Beyond Minutiae: A Fingerprint Individuality
Model with Pattern, Ridge and Pore Features
Yi Chen1,2 and Anil. K. Jain1
1 Michigan State University
2 DigitalPersona Inc.
Abstract. Fingerprints are considered to be unique because they con-
tain various distinctive features, including minutiae, ridges, pores, etc.
Some attempts have been made to model the minutiae in order to get
a quantitative measure for uniqueness or individuality of fingerprints.
However, these models do not fully exploit information contained in non-
minutiae features that is utilized for matching fingerprints in practice.
We propose an individuality model that incorporates all three levels of
fingerprint features: pattern or class type (Level 1), minutiae and ridges
(Level 2), and pores (Level 3). Correlations among these features and
their distributions are also taken into account in our model. Experimen-
tal results show that the theoretical estimates of fingerprint individuality
using our model consistently follow the empirical values based on the
public domain NIST-4 database.
Key words: fingerprint individuality, statistical models, probability of
random correspondence (PRC), minutiae, ridges, pores
1 Introduction
A number of court challenges have brought into question the validity and re-
liability of using fingerprints for personal identification [1–3]. These challenges
are based on, among other factors, the lack of (i) conclusive evidence to sup-
port the claim of fingerprint uniqueness, and (ii) scientific evaluation of criteria
used to determine a match between two fingerprints. These legal challenges have
generated substantial interest in studies on fingerprint individuality.
Fingerprint individuality can be formulated as the probability that any two
prints from different fingers will be “sufficiently” similar. Because similarity be-
tween fingerprints is often quantitatively defined based on the similarity of fin-
gerprint features, it is equivalent to finding the probability of random correspon-
dence (PRC), or the probability of matching k features, given that two impostor
fingerprints contain m and n features, respectively. The key to calculating PRC
is to model the distribution of fingerprint features, as it is not only the number
of features, but also their spatial distributions that account for individuality.
The variety of features that are incorporated in the individuality model is also
important. Latent experts use three levels of features, namely, Level 1 (e.g., pat-
tern), Level 2 (e.g., minutiae and ridges) and Level 3 (e.g., pores) in fingerprint
matching [4]. All of these features are believed to be distinctive and unique and
should be considered in the individuality model (see Fig. 1).
To appear at the International Conf. on Biometrics (ICB), June 2009
2 Yi Chen and Anil. K. Jain
pore
ridge
minutiae
rige loop type whorl type
(a) (b)
Fig. 1. Fingerprint features (a) two different pattern types (b) minutiae, ridges and
pores.
To learn the spatial distribution of fingerprint features, a couple of generative
models have been proposed [5, 6]. Pankanti et al. [5] modeled the minutiae as
uniformly and independently distributed, however, it has been empirically deter-
mined that minutiae tend to cluster where ridge directions change abruptly, with
higher density in the core and delta regions [7, 8]. To account for this clustering
tendency of minutiae, Zhu et al. [6] proposed a finger-specific mixture model,
in which minutiae are first clustered and then independently modeled in each
cluster. In modeling features other than minutiae, Fang et al. [9] modeled ridges
by classifying the ridge segment associated with each minutia into one of sixteen
ridge shapes. However, most of the ridge shapes used in this model were quite
rare as the ridge segments associated with minutiae either terminate or bifur-
cate. Roddy and Stosz [10] proposed to approximate the spatial distribution of
pores using a discrete grid, and consequently, their formulation is sensitive to
distortion as well as the size and the position of the grid cells.
2 Proposed Model
We propose to evaluate fingerprint individuality by modeling the distribution
of minutiae, ridge and pore features. In particular, the minutiae and ridge fea-
tures are separately modeled for each fingerprint pattern type (whorl, left loop,
right loop, arch and tented arch). This is because minutiae and ridge feature
distributions are highly correlated with the fingerprint pattern [8], which was
also demonstrated by Zhu et al [6]. Compared to finger-specific models, pattern-
specific models can be easily generalized to different target populations and are
less prone to overfitting, especially when the number of minutiae in a finger-
print is small. To model ridges and pores, transform-invariant features such as
ridge period, ridge curvature, and pore spacing associated with each minutia are
derived.
