Can t Qute Put Our Fnger On It 231 Can t Qute Put Our Fnger On It Séamus Ó Ceallagh Alva Sheeley Adan Crangle Unversty College Cork Cork, Ireland Advsor: James J. Grannell Summary There are two man paths to dentfcaton through fngerprnts. Global analyss reles on the specfc arrangement and characterstcs of the rdges of a prnt. Local analyss assumes that the ndvdualty of a prnt s based on the poston and orentaton of the two basc types of mnutae. We subdvde a prnt nto a grd of square cells, consder the dstrbuton of rdge features, and calculate probabltes from combnatoral analyss. We make predctons by refnng parameters of the model, such as the number of mnutae requred for a postve match between two generc prnts, and the sze of a cell n the man grd. We compare our results to prevous studes and dscuss the relaton to DNA proflng. The smplcty of our model s ts key strength. We conclude that t s extremely unlkely that any two randomly selected people have, or have ever had, the same set of fngerprnts. Despte the apparently smplstc nature of fngerprntng, t s vastly more relable n dentfcaton than a DNA comparson test. Introducton In recent years, the scentfc bass of fngerprnt analyss has been questoned, n the U.S. Supreme Court rulng n the Daubert case that the relablty of expert scentfc testmony must be establshed along the followng fve crtera: The UMAP Journal 25 (3) (2004) 231 244. c Copyrght 2004 by COMAP, Inc. All rghts reserved. Permsson to make dgtal or hard copes of part or all of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce. Abstractng wth credt s permtted, but copyrghts for components of ths work owned by others than COMAP must be honored. To copy otherwse, to republsh, to post on servers, or to redstrbute to lsts requres pror permsson from COMAP.
232 The UMAP Journal 25.3 (2004) 1. whether the partcular technque or methodology n queston has been subject to a statstcal hypothess testng; 2. whether ts error rate has been establshed; 3. whether the standards controllng the technque s operaton exst and have been mantaned; 4. whether t has been peer-revewed and publshed; and 5. whether t has a general wdespread acceptance. Our model tres to address the frst two ssues. We am to produce a probablstc method to measure the unqueness of a partcular prnt. Assumptons A thumbprnt s defned globally by rdge patterns and locally by a dstrbuton of mnutae, whch we refer to also as features. The area of nterest typcal thumbprnt s a 20 mm 20 mm square grd. There are two sgnfcant types of mnutae, the bfurcaton and the rdge endng: all other mnutae are compostons of these [Osterburg et al. 1977]. The probablty of a mnuta occurrng n a grd box s.234 [Osterburg 1977]. The orentaton of the mnutae was not taken nto account by Osterburg; we assgn a mnutae one of eght angles, from 0 to 157.5, n steps of 22.5. When comparng two prnts, we know one prnt arbtrarly well. The number of people who have ever lved s 1.064 10 11 [Haub 1995]. The Model Global Analyss Rdge Patterns And Orentaton Felds Global analyss concerns rdge patterns, whch dstngush prnts nto sx man pattern groups: Arch, Tented Arch, Left Loop, Rght Loop, Twn Loop and Whorl. Each pattern s determned by an orentaton feld, whch may have specfc statonary ponts, known as the the delta and the core. If a prnt contans 0 or 1 delta ponts and 0 or 1 core ponts, then t s classfed as Lasso, and Wrbel otherwse. The Lasso class conssts of arch, tented arch, rght loop, and left loop.
