A Variational Formulation for Fingerprint Orientation...

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Abstract—Orientation pattern is an important feature for characterizing a fingerprint and has played critical

roles in fingerprint recognition and fingerprint classification. This paper proposes a framework for modeling the

fingerprint orientation field based on the variational principle, where the orientation pattern can be estimated

through solving the associated Euler-Lagrange equation. Compared with existing methods, our proposed

method has the following features. Firstly, it does not require any prior information about the structure of the

acquired fingerprint, like the knowledge of location of singular point(s). Secondly, it explicitly provides freedom

for modeling the singularity in the orientation field. Thirdly, it has fewer number of parameters. Comparison

has been made with respect to state-of-the-arts in fingerprint orientation modeling in terms of modeling

accuracy, fingerprint enhancement and singular point detection.

Index Terms—Fingerprint, Modeling, Orientation, Variational principle.

I. INTRODUCTION

N the recent years, with the increasing concern on security, the pace of developing and deploying biometrics

technology, in particular the fingerprint based technology, has been accelerated tremendously in a wide range of areas

from government, defense, airport security, to commercial services. In spite of the wide application of fingerprint

technology to our daily life, there remain several issues which have not been adequately addressed. Among them, how

to recognize fingerprints acquired with poor quality is still a challenging problem. Key to this problem is to enhance the

Manuscript received October 11, 2010. A preliminary version of this paper has been presented in ICPR 2010, Istanbul, Turkey. Zujun Hou is the corresponding author (phone: +65-6408-2484; fax: +65-6776-1378; e-mail: zhou@ i2r.a-star.edu.sg).

A Variational Formulation for Fingerprint Orientation Modeling

Zujun Hou1, Hwee-Keong Lam1, Wei-Yun Yau1, Yue Wang1 and Ting Hu2 1Institute for Infocomm Research, A*STAR, Singapore

2Faculty of Engineering, National University of Singapore

I


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fingerprint image before the processes of feature extraction and fingerprint matching. A number of research efforts

have been put on this topic and a wealth of techniques has been proposed [1], [2]. Instead of directly employing generic

methods in image enhancement for improving the fingerprint image quality, most fingerprint enhancement methods are

based on the characteristic structure within the fingerprint, which have been proven to be more effective in practice.

One of the most important features in the fingerprint is the highly parallel oriented pattern, which is not only useful for

fingerprint enhancement, but also applicable to fingerprint recognition, classification, indexing, and inference of other

features such as fingerprint singular points. For these reasons, increasing attention has been paid to fingerprint

orientation estimation, which can roughly be classified into the following two categories:

1. Local estimation methods [3-9]: to estimate the orientation based on the intensity variation in a local region.

Typical example is the gradient method, which uses the direction orthogonal to the gradient as an estimate of the

ridge orientation.

2. Global modeling methods [10-17] [19-22]: to construct a global model based on the orientation field derived

from a local estimation method, followed by recalculating the orientation field using the constructed model.

Popular methods of this type include the zero-pole model, the polynomial model, and the Fourier series model.

In this study we will present a new framework for modeling the fingerprint orientation field. The proposed method is

based on the variational principle and need not to have the knowledge of the explicit functional form or the prior

topology of the modeling data. In addition, fingerprint singularities can be modeled seamlessly in the proposed

framework.

The rest of the paper is organized as follows. Section II reviews the work related to fingerprint orientation modeling.

Then, Section III details the proposed fingerprint orientation modeling based on the variational principle, where a brief

account on variational principle is given. After that, validation of the performance of the proposed method as well as

comparison with state-of-the-arts is presented in Section IV. Finally, the paper is concluded in Section V.

II. RELATED WORKS

In general, fingerprint orientation modeling starts from a local estimation of the orientation, followed by a refining


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process where prior knowledge or information from a larger scale will be employed. The local estimation can simply be

based on the image gradient [3], or be derived from more sophisticated methods such as statistical techniques [3], [4],

filter-bank [5], ridge projection [6], structure tensor [7-8], integration operator [9] or local voting [10]. After that, a

global model can be built and applied in turn to local prediction. A pioneered work in this direction was presented by

Sherlock and Monro in 1993 [11], where the orientation field is described by a zero-pole model. The model is

formulated in the complex plane with the core point as zero and the delta point as pole, and the orientation at a point is

simply the sum of the influences of all singular points (core and delta). The zero-pole model is simple and almost

perfect in regions near singular points, but often unsatisfactory in other regions. An improvement was made by

Vizcaya and Gerhardt [12] using a piecewise linear approximation model around singular points to adjust the zero and

pole’s influence. A further improvement has been presented in [13] where rational complex functions are used and with

this modification orientation field in fingerprints without singular points (such as arch type fingerprint) can also be well

modeled. These models improve the zero-pole model in describing global orientation, but at cost of characterizing the

orientation pattern near singular points. To overcome this drawback, a combination model was presented in [14-15],

where the polynomial model was utilized for capturing the orientation pattern in regions with small variations, and a

point-charge model was proposed to approximate the orientation pattern in regions close to fingerprint singular points.

