DSpace at VNU: An efficient method for fingerprint matching based on local point model

These points are associated with the known corresponding point pairs to select a suitable landmark point set for using Local Thin Plate Spline deformation model over 9 partial areas o

An Efficient Method for Fingerprint Matching

Based on Local Point Model

1

General Department of Technique - Logistic,

Vietnam Ministry of Public Security

huongthuykta@yahoo.com, kynguyen22@gmail.com

2

Information Technology Faculty, Vietnam National University - Hanoi College of Technology,

huanhx@vnu.edu.vn

Abstract - This paper proposes a fingerprint matching method

based on local Thin-Plate-Spline (TPS) deformation model, a

warping technique, to deal with non-linear distorted fingerprints

After determining the set of corresponding minutiae pairs

between two fingerprints by using an affine transformation, a set

of corresponding pseudo-minutiae pairs are created based on

their local ridge-valley structure comparison These points are

associated with the known corresponding point pairs to select a

suitable landmark point set for using Local Thin Plate Spline

deformation model over 9 partial areas of fingerprint images in

order to find new corresponding minutiae pairs This procedure

is repeated until no more new corresponding minutiae pairs are

distinguished or the number of corresponding point pairs is large

enough The experimental results on the database FVC2004 show

that the proposed method significantly improves matching

performance compared to the global TPS warping method

Keywords: Fingerprint verification; Minutiae matching; Elastic

deformations; Thin-Plate-Spline models; Local deformation

models

I INTRODUCTION

Automatic Fingerprint Identification and Authentication

(AFIA) is a biometric technology which had been considered

for a long time [13] but it is still an active research area [4, 7,

8, 13, 15] In the fingerprint identification system, the

matching algorithm is the most important stage Many

fingerprint matching algorithms have been published [4, 7, 8,

13, 14, 15], and can be classified into three main categories

[12] based on: minutiae sets, texture and greyscale correlation

The minutiae based [5, 7-11, 15, 19], the most widely used

algorithm, is of concern in this paper

In a fingerprint image, minutiae such as end points or

bifurcation points are called the minutiae In order to match

the query fingerprint image Iq with the template fingerprint

image It in a database, a general approach of the fingerprint

matching algorithm based on minutiae attempts to estimate the

affine transformation to align two minutiae sets of two

fingerprint images originated from the same finger so that the

respective minutiae pairs “match fit” together in pairs The

most popular but simple deformation is affine deformation

(combining translation, rotation and scale) However, in

general, distorted fingerprint is non-linear therefore the

mathematical efficiency using affine transformation is limited

Nowadays, it usually used for simple matching as an initial

stage and then warping methods are applied [10, 12] to

determine the similarity scores of two fingerprints Amanssa

et al [1] and other authors [2, 5, 11, 12, 17] used additional

sheets tuning adapter model (thin-plate spline: TPS) in this

procedure Particularly, after using affine transformation to

calculate the number of corresponding minutiae pairs, Li et al

[12] used these corresponding minutiae pairs for landmark

points which help warping TPS and combined with greyscale

correlation to decide the similarity This method is called global TPS

In global TPS, we have to calculate local correlation and solve simultaneous linear equations The size of these linear equations directly proportionates to the number of landmark points It normally takes a considerable amount of calculation time when landmark points are large Moreover, this method cannot process distorted fingerprints in image’s area which is far from landmark point pairs

In facts, the efficiency of warping TPS depends on the distribution of landmark points on fingerprint images Points, which are far from landmark points, will be variously affected

by non-linear distortion and give different matching results Furthermore, by using affine transformations, the originally detected landmark points are normally focused on the little nonlinear deformed area It makes a paradox: the no need warping and less distorted areas contain too much landmark points whereas the far and more deformed areas have no landmark point for warping

In order to improve this approach, we propose a warping method P-TPS with two steps The first step is to determine the set of initial corresponding minutiae pairs by applying an affine transformation then using the sequence of quantized points of their ridge-valley pair associated to choose the set of landmark points with higher reliability The second step is based on convex hull of all known corresponding pairs, fingerprint images are divided into 9 areas New created corresponding points are combined with available corresponding minutiae pairs to choose suitable landmark points on each area for warping in order to calculate new set

of corresponding minutiae pairs The new set of corresponding minutiae pairs is able to reuse iteratively until there are no more corresponding point pairs or the number of corresponding pairs is large enough to distinguish

