Improving Fingerprint Verification Using Minutiae Triplets

Abstract

Improving fingerprint matching algorithms is an active and important research area in fingerprint recognition. Algorithms based on minutia triplets, an important matcher family, present some drawbacks that impact their accuracy, such as dependency to the order of minutiae in the feature, insensitivity to the reflection of minutiae triplets, and insensitivity to the directions of the minutiae relative to the sides of the triangle. To alleviate these drawbacks, we introduce in this paper a novel fingerprint matching algorithm, named M3gl. This algorithm contains three components: a new feature representation containing clockwise-arranged minutiae without a central minutia, a new similarity measure that shifts the triplets to find the best minutiae correspondence, and a global matching procedure that selects the alignment by maximizing the amount of global matching minutiae. To make M3gl faster, it includes some optimizations to discard non-matching minutia triplets without comparing the whole representation. In comparison with six verification algorithms, M3gl achieves the highest accuracy in the lowest matching time, using FVC2002 and FVC2004 databases.

Full text

Sensors 2012,12, 3418-3437; doi:10.3390/s120303418
OPEN ACCESS
sensors
ISSN 1424-8220
www.mdpi.com/journal/sensors
Article
Improving Fingerprint Verification Using Minutiae Triplets
Miguel Angel Medina-P´
erez 1,2,⋆, Milton Garc´
ıa-Borroto 1,
Andres Eduardo Gutierrez-Rodr´
ıguez 1and Leopoldo Altamirano-Robles 2
1Centro de Bioplantas, Universidad de Ciego de ´
Avila, Carretera a Mor´
on km 9, Ciego de ´
Avila ,
C.P. 69450, Cuba
2Instituto Nacional de Astrof´
ısica, ´
Optica y Electr´
onica. Luis Enrique Erro No. 1, Sta. Mar´
ıa
Tonanzintla, Puebla, C.P. 72840, M´
exico
⋆Author to whom correspondence should be addressed; E-Mail: migue@bioplantas.cu;
Tel.: +53-33-224-016.
Received: 25 January 2012; in revised form: 28 February 2012 / Accepted: 28 February 2012 /
Published: 8 March 2012
Abstract: Improving fingerprint matching algorithms is an active and important research
area in fingerprint recognition. Algorithms based on minutia triplets, an important matcher
family, present some drawbacks that impact their accuracy, such as dependency to the
order of minutiae in the feature, insensitivity to the reflection of minutiae triplets, and
insensitivity to the directions of the minutiae relative to the sides of the triangle. To alleviate
these drawbacks, we introduce in this paper a novel fingerprint matching algorithm, named
M3gl. This algorithm contains three components: a new feature representation containing
clockwise-arranged minutiae without a central minutia, a new similarity measure that shifts
the triplets to find the best minutiae correspondence, and a global matching procedure
that selects the alignment by maximizing the amount of global matching minutiae. To
make M3gl faster, it includes some optimizations to discard non-matching minutia triplets
without comparing the whole representation. In comparison with six verification algorithms,
M3gl achieves the highest accuracy in the lowest matching time, using FVC2002 and
FVC2004 databases.
Keywords: fingerprint verification; minutiae descriptor; minutiae triplet

