Nonlinear phase portrait modeling of fingerprint orientation

Wei-Yun Yau1, Jun Li2, Han Wang3 Institute for Infocomm Research, Agency for Science Technology & Research, Singapore1

School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore2,3 wyyau@i2r.a-star.edu.sg1, leejun@pmail.ntu.edu.sg2, hw@ntu.edu.sg3

Abstract

Fingerprint orientation is crucial for automatic fingerprint identification. However, recovery of orientation is still difficult especially in noisy region. A way to aid recovery of the orientation is to provide an orientation model. In this paper, an orientation model for the entire fingerprint orientation using high order phase portrait is suggested. Proper analysis of the orientation pattern at the singular point regions is provided. Then a low-order phase portrait near each of the singular point is added as constraint to the high-order phase portrait to provide accurate orientation modeling for the entire fingerprint image. The main advantage of the proposed approach is that the nonlinear model itself is able to model all type of fingerprint orientations completely. Experiments and visual analysis show the effectiveness of the proposed model.

Keywords : fingerprint, orientation field, phase portrait model, nonlinear 1. Introduction

The most widely used biometrics at present is

fingerprint because it is accurate and affordable. With the increasing emphasis on identity management, automatic fingerprint recognition has received wide attention commercially. Nevertheless there still exist critical research issues such as the long processing time in large databases and dealing with poor quality images. Solving these two problems will require improvement to fingerprint classification and identification. In both topics, orientation pattern, which can be defined by the local direction of ridge-valley structure, plays an important role [6, 7, 8, 9].

However there are many fingerprint images that are noisy caused by dust, oil, moisture, scars or excessively wet or dry fingers. As such, we cannot always obtain clear orientation patterns. Several methods have been proposed to improve the estimation of the orientation field which can broadly be categorized as filtering-based [6, 7] and model-based [1, 2, 3, and 4]. Filtering-based methods only operate at the local region and thus it cannot solve large noisy or missing patches in the image. On the other hand, model-based methods consider the global constrain and regularity of orientation field except for the area near the singular point. This provides a

means to overcome the shortage of filtering-based methods. This paper will focus on the model-based methods.

Several model-based methods have been proposed in the literature. Sherlock and Monro [1] presented the pole-zero model approach to model the fingerprint orientation topology using:

−−−+= ∑∑

==

K

kd

L

lcm kk

zzzzOzO11

0 )arg()arg(21

)( (1)

where kdz ,

kcz are the positions of the deltas and cores.

However, it is clear that for any two fingerprints with the same singular points, both will be modeled by the same function even though their local ridge orientation values may differ significantly. Furthermore, it cannot handle the fingerprints where there are no singular points. In other words, the zero-pole model cannot model all possible fingerprint orientation patterns accurately. [2] revises the zero-pole model to a nonlinear orientation model as in Eqn.(2):

−−−+= ∑ ∑= =

K

k

L

lccddm kkkk

zzgzzgOzO1 1

0 ))(arg())(arg(21)( (2)

where g is the correction term which essentially is any nonlinear function that preserves the singularity at the given point. However, this additional correction is not precise enough to approximate orientation fields. Gu , Zhou and Zhang [3] then proposed a combination model to reconstruct the fingerprint orientation fields. The combination consists of a polynomial model and a point charge model, where the polynomial model is used to approximate the global orientation fields and the point charge model is used to correct the orientation fields formed by polynomial model near the singular points. This combination model gives better estimation of the orientation compared to the other previous models . However, the orientations reconstructed by the polynomial model near to the singular point regions fail in some cases. This is because the point charge model used to model the complex orientation at the singular region is too simple to reflect the complex patterns of the singular points since the point charge model is independent from the polynomial model used to model the overall orientation. Zhou and Gu [4] also presented another complex model which is quite similar to [2] except that high order rational function is used as the nonlinear correction instead. This proposed approach is also not able to precisely reflect the patterns near the singular points.

We found that there have been several works in orientation modeling not directly related to fingerprint. Jain and Rao [10] used the geometric theory of differential equations to derive a symbol set based on the visual appearance of phase portraits. This approach is able to provide a reconstruction of salient features of the original texture based on the symbolic descriptor. The work is expanded in [11] with the use of a weighted estimation algorithm for single linear oriented pattern, which is robust to noise in the orientation pattern. Ford et al.[12] used the phase portraits to model the flow-like orientation patterns. They suggested that flows are modeled as a superposition of primitives, where their associated strengths are determined from the orientation patterns. Compared with [10], such an approach provides a framework to describe and reconstruct global complex orientation patterns. In addition, the work also showed that even when the data is occluded up to 40%, the proposed phase portrait approach is still able to approximate the orientation quite accurately. Ford et al expanded their work in [13] and [14] by proposing nonlinear phase portrait models. In these papers, they decompose the flow field into simple component flows based on the singular point pattern. Then a Taylor’s model is assumed for the velocity components, and the model coefficients are computed by considering both local singular points and global flow field pattern. This model’s advantage is that it can provide a better approximation in the singular point regions than the linear phase portrait approach. However, their method is only aimed at orientation field with symmetric orientation structure. In addition, only two and three order phase portraits have been reported with good performance. At such low orders, the proposed approach is not able to cater for complex patterns.