Our new method is suitable for both roll/full and plain/flat fingerprint images Experimental results on the database of the fingerprint FVC2004 (DB1 and DB3) [20] show that our proposed method provides better results than global TPS method Calculation time and memory usage also has been significantly reduced

The rest of the paper is presented as following: Section 2 introduces the method of fingerprint matching based on minutiae and some methodologies which will be used New proposed algorithm is described in Section 3 Section 4 presents experimental results and the final section is reserved for evaluation and conclusion

II MINUTIAE BASED FINGERPRINT MATCHING AND RELATED

WORKS

This section briefly introduces minutiae based matching method, distorted TPS model and global TPS method

A Fingerprint matching problem and matching scheme

based on minutiae

Fingerprint matching problems can be described as

following: Give a query fingerprint Iq and a template

fingerprint It, it is to determine whether the two images are

originated from the same finger or not The answer is firstly

determined by calculating the similarity scores and then by

decision yes or no base on the similarity threshold

Fingerprint matching algorithms can be classified into three

main categories [12] based on: minutiae, texture and greyscale

correlation Within these categories, the method based on

minutiae is simple but yet efficient therefore it is the most

widely used in [5, 8-10, 15, 19] A short scheme of this

method is outlined below and the detail is in [10, 14]

1) Minutiae based method

In a fingerprint image, points, which represent

discontinuities of fingerprint local structure such as end

points, bifurcation points, are called minutiae

From two fingerprint images Iq, It, we define two sets of

minutiae Mq and Mt as follows:

M q = {m 1 , m 2 , …, m M }; with mi= (xi,yi,θi), i = 1, ,M (1)

where (xi, yi) are coordinates of mi on the image plane R2 of Iq,

and θi is the fingerprint direction at mi

M t = {m 1 ’, m 2 ’, …, m N '}; with mi'= (xi’,yi’,θi’), i = 1, ,N (2)

where (xi’,yi’) are coordinates of mi' on the image plane R2 of

It, and θi’ is the fingerprint direction at mi'

2) Matching scheme based on minutiae

A suitable aligned transformation from image plane of Iq to

image plane of It is required to determine corresponding point

pairs of Mq and Mt Point pair mi (Mq) and mj'(Mt) are called

correspondent if image mi

”

of mi via this transformation belongs to the neighbourhood (with a small values of r and θ)

of mj' In this case, it is said that mi matched with mj' When

two images Iq and It are genuine (from the same finger), the

number of corresponding minutiae pairs is much more than in

the case they are impostor (not from the same finger) In fact,

due to non-linear distorted fingerprints and noise, even two

images are genuine, it is not simple to find corresponding

minutiae pairs

Suppose that there are n corresponding minutiae pairs

found from two images, M is the number of minutiae on the

query fingerprint and N is the number of minutiae on the

sample fingerprint, the similarity of two fingerprint images is

characterized by the measurement S(It,Iq) given by the

following formula:

S(I t ,I q ) = n 2 /M× N (3)

In the general algorithm, one is interested only in the

similarity S(It,Iq), which is also the output

Based on predefined thresholds of scores St max and St min, we

conclude that two fingerprints are matched if S(I t ,I q ) > St max,

otherwise they are mismatched (S(It,Iq) < St min) For the pairs

of fingerprints having the similarity scores in interval [Stmin,

Stmax] the warping stage is needed

The simplest and most general transformation used in

matching methods to align two images is affine transformation

(combined with linear transformations: translation, rotation and scale) However, due to the nonlinear distorted fingerprints when capturing fingerprint images, the efficiency

of this method is not enough and is usually employed to determine initial corresponding minutiae pairs for advanced

transformations is TPS deformation model in [1, 2, 8 -10, 15,

19]