Sensors 2012,12 3419
1. Introduction
Fingerprints are the marks made by ridges and furrows in the fingers. They are formed during the sixth
month of intrauterine life of human beings and do not disappear until some time after death. Since there
are not two persons with the same fingerprints and they do not change naturally, they are an important
element to identify people.
Due to the high complexity of fingerprint matching and the huge amount of existing fingerprints, it is
necessary to build computer systems that automatically process fingerprints with high accuracy and low
computational costs. Historically, fingerprint recognition systems were mostly used in forensic sciences,
but the current popularity of these systems is mainly due to civilian applications such as the control of
physical access to facilities, the control of logical access to software, and the control of voters during
elections. A key component in fingerprint recognition systems is the fingerprint matching algorithm.
To measure the quality of a fingerprint matching algorithm, the evaluator must consider the context
in which it is applied. For example, the algorithms based on microscopic characteristics are suitable
for applications where fingerprints are acquired with high resolution sensors (1,000 dpi or higher); the
quality of these algorithms dramatically decreases when the resolution is low [1]. The following quality
parameters are proved to be important for evaluating general matchers:
•Low computational cost: The algorithm satisfies memory and time restrictions for its application
context [2].
•Invariance to translation: The algorithm returns a high similarity value when comparing
fingerprints from the same finger notwithstanding that fingerprints be translated horizontally and/or
vertically [3].
•Invariance to rotation: The algorithm returns a high similarity value when comparing fingerprints
from the same finger in spite of fingerprints rotation [3].
•Tolerance to non-linear distortion: The algorithm returns a high similarity value when comparing
fingerprints from the same finger even when fingerprints are affected by non-linear distortion, as a
result of fingerprint creation mechanisms [4].
•Sensitivity to the individuality of fingerprints: The algorithm returns a high similarity value when
comparing fingerprints from the same finger and returns a low similarity value when comparing
fingerprints from different fingers [5].
•Insensitivity to select a single alignment: The algorithm does not perform a single global alignment
from the best local alignment. Maximizing the local similarity value does not guarantee to find a
true matching local structure pair. Even if the selected local structure pair is a true matching pair,
it is not necessarily the best pair to carry out fingerprint alignment [3].
•Tolerance to partial fingerprints: The algorithm returns a high similarity value when comparing
fingerprints from the same finger even when the fingerprints are not complete [5]. Partial
fingerprints can be produced by the restrictions of the sensors, latent fingerprints in crime scenes,
and different fingerprint creation mechanisms.
•Tolerance to the low quality of fingerprints: The results of the algorithm are not significantly
affected by low fingerprint quality [6]. Due to different skin conditions and/or the different
fingerprint creation mechanisms, sometimes many details of the fingerprints do not appear clearly.

Sensors 2012,12 3420
•Tolerance to errors of the feature extractor: The algorithm returns a high similarity value when
comparing fingerprints from the same finger even when the feature extractor has missed some
features and/or has extracted some non-existent features [3].
•Determinism: Two executions of the algorithm with the same parameters return the same results.
Modern technologies impose new challenges to fingerprint matching algorithms; new systems reside
on light architectures, need standards for systems interoperability, and use small area sensors [3,4,7].
In these contexts, the algorithms with higher quality are those based on minutiae [2]. Minutiae are the
points where the ridge continuity breaks and they are typically represented as (x, y, θ); where (x, y)
represent the 2D point coordinates, and θthe ridge direction at that point.
As a minutia-based matcher should be invariant to translation and rotation, the process of minutia
pairing is ambiguous [8]. Thus, most matchers in this family use local minutia structures (minutiae
descriptors) to quickly establish the minutiae correspondences [4].
A simple and accurate minutiae descriptor is based on minutiae triplets [3]. Minutiae triplets are
local structures represented by three minutiae. Algorithms based on minutiae triplets have the following
advantages, which make them of higher quality than algorithms based on other representations:
•They are tolerant to fingerprint deformations [9].
•They are faster and more accurate, compared to algorithms based on other representations [10,11],
especially in applications with partial fingerprints [12].
•They are suitable for applications based on interoperability standards because the most popular
standards are based only on minutiae [2].
•They are appropriate for systems embedded on light architectures because the representation and
comparison of minutiae triplets can be performed efficiently [13].
•Minutiae triplets have higher discriminative power than minutiae pairs and single minutiae [10].
Some important quality parameters related to fingerprint matching algorithms based on minutiae
triplets are:
•Invariance to the order of minutiae in the feature: No matter the minutiae order in the triplet, the
algorithm finds the correct correspondences of minutiae when matching similar triplets (Figure 1).
•Sensitivity to the reflection of minutiae triplets: The algorithm does not match a triplet with its
reflected version (Figure 2).
•Sensitivity to the directions of the minutiae relative to the sides of the triangle: In order to find
similar triplets, the algorithm takes into account the directions of the minutiae relative to the sides
of the triangles formed by the triplets (Figure 3).

Sensors 2012,12 3421
Figure 1. Similar minutiae triplets that were not classified as true matching by some
algorithms because in image (a) the features are arranged according to the length of the
sides, in image (b) the algorithms try to match the main minutia q1(left triplet) with the
main minutia p1(right triplet).
(a) (b)
Figure 2. Minutiae triplets that do not match because (p1,p2,p3)is a reflected version of
(q1,q2,q3).
Figure 3. Minutiae triplets that do not match because minutiae pairs (q1,p1),(q2,p2)and
(q3,p3)highly differ in the directions of the minutiae relative to the sides of the triangles.
State-of-the-art algorithms based on minutiae triplets do not fulfil all the quality parameters, which
has a negative impact on their accuracy. Table 1shows the lacking quality parameter of all the reviewed
matchers, according to the following parameters:
I Invariance to the order of minutiae in the feature.