In this paper, we present a nonlinear phase portrait model to reconstruct the orientation field of a fingerprint. Model-based method is chosen since it considers the global constrains and thus provides accuracy and robustness to noise. Compared with previous works, we introduced high order phase portrait to model the global orientation pattern and proposed the addition of first order phase portrait near the singular points as the local constraint.

The rest of this paper is arranged as follows: In section 2, the nonlinear phase portrait model will be described. Section 3 then describes our proposed algorithm. Experiments and discussion follow at Section 4 before the paper ends with a conclusion.

2. Nonlinear Phase Portrait Models

a) Brief introduction of phase portrait First-order phase portrait has been proposed to

describe the local orientation fields [10-14]. It can be expressed by the following two equations in the Cartesian coordinate system:

BxAfe

yx

dcba

yx vr&&

+=

+

=

(3)

Therefore the angle is given by,

)(tan)(tan),( 11

fdycxebyax

xy

yx++++

== −−

&&

θ (4)

where x& and y& are the x, y components of the orientation. (x, y) is the coordinate, (a, b, c, d, e, and f) are the coefficients of the phase portrait. The form of the A matrix determines the qualitative behavior of the system, and a family of curves in the phase plane. The vector B

v allows for translation of the origin. The

qualitative behavior of the system is classified into one of six canonical forms according to the eigenvalues of the nonsingular A matrix. The classes are node, star node, improper node, saddle, center, and spiral. Singular (critical) points of the system occur where 0== yx && . In Fig 1 we show the example of spiral and saddle.

(a) saddle (b) spiral

Fig 1 examples of first order phase portrait In particular, we assume that the x and y components

are represented by a Taylor series expansion as

=

=

∑ ∑

∑ ∑

= =

−−

= =

−−

n

i

i

j

jjijji

n

i

i

j

jjijji

yxby

yxax

0 0)(

0 0)(

&

& (5)

This is a nonlinear phase portrait equation. The nonlinear description provides a more accurate representation of the orientation field than the linear description. More importantly it has the ability to describe multiple singular points. Since there is a possibility that several singular points are present in one fingerprint image, we adopt the nonlinear phase portrait approach to model the fingerprint orientation field.

b) Orientation patterns analysis near

singular points By observing the original fingerprint orientation

fields as shown in Fig 2a & Fig. 2b, there is no simple canonical form suitable to model the orientation fields near the singular points. However, the square of the orientation fields (Fig 2(c, d)) show the pattern of spiral at the core and saddle at the delta [5]. The orientation of squared orientation fields is denoted by θ2 . It expands the original orientation range from ]2/,2/[ ππ− to ],[ ππ− , as its x, y components will be continuous

when the transition from πθ −=2 to πθ =2 . This will allow us to acquire the coefficients of the phase portrait with the least square algorithm. Fig 2 (e -f) gives the

examples of the reconstructed orientation by the first order phase portrait near singular points.

(a) (b)

(c) (d)

(e) (f)

Fig 2 orientation patterns near the singular points (a) original orientation near the core (b) original orientation near delta (c) squared orientation fields near core (d) squared orientation fields near delta (e) reconstructed orientation near core (f) reconstructed orientation near delta

c) Constrained nonlinear model A low-order nonlinear phase portrait is not able to

provide accurate approximation of the orientation field beyond the region near to the singular points. This can be solved using a higher order nonlinear phase portrait, but at the expense of higher computation and storage requirements. However, when the order is beyond a certain value, the pattern of its phase portrait will be ill due to numerical approximation, making it sensitive to noise. To resolve this problem, some constrains will be added to the nonlinear phase portrait model for correction of the orientation near the singular point.

Since using the high order phase portrait will result in poor approximation of the phase portrait near the singular points while at these regions the orientation field can be reconstructed by the first order phase portrait, we used this characteristic to constrain the high order phase portrait.