The thresholds St max and St min together with the false rejected rate FRR(St) and the false accepted rate FAR(St) as function of St are commonly determined by learning on the two sets of genuine and impostor fingerprint images In this

case, St max is the largest value for FAR = 0 and St min is the

smallest value for FRR=0

B Thin-Plate Spline deformation model and global TPS matching

1) Thin-Plate-Spline deformation model

After determining the n pairs of corresponding minutiae by using affine transformations for creating an initial set of landmark points, the image is warped by TPS model [1], [11] This method uses affine transformations and radial interpolation functions of two-dimensional as follows:

=

−

n i

i

w

1

) , ( (4) Where, U ( r ) = r2log r , ( v u , ) = u +2 v2 and

by convention U(0)=0; Pi are landmark points, the first three terms describes full affine transformations with affine parameters a1, ax, ay and the rest term describes the non linear deformation with weight values wi which need to find so that the landmark points matched

The coefficients of f(x,y) transformation are determined by solving linear equations (4) Based on this warping method, if two fingerprints are genuine, it is expected to detect more new corresponding point pairs for calculating similarity Actually, while determining corresponding minutiae pairs based on tolerance, there are some ambiguity correspondence cases In order to overcome this issue, the two-way graph and Ford-Fulkerson algorithm are applied to determine this correspondence [16, 19] Occasionally, additional local information of minutiae pairs is used as secondary feature [9], [10] Based on this technique, the similarity between corresponding pairs could be calculated by using local greyscale correlation [12]

2) Global TPS matching

Li et al [12] proposed fingerprint matching method known

as global TPS In this method, after determining TPS transformation, they calculate local greyscale correlation in correspondingpairs Based on the similarity of corresponding pairs via this greyscale correlation, the decision for identification is then determined The detail of this method is given in [12]

III NEW METHOD: PARTIAL TPS FINGERPRINT MATCHING

335

The TPS method is a good solution to overcome non

distorted fingerprints but it still has some following

disadvantages:

i) By considering all sets of initial corresponding minutiae

pairs as landmark points without using local info

verification, it is easily to introduce some false landmark

points However, using local greyscale correlation before

warping is not reliable because it is highly sensitive with

distortion, noise, and incomplete information

ii) Because landmark points are determined by global affine

transformation, therefore they generally focus on areas with

only few non-linear distortions and not distributed equally

over the entire overlapping area of two fingerprints I

Therefore, minutiae which are far away from non

distorted landmark points, are often disregarded

iii) Warping global fingerprint one requires solving large

linear equations that takes a lot of calculation time but the

result normally has high variation In reality, fingerprint

distortion is local natured and closer points have fewer

distortions

In order to overcome the first disadvantage, we propose

using local structure known as ridge-valley pair associated

with each minutia to verifying and select a set of

corresponding minutiae pairs with higher reliability In order

to deal with the second disadvantage, our method proposes an

additional technique to strengthen the set of landmark points

by utilizing more pseudo-minutiae belonged to this ridge

valley pair associated Based on the distribution of pseudo

minutiae on fingerprint image, a subset with more equal

distribution on the overlapping area of two fingerprints is

chosen as additional landmark points for warping Finally, our

method uses partial TPS warping method instead of

TPS method to overcome the third disadvantage

A Associated ridge-valley structure and pseudo

1) Associated ridge-valley structure

As aforementioned, components of the minutiae including

only positions and orientations do not have enough

information to present the local structure, therefore Ross

[18] proposed to use associated ridge information to match

corresponding minutiae pairs In order to get more local

information at these points, we also considered the valley

which associated with dual minutiae on the valley image Our

consideration is based on the duality that can be formulated as

follows: Opposing with a minutiae "bifurcation/end" point on

the ridge image, there always exist a dual minutiae

"end/bifurcation" point on valley image It is also true for

valley images (negative image) Thus ridge

associated with one minutiae composed from associated ridge

with this minutiae and associated valley associated with its

dual one on the valley image Fig 1 describes some minu

on the ridge image and its dual minutiae on the valley image

The TPS method is a good solution to overcome non-linear

distorted fingerprints but it still has some following

i) By considering all sets of initial corresponding minutiae

pairs as landmark points without using local information for

verification, it is easily to introduce some false landmark

points However, using local greyscale correlation before

warping is not reliable because it is highly sensitive with

k points are determined by global affine transformation, therefore they generally focus on areas with