Sensors 2012,12 3422
II Sensitivity to the reflection of minutiae triplets.
III Sensitivity to the directions of the minutiae relative to the sides of the triangle.
IV Insensitivity to select a single alignment.
V Tolerance to errors of the feature extractor.
VI Determinism.
Table 1. Summary of the lacking quality parameter on fingerprint matching algorithms based
on minutiae triplets.
Lacking quality parameter
Algorithms I II III IV V VI
JY [9] X X X
KV [14] X X X
PN [11] X X
JG1 [12] X X X X
JG2 [5] X X
RUR [13] X X
TB [15] X X
FFCS [16] X X
CTYZ [17] X X
XCF [18] X X X X
ZGZ [19] X X
HK [20] X X
GSM [21] X X
In this research, we introduce M3gl, a new fingerprint matcher that fulfils most quality parameters.
M3gl is based on a new representation and a new comparison function for minutia triplets. An
experimental comparison with algorithms based on different types of minutiae descriptors shows that
M3gl is highly accurate and it has acceptable computational costs.
2. Definitions
This section defines some basic functions that we use throughout the paper. Given two minutiae
pi= (xi, yi, θi)and pj= (xj, yj, θj),ed(pi,pj)represents the Euclidean distance between the
coordinates of piand pjEquation (1).
ed(pi,pj) = (xi−xj)2+ (yi−yj)2(1)
For two given angles αand β,adπ(α, β)computes the minimum angle required to superpose two
vectors with the same origin and angles αand βrespectively using Equation (2), while ad2π(α, β)

Sensors 2012,12 3423
computes the angle required to rotate a vector with angle βin clockwise sense to superpose it to another
vector with the same origin and angle αusing Equation (3).
adπ(α, β) = min |α−β|,2π− |α−β|(2)
ad2π(α, β) = β−αif β > α
β−α+ 2πotherwise (3)
Finally, for a given pair of minutiae piand pj,ang(pi,pj)computes the angle of the vector with
initial point at piand terminal point at pjusing Equation (4).
ang(pi,pj) =















arctan(∆y/∆x)if ∆x > 0∧∆y≥0
arctan(∆y/∆x)+2πif ∆x > 0∧∆y < 0
arctan(∆y/∆x) + πif ∆x < 0
π/2if ∆x= 0 ∧∆y > 0
3π/2if ∆x= 0 ∧∆y < 0
(4)
where ∆y=yi−yjand ∆x=xi−xj.
3. Feature Representation
In this section, we introduce m-triplets, a robust feature representation based on minutiae triplets.
Provided that a fingerprint is described by the minutia set P, our representation is a tuple with the
following components (see Figure 4):
•minutiae pi∈Pare clockwise arranged starting on p1.
•di∈1...3, where diis the Euclidean distance between the minutiae different than pi.
•dmax,dmid and dmin are the maximum, middle and minimum distances in the triplet, respectively.
Although these components can be calculated based on distance values, we store them because they
are a key point for the algorithm optimizations.
•αi∈1...6are the angles ad2πang(p,q), θrequired to rotate the direction θof a minutia to superpose
it to the vectors associated with the other two minutiae in the triplet.
•βi= ad2πθj, θkis the angle required to rotate the direction of the minutia pkin order to
superpose it to the direction of the minutia pj.
The m-triplets are sensitive to the directions of the minutiae relative to the sides of the triangle
(angles α). The minutiae in this representation are arranged in clockwise direction without a central
minutia. Therefore, in order to compare m-triplets, the similarity function considers the three minutiae
rotations in clockwise sense (next section contains a formal definition of this procedure). This
representation and the comparison function guarantee the invariance to the order of minutiae in the
feature, and the sensitivity to the reflection of minutiae triplets.