Can t Qute Put Our Fnger On It 233 If the fngerprnt has 0 delta ponts or 0 core ponts, then t s an arch. Otherwse, f the core pont and the delta pont are algned n the vertcal drecton, then the fngerprnt s an arch f the length between the core pont and the delta pont s less than 2.5 mm and a tented arch otherwse. Otherwse, f the core pont s to the rght of the delta pont, the fngerprnt s a rght loop. Otherwse, the fngerprnt s a left loop. The Wrbel class conssts of the whorl and the twn loop classes: If there are exactly two core ponts and exactly two delta ponts, then the fngerprnt s a whorl f the two core ponts are algned horzontally and a twn loop otherwse. Otherwse, the fngerprnt s a whorl. The man am of global analyss s a vector feld or orentaton feld to the rdge lnes of a fngerprnt. We must fnd sutable parameters for such functons that gve rse to the dfferent classes of rdge pattern. The most basc pattern wthout statonary ponts s the arch Fgure 1, modeled by the smple system dx dt = μy, dy dt = ν, wth parameters μ and ν. The orentaton felds for other rdge patterns are more complex, so the bulk of our model s drected at the prnt s local features. Local Analyss Estmates Fgure 1. Arch orentaton feld. For an ntal estmate of the probablty of any two people havng the same thumbprnt, we must consder:
234 The UMAP Journal 25.3 (2004) the total number of people who have ever lved, estmated to be 1.064 10 11 [Haub 1995] (that s, about 6% of all people are alve rght now); and the total number of possble thumbprnts that can be classfed as dfferent. To decde whether or not one thumbprnt s the same as another, one must frst decde on what exactly s a thumbprnt. In our model, we take a typcal area of the prnt as 20 mm 20 mm, whch s large enough to encompass the area of nterest on any prnt. We dvde ths area nto boxes 1 mm on a sde, thus gvng 400 boxes, each of area 1 mm 2. In prncple, each box can be examned to determne whether or not t contans a mnuta. Mnutae are the features on a thumbprnt that are used by almost all dentfcaton technques to dstngush between prnts. There are from 10 [Galton 1892] to 16 [Optel Ltd. 2003] dfferent types of mnutae, but they are all composed of two fundamental types: rdge endngs and rdge bfurcatons. In our analyses, we consder only these two types. Also, f there are two mnutae n the same cell, t s mpossble to resolve them separately. Say that there are resolvable features on the prnt. The number of ways that we can nsert these features nto the 400 spaces n the grd s ( ) 400. Rememberng that there are two possbltes for each feature, the total number of combnatons s then ( ) 400 2. What s the value of? The probablty that any box contans a feature we take as.234 [Osterburg et al. 1977]. Then has a bnomal dstrbuton: ( ) 400 P ( = x) = (.234) x (1.234) 400 x, x wth mean μ 94 and standard devaton σ 8, so that the average number of cells contanng features s 94. Thus, the total number of possble thumbprnts s 400 ( ) 400 N = 2. =0 The bnomal dstrbuton for, however, s concentrated manly n the regon μ σ<<μ+ σ,or94 8 <<94 + 8. To be conservatve, we consder only ths range of numbers of mnutae; thus, there are approxmately N 102 =86 ( 400 ) 2 1.19 10 128
Can t Qute Put Our Fnger On It 235 dfferent thumbprnts avalable for any actual thumb to hold. So very roughly, P (two people ever havng the same thumbprnt) = 1.064 1011 1.19 10 128 10 117. Comparson Ths fgure s the (approxmate) probablty that there have ever been two people who have had the same thumbprnt. How mght we take nto account the chance that, when compared, two prnts wll be judged to be the same? To do ths, we consder two hypothetcal prnts: A control prnt: an deal known prnt, n whch all features are seen. A sample prnt: a prnt wth more features than the n avalable for comparson. For two prnts that are compared n a realstc crcumstance, there wll be at least ( n) features that are not ncluded n the comparson. These features, n theory, could be n any combnaton of postons n the grd. The man queston s: How many prnts have n<features correspondng to a match? In other words, how many dfferent ways can the remanng ( n) features be nserted nto the grd and stll produce a match wth the control prnt? Knowng that, we can estmate how lkely t s that two thumbprnts not actually the same wll match. Incorrect Matchng We have ( n) features to dstrbute among (400 n) grd elements. The number of dfferent ways to do ths s, by prevous reasonng, 102 ( ) 400 n 2 n. n 86 In crmnal proceedngs, a matchng number of mnutae of anythng from 8 [Collns 1992] to 15 [Vacca 2002] are accepted as conclusve proof of dentfcaton. Our model predcts that for n =12, the total number of thumbprnts that could have the same set of matchng mnutae whle not beng the same prnt s N =1.3 10 117. But expressed as a fracton of the total number of possble prnts, the probablty of the prnt beng one of these, f t selected from them, s P (false match) = Ths s an extremely low probablty. 1.3 10117 1.19 10 128 1.09 10 11. (1)