linear distortions and not distributed equally

over the entire overlapping area of two fingerprints It and Iq

far away from non-linear distorted landmark points, are often disregarded

iii) Warping global fingerprint one requires solving large

linear equations that takes a lot of calculation time but the

result normally has high variation In reality, fingerprint

distortion is local natured and closer points have fewer

In order to overcome the first disadvantage, we propose

valley pair associated

to verifying and select a set of iae pairs with higher reliability In order

to deal with the second disadvantage, our method proposes an

additional technique to strengthen the set of landmark points

minutiae belonged to this ridge-the distribution of pseudo-minutiae on fingerprint image, a subset with more equal

distribution on the overlapping area of two fingerprints is

chosen as additional landmark points for warping Finally, our

warping method instead of global TPS method to overcome the third disadvantage

valley structure and pseudo-minutiae

As aforementioned, components of the minutiae including

only positions and orientations do not have enough

rmation to present the local structure, therefore Ross et al

[18] proposed to use associated ridge information to match

corresponding minutiae pairs In order to get more local

information at these points, we also considered the valley

th dual minutiae on the valley image Our consideration is based on the duality that can be formulated as

follows: Opposing with a minutiae "bifurcation/end" point on

the ridge image, there always exist a dual minutiae

ge It is also true for valley images (negative image) Thus ridge-valley pair

associated with one minutiae composed from associated ridge

with this minutiae and associated valley associated with its

dual one on the valley image Fig 1 describes some minutiae

on the ridge image and its dual minutiae on the valley image

(a) Fig 1: The minutiae on the ridge (thick line) have it's dual minutiae on the valleys (thin line).

Some configurations of figured minutiae with their ridge valley pair associated are described in Fig 2

Fig 2 (a) A "short ridge or island" and its dual valley form "lake"; (b) A

"spur form" ridge and its dual spur form valley reverted; (c) A "Crossover" and its dual "two meeting ridges"; (d) Not exist the dual for line break, the extracted minutiae is rejected

2) Sampling ridge-valley pair associated and appending

corresponding point pairs

Note that l0 is the length (usually chosen as the double distance of ridges space) and lmax =4l

minutiae is generated on associated ridge corresponding minutiae pairs as follows: From each minutiae,

on ridge image and from dual minutiae on valley image we used the contour tracing algorithm proposed in [10] to following along ridge and valley to sampling quantized points with equidistant step l0 until meeting the border or the length overcome the threshold lmax These quantized po

corresponding minutiae are characterized by positions and directions hence called pseudo-minutiae For each genuine corresponding minutiae pairs, ridge-valley pair associated is also corresponding, but it is not true for impostor case Therefore, after checking the coordination and the orientation

of these pseudo-minutiae, we select only corresponding pseudo-minutiae to append into set of landmark points Fig 3 described pseudo-minutiae sampled with equidistance of l

Fig 3 The minutiae m j and its sampled points with equidistance l

B Area Division and choice the landmark points set for local deformation model

In order to warp the image using partial TPS, it is required

to divide an image into 9 areas and choose for TPS transformation of each area

1) Area Division for TPS warping Pseudo point

(b) Fig 1: The minutiae on the ridge (thick line) have it's dual minutiae on the

minutiae with their ridge-valley pair associated are described in Fig 2

Fig 2 (a) A "short ridge or island" and its dual valley form "lake"; (b) A

"spur form" ridge and its dual spur form valley reverted; (c) A "Crossover" its dual "two meeting ridges"; (d) Not exist the dual for line break, the

valley pair associated and appending

is the length (usually chosen as the double

=4l0 The set of pseudo minutiae is generated on associated ridge-valley structure of corresponding minutiae pairs as follows: From each minutiae,

minutiae on valley image we used the contour tracing algorithm proposed in [10] to following along ridge and valley to sampling quantized points

until meeting the border or the length These quantized points of two corresponding minutiae are characterized by positions and

minutiae For each genuine valley pair associated is also corresponding, but it is not true for impostor case