Sensors 2012,12 3424
Figure 4. The components of the new feature representation proposed in this paper.
4. M-Triplets Similarity
In this section, we introduce a new m-triplets similarity. It is designed to accurately distinguish
between similar and non-similar m-triplets. Given two m-triplets tand r, we propose sinv(t,r)in
Equation (5) to compare m-triplets.
sinv(t,r) = max spart(t,r),spart t, shif t(r),spart t, shiftshift(r)(5)
where: shift(r)is the clockwise-shifted m-triplet rand spart(t,r)is the base similarity function in
Equation (6).
spart(t,r) = 0 if sθ(t,r)=0 ∨sd(t,r)=0 ∨sα(t,r) = 0 ∨sβ(t,r)=0
1-1- sd(t,r)1−sα(t,r)1- sβ(t,r),otherwise (6)
The base similarity function spart is defined using functions sθ,sd,sα, and sβ, which consider different
components of the m-triplets. According to Equation (6), two m-triplets are totally dissimilar if they
have at least one component totally dissimilar. If all component similarities are above zero, the product
rule makes the similarity hight if at least one component is close to 1.
The invariance to rotation is an important quality parameter for fingerprint recognition, especially
for fingerprint identification. Nevertheless, for fingerprint verification, the rotation is usually restricted
by sensors. The function sθEquation (7) incorporates this information into the m-triplets similarity to
increase minutia discrimination for such problems. Consequently, two m-triplets are dissimilar if their
minutiae directions differ more than π/4.
sθ(t,r) = 


0if ∃iadπ(θt
i, θr
i)> π/4
1otherwise
(7)

Sensors 2012,12 3425
The function sdEquation (8) compares m-triplets in terms of the side lengths of the triangle formed
by the triplet minutiae. sdreturns values in the interval [0,1], returning 0if at least one length difference
is greater than threshold tl.sdreturns 1if all length differences are 0; that is, the triangles formed by
both m-triplets are identical.
sd(t,r) = 


0if ∃i|dt
i−dr
i|> tl)
1−maxi=1...3{|dt
i−dr
i|}/tlotherwise
(8)
The function sαEquation (9) compares m-triplets based on the angles formed by minutiae directions
and the sides of the triangles (angles αon Figure 4).
sα(t,r) = 


0if ∃iadπ(αt
i, αr
i)> ta)
1−maxi=1...6{adπ(αt
i, αr
i)}/taotherwise
(9)
The function sβEquation (10) compares m-triplets based on relative minutiae directions (angles βon
Figure 4
sβ(t,r) = 


0if ∃iadπ(βt
i, βr
i)> ta)
1−maxi=1...3{adπ(βt
i, βr
i)}/taotherwise
(10)
Equations (9) and (10) return values in the interval [0,1]. They return 0if at least two compared
angles differ more than threshold ta. The less the angles differ, the higher is the value returned by the
equations; therefore, they return 1if the compared angles are identical.
In order to detect dissimilar m-triplets in advance, avoiding all costly shifts, we use Theorems 1,2,
and 3. Proofs for Theorems 1and 2appear in the appendixes. Proof of Theorem 3is similar to the proof
of Theorem 1and is not stated.
Theorem 1. Given two m-triplets tand r, if |dt
max −dr
max|> tlthen sinv(t,r) = 0.
Theorem 2. Given two m-triplets tand r, if |dt
mid −dr
mid|> tlthen sinv(t,r) = 0.
Theorem 3. Given two m-triplets tand r, if |dt
min −dr
min|> tlthen sinv(t,r) = 0.
Based on these theorems, we can modify sinv Equation (5). This way, our m-triplets similarity can be
re-written as Equation (11).
sinv(t,r) = 0if (|dt
max −dr
max|> tl)∨(|dt
mid −dr
mid|> tl)∨(|dt
min −dr
min|> tl)
max{spart(t,r),spart (t, shif t(r)),spart(t, shif t(shift(r)))}, otherwise (11)
Local distance threshold tland angle threshold taare parameters of the algorithm and must be tuned
according to the image characteristics.
The function sinv, by means of spart, achieves the invariance to translation and the invariance to
restricted rotation, the tolerance to non-linear distortion and the sensitivity to the directions of the
minutiae relative to the sides of the triangle. The clockwise shifting makes function sinv invariant to
the order of minutiae in the feature and sensitive to the reflection of minutiae triplets.

