Fingerprint Identification Problem: Using Delaunay
Triangulation Technique for Model Database Indexing
Michael Khachay
1,2,3?
, Alexander Dremin
1
and Anton Leshko
1
1
Krasovsky Institute of Mathematics and Mechanics,
16 S. Kovalevskoy st., Ekaterinburg, Russia
2
Ural Federal University, 15 Mira st., Ekaterinburg, Russia
3
Omsk State Technical University, 11 Mira ave, Omsk, Russia
Abstract. A new modification of Delaunay triangulation of finite minutiae sub-
sets extracted from fingerprints is proposed. Stability examining and numerical
analysis results are presented as well.
1 Introduction
Recently developing activity of automatic biometric verification and identification is
steadily increasing. A variety of biometric technologies have been developed for the
last fifty years. Among these, fingerprints, face, iris, speech, and so on. Each technol-
ogy has its strength and weakness. The main criteria used for comparative analysis of
several biometric technologies are universality, uniqueness, collectability, permanence
etc. Fingerprint biometric technology (also known as dactyloscopy) is the oldest and
the most popular due to the high personality and stability of fingerprint images. For a
given finger and a given person, fingerprint is just a digital gray-scale image obtained
by optical scanner and containing a picture of papillary ridges and valleys. So, finger-
print verification / identification systems are based on special image processing and
image analysis algorithms. While the former, verification problem seems to be solved
for the most part, the later remains a great challenge for researchers and developers.
Along with accuracy, scalability becomes one of the first-priority issues of developing
identification methods and algorithms.
Although there are known several fingerprint verification systems comparing raw
fingerprint images using correlation analysis, usually [1] the verification / identification
stage is preceded by some feature extraction one. Among them, minutiae feature ex-
traction technique is the most popular. Geometrically, minutia is an irregular point on
a fingerprint image (termination, bifurcation, crossover etc. of papillary lines). These
points can be considered as finite collection (template) on the plane, and many modern
fingerprint analysis algorithms are based on this consideration. Unfortunately, a reg-
ular fingerprint image usually contains several dozens of minutiae and analysis of all
their sub-combinations can be very computationally expensive. Several geometric tech-
niques are developed [2] to reduce this combinatorial complexity, and triangulation of
the minutiae set is one of them.
?
This research was partially supported by RFBR, grants no. 13-07-00181 and 13-01-210, and
Ural Branch of RAS, grants no. 12-P-1-1016 and 12-S-1017/1
Khachay M., Dremin A. and Leshko A..
Fingerprint Identification Problem: Using Delaunay Triangulation Technique for Model Database Indexing.
DOI: 10.5220/0004394400940100
In Proceedings of the 4th International Workshop on Image Mining. Theory and Applications (IMTA-4-2013), pages 94-100
ISBN: 978-989-8565-50-1
Copyright
c
 2013 SCITEPRESS (Science and Technology Publications, Lda.)

In the paper, an improved version of the Delaunay triangulation technique for fin-
gerprint identification is presented.
2 A Structure of Identification System
From conceptual point of view, any automated fingerprint identification system consists
of two subsystems. First of them (we call it offline) is used in the preparation stage
during a model database construction, and the second one makes an identification of
query fingerprints on the basis of the prepared database.
To describe the first subsystem briefly, it is convenient to use the well known black
box model. By virtue of any standard minutiae extracting algorithm, an offline subsys-
tem maps the initial model fingerprint set I = {I
j
: j = 1, . . . , N} into the family of fi-
nite subsets of Z
3
+
(cube of the set of nonnegative integer numbers). Actually, any model
image I
j
∈ I is mapped to subset T (I
j
) = {(x
i
, y
i
, w
i
) ∈ Z
3
+
: i = 1, . . . , N
j
} that is
called a template. For any triple, x
i
and y
i
coordinates coincide with appropriate geo-
metric coordinates of the i-th minutia detected on the image, and w
i
is a confidence level
of this detection. So, in the beginning of the first stage we have a set B = {T
j
= T (I
j
)}
of templates of the initial images (which is called a model database). Further, any cou-
ple (T
j
, q) we assign a minutiae subset T
j,q
= {(x
i
, y
i
, w
i
) ∈ T
j
: w
i
≥ q} of the
model template consisting of minutiae having sufficiently high accuracy level. In the
sequel, we consider projections of these subsets onto planes H
q
= {(x, y, w) : w = q}
which are parallel to the coordinate plane xOy.
On the second (online) stage, the query image I is processed by the similar way,
and a template T = T (I) is produced, after that the final identification decision is made
by one-to-one matching T with corresponding candidates subset B
T
⊂ B of the model
database. Time complexity of this procedure (for a fixed template T ) is O(M|B
T
|),
where M is complexity of the inner matching algorithm. So, the problem is to construct
the reducing algorithm R constructing for any T a subset R(T ) = B
T
satisfying the
following additional constraints.
1. |R(T )|  N = |B|.