These two models are combined for more accurate modeling of the orientation pattern in the whole fingerprint region.

A similar work was presented in [16]. Very recently, a unified framework based on quadratic differentials was

presented in [17] and most of aforementioned models can be regarded as special cases.

For most of the above global estimation methods, they have one common feature; that is the dependency on the

knowledge of fingerprint singular points, which characterize the discontinuity of fingerprint global orientation flow. An

example is shown in Fig. 1, where triangle denotes delta point and circle denotes core point. The location and the type

of fingerprint singular point are important feature for fingerprint characterization. However, the detection of singular

points is never a trivial issue and the success of the detection strongly relies on the quality of the derived fingerprint

orientation field. Manual detection would prevent the fingerprint orientation modeling from being applied to automatic

fingerprint recognition systems. Simple methods such as the Poincaré Index method [18] are sensitive to noise or other

perturbations. As a result, the problem will be as complicated as the chicken-egg paradox. To circumvent this problem,


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Wang et al [19] presented an orientation estimation method in another way, where the modeling problem is formulated

as a data fitting problem and trigonometric polynomial is utilized to fit the orientation data as estimated by local

methods. A remarkable feature of this model is that it does not require any prior knowledge of singular points. The

method has been demonstrated to outperform the singular points dependent method like the combination model in terms

of fingerprint image enhancement. The method has recently been extended in [20], where Legendre polynomial is

employed as the functional basis. Moreover, as aforementioned, fingerprint singular point is an important fingerprint

feature and true fingerprint singularity should be preserved in the process of fingerprint orientation modeling. In [20],

the issue of singularity preservation is addressed through minimizing a cost function between the estimated data and

the original one. A further improvement was presented very recently [21], where a lower order model is utilized to

preliminarily model the global orientation pattern. Then singular regions around fingerprint singular points are detected

and a higher order model is employed for refining the orientation model in these regions. The process of higher order

modeling and that of singular region detection are alternated and iterated till the modeling accuracy is sufficient.

Another interesting work was presented in [22] where the orientation is modeled in a Bayesian framework and

fingerprint singularities are described using a singularity template model. For a recent review on fingerprint orientation

estimation, interested readers can refer to [23].

Fig. 1. This plot exemplifies the fingerprint singular points (core and delta), which should be preserved in the process of orientation modeling.


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III. METHODS

A. Variational Principle

Originated by Leibniz and founded by Euler and Lagrange, variational principle is an important method in physics

for determining the state or the dynamics of a physical system. As stated by Euler [24], “Since the fabric of the

Universe is most perfect and is the work of a most wise Creator, nothing whatsoever takes place in the Universe in

which some relation of maximum and minimum does not appear”. Variational principle seeks the solution through

finding the extremum (minimum, maximum or saddle point) of a functional. The method can be expressed using the

calculus of variations, which is a branch of mathematics dealing with integral minimization. The functional to be

minimized can be formed as an integral involving unknown function f or its derivatives as follows

.',,2

1

dxffxLfJx

x (1)

Then the problem is to find the extremal function *f where the rate of change of the functional fJ is zero, i.e.,

.0fJ (2)

Some classical examples include geodesics, the principle of least time (Fermat’s Principle), and the principle of least

action (which plays central roles in modern physics). Besides enormous applications of the variational principle to

physics and chemistry, the method has also been employed frequently to investigate problems in computer vision, such

as edge detection, image denoising, super-resolution image reconstruction, optical flow, surface reconstruction, shape

from shading, stereo and image inpainting [25-26]. For more details on the variational principle, interested readers can

refer to [27]. Readers with interest in its application to computer vision can refer to [28] or [29].

In the following, we present another application of the variational principle, i.e., a formulation of fingerprint

orientation modeling using the variational principle.

B. Variational Approach to Orientation Modeling

For the convenience of description, let θ denote an orientation field in image domain . Then, the problem in

fingerprint orientation modeling is to reconstruct an orientation field such that


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(C1) is smooth and

(C2) true singularities in θ are preserved in .