Sensors 2012,12 3426
5. M3GL Algorithm
In this section, we introduce M3gl, a new fingerprint verification algorithm based on the proposed
minutiae triplet representation and similarity measure.
Given a fingerprint described by the minutia set Pwe compute the m-triplets as follows. For each
p∈P, find its cnearest minutiae in Pand build all m-triplets that include pand two of its nearest
minutiae, discarding duplicates. This way of computing m-triplets makes M3gl tolerant to the low quality
of fingerprints and tolerant to errors of the feature extractor. Additionally, m-triplets in the fingerprint
are sorted according to the length of the largest side to perform a binary search when looking for similar
m-triplets; Theorem 1guarantees the safety of this procedure.
M3gl consists of the following major steps: local minutiae matching, global minutiae matching, and
similarity score computation.
5.1. Local Minutiae Matching
This step finds the similar m-triplets in the template fingerprint using binary search. Then, it sorts
all matching pairs according to the similarity value and finds the local matching minutiae. Formally, the
algorithm is the following:
1. Let Qand Pbe the query and template fingerprint minutiae sets respectively. Let Rand Tbe the
query and template fingerprint m-triplets sets respectively. Let A← {} be the set that will contain
local matching m-triplets pairs.
2. For each query m-triplet ri∈Rperform binary search looking for the template
m-triplets {t1,t2, . . . , tu} ⊂ Twith similarity value higher than 0 and add the pairs
(ri,t1),(ri,t2),...,(ri,tu)to A.
3. Sort in descendant order all matching pairs (r,t)in Aaccording to the similarity value.
4. Let M← {} be the set containing local matching minutiae pairs.
5. For each (r,t)∈Ado
(a) Let B←(q1,p1),(q2,p2),(q3,p3)be the matching minutiae that maximizes sinv(r,t)
where q1,q2,q3∈Qand p1,p2,p3∈P.
(b) For each (qi,pi)∈Bdo:
i. If there is not any pair (qj,pj)∈Mthat qj=qior pj=pithen M←M∪ {(qi,pi)}.
5.2. Global Minutiae Matching
This step uses every minutiae pair as a reference pair for fingerprint rotation and performs a query
minutiae transformation for each reference pair. Later, it selects the transformation that maximizes the
amount of matching minutiae. This strategy overcomes the limitations of the single alignment based
matching. M3gl uses three criteria to determine if two minutiae match at global level and to achieve the
tolerance to non-linear distortion. First, the Euclidean distance must not exceed threshold tg. Second,
minutia directions must not exceed threshold ta. Third, the directions differences relative to reference
minutiae pair must not exceed threshold ta. The formal algorithm is the following:

Sensors 2012,12 3427
1. Let n←0be the maximum number of matching minutiae computed after performing all
fingerprint rotations.
2. For each (qi,pi)∈Mdo:
(a) Let E← {} be the set containing global matching minutiae pairs for the current iteration.
(b) For each (qj,pj)∈M, if ∀(qk,pk)∈E(qj̸=qk)∧(pj̸=pk)
i. Compute q′= (x′, y′, θ′)as 


x′
y′
θ′

=


cos(∆θ)−sin(∆θ) 0
sin(∆θ)cos(∆θ) 0
0 0 1





x3−x1
y3−y1
θ3−θ1


+


x2
y2
θ2



where qi= (x1, y1, θ1),pi= (x2, y2, θ2),∆θ=θ2−θ1,qj= (x3, y3, θ3).
ii. Let pj= (x4, y4, θ4); if ed(q′,pj)≤tg∧adπad2π(θ2, θ1),ad2π(θ4, θ3)≤ta∧
adπ(θ′, θ4)≤ta, then E←E∪ {(qj,pj)}.
(c) if n < |E|then n← |E|.
5.3. Similarity Score Computation
The similarity value is computed using the formula n2
|P||Q|; where Pand Qare the template and query
fingerprint minutiae sets respectively, and nis the amount of matching minutiae pairs.
Global distance threshold tgand angle threshold taare parameters of the algorithm, and must be tuned
based on the image characteristics.
6. Experimental Results
In order to evaluate the new fingerprint matching algorithm, we use databases DB1 A, DB2 A,
DB3 A and DB4 A of FVC2002 and FVC2004 competitions. These databases are commonly used
as benchmarks for evaluating fingerprint matchers in the context of fingerprint verification. We also use
the FVC evaluation protocol [22]. Performance is measured by indicators EER, FMR100, FMR1000
and ZeroFMR. The indicator Time refers to the average matching time in milliseconds. We carry out all
the experiments on a laptop with an Intel Core i7 740QM processor (1.73 GHz) and 4 GB of RAM.
We use the same parameters values for M3gl in all databases (tl= 12,tg= 12,ta=π/6,c= 4).
We estimate these parameters through a few experiments using fingerprint databases DB1 B, DB2 B,
DB3 B and DB4 B of FVC2004 competition.
In the experimental comparisons, we include the algorithm proposed by Jiang and Yau [9] (JY)
because it is the most popular algorithm based on minutiae triplets in the literature. We implement
the algorithm proposed by Parziale and Niel [11] (PN) because it is the algorithm based on minutiae
triplets more similar to our proposal and we are interested in showing the difference in accuracy and
speed. We also compare our proposal with algorithms based on other types of minutiae descriptors, that
is why we implement the algorithms proposed by Wang et al. [23] (WLC), Qi et al. [24] (QYW), Tico
and Kuosmanen [25] (TK), and Udupa et al. [26] (UGS).
Figures 5and 6show that M3gl achieves lower FNMR for most of the FMR values. Tables 2and
3show that our algorithm achieves the best results for most of the performance indicators. Figure 7
shows examples where our algorithm is able to find true matching minutiae in difficult cases (partial