We expand the high order phase portrait (3) a t a singular point ),( ss yx as follows:

),(

)()(

)()(

)()(

2

0 0

1)(

0 0

1)(

0 0)(

yxO

yyxa

xyxa

yxax

x

n

i

i

j

js

jisjji

n

i

i

j

js

jisjji

n

i

i

j

js

jisjji

&

&

+

+

−−+

+

−−+

+−−=

∑∑

∑∑

∑∑

− =

−−−

− =

−−−

= =

−−

(6)

),(

)()(

)()(

)()(

2

0 0

1)(

0 0

1)(

0 0)(

yxO

yyxb

xyxb

yxby

y

n

i

i

j

js

jisjji

n

i

i

j

js

jisjji

n

i

i

j

js

jisjji

&

&

+

−−+

+

−−+

+−−=

∑∑

∑∑

∑∑

− =

−−−

− =

−−−

= =

−−

(7)

As mentioned above, a first order phase portrait can represent the patterns near the singular points by the following equations.

lykxx +=& (8) nymxy +=& (9)

By neglecting the nonlinear terms ),(2 yxOx in (6), (6) and (8) should have the s imilar pattern. This gives:

=−−

=−−

∑∑

∑∑

− =

−−−

− =

−−−

lyxa

kyxan

i

i

j

js

jisjji

n

i

i

j

js

jisjji

0 0

1)(

0 0

1)(

)()(

)()( (10)

At the same time, the x components should be zero at the singular points.

00 0

)( == ∑∑= =

−

−

n

i

i

j

j

s

ji

sjji yxax& (11)

The same analysis applies to equations (7) and (9)

=−−

=−−

∑∑

∑∑

− =

−−−

− =

−−−

nyxb

myxbn

i

i

j

js

jisjji

n

i

i

j

js

jisjji

0 0

1)(

0 0

1)(

)()(

)()( (12)

00 0

)( == ∑∑= =

−

−

n

i

i

j

j

s

ji

sjji yxby& (13)

Therefore for one singular point, 6 constraints will be added to coefficients ija and ijb of the high order phase

portrait to preserve to singular point patterns.

3. Details of Proposed Algorithm

a) Coarse orientation and singular points The computation of coarse orientation fields can be

found in many references [5, 7, 9, and 10]. Here we adopt the method by [5]. Its coarse orientation field O, and its coherence, C, can be estimated by

( )( ) ( )( )( )

+

+−=

+−

=

∑∑∑

∑∑−

222

222

221

4),(

2)(2

tan21),(

W yx

W yxW yx

yx

W yx

GG

GGGGyxC

GGGG

yxO π

(14)

Where W is a small window at point, ),( yx and

)( yxGG is the gradient vector at ),( yx .

Subsequently, the proposed nonlinear phase portrait model is imp lemented on the x and y components of the squared orientation fields O(x,y), giving:

==

)),(*2cos(),()),(*2sin(),(

yxOyxdyyxOyxdx

(15)

From the coarse orientation field, we can also obtain the singular points using the Poincare index method [5].

b) Estimation of firs t order phase portrait

near the singular points Once we obtained the coarse orientation fields and its

singular points, the orientation patterns near the singular points can be modeled by using the first order phase portrait. The coefficients of the phase portrait at the region surrounding the singular points are estimated using the algorithm by Shu ([11]).

From equations 8 and 9, the tangent of a planar first order system is given by:

)()(tan

flykxenymx

xy

++++==

&&

θ (16)

Let ζθ =tan , then Equation (16) becomes 0)()()( =−+−+− ζζζ feylnxkm (17)

We can directly estimate the parameters by using the triplet data ),,( iii yx ζ , and (16), where ),( ii yx is the coordinate of a pixel. The optimal weighted least square estimator is one that minimizes the following cost function.

∑−

=−+−−+

1

0

22 )(n

iiiiiiiii feylxknymx ζζζω (18)

Where ),( iii yxC=ω and subject to the constraint

1)( 2222 =+++ lknm .

N is the total number of triplet data used to estimate the parameter set ),,,,,( felknm .

Let

=

lknm

L4,

−

−−

=Ω

nnn ωζω

ωζωωζω

MM222

111

2 ,

=

fe

L2

−−

−−−−

=Ω

nnnnnnnnnn yxyx

yxyxyxyx

ωζωζωω

ωζωζωωωζωζωω

1

2222222222

1111111111

4 MMMM,

)()(42

1222444

ΩΩΩΩΩΩ+ΩΩ−= − TTTTψ (19)

Then the eigenvector of ψ with the largest eigenvalue

gives the best estimation of 4Lr

. We can further compute

2Lr

by using (19).

4421

222 )( LL TTrr

ΩΩΩΩ−= − (20) c) Construction of nonlinear phase portrait The nonlinear phase portrait will be estimated along

the x and y components respectively. Weighted least squared algorithm is used to compute the coefficients ija

and ijb so that the least squared errors between dx (or dy )

of Equation (16) and the phase portrait model is minimized. Coherence matrix will act as the weighting factor.

