
In this section, we analyze the data presented in Table 4.1 using the methods described in chapter three and equations (11) and (12) respectively with and present the results in table 4.2.
Table 4.2 Determination of unique Probabilities of thumbprints, prior and posterior probabilities
3.41E-38
8.32E+81
2.5E-71
6.34E-62
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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