236 The UMAP Journal 25.3 (2004) Varyng Parameters The result (1) depends on the parameters, whch can be vared accordng to crcumstance and also as a way of refnng the model: p: the probablty of fndng a feature n a grd cell. We take p =.234 [Osterburg et al. 1977]. Others [Tha 2003; Kngston 1964; Stoney and Thornton 1987; Dankmejer et al. 1980 ] gve values n the range.19 <p<.25. N: the number of cells n the grd. If there are more cells, on average, then more features wll be observed, snce p(feature) for a cell remans the same. n: the number of mnutae that one takes for comparson. We take n =12. : a varable, determned by p and N, that gves reasonable bounds for the summaton. F : the number of dfferent features that can appear n a grd cell. In our ntal estmate, we take F =2. L, A: the length of a sde, and the area, of a grd cell. We take L 1 mm, the average dstance between features [Tha 2003]. If we wsh to examne a thumbprnt more closely, we should consder smaller and smaller areas of the prnt. It s not meanngful, however, to take L less than 0.1 mm, snce ths s the typcal rdge wdth. The Dependence on L We rework the model, takng the wdth of the generc grd cell to be 0.5 mm. Takng the overall area of the prnt to be the same, there are now 1,600 grd cells to consder, each wth the same probablty of havng a feature. The bnomal expresson for, the number of features observed on the whole prnt, s now ( ) 1600 P ( = x) = (.234) x (1.234) 1600 x, x wth mean μ = 374 and standard devaton σ =17. Thus, the regon of relevance when summng s now 374 17 = 357 <<391 = 374 + 17. Ths means that when the thumbprnt s examned on a scale half that of the ntal, about four tmes as many mnutae wll be observed. Intutvely, the lkelhood of a false match wll decrease, snce there are more possbltes for the number of prnts: N 1600 = 391 =357 ( 1600 ) 2 3.5 10 502.
Can t Qute Put Our Fnger On It 237 The probablty that any of the 100 bllon people who have ever lved have had the same thumbprnt s 1.064 1011 P (2 people ever havng the same thumbprnt) = 3.5 10 502 10 492. We now determne the number of ways n whch, when a certan number of mnutae are selected for comparson, the remanng mnutae can be arranged. Followng the same logc as before, ths fgure s 391 ( ) 1600 n 2 n, n 357 whch evaluates to 3.4 10 491 for n =12. The probablty that any two compared thumbprnts, judged to be dentcal by the standards of comparson, are actually dfferent s therefore 3.4 10491 P (false match n =12,N = 1600) = 3.5 10 502 10 11. Ths, nterestngly, s not much greater than the probablty for the prevous estmate. The result s not therefore acutely dependent on the value of L, nor, by assocaton, on the number N of grd cells. That sad, t easy to examne detals n any prnt to a scale of 0.5 mm. The Dependence on p The probablty of a feature n a cell s, as of yet, a purely emprcal fgure. The formaton of fngerprnts, and ther assocated characterstcs, s known: The foetus, at about 6.5 weeks, grows eleven volar pads pouches on varous locatons of the hand [Anonymous 2001]. These shrnk at about 11 weeks; and when they are gone, beneath where they lay are fngerprnts. However, the mechansm of formaton of the specfc features s unknown. Genetc nfluences are present, but the envronment s crucal also, evdenced by the fact that dentcal twns who have the same DNA genotype do not have the same fngerprnts. There s no way yet determned of predctng the frequency of occurrence of any type of mnuta on the prnt of a partcular person. Prevous studes, cted n Table 1, show varaton about 0.2 mnutae/mm 2. It s not unreasonable to propose that the densty depends on the prnt classfcaton (.e., whorl, loop, arch, etc.). The range s.204 <p<.246. For 1,600 boxes, we have P (false match) 1.89 10 12 for p =.204, P (false match) 1.78 10 11 for p =.246. The varaton of p changes the fnal predcton by no more than an order of magntude.