If there is a singular point in the fingerprint, we can obtain its first order phase portrait. 3 constraints will then be added into the nonlinear model as given by the equations (10)-(11) for coeffic ients ija and equations

(12)-(13) for coefficients ijb . This will reduce the number

of coefficients in the Eqn 5.

4. Experiments

As pointed out in [5], there is no ground truth for the orientation field of fingerprints and a truly objective error measurement cannot be constructed. In order to evaluate the effectiveness of the proposed orientation prediction algorithm quantitatively, we computed the original orientation field using a Gabor filter bank (64 filters). Then the mean absolute error (MAE) between the predicted orientation and the original orientation is computed as follows:

∑Ω∈

−=),(

mod )),(),((1yx

gaborel yxOyxOdN

MAE (21)

Here Ω is the region of comparison, which contains a total of N points.(x, y) is the coordinate of a point in Ω , elOmod and gaborO denote the predicted orientation

using the proposed algorithm and the orientation field computed using the Gabor filter bank respectively. Since the orientation is in ],0( π , the function )d(⋅ is defined as

−<

=otherwise

d||

2/|||,|)(

θππθθ

θ (22)

We randomly select 100 images from the NIST special database 4 where the image tends to cover the full fingerprints. We set the order of nonlinear phase portrait to 8 as a trade-off between accuracy and conciseness of the model. The MAE of the 8-order model is 5.02o with a standard deviation of 2.5o. This shows that the reconstructed orientation is very close to the original orientation since the orientation resolution of the Gabor filter bank is 2. 8125o. Fig. 3 gives some

examples of the orientation model obtained using the proposed nonlinear phase portrait approach.

(a)whorl

(b)double loop

(c)left loop

(d)right loop

(e)arch

Fig 3 reconstructed orientation by nonlinear phase portrait

From the experiments, we can see that the proposed nonlinear model has similar accuracy as the combination approach proposed in [3] in most areas and better accuracy around the singular points. The key difference between the nonlinear phase portrait and the combination model is that in the phase portrait model we used the first order phase portrait to reconstruct the orientation fields near the singular points. This description is used as constraints to the high order phase portrait. This will allow the fingerprint orientation patterns, especially near the singular points, to be expressed entirely by the two sets of coefficients of the high order phase portrait. However, in the combination model there are two independent models: polynomial model and point charge model. Both these two models are weighted to describe the fingerprint orientation fields and the choice of the weighting is somewhat ad-hoc. Point charge model (described by equation 23) is relatively simple to be able to reconstruct all the possible complex orientation patterns near the singular points.

Qyyxx

xxiQ

yyxx

yyPC

Qyyxx

xxiQ

yyxx

yyPC

Delta

Delta

20

20

0

20

20

0

20

20

0

20

20

0

)()()()(

)()()()()(

)(

−+−

−−

−+−

−−=

−+−

−−

−+−

−−=

(23)

As for the polynomial model, it has the capability to model complex orientation patterns. However, the area covered by the coarse orientation patterns near the singular points is much less than the total area covered by the entire fingerprint orientation. In addition, the orientation pattern of the entire fingerprint is somewhat smooth with characteristic difference exhibited mainly at the small region surrounding each singular point. Therefore, the errors of the high order phase portrait will occur mainly at the singular point region when the least squared algorithm is used to estimate the coeffic ients without any constraint. The constraints introduced in this paper showed that it is able to reduce the errors at the singular points so that only one nonlinear phase portrait model is needed to reconstruct the orientation fields of fingerprints, producing accurate description of the singular point regions. Fig. 4 shows the comparison between the polynomial approach and the proposed nonlinear phase portrait approach with constraints . It

clearly shows that the proposed approach has better orientation modeling around the core point.

.

(a) reconstructed orientation by nonlinear phase portrait (b)reconstructed orientation by polynomial model

Fig 4 comparative result of reconstructed orientation fields using the proposed approach of nonlinear phase portrait with constraints against the polynomial approach with point charge.

Similar to the polynomial approach, the main limitation of the proposed approach is that it will not perform well for very poor quality fingerprint images where the singular points cannot be reliably extracted. 5. Conclusions In this paper, we proposed a nonlinear phase portrait model to reconstruct fingerprint orientation fields. A first order phase portrait model is used to reconstruct the orientation fields near the singular points. Subsequently this is used to constrain the coefficients of the high order phase portrait which is used to model the orientation pattern of the entire fingerprint. The proposed approach allows us to reconstruct the full orientation field using only one model. That is to say, the constrained model completely expresses the entire orientation field, including the orientation patterns near the singular points. No ad-hoc weighting is needed to combine results from multiple models to form the entire model. The experimental results show the effectiveness of the proposed algorithm.

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