2.1 Pattern-specific Minutiae Modeling
Given a large database of fingerprints, we assign each image to one of the five ma-
jor pattern types [11]: whorl, left loop, right loop, arch and tented arch. For each
fingerprint class, minutiae from fingerprints in that class are consolidated and
their spatial distribution is estimated. These minutiae are grouped into clusters
To appear at the International Conf. on Biometrics (ICB), June 2009
Title Suppressed Due to Excessive Length 3
based on their positions (X) and directions (O) using the EM algorithm [12]. In
each cluster, the minutiae position X is modeled by a bivariate Gaussian distri-
bution and the minutiae direction O is modeled using a Von-Mises distribution
[6]. Each minutia m(X,O) in a fingerprint of class G has the following mixture
density:
f(x, o|ΘG) =
NG∑
g=1
τg · fX(x|µg, Σg) · fO(o|νg, κg), (1)
where NG is the number of clusters in the mixture for class G, τg is the weight
for the g-th cluster, ΘG is the set of parameters describing the distributions in
each cluster, fX(x|µg, Σg) is the p.d.f. over minutiae position in the g-th cluster,
and fO(o|νg, κg) is the p.d.f. over minutiae direction in the g-th cluster. Minu-
tiae in the same cluster have the same distribution of directions, establishing a
dependence between minutiae position and direction.
Let T be a fingerprint belonging to class G with minutiae density f(x, o|ΘG).
Similarly, let Q be a fingerprint belonging to class H with minutiae density
f(x, o|ΘH). Let m(XT , OT ) and m(XQ, OQ) be two minutiae from T and Q,
respectively. The probability that these two minutiae would match is defined as
P (|XT −XQ| ≤ x0, |OT −OQ| ≤ o0|ΘG, ΘH) (2)
=
NG∑
g=1
NH∑
h=1
τg · τh · P (|XT −XQ| ≤ x0|µg, µh, Σg, Σh) · P (|OT −OQ| ≤ o0|νg, νh, κg, κh),
where parameters x0 = 15 pixels and o0 = 22.5 degrees are used as tolerances.
Note that this probability can be directly computed since (XT −XQ) follows a
2D Gaussian distribution with mean (µg − µh) and covariance (Σg + Σh); and
(OT−OQ) can be approximated by a Von-Mises distribution with mean (νg−νh)
and variance κg,h defined as [13]:
A(κg,h) = A(κg)A(κh), (3)
A(x) = 1− 12 − 18x2 − 18x3 + o(x−3). (4)
Finally, the PRC, or the probability of matching k pairs of minutiae between
Q and T , is calculated as [6]
p(m,n, k) =
e−λ · λk
k!
, λ = mnl, (5)
where m and n are the number of minutiae in Q and T , respectively, and
l = P (|XT − XQ| ≤ x0, |OT − OQ| ≤ o0|ΘG, ΘH) is the probability calcu-
lated in Eq. (2). This PRC calculation corresponds to the Poisson probability
mass function with mean λ = mnl, which can be interpreted as the expected
number of matches from the total number of mn possible pairings between Q
and T with the probability of each match being l.
To appear at the International Conf. on Biometrics (ICB), June 2009
4 Yi Chen and Anil. K. Jain
2.2 Pattern-specific Ridge Modeling
To incorporate ridge features in the model, we extend the density function in
Eq. (1) to the following mixture density for class G:
f(x, o, r, c|ΘG) =
NG∑
g=1
τg · fX(x|µg, Σg) · fO(o|νg, κg) · fR(r|ωg, σ2g) · fC(c|λg), (6)
where fR(r|ωg, σ2g) and fC(c|λg) are the probability density functions of ridge
period and ridge curvature associated with each minutia in the g-th cluster, re-
spectively. We use a Gaussian distribution to model ridge period because ridge
period only fluctuates towards the tip (smaller) and flexion crease (larger) re-
gions. Ridge curvature, on the other hand, is usually low except near the core
and delta regions, and is modeled by a Poisson distribution. Note that minutiae
in the same cluster have the same distribution of direction, ridge period and
curvature, hence, establishing a dependency among these features.
1
2
3
1
2
3
Fig. 2. Converting local ridge structure to a canonical form such that matching two
minutiae and their associated ridges is equivalent to matching the position and direction
of minutiae and the period and curvature of the ridges.
Let T be a fingerprint belonging to classG with minutiae density f(x, o, r, c|ΘG).