e, after checking the coordination and the orientation

minutiae, we select only corresponding minutiae to append into set of landmark points Fig 3

minutiae sampled with equidistance of l0

and its sampled points with equidistance l 0

Area Division and choice the landmark points set for local

In order to warp the image using partial TPS, it is required

to divide an image into 9 areas and choose landmark points

m j

336

For each known corresponding minutiae pairs set, it is

possible to estimate the aligned region by determination their

convex hull on the image plane based on the proposed

technique in [6] The largest diameter of convex hull is chosen

as the horizontal axis of corresponding image plane and the

centeris its midpoint We divided the plane into 8 areas by 8

rays from center with an angles separated regularly of 450

Taking central area with radius R0 (predefined as 4mm), the

outside is 8 areas separated by rays So fingerprint image is

divided into 9 areas to compare minutiae

2) Choice the landmark points set for local TPS deformation

model

For each considered area, it is possible to determine square

grids with the size step of 4mm on a image with standard

resolution On each grid square, we choose at most one

landmark point (if it is available) from the set of

corresponding minutiae or pseudo minutiae according to the

following priorities: The minutiae nearest grid’s central

square, left to right and top to bottom Based on the method of

sampling in III-A, this set of landmark points is determined by

pairs If there are only few landmark point pairs on this area, it

is possible to reuse nearest points in adjacent areas to find

TPS transformation

Now we describe our P-TPS algorithm to match

fingerprints by using the above model

C P-TPS algorithm

Given two set of minutiae Mt and Mq of the sample ridge It

and the inquiry ridge Iq; applying affine transformation to

obtain corresponding minutiae pairs set M1 and all its

rige-valley pairs associated, the fingerprint matching method

applying P-TPS is described in Fig 4

In this algorithm, based on predefined thresholds of scores

St max and St min, we conclude that two fingerprints are matched

if S(I t ,I q ) > St max, otherwise they are mismatched (S(It,Iq) <

St min) For the pairs of fingerprints having the similarity scores

in interval [Stmin, Stmax] the warping stage is needed

In the warping stage of algorithm, based on the initial

corresponding minutiae pairs set and initial similarity score, a

set of corresponding pseudo minutiae pairs is generated by

their ridge-valley pairs associated to increase the selection

space to choice only the small subset of landmark points more

regularly distributed in each area for using a local warping

model According to warped coordination results, the

algorithm detects more new genuine corresponding minutiae

pairs over each partitioned area and recalculating the global

similarity between two images

In this algorithm we also used a newer version of the

similarity formula in 3 Replacing M and N in (3), we only

considered nq, nt, the minutiae number respectively belonged

to the aligned region of two query and template fingerprints

estimated at each step from convex hull of the set of expanded

landmark points This procedure is repeated until the

correspondent set is large enough or new corresponding

minutiae pairs are not found

Procedure P-TPS Fingerprint Matching

Input: Two sets of minutiae M t , M q with M, N; The initial

corresponding minutiae pairs set M1 with n pairs [21], Set M1* with n* of quantized points (pseudo minutiae) Warping threshold

St min , St max ; ridge step: step;

Output: The corresponding minutiae pairs set with n" pairs found

after warping and overall similarity of the two fingerprints S

1 Verification the initial set of corresponding minutiae pairs by comparing two pairs vectors of quantized points sampled from its the ridge-valley pairs associated (n – the original minutiae pair), The output result of this step is the number n' (<= n) of corresponding pairs of minutiae

2 Similarity calculation S = n'2/(nt*nq) If If S ∉ [St min , St max], passing to 6;

3 Increasing the number of landmark points by adding more quantized points (pseudo minutiae)

3.1 Adding pseudo minutiae on the ridge-valley pairs associated to the set M1 , the result: M1 + M1* (n*+n' is the total number of corresponding minutiae and pseudo minutiae pairs)

3.2 Calculating the convex hull of M1 + M1* using algorithm [5]

4 Spatial Filtering on M1 + M1* and landmark points selection

4.1 Spatial Filtering using square grid by selecting a set of m corresponding points on the m square grid;

4.2 Selecting the sets of landmark points for 9 divided areas; 4.3 Warping partial using TPS deformation model on each divided area and calculating new set NewM1 of corresponding minutiae pairs with cardinal (NewM1)= n"