In fingerprint image processing, directions of gradients with difference of have the same effect on inference of

ridge orientation or the design of filters. However, direct operation (integration/summation) will cancel out these

directions. To avoid this problem, a common practice in fingerprint orientation modeling is to project θ to the

complex domain and take the square as follows:

2exp~ i (3)

or to double the angle, take the sine and cosine transform and project into a vector space as

,2sin,2cos~ (4)

which has been termed the double angle approach in literature [3]. Similarly, we have

2exp~ i (5)

or

.2sin,2cos~ (6)

As will be shown in the following, the estimation of the orientation field can be formulated in the framework of the

variational principle. To model the constraint (C1), it is usually accomplished through minimizing the derivatives of the

function. In this study, it is modeled as

21

~L (7)

where is the magnitude of the complex number. As for the modeling of the second constraint, it is given by

2

2~~~ L . (8)

Here ~

stands for a function to capture the singularity in the original orientation field and its value increases

with respect to the increase of the saliency of ~

, thus singularities in ~

have higher weights in determining the

function .~ Combined these constraints together, the functional ~J can be defined as:


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dxdyLLJ 21~ (9)

where 0 is a regularization parameter. In the language of variational principle, the first term is called the kinetic

energy and the second is the potential energy. It should be noted that the design of these two energy terms here is

merely for the purpose to illustrate the feasibility of fingerprint orientation modeling using the variational principle.

C. Numerical Implementation

For notational convenience, let us write

2cosu (10a)

2sinv (10b)

Then we have

22221 yxyx vvuuL (11a)

,2sin2cos~ 222 vuL (11b)

which is very similar to the gradient vector flow formulation for object boundary detection [30]. Substituting (10) and

(11) to (9), we can derive the solution of the variational problem by solving the associated Euler-Lagrange equations:

02cos~ uu (12a)

02sin~ vv (12b)

where stands for the Laplacian operator. Furthermore, regarding the left hand side of the Euler-Lagrange equation

as an infinite dimensional gradient, the equations (12a) and (12b) can be solved using the gradient descent method,

which leads to the following equations:

2cos~ uuut (13a)

.2sin~ vvvt (13b)

If negating the second term, Eq. (13) is the heat equation, a special case of the more general diffusion equation, and

is called thermal diffusivity. The derived partial differentiation equation can be solved using numerical methods. In

this study, it is solved by the method of finite difference. In the following subsection, an analysis on numerical stability


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is given, from which stability requirement on the value of regularization parameter can be derived.

D. Numerical Stability

Due to truncation error in computer numerical calculation, an important issue in solving a partial diffusion equation

numerically is the stability or the convergence of the numerical algorithm. A numerical scheme is stable if the errors do

not increase with respect to the increase of the time. As the equations in (13) are linear, a Von Neumann stability

analysis [31] is carried out to find the values of the regularization parameter for which the numerical scheme is stable.

With forward difference in time and central difference in space, a finite difference approximation of (13a) is given by

2cos~4

,

,1,1,,1,1,

1,

tji

tji

tji

tji

tji

tji

tji

tji

u

uuuuutuu

, (14)

where t stands for the time discretization step. And it is supposed that the image is of dimension NM and we

take the space discretization steps as 1 (i.e., 1 yx ). If let tji, denote the computation error of t

jiu , at time t ,

then it can be shown that tji, satisfies

t

yx

tyx

tyx

tyx

tyx

tyx

tyx

tyx

t

,

,1,1,,1,1,

1,

~4

(15)

Von Neumann stability analysis is also known as Fourier series stability analysis, with name coming from the

Fourier series expansion of the error

M

m

N

n

yjkxikat nmeeyx1 1

, , (16)

where an assumption of the exponential variation of the amplitude with respect to time for a linear differential equation

has been applied. Since the difference equation is linear, it is enough to consider the growth of error for a typical

component

yjkxikatnm

nmeetyx ,,, . (17)


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On top of this, it is noted that

ynkm jxikatyx ee

1,1 (18a)

yjkxikatyx

nmee 1

,1 (18b)

11,

yjkxikat

yxnmee (18c)

11,

yjkxikat

yxnmee (18d)

Substituting (17) and (18) to (15) and after simplification, we can derive

4

~1

nnmm jkjkikik

ta

eeeett

(19)

Using the identities

mm ikikm eek

21cos (20)

and

mm kk

cos121

2sin 2 , (21)

we can write equation (19) as

2sin

2sin4~1 22 nmta kktt (22)

In order for the error to be bounded, the error amplification must be less than 1, that is to say

1

,,,,

,,1,,

,

,

,

,

ta

nm

nmta

nm

nm etyx

tyxetyx

tyx

. (23)