238 The UMAP Journal 25.3 (2004) Table 1. The multplcaton table of D 10. Source Number Mean densty of prnts (mnutae/mm 2 ) Osterburg et al. [1977].234 Dankmejer et al. [1980] 1,000.19 Stoney and Thornton [1987] 412.223 Kngston [1964] 100.246 Tha [2003] 30.204 The Dependence on F In out ntal estmate, we take the number F of degrees of freedom of a feature n a prnt to be two: ether a rdge endng or a rdge bfurcaton. However, one can also consder the orentaton of a feature. Each mnuta les on a rdge, whch has a well-defned drecton. We dscretzed ths varable to one of eght possble drectons, angles from 0 to 157.5. Thus each feature, nstead of havng 2 degrees of freedom, now has 16. The probablty of a false match, takng 1,600 grd cells and a probablty of occurrence p =.234, snow P (false match) = Takng n =12as before, we fnd that 391 =357 ( ) 1600 n 16 n n ( ). 400 16 391 =357 P (false match) 2.6 10 22. Ths s an astonshngly smaller probablty than the prevous estmate of 10 11. The orentaton of a feature s no more dffcult to determne n practce than ts nature, so ncludng t n the comparson process s a great mprovement n effcacy wth a modest ncrease n effort. The Dependence on n It s crucal to determne how many matchng mnutae are necessary for a postve comparson. We have taken n =12n the precedng analyses; t s nstructve to consder the varaton of the probablty of a false match wth n. The graphs n Fgure 2 show that the probablty falls off sharply, even as n ncreases beyond 1. A value of n 5 s qute suffcent.
Can t Qute Put Our Fnger On It 239 Fgure 2. Probablty of false match (lnear and logarthmc scales) vs. n. Concluson The Model where Our model, n the most general form, s P (false match) = μ+σ =μ σ μ+σ =μ σ ( ) N n F n n (, N )F F s the number of degrees of freedom and N s the number of grd cells, μ and σ are the mean and standard devaton of the bnomal dstrbuton determned by N and p, p s the probablty of there beng a feature n a cell, and n s the number of mnutae beng used to make a comparson. The prelmnary result returned from our model, for n = 12 (a typcal threshold for postve dentfcaton n many countres), s P 10 11. Further refnement of the parameters reduces ths to P 7 10 22. We conclude that 12 s a very reasonable comparson crteron, and that n =5or 6 s qute damnng for any suspect so compared. Thus, we conclude that to a very hgh degree of certanty, not only that no two people, now lvng or havng ever lved n the past, have had the same thumbprnt, but also that there s a vanshngly small chance that two prnts are even close enough to be confused, gven a small fracton of mnutae from ther patterns to compare.
240 The UMAP Journal 25.3 (2004) DNA Analyss DNA dentty testng s based on aspects of the DNA patterns called loc. For a 100 % match, the FBI [Thompson et al. n.d.] recommends that 13 loc be used. Usng STR (Short Tandem Repeat) markers ensures that the nhertance profle at one locaton does not nfluence the nhertance at other locatons. Each loc has two alleles, so 26 alleles must match. The FBI says that the possblty of a false match s 2.60 10 9 whle other sources quote between 10 9 and 10 12. For two people chosen at random, the probablty of a match based on the four most frequently analyzed alleles s between 1 10 5 and 1 10 8. Ths s sgnfcantly hgher than our estmated probablty or a match for thumbprnts. Hence, thumbprntng remans the most accurate form of bometrc securty known. Strengths and Weaknesses Strengths of the Model Smplcty. Our model s based on easly understood prncples and smply expressed assumptons. Realstc assumptons. Parameters. The parameters n the model, such as the sze of the grd-box, the total area, and the number of mnutae needed to match two thumbprnts, can be easly vared. Degrees of freedom. In specfyng two dfferent knds of possble mnutae and 8 orentaton ranges for each one, the number of degrees of freedom s 16, greatly ncreasng the number of possble confguratons of thumbprnts and so mnmsng the probablty of msdentfcaton. Other studes [Osterburg et al. 1977; Galton 1892] do not take nto account the orentaton of the mnutae. By dscretzng the drectons of the features, we agan keep the model smple. Corroboraton. The probabltes returned by our model te n wth those gven by prevous studes by experts n the feld (Table 2). Table 2. Comparson probabltes of studes. Galton [1892] Osterburg et al. [1977] Stoney and Thornton [1987] our model 1.45 10 11 1.33 10 27 3.5 10 26 10 11 to 10 22