Sensors 2012,12 3428
fingerprints with low overlapping, non-linear distortion and low quality) where the other algorithms fail.
M3gl is also the fastest algorithm due to the following reasons:
•The m-triplets similarity function includes the properties demonstrated in Theorems 1,2and 3to
discard comparisons without performing all its operations.
•The algorithm performs binary search when looking for similar m-triplets in the local minutiae
matching step.
Figure 5. ROC curves with the performance of the compared algorithms in FVC2002.

Sensors 2012,12 3429
Figure 6. ROC curves with the performance of the compared algorithms in FVC2004.
The similarity score computation of M3gl is simple but robust. In order to test this robustness we
substitute the similarity score computation of M3gl for the strategies proposed by Wang et al. [23],
Qi et al. [24], Jiang and Yau [9], and Tico and Kuosmanen [25]; we name the new algorithms M3gl1,
M3gl2, M3gl3 and M3gl4 respectively. We test M3gl and its variations in the testing databases DB1 B,
DB2 B, DB3 B and DB4 B of FVC2002 and FVC2004. The experimental results (Tables 4and 5) do not
show a clear superiority of any algorithm; nevertheless, when comparing M3gl with the other algorithms
by pairs, we find that M3gl wins in most of the accuracy indicators.

Sensors 2012,12 3430
Table 2. Experimental results on databases DB1 A, DB2 A, DB3 A and DB4 A of
FVC2002.
Database Algorithm EER FMR100 FMR1000 ZeroFMR Time(ms)
WLC 29.5 57.2 63.6 66.9 123.0
D QYW 22.8 46.8 52.6 55.9 13.6
B JY 5.1 10.6 23.9 31.9 3.3
1 TK 4.0 4.9 7.0 8.9 12.4
- PN 1.9 2.5 3.4 5.8 20.3
A UGS 2.2 2.8 4.1 5.9 1,239.4
M3gl 1.1 1.3 2.3 3.8 1.4
WLC 34.0 63.6 69.3 72.6 269.6
D QYW 22.8 47.8 53.7 59.2 27.2
B JY 4.5 8.3 17.3 27.6 4.6
2 TK 3.6 4.6 6.3 23.1 19.1
- PN 1.4 1.6 2.4 3.5 44.4
A UGS 1.9 2.3 4.9 5.6 2,846.1
M3gl 1.3 1.4 1.9 2.2 1.6
WLC 29.8 57.4 62.7 65.0 27.1
D QYW 30.0 55.7 63.5 77.9 6.3
B JY 9.4 16.4 26.1 33.1 1.5
3 TK 7.7 9.8 12.6 16.4 5.8
- PN 5.6 6.9 10.2 12.8 5.6
A UGS 5.3 8.0 12.2 26.4 96.0
M3gl 3.1 4.5 7.5 9.4 0.6
WLC 22.9 51.7 61.0 63.5 37.0
D QYW 24.3 57.3 63.2 67.6 8.5
B JY 7.4 13.0 23.0 28.3 2.1
4 TK 5.1 7.1 9.4 12.1 8.4
- PN 3.1 3.9 5.6 10.3 10.3
A UGS 4.2 7.1 12.6 16.8 463.0
M3gl 2.4 3.4 5.6 11.2 1.0