Therefore, from (22) and (23), the condition of stability is given by

12

sin2

sin4~1 22

nm kktt , (24)

which implies

8

~

41

t

. (25)


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If the potential energy is zero, the stability condition will simply be

t

41

(26)

IV. EXPERIMENTS

To validate the performance of the proposed method, experiments have been carried out using publicly available

fingerprint database, FVC 2000 Db1a [32] and FVC 2004 Db1a [33]. Each database consists of 100 fingers with 8

impressions per finger. FVC 2000 Db1a was collected using a low-cost optical sensor with image size of 300300 ,

while FVC 2004 Db1a was collected using an optical sensor with image size of 480640 . The latter contains

fingerprints that were dried and moistened, and distorted and rotated artificially, thus representing more challenging for

fingerprint recognition.

For illustration, the function is simply taken as the saliency of the original orientation field

Fig. 2. An illustration of the convergence speed, where upper left is the orientation before iteration, upper right is the orientation after step 1, bottom left is orientation after step 4 and bottom right is orientation after step 6.


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xNxx

xxN

x'

.'sin1)( (27)

Parameter is set as 0.25 and 1t . As the proposed method needs to solve the partial diffusion equation, the

speed of convergence of the iterative process would be of practical importance. It is found from our experiment that the

iteration converges pretty fast, mostly in 4 to 6 steps. An example is given in Fig. 2. It can be seen that the orientation

field is initially pretty noisy (upper left). After 6 iterations, almost all noisy orientations have been satisfactorily

reconstructed. The method is implemented in Matlab and run in a desktop PC with 2.66 GHz CPU and 2GB RAM. On

average it takes about 16ms to reconstruct the orientation field for an image of size 300300 .

The proposed method is compared with the Fourier series expansion method [19], which fits the transformed

orientation data 2sin,2cos~ using a set of trigonometric functions. The parameters are set as suggested in

[19]. Both methods are calculated in blockwise with 88 pixels per block. The comparison is composed of qualitative

comparison of modeling accuracy, comparison of error rates in fingerprint recognition, and comparison of accuracy in

fingerprint singular points detection, as presented in the following respectively.

A. Qualitative Comparison of Modeling Accuracy

Fig. 3 shows an example of low-quality fingerprint image and the estimated orientation field is overlain onto the

fingerprint image, where on the left is Fourier series solution and on the right is our proposed solution. Comparing

Fig. 3. A comparison of orientation estimation by Fourier Series expansion (left) and variational formulation (right), where the orientation field is overlaid with the fingerprint structure.


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these two solutions from the Figure, it can be observed that there are two regions with significant differences. One is on

the lower left part of the fingerprint structure where the original fingerprint presents a number of fragmented

structures, possibly due to moisture. The Fourier series solution fails to correct the orientation in this region, whereas

our proposed solution is almost perfect. Another region is located on the right side of the central part (where the

singularity takes place). It can be seen that the actual fingerprint ridge flow is southeastern, however the Fourier

solution is apparently more eastern oriented. On the contrary, the proposed solution is pretty consistent with respect to

the actual flow of the fingerprint ridges.

Similar observation can be noticed in Fig. 4, where a zoomed-in view of the region around fingerprint singular points

is shown on the right. Compared with the Fourier series solution (Row 1), the orientation estimated by the proposed

method is more faithful to the ridge flow. Figs. 5 and 6 present another two examples, from which one can see that our

proposed method is superior to the Fourier series expansion method in terms of correcting the perturbation and

preserving the true singularities. Of particular interest is the fingerprint in Fig. 6, where the location of the core point is

very close to the position of the delta point and this would impose difficulty for accurate orientation modeling. The

influence is obvious in the Fourier series solution (Row 1), where the orientation singularity corresponding to the core

point shifts towards the east and it is almost indistinguishable with the orientation singularity corresponding to the delta

Fig. 4. A comparison of orientation estimation by Fourier Series expansion (Row 1) and variational formulation (Row 2), where the orientation field is overlaid with the fingerprint structure and on the right shows a zoomed-in view.


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point. In contrast, the solution by our proposed method is nearly perfect.




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B. Singular Point Detection

As aforementioned, fingerprint singularity is an important feature in characterizing the fingerprint structure, thus the

ability of preserving true singularity and smoothing out false singularity due to perturbation can be utilized to evaluate

the performance of a fingerprint orientation modeling method [20-21]. To this end, after the end of orientation

modeling, the Poincaré index method is applied to the orientation field for extracting the core and the delta point. Apart

from that, all fingerprint images in FVC 2004 Db1a are manually processed and the manually labeled core and delta

points will serve as the ground truth for comparison. In total, there are 976 core points and 541 delta points.