Can t Qute Put Our Fnger On It 241 Weaknesses of the Model Multple entres. We assumed that n any gven grd-box only one mnuta can be present, whch s suffcently accurate for most types of mnutae. For example, both the brdge (consstng of two rdge bfurcatons) and the spur (consstng of a rdge bfurcaton and a rdge endng) have been defned [Osterburg et al. 1977] as beng less than 2 mm n length. Thus f the brdge or spur s more than 0.707 mm n length, ther consttuent endngs and bfurcatons appear n dfferent boxes and are counted as two separate mnutae. However, for mnutae consstng of rdge endngs and rdge bfurcatons n very close proxmty, there s a chance that each wll not be caught n a dfferent box. An example s a dot. The dstance between the two rdge endngs that make up a dot s so small that t s unlkely that our model would catch these two occurrences of rdge endngs n dfferent boxes. A dot has been defned [Osterburg et al. 1977] as beng large enough to encompass one pore, whose sze ranges from 0.088 mm to 0.22 mm [Roddy and Stosz 1997]. Therefore, the two rdge endngs wll not appear n dfferent boxes but wll nstead be msdentfed as a sngle mnuta. Independence of mnuta occurrence. We assume that the placement of a mnuta s completely unrelated to the placement of any others. Ths s not qute the case; there s a slght tendency for mnutae not to occur n drect proxmty to each other. Global analyss. The overall rdge pattern of a thumbprnt s entrely dstnctve n ts own rght. We have not quantfed ths factor n our model. Appendx: Classfcaton Mnutae The rdges n a fngerprnt or thumbprnt form varous patterns, those patterns beng called mnutae. Ten dfferent types are shown n Fgure A1. Rdge Endng. A rdge endng occurs when a rdge ends abruptly. We defne the orentaton of a rdge endng as the drecton the rdge came from. Bfurcaton. A bfurcaton s formed when two dfferent rdges merge. We defne the orentaton as beng the drecton n whch the merged rdge came from. Island. An sland s a short rdge, comprsed of two rdge endngs whose orentatons are n opposng drectons. Two rdge endngs occurrng n neghbourng boxes wth opposte confguraton ndcate the presence of an sland.
242 The UMAP Journal 25.3 (2004) Fgure A1. Ten types of mnuta. Dot. A dot s an sland but on a smaller scale. Brdge. A brdge s when a rdge branches out and merges wth another rdge wthn a short regon. It s composed of two bfurcatons. Spur. A spur s when a rdge branches out and does not merge wth another rdge. It s composed of one bfurcaton and one rdge endng. Eye. An eye s formed by a rdge branchng out nto two rdges, and then recombnng agan a short dstance later. It conssts of two bfurcatons. Double Bfurcaton. As the name suggests, ths type of mnuta contans two bfurcatons n successon. Delta. Ths type of mnuta s composed of a dot and bfurcaton, where the dot s between the mergng rdges. Trfurcaton. A trfurcaton s a rdge that splts nto 3 separate branches. It can be thought of as two bfurcatons occurrng n the same place. Rdge Patterns Fgure A2 shows examples of the dfferent classfcatons of a fngerprnt, ncludng the presence of cores and delta ponts.
Can t Qute Put Our Fnger On It 243 Fgure A2. Features n a fngerprnt. References Anonymous. 2001. Frcton skn growth. http://www.rdgesandfurrows. homestead.com/frcton_skn_growth.html. Ballan, Meltem, F. Ayhan Sakarya, and Bran L. Evans. 1997. A fngerprnt classfcaton technque usng drectonal mages. In Proceedngs of the IEEE Aslomar Conference on Sgnals, Systems, and Computers (Nov. 3 5, 1997, Pacfc Grove, CA), vol. 1, 101 104. http://www.ece.utexas.edu/~bevans/ papers/1997/fngerprnts/. Beeton, Mary. 2002. Scentfc methodology and the frcton rdge dentfcaton process. Identfcaton Canada 25 (3) (September 2002). Reprnted n Georga Forensc News 32 (3) (November 2002): 1, 6 8. http://www. rdgesandfurrows.homestead.com/fles/scentfc_methodology.pdf. Collns, Martn W. Marty. 1992. Realzng the full value of latent prnts. Calforna Identfcaton Dgest (March 1992): 4 11. http://www.latent-prnts. com/realzng_the_full_value_of_late.htm. Dankmejer, J., J.M. Waltman, and A.G. de Wlde. 1980. Bologcal foundatons for forensc dentfcaton based on fngerprnts. Acta Morphologca Nederlando-scandnavca 18 (1) (March 1980): 67 83. http://www.ncb.nlm. nh.gov/entrez/query.fcg?cmd=retreve&db=pubmed&lst_uds=7395553&dopt= Ctaton. Galton, F. 1892. Fnger Prnts. London: Macmllan.
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