Sensors 2012,12 3431
Figure 7. Two examples where all matching algorithms fail but our algorithm finds true
matching minutiae. The first row contains fingerprints db1 36 1 and db1 36 4 of database
DB1 A (FVC2002); the second row contains fingerprints 85 6 and 85 8 of database DB1 A
(FVC2004).
Table 3. Experimental results on databases DB1 A, DB2 A, DB3 A and DB4 A of
FVC2004.
Database Algorithm EER FMR100 FMR1000 ZeroFMR Time(ms)
WLC 27.3 64.8 73.9 77.3 150.0
D QYW 24.3 60.6 80.3 97.3 15.5
B JY 13.5 28.5 42.8 55.9 4.2
1 TK 15.9 29.1 41.8 51.0 11.7
- PN 11.4 17.7 24.4 25.9 20.9
A UGS 7.9 14.8 24.9 31.3 1,649.4
M3gl 6.3 11.4 19.3 21.7 1.3
WLC 28.1 62.1 68.9 78.0 103.0
D QYW 24.8 52.1 58.9 73.7 12.9
B JY 11.0 19.4 28.4 39.2 3.0

Sensors 2012,12 3432
Table 3. Cont.
Database Algorithm EER FMR100 FMR1000 ZeroFMR Time(ms)
2 TK 7.8 12.0 18.7 24.9 11.6
- PN 10.0 12.1 15.1 16.9 14.9
A UGS 6.4 10.5 16.7 19.9 1,210.8
M3gl 6.2 9.1 13.6 15.3 1.1
WLC 24.5 57.9 64.9 69.5 312.0
D QYW 19.7 47.7 65.9 87.5 22.6
B JY 12.0 22.1 32.3 41.4 6.4
3 TK 9.6 19.7 32.9 37.3 17.0
- PN 7.1 10.7 17.6 24.9 38.0
A UGS 5.1 8.6 14.1 22.4 6,510.9
M3gl 6.1 8.6 14.4 16.4 1.9
WLC 23.6 54.9 65.0 70.0 108.8
D QYW 25.5 55.5 65.8 74.7 13.3
B JY 9.7 16.3 25.3 28.6 3.2
4 TK 7.6 11.4 16.6 39.4 11.8
- PN 5.2 6.9 9.3 11.9 17.2
A UGS 5.1 7.6 11.2 13.1 1,687.1
M3gl 3.0 4.0 6.9 10.3 1.2
Table 4. Experimental results on databases DB1 B, DB2 B, DB3 B and DB4 B of
FVC2002.
Database Algorithm EER FMR100 FMR1000 ZeroFMR Time(ms)
D M3gl1 1.7 2.5 3.6 3.6
B M3gl2 2.5 1.6 3.2 3.2
1 M3gl3 1.4 2.9 3.6 3.6
- M3gl4 1.6 2.9 3.2 3.2
B M3gl 2.0 2.5 3.2 3.2
D M3gl1 0.7 1.1 1.8 1.8
B M3gl2 1.9 2.5 3.2 3.2
2 M3gl3 0.8 0.8 1.8 1.8
- M3gl4 1.9 2.9 2.9 2.9
B M3gl 0.7 1.1 1.8 1.8
D M3gl1 6.5 9.4 12.1 12.1

Sensors 2012,12 3433
Table 4. Cont.
Database Algorithm EER FMR100 FMR1000 ZeroFMR Time(ms)
B M3gl2 7.7 13.9 18.6 18.6
3 M3gl3 6.9 9.3 11.8 11.8
- M3gl4 6.9 11.4 27.9 27.9
B M3gl 6.4 9.3 11.1 11.1
D M3gl1 2.3 3.6 19.6 19.6
B M3gl2 2.8 3.9 11.1 11.1
4 M3gl3 2.2 3.2 16.4 16.4
- M3gl4 4.0 32.5 78.2 78.2
B M3gl 2.4 3.6 19.6 19.6
Table 5. Experimental results on databases DB1 B, DB2 B, DB3 B and DB4 B of
FVC2004.
Database Algorithm EER FMR100 FMR1000 ZeroFMR Time(ms)
D M3gl1 4.0 5.4 10.4 10.4
B M3gl2 5.8 24.3 30.7 30.7
1 M3gl3 3.8 6.1 13.6 13.6
- M3gl4 6.4 33.6 50.4 50.4
B M3gl 2.5 5 9.3 9.3
D M3gl1 9.7 21.1 24.3 24.3
B M3gl2 12.9 25.0 28.6 28.6
2 M3gl3 8.9 18.9 20.7 20.7
- M3gl4 11.4 25.0 31.1 31.1
B M3gl 10.5 20.7 25.7 25.7
D M3gl1 0.9 1.4 2.5 2.5
B M3gl2 3.1 6.1 24.6 24.6
3 M3gl3 1.0 1.4 2.5 2.5
- M3gl4 3.7 48.2 70.4 70.4
B M3gl 0.6 0.7 1.4 1.4
D M3gl1 3.4 6.1 7.5 7.5
B M3gl2 4.6 11.4 13.2 13.2
4 M3gl3 3.8 6.1 7.9 7.9
- M3gl4 5.3 20.0 72.9 72.9
B M3gl 4.2 6.1 7.5 7.5