Similarly, let Q be a fingerprint belonging to class H with minutiae density
f(x, o, r, c|ΘH). Let m(XT , OT , RT , CT ) and m(XQ, OQ, RQ, CQ) be two minu-
tiae from T and Q, respectively. The probability that these two minutiae would
match is then defined as
P (|XT −XQ| ≤ x0, |OT −OQ| ≤ o0, |RT −RQ| ≤ r0, |CT − CQ| ≤ c0|ΘG, ΘH) =
NG∑
g=1
NH∑
h=1
τg · τh · P (|XT −XQ| ≤ x0|µg, µh, Σg, Σh) · P (|OT −OQ| ≤ o0|νg, νh, κg, κh)
·P (|RT −RQ| ≤ r0|ωg, ωh, σ2g , σ2h) · P (|CT − CQ| ≤ c0|λg, λh), (7)
To appear at the International Conf. on Biometrics (ICB), June 2009
Title Suppressed Due to Excessive Length 5
where r0 = 2 and c0 = 2. Here, matching the ridges associated with two minutiae
is equivalent to matching the ridge period and curvature after converting the
ridge structure associated with each minutia to a canonical form centered at the
minutia and rotated by the minutia direction (see Fig. 2). Similar to Eq. (2),
this probability can be directly computed since (RT − RQ) follows a Gaussian
distribution with mean (ωg−ωh) and variance (σ2g +σ2h) and (CT −CQ) follows
a Skellam distribution [14] with mean (λg − λh) and variance (λg + λh). The
final PRC is again computed as in Eq. (5), except that l is replaced by Eq. (7).
2.3 Pore Modeling
Since pores are almost evenly spaced along the ridges [10], modeling pores can be
approximated by modeling the intra-ridge spacing of pores on modeled ridges.
As a result, each minutia and its local features (position X, direction O, ridge
period R, ridge curvature C and pore spacing S) have the following mixture
density for fingerprints from class G:
f(x, o, r, c, s|ΘG) = (8)
NG∑
g=1
τg · fX(x|µg, Σg) · fO(o|νg, κg) · fR(r|ωg, σ2g) · fC(c|λg) · fS(s|µp, σ2p),
where fS(s|µp, σ2p) is the probability density function for pore spacing. Because
there is no evidence that intra-ridge pore spacing is dependent on minutiae
location or ridge flow patterns, pore spacing is not clustered.
Let T be a fingerprint belonging to classG with minutiae density f(x, o, r, c, s|ΘG).
Similarly, let Q be a fingerprint belonging to class H with minutiae density
f(x, o, r, c, s|ΘH). Let m(XT , OT , RT , CT , ST ) and m(XQ, OQ, RQ, CQ, SQ) be
two minutiae from T and Q, respectively. The probability of matching the two
minutiae features can be calculated as follows:
P (|XT −XQ| ≤ x0, |OT −OQ| ≤ o0, |RT −RQ| ≤ r0, |CT − CQ| ≤ c0, |ST − SQ| ≤ s0
|ΘG, ΘH) =
NG∑
g=1
NH∑
h=1
τg · τh · P (|XT −XQ| ≤ x0|µg, µh, Σg, Σh)
·P (|OT −OQ| ≤ o0|νg, νh, κg, κh) · P (|RT −RQ| ≤ r0|ωg, ωh, σ2g , σ2h)
·P (|CT − CQ| ≤ c0|λg, λh) · P (|ST − SQ| ≤ s0|σ2p), (9)
where s0 = 2. Again, this probability can be directly calculated since (ST −SQ)
follows a Gaussian distribution with mean 0 and variance 2σ2p. Again, PRC is
still computed as in Eq. (5), except that l is now replaced by Eq. (9). Note that
after incorporating ridge and pore features in our individuality model, PRC
represents the probability of matching k minutiae not only in their positions
and directions, but also with respect to the local ridge period and curvature as
well as pore spacing in the neighborhood of the matching minutiae.
3 Model Evaluation and Validation
In order to demonstrate the utility of our individuality model, we perform the
following evaluation protocol: (i) learn the mixture density of minutiae and their
To appear at the International Conf. on Biometrics (ICB), June 2009
6 Yi Chen and Anil. K. Jain
local ridge and pore features, (ii) compute the theoretical probability of random
correspondence (PRC), and (iii) compare the theoretical PRC with the empir-
ical values obtained on a public database. The database used for evaluation is
the NIST Special Database 4 (NIST-4) [16], which contains 2,000 pairs (4,000
images) of inked rolled prints at 500 ppi. Rolls are appropriate for fingerprint in-
dividuality study because they provide a complete representation of fingerprints.