2. Let a fingerprint I producing the query template T belong to some known person
and the model database contain templates produced by another him (or her) finger-
prints. Denote the subset of these templates by B
0
T
and the conditional probability
of the event B
0
T
∩R(T ) = ∅ by P
T
. Then P
T
≤ α for some given value α ∈ (0, 1).
Mathematically, this problem is equivalent to the construction problem of the effi-
cient computable mostly powerful test statistic of the significance level α for the null
hypothesis ’known person’. For any query template T the test produces a ’candidates’
subset R
α
(T ) for the subsequent one-to-one matching (with T ).
Several ways to constructing a solution of this problem are known. The approach
based on preliminary clusterization of the model database by the type of core of the
initial fingerprint images seems [1] to be the earliest. According to this approach, at the
online stage a core type is for the query template is computed after that the search can
be narrowed to the corresponding cluster. Unfortunately, number of known core types
is small and distribution of the real fingerprints (among them) is far from the uniform.
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The another approach based on indexing of the model database supposed [2] to
be more promising. Indexing procedures improve the classical two-stage identification
scheme in the both stages. At offline stage, a model database is indexed using some
hash functions. At the online stage the required subset R(T ) is constructed from the
models with the most similar hash values to ones calculated from the query template T .
During the indexing substage, for any model template, several partial invariants
(that of values of the geometrical nature that are almost invariant to the given trans-
formation group on the plane) are computed and quantized. For instance, let some
numerical features f
1
, f
2
, . . . , f
k
of the geometrical shapes of some kind formed by
the fingerprint minutiae are used as partial invariants, then for any model T
i
and for
any shape S of interest the following record g
1
(S), . . . , g
k
(S), r
i
will be included into
the indexing table. Here g
j
is quantized value of the feature f
j
and r
i
is reference to
model T
i
. Such a way, any model template T
i
is transformed to some finite subset in
k-dimensional indexing space.
The online identification stage is started on calculating the similar invariant values
for the query template. The computed k-dimensional vectors are filtered using some
system of the additional constraints that of control parameters of the algorithm. Further,
the separated vectors are used for search in index table and posterior probabilities of
models T
i
computing. The result ordered subset R(T ) is constructed from the most
probable models according to this posterior measure.
Performance of indexing algorithms is suggested [2] to estimate by values of correct
index power (CIP).
Suppose, we have a couple (T
i
, T
0
i
) of fingerprints for any respondent obtained from
the same finger. Construct the model database B from the former elements of each
couple, and test database C from the later (|B| = |C| = N, by construction). The
model T
i
∈ B is said to be correctly indexed by the algorithm R if T
i
∈ R(T
0
i
). Let
N
ci
(R) be the number of correctly indexed elements of B then
CIP(R) =
N
ci
(R)
N
. (1)
It’s clear, that CIP(R) is a stochastic variable which depend not only on the algorithm
R in question but also on random choice of (B, C). Its population value can be sta-
tistically estimated on representative fingerprint samples. In this paper the well known
’NIST Special Fingerprint Database 4’ [3] is used for such estimations.
3 Partial Invariants
The system of invariants constructed in this paper generalizes the system developed
in [4] and contains quantities that are invariant to the similarity transformations group
(over the plane). For a fixed accuracy value q of detected minutiae and any template
T a projection Π
q
(T ) of the set T
q
= {(x
i
, y
i
, w
i
) ∈ T : w
i
≥ q} onto the plane
H
q
= {(w, y, w) : w = q} is assigned and Delaunay triangulation [5] of the set Π
q
(T )
is constructed. The choice of Delaunay triangulation method is due to the following
reasons
– such a triangulation is unique for any non degenerate finite set on the plane;
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– the resulting triangulation consists of O(n) facets that is substantially smaller than
the number O(n
3
) of all possible triangles with the vertices of the given n-point
set;
– this triangulation can be constructed efficiently, we use an algorithm [6] with time-
complexity O(n log n);
– a topological structure of the resulting triangulation is stable [7] to small perturba-
tions of the initial data.
Suppose, a triangle ∆ is a triangulation facet with edges a ≤ b ≤ c. Assign to this
triangle a vector ν(∆) = [α, β, γ] by the formulas α = b/c, β = a/b, and γ = cos C.
This vector is invariant to any similarity (e.g. translation, rotation and scaling) transform
on the plane and satisfies the following inequalities
1
2
< α ≤ 1, 0 < β ≤ 1, −1 < γ ≤
1
2
.
Suitable discretized (particularly, to automatically distinguish the triangles-isomers)
these parameters are used in both stages, offline and online.
4 Proposed Algorithm
4.1 Indexing Stage
Input.
1. Model database B = {T
j
: j = 1, . . . , N}.
2. Minimum accuracy level q for detected minutiae.
3. Maximum index values n
1
, n
2
, n
3
.
Output. Set-valued map h : Z
3
q
→ 2
B×D
(index table) defined on integer lattice
[0, . . . , n
1
] × [0, . . . , n
2
] × [0, . . . , n
3
]
as follows: any triple (i, j, k) is assigned to set of couples (T
o
, ∆
t
), where T
p
is some
model template and the triangle ∆
t
is a facet of Delaunay triangulation of Π
q
(T
p
) such
that discretized value of the vector ν(∆) is equal to (i, j, k).