5 If (n' = n") passing to 6, if not:

5.1 n'← n"; M1 ← NewM1 5.2 come back 2;

6 Return Output: M1 and S;

Fig 4 Description of P-TPS matching algorithm

IV EXPERIMENTAL RESULTS

Although our experimental results on rolled/full fingerprint images database show that proposed method give significantly better results than global TPS method, for more objective evaluation, this section presents experimental results on FVC2004 (DB1, DB3) [20] Each database in FVC2004 has several significant deformation fingerprints Each set contains

800 fingerprints images of 100 different fingers, 8 different images for each finger DB1 includes fingerprints collected by the optical sensor “CrossMatch V300”, DB3 is collected by the thermal sensor “FingerChip FCD4B14CB” of Amtel Performing matching on each DB, we examine 2800 iterations for matching fingerprints of same fingers and 4950 test iterations to match fingerprints of different fingers to estimate the Genuine and Impostor distribution Based on estimated distributions, the errors FAR, FRR and FAR100, FAR1000, zeroFAR, zeroFRR as requirements by protocol of FVC2004 [20] are caculated Threshold values Stmin, Stmax

then determined from zeroFRR and zeroFAR

The results of two warping method are described in Table

1, the comparisonof two curves ROC on DB1, DB3 databases shown on Fig 5.(a) and 5.(b) respectively

The comparison of experimental results based on time and memory usage of our proposed method P-TPS with TPS Global method using DB1, DB3 database is shown in Table 2

337

The results of two warping methods applied to the same

database show that P-TPS warping method achieves higher

accuracy than global warping method on all evaluated parameters

Fig 5 Comparison the ROCs of global TPS method and P-TPS method on FVC2004 DB1, DB3

T ABLE 1: C OMPARING THE PERFORMANCE OF TWO MATCHING ALGORITHM ON FVC 2004 DB1, DB3

FVC2004

DB

Global-TPS P-TPS Global-TPS P-TPS

Global-TPS

P-TPS Global-TPS P-TPS

TABLE 2: COMPARISON THE TIME AND MEMORY CONSUMPTION OF TWO METHODS ON FVC 2004 DB1, DB3

FVC2004

DB

Global-TPS P-TPS Global-TPS P-TPS Global-TPS P-TPS Global-TPS P-TPS

In more details, the average EER of P-TPS (2.32%) is

smaller than Global TPS (3.43%) The average comparing

time of P-TPS is 62.49 ms with Genuine matching and

40.62ms with Impostor matching, smaller than Global TPS’s

average is 72.05ms with Genuine matching and 68.31ms with

Impostor matching Regarding the memory usage, P-TPS

method uses only 2.07 Kbytes with Genuine matching and

0.95 Kbytes with Impostor matching, a lot of reduction

comparing with Global TPS’ result is 226.02 Kbytes with

Genuine matching and 224.39 with Impostor matching can be

explained by fact that the Global TPS' method requires a lot of

memory to store the original fingerprint image for calculation

of correlation while our proposed method does not require it

All experiments are performed on the same computer with

CPU Intel Pentium 4, 2.8 GHz

V CONCLUSION

In this article, we have proposed a new method to match

fingerprints using local TPS deformation model The

advantage of our new method is the introduction to use pseudo-minutiae extracted from ridge-valley pairs associated

as a highly reliable local information without complicated calculations These new aided pseudo minutiae set together with initial set of corresponding minutiae give us more alternatives to choose the set of landmark points which reflect better the local distortion and distributed more regularly on he image plane Thus they give better warping performance when the fingerprints are non-linear distorted By modifying global warping method, our proposed local warping method therefore provide more reliable answers

Experimental results on FVC 2004 (DB1, DB3) databases show that comparing with the global TPS method, our new method significantly increases matching efficiency by improving both genuine and impostor distribution, therefore reducing simultaneously both FAR and FRR parameters Furthermore, our proposed method reduces the calculation time and memory usage