Each fingerprint in NIST-4 comes with a class label (28.2% right loop, 26.6%
left loop, 21.5% whorl, 19% arch and 4.7% tented arch) assigned by forensic
experts. These prints are manually aligned by the author based on the locations
of core(s). For example, fingerprints of the left loop, right loop and tented arch
classes are aligned at the core point; fingerprints of the whorl class are aligned
at the centroid of the two cores; and fingerprints of the arch class are aligned
at the highest curvature point on the most upthrusting ridge. Figure 3 shows
the empirical distribution of minutiae positions from fingerprints in each of the
five classes. It can be observed that the minutiae distribution is highly correlated
with the fingerprint class. Note the higher minutiae density in the core and delta
regions, which is consistent with Champod’s finding [8].
100 200 300 400 500 600
100
200
300
400
500
600
100 200 300 400 500 600
100
200
300
400
500
600
100 200 300 400 500 600
100
200
300
400
500
600
100 200 300 400 500 600
100
200
300
400
500
600
100 200 300 400 500 600
100
200
300
400
500
600
(a) (b) (c) (d) (e)
Fig. 3. Distribution of minutiae positions (empirically extracted [20]) in each of the five
fingerprint classes (a) arch, (b) tented arch, (c) left loop, (d) right loop, and (e) whorl,
obtained from 4,000 rolled fingerprints in NIST-4. The darker the area, the higher the
minutiae density. These plots (with size 600 × 600) were smoothed using a disk filter
with a radius of 25 pixels. The center of each plot corresponds to the alignment origin.
The theoretic probability of matching a minutiae pair between impostor fin-
gerprints of each class is calculated using Equation 2. As shown in Table 1,
impostor fingerprints from the same class have a higher matching probability
(highlighted in gray) than those from different classes. This is consistent with
the results of Jain et al.’s study [18], which revealed that fingerprints from the
same class are more likely to be matched (have higher False Accept Rate) than
fingerprints from difference classes.
For comparison, we also calculate the empirical probabilities of matching
a minutiae pair between impostor fingerprints based on NIST-4. An in-house
fingerprint matcher [15] was used to automatically establish minutiae correspon-
dences between the impostor pairs in NIST-4. To be compatible with our model,
the matcher is restricted to register fingerprints within a 50× 50 neighborhood
of the manually aligned origin. A total of 7, 998, 000 impostor matches were
conducted and the empirical probability of matching a minutiae pair between
To appear at the International Conf. on Biometrics (ICB), June 2009
Title Suppressed Due to Excessive Length 7
Table 1. Theoretical probabilities of matching a minutiae pair between impostor fin-
gerprints belonging to class A=arch, TA=tented arch, L=left loop, R=right loop and
W=whorl based on the proposed model.
Type A TA L R W
A 13.20× 10−4 6.05× 10−4 6.33× 10−4 3.95× 10−4 4.65× 10−4
TA 6.05× 10−4 12.76× 10−4 6.85× 10−4 5.01× 10−4 6.40× 10−4
L 6.33× 10−4 6.85× 10−4 10.95× 10−4 7.44× 10−4 4.92× 10−4
R 3.95× 10−4 5.01× 10−4 7.44× 10−4 11.59× 10−4 4.74× 10−4
W 4.65× 10−4 6.40× 10−4 4.92× 10−4 4.74× 10−4 10.01× 10−4
Table 2. Empirical probabilities of matching a minutiae pair between impostor fin-
gerprints belonging to class A=arch, TA=tented arch, L=left loop, R=right loop and
W=whorl based on NIST-4.