Several examples on the detected singular points are shown in Fig. 7, where Row 1 is based on the Fourier series

modeling method and Row 2 based on our proposed method. It can be seen that when the fingerprint quality is low

(like Columns 3 and 4), there could be some singularity due to perturbation which is falsely preserved after the

orientation modeling. For a quantitative evaluation, the following three quantities are calculated for each modeling

method: the number of total detection points, the number of False Positive (FP) points, and the number of False

Negative (FN) points. The results are summarized in Table 1. It can be noted from the table that two methods are close

in the detected number of FN core or delta points, but the detected number of FP singular points (either core or delta)

by our proposed method is remarkably smaller than that by the Fourier series expansion method. For example, the

Fig. 7. Examples of detected singular points based on the orientation field modeled by the Fourier Series method (Row 1, red color) and the proposed method (Row 2, blue color) respectively, where green color indicates the manually labeled ground truth.


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falsely detected delta points are 106 for the latter, but only 14 for the proposed method. Noting that the number of false

negative detection results is relatively large, we have carried out manual investigation on those fingerprints and it turns

out that it is mostly due to the following two reasons. Firstly, the singular point (very often the delta point) is located

near the boundary of the fingerprint region (as illustrated in Fig. 8a). Then the singular point could be missing after

segmentation or over-smoothing. Secondly, the fingerprint is of arch type or close to the structure of an arch type (Fig.

8b, for example). For arch type fingerprints, there is no singular point by the definition of singular points in the

Poincaré index method.

From these data, we can further compute the Precision (the rate of detected singular points that are true ones), the

Recall (the rate of true singular points that are detected) and the F-measure (harmonic mean of Precision and Recall,

which represents an aggregated performance score between Precision and Recall). These three measures are also listed

in Table 1. The superiority of our proposed method is evident, in particular, in terms of Precision.

(a) (b)

Fig. 8. Examples to illustrate when singular points could be missing, (a) a delta point is located near the boundary, (b) the fingerprint structure is close to that of an arch-type fingerprint.

TABLE I SINGULAR POINTS DETECTION RESULTS ON FVC 2004 DB1A.

Fourier Series

method Proposed method

Core Point Total detected 1018 943 False Positive 108 25

False Negative 66 58 Precision 89.4% 97.3% Recall 93.2% 94.1% F-measure 91.3% 95.7% Delta point Total detected 480 393 False Positive 106 14 False Negative 167 162 Precision 77.9% 96.4% Recall 69.1% 70.1% F-measure 73.2% 81.2%


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In addition to Precision and Recall, accuracy is another instrumental measure for characterizing the performance of

singular point detection. For a good singular point detection method, the detected singular point should be near the

location of true one. Table 2 gives the average distance for singular points detected using the orientation map by the

Fourier series expansion method and our proposed method respectively. Evidently, our proposed method not only yield

smaller value of average distance, but also smaller value of deviation, which indicates that singular points derived from

the proposed method are more accurate and more stable. Moreover, it can be noted from Table 2 that the mean

distance of detected delta point by our proposed method is around 7 pixels. Considering that the calculation is on a

coarse scale ( 88 pixels per block), this result is actually very promising, in particular when the deviation is also

small.

To have a more intuitive observation, we further divide the distance into different levels and compare the two

methods how the distances of detected singular points are distributed. A coarse division (2-level) is given in Table 3

and a finer one in Table 4. Comparing with the Fourier series method, our proposed method has more distances falling

in bins closer to 0. For bins away from 0, the number of distances is significantly smaller, for example, the Fourier

series method has 16 delta points with distance >40 pixels, whereas our proposed method has only 1 delta point falling

in this bin. This fact further justifies the accuracy and the reliability of our proposed method in preserving the location

TABLE II COMPARISON OF THE AVERAGE DISTANCE (IN UNIT OF PIXELS) FROM THE TRULY DETECTED SINGULAR POINT TO THE GROUND TRUTH.

Fourier Series method Proposed method

Core Point 18.7 ± 10.2 14.6 ± 8.5 Delta point 14.9 ± 38.8 6.8 ± 6.7

TABLE III

DISTANCE DISTRIBUTION OF THE TRULY DETECTED SINGULAR POINTS WITH TWO-LEVEL BREAKDOWN OF DISTANCE (IN PIXELS). Fourier Series


Core Point ≤15 377 550 >15 533 368

Delta point ≤15 335 367 >15 39 12

TABLE IV

DISTANCE DISTRIBUTION OF THE TRULY DETECTED