Sensors 2012,12 3434
Experimental results confirm that fulfilling the quality parameters discussed in Section 1has a direct
impact in the matcher accuracy.
7. Conclusions
Fingerprint matching algorithms based on minutiae triplets have proved to be fast and accurate.
They are commonly used on light architectures and in systems based on interoperability standards for
fingerprints represented by minutiae. However, existing algorithms have several limitations that affect
their accuracy. In this paper, we identify the quality parameters that have a more significant impact on
fingerprint matching accuracy in order to create M3gl, a more accurate matcher.
M3gl uses a new representation based on minutiae triplets and a new comparison function that
achieves the invariance to the translation and rotation of fingerprints, and achieves the sensitivity to
the directions of the minutiae relative to the sides of the triangle. The new representation arranges the
minutiae in clockwise sense and the comparison function performs the three possible rotations of the
triplets achieving the invariance to the order of minutiae in the feature and the sensitivity to the reflection
of minutiae triplets. The components of the proposed representation are compared using thresholds that
allow the tolerance to the non-linear distortion of fingerprints. Besides, the new comparison function
includes optimizations that avoid comparing all the components of the triplets, increasing the matching
speed in many cases.
M3gl is deterministic, insensitive to select a single alignment, and sensitive to the individuality of
fingerprints. It uses a simple and effective procedure to compute minutiae triplets which makes the
algorithm tolerant to the low quality of fingerprints and to errors of the feature extractor.
Experimental results in databases of the FVC2002 and FVC2004 competitions show that M3gl has
low computational costs and is more accurate than other algorithms based on minutiae triplets and other
algorithms based on different representations.
In the near future we plan to study the behaviour of the new algorithm for latent fingerprint
identification.
Acknowledgment
The authors would like to thank Dania Yudith Su´
arez Abreu for her valuable contribution improving
the grammar and style of this paper.
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Appendix Theorem Proofs
Theorem A1. Given two m-triplets tand r, if |dt
max −dr
max|> tlthen sinv (t,r) = 0.
Proof. From the definition of m-triplet we have:
dr
max ≥dr
mid (A1)
dr
max ≥dr
min (A2)
dr
mid ≥dr
min (A3)
Assume, without loss of generality, that dt
max > dr
max; then, from the hypothesis of Theorem A1, we
infer that:
dt
max −dr
max > tl(A4)
which is equivalent to:
dt
max −tl> dr
max (A5)
From Equations (A5) and (A1) we have dt
max −tl> dr
mid; which is equivalent to:
dt
max −dr
mid > tl(A6)
Similarly, from Equations (A5) and (A2) we obtain:
dt
max −dr
min > tl(A7)
From Equations (A4), (A6) and (A7) we conclude that dt
max differs more than threshold tlwith respect
to dr
max,dr
mid and dr
min respectively. Therefore, sd(t,r)returns 0for every clockwise shifting of rand
consequently sinv(t,r) = 0.

Sensors 2012,12 3437
Theorem A2. Given two m-triplets tand r, if |dt
mid −dr
mid|> tlthen sinv(t,r) = 0.
Proof. From the definition of m-triplet we have:
dt
max ≥dt
mid (A8)
Assume, without loss of generality, that dt
mid > dr
mid; then, from the hypothesis of Theorem A2, we
infer that:
dt
mid −dr
mid > tl(A9)
which is equivalent to:
dt
mid −tl> dr
mid (A10)
From Equation (A10) and (A3) we have dt
mid −tl> dr
min; which is equivalent to:
dt
mid −dr
min > tl(A11)
Similarly, from Equations (A8) and (A9) we obtain:
dt
max −dr
mid > tl(A12)
Additionally, from Equations (A8) and (A11) we obtain:
dt
max −dr
min > tl(A13)
From Equations (A9), (A11), (A12) and (A13) we conclude that dt
max and dt
mid differ more than
threshold tlwith respect to both dr
mid,dr
min. Therefore, sd(t,r)returns 0for every clockwise shifting of
rand consequently sinv(t,r) = 0.
c
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