Type A TA L R W
A 14.33× 10−4 12.25× 10−4 7.73× 10−4 8.31× 10−4 2.35× 10−4
TA 12.25× 10−4 12.54× 10−4 8.33× 10−4 9.17× 10−4 2.61× 10−4
L 7.73× 10−4 8.33× 10−4 11.46× 10−4 5.92× 10−4 4.36× 10−4
R 8.31× 10−4 9.17× 10−4 5.92× 10−4 11.54× 10−4 4.64× 10−4
W 2.35× 10−4 2.61× 10−4 4.36× 10−4 4.64× 10−4 6.36× 10−4
two impostor prints is computed by the average of k/(m × n), where k is the
number of matched minutiae and m and n are the number of minutiae in the
two fingerprints. This probability is tabulated by the fingerprint class informa-
tion, resulting in all possible intra-class and inter-class probability values (see
Table 2). Note that the theoretical probabilities based on our model are closer
to empirical probabilities (with correlation coefficient 0.59) compared to those
calculated by Zhu et al.’s model (empirical probability = 1.8× 10−3, theoretical
probability = 6.50× 10−4) [6].
h
r
(a) (b)
Fig. 4. Ridge features in a neighborhood (30 × 30 pixels) of a minutia. (a) ridge period
(average of inter-ridge distance h), and (b) ridge curvature (inverse of radius r).
To incorporate ridge features, we extract ridge period and curvature in a 30
× 30 neighborhood of each minutia (see Fig. 4). Ridge period is calculated as
the average inter-ridge distances, and ridge curvature, defined as the inverse of
its radius, is calculated by the second derivative of sampled ridge points. In the
model, these features are used to retrain the clustering algorithm and reevaluate
the theoretical probability in Equation 7. In the empirical case, minutiae cor-
To appear at the International Conf. on Biometrics (ICB), June 2009
8 Yi Chen and Anil. K. Jain
respondences that disagree in ridge period or curvature (with difference larger
than r0 and c0, respectively) are removed. The resulting number of matched
minutiae pairs k′ is used to recalculate the empirical probability as the aver-
age of k′/(m× n). The theoretical and empirical matching probability matrices
tabulated by fingerprint class information after incorporating the ridge features
are shown in Tables 3 and 4, respectively. As we can see, higher probabilities
are still observed among impostor fingerprints from the same class than those
from different classes. The use of ridge features reduces the probability of ran-
dom minutiae correspondence both empirically and theoretically; the correlation
coefficient between the two probability matrices is increased to 0.67.
Table 3. Theoretical probabilities of matching a minutiae pair and their local ridge
features between impostor fingerprints belonging to class A=arch, TA=tented arch,
L=left loop, R=right loop and W=whorl based on the proposed model.
Type A TA L R W
A 6.21× 10−4 2.59× 10−4 2.62× 10−4 1.74× 10−4 1.88× 10−4
TA 2.59× 10−4 6.06× 10−4 3.36× 10−4 2.40× 10−4 2.92× 10−4
L 2.62× 10−4 3.36× 10−4 5.52× 10−4 3.71× 10−4 2.24× 10−4
R 1.74× 10−4 2.40× 10−4 3.71× 10−4 5.99× 10−4 2.21× 10−4
W 1.88× 10−4 2.92× 10−4 2.24× 10−4 2.21× 10−4 4.56× 10−4
Table 4. Empirical probabilities of matching a minutiae pair and their local ridge
features between impostor fingerprints belonging to class A=arch, TA=tented arch,
L=left loop, R=right loop and W=whorl based on NIST-4.
Type A TA L R W
A 5.86× 10−4 4.70× 10−4 2.91× 10−4 3.10× 10−4 0.84× 10−4
TA 4.70× 10−4 5.68× 10−4 3.68× 10−4 3.89× 10−4 1.07× 10−4
L 2.91× 10−4 3.68× 10−4 5.05× 10−4 2.39× 10−4 1.79× 10−4
R 3.10× 10−4 3.89× 10−4 2.39× 10−4 5.12× 10−4 1.93× 10−4
W 0.84× 10−4 1.07× 10−4 1.79× 10−4 1.93× 10−4 2.78
本文档为【ChenJainIndividuality_ICB09】,请使用软件OFFICE或WPS软件打开。作品中的文字与图均可以修改和编辑,
图片更改请在作品中右键图片并更换,文字修改请直接点击文字进行修改,也可以新增和删除文档中的内容。
该文档来自用户分享,如有侵权行为请发邮件ishare@vip.sina.com联系网站客服,我们会及时删除。
[版权声明] 本站所有资料为用户分享产生,若发现您的权利被侵害,请联系客服邮件isharekefu@iask.cn,我们尽快处理。
本作品所展示的图片、画像、字体、音乐的版权可能需版权方额外授权,请谨慎使用。
网站提供的党政主题相关内容(国旗、国徽、党徽..)目的在于配合国家政策宣传,仅限个人学习分享使用,禁止用于任何广告和商用目的。