Acknowledgement

338

This work is partially supported by Vietnams National

Foundation for Science and Technology Development

(NAFOSTED, Project:102.01-2011.21 )

REFERENCES [1] A Almansa and L Cohen, Fingerprint image matching by

minimization of a thin-plate energy using a two-step algorithm with

auxiliary variables, in Applications of Computer Vision, 2000, Fifth

IEEE Workshop on., 2000

[2] A M Bazen and S H Gerez, Fingerprint matching by thin-plate spline

modeling of elastic deformations, Pattern Recognition, Vol 36 no 8,

1859–1867 (2003)

[3] T H Cormen, E L Charles, and L R Ronald Introduction to

algorithms McGraw-Hill Book Company (1998)

[4] S A Daramola and C N Nwankwo, Algorithm for Fingerprint

Verification System, Journal of Emerging Trends in Engineering and

Applied Sciences (JETEAS), Vol 2 no 2, 355-359 (2011)

[5] K Divya, S Jeyalatha, Distorted Fingerprint Verification System,

Informatica Economică, vol 15, 13-21 (2011)

[6] R.L Graham, An efficient algorithm for determining the convex hull of

a finish planar point set, Information Processing Letter, Vol 1,

226-239 (1998)

[7] K Hoyle Minutiae Triplet-Based Features with Extended Ridge

Information for Determining Sufficiency in Fingerprints, Master

Thesis, Virginia Polytechnic Institute and State University, Burruss

Hall Blacksburg, VA, USA (2011)

[8] A K Jain, J Feng, K Nandakumar, Fingerprint Matching, IEEE

Computer, Vol 43 no 2, 36-44 (2010)

[9] T Y Jea and V Govindaraju, A minutia-based partial fingerprint

recognition system, Pattern Recognition, Vol 38 no 10, 1672 – 1684

(2005)

[10] T Y Jea, Minutiae based partial fingerprint recognition PhD thesis of the University at Buffalo, the University of NewYork November (2005)

[11] D Kwon, I D Yun, D H Kim, S U Lee, Fingerprint Matching Method Using Minutiae Clustering and Warping, Proc of The 18th International Conference on Pattern Recognition (ICPR'06), Vol 4,

525 – 528 (2006)

[12] J Li, S Tulyakov, Z Zhang, V Govindaraju, Fingerprint Matching Using Correlation and Thin-Plate Spline Deformation Model, 2nd IEEE Conference on Biometrics: Theory, Applications, and Systems (BTAS 08), Washington, 1 – 4 (2008)

[13] S U Maheswari, Dr E Chandra, A Review Study on Fingerprint Classification Algorithm used for Fingerprint Identification and Recognition, IJCST Vol 3 no 1, 739-744 (2012)

[14] D Maltoni, D Maio, A K Jain, S Prabhakar, Handbook of fingerprint recognition, Second ed, Springer-Verlag (2009)

[15] M A Medina-Pérez, M García-Borroto, A E Gutierrez-Rodriguez, L Altamirano-Robles, Improving Fingerprint Verification Using Minutiae Triplets, Sensors, vol 12, 3418-3437 (2012)

[16] A Rattani, G L Marcialis, F Roli, Biometric Template Update using the graph min cut algorithm, The 2008 Biometrics Symposium (BSYM), 23-28 (2008)

[17] A Ross, S Dass, A Jain, A Deformable Model for Fingerprint Matching, Pattern Recognition, Vol.38 no 1, 95–103 (2005)

[18] A Ross, S Dass, A Jain, Fingerprint warping using ridge curve correspondences, IEEE Transactions on Pattern Analysis and Machine Intelligence Vol 28 no 1, 19–30 (2006)

[19] C Wang, M Gavrilova, Y Luo, J Rokne, An efficient algorithm for fingerprint matching, Proc of The 18th International Conference on

Pattern Recognition (ICPR'06), Vol 1, 1034– 1037 (2006)

[20] http://bias.csr.unibo.it/FVC2004/ (2004), FVC2004, The Third International Fingerprint Verification Competition

[21] http://www.neurotechnology.com , Neurotechnology, Inc, Verifinger 4.2 SDK.

339