4.2 Identification (Query) Stage
Input.
1. A query template T and minutiae extracting accuracy level q.
2. Index (hash) table h.
3. Maximum number L of the resulting hypotheses.
4. Discretization parameters n
1
, . . . , n
4
for similarity transformations.
Output. A sequence of couples (T
p
, r
p
, S
p
), where T
p
is extracted model (identification
hypothesis), r
p
is its rating, and S
p
is affine similarity transform assigning T to T
p
. The
sequence is ordered by r
p
by descending.
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Scheme.
1. Similar to the above considerations, for any Delaunay triangulation facet ∆
t
of the
projection Π
q
(T ) (for the query template T ) an appropriate index cell (i
t
, j
t
, k
t
)
and the set h(i
t
, j
t
, k
t
) are assigned.
2. For any triangle δ
v
∈ h(i
t
, j
t
, k
t
) an appropriate similarity transform S
tv
(map-
ping the vertexes of ∆
t
into corresponding vertexes of δ
v
) is computed. Scaling
parameter λ
tv
, cosine cos ϕ
tv
of the rotation angle, and translation vector b
tv
are
discretized, and the corresponding model template T
v
is added as entry of the sec-
ondary index table (by the increasing of its rating).
3. The required L hypotheses and the appropriate similarity transforms are found by
searching over the secondary table (Fig. 1).
5 Testing of the Algorithm
Training (tuning the controllable parameters) and testing of the algorithm were made on
the well known NIST-4 Special Fingerprint Database, the respectable testing polygon
for modern fingerprint verification/identification heuristics. By structure, this database
consists of 2K fingerprint pairs, for each of them both images (denoted by ’fXXXX’
and ’sXXXX’) are obtained from the same finger.
We use this database for solving the following additional problems.
1. Proving the stability of the proposed indexing scheme to small perturbations of
the initial data, such as addition (deletion) of minutiae and modifications of their
geometrical locations.
2. Discretization parameters tuning for primal and secondary indexing tables.
Fig. 1. Query fingerprint S1678 and the most valuable hypothesis F1678.
Accuracy level for detected minutiae was fixed to q = 64. In both problems, the
subset the dataset of 1923 pairs (96%), where f-image produced a template with at least
50 minutiae, was chosen.
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5.1 Stability Examining
This kind of testing proceed on special synthetic dataset obtained from the mentioned
above NIST-4 database. According to the well known ”white noise” model, for any
f-image from the initial dataset, geometrical locations of the minutiae were modified
using independent identically, N(0, σ)-distibuted normal noise
x
0
i
= x
i
+ ξ
i
, y
0
i
= y
i
+ η
i
.
For an additional parameter r ∈ [1, 2], a minutiae (x
0
i
, y
0
i
, w
i
) was included to perturbed
template iff ξ
2
i
+ η
2
i
≤ r
2
σ
2
Such a way, for any initial model template a 20-element
stochastic perturbed sample was assigned.
Fig. 2. CIP-index for σ = 3 and r = 1.5.
Further, the initial templates were used as models in the indexing stage and any
constructed template was identified using the algorithm proposed. Obtained numerical
data confirm the known theoretically proved [7] stability result. Particularly, for σ = 1
and r ∈ [1.5, 2] (from 33% to 13% of excluded minutiae in average) 100% perturbed
templates were classified correctly within L = 1. Increasing of σ leads to increasing of
the L-value, as expected. But stability of the entire algorithm remains high. For instance,
for σ = 3 and r = 1.5, CIP-value for L = 1 was 77%, and for L = 10 (0.5% of the
initial database) - more then 89% (Fig. 2).
5.2 Tuning and the Final Testing
For training (parameter tuning) a subset of 430 (21.%) fingerprint pairs is used, where
f-image possessed at least 100 minutiae on the accuracy level 64, while the for the
testing the complement (of this subset) to the initial database were taken. During the
training stage the parameters were tuned by local search heuristic. The optimal values
of parameters are 8 × 8 × 8 for a primal index table (hash) and 17 × 17 × 17 × 47 for
the secondary. The final results for several voting strategies are presented in Fig. 3.
99

Fig. 3. CIP-analysis of several voting strategies.
6 Discussion
A detailed comparative analysis of the known indexing techniques for fingerprint iden-
tification problem is presented in the book [2]. The algorithm with 85% CIP value for
L = 0.1N is recognized optimal among them. The indexing scheme used in this al-
gorithm was based on considering the all possible triangles with vertexes in minutiae
locations, and time-complexity of its online stage was O(n
3
). Our method has CIP
value of 82 ± 5% for the same L, while its time-complexity is O(n log n) thanks to
Delaunay triangulation.
7 Conclusions
A new indexing technique for the model database in fingerprint identification is pro-
posed. The technique is based on Delaunay triangulation of finite minutiae set on the
plane and original indexing parameters adjusting heuristic. Computational experiments
proved high performance of the proposed algorithm in relation to the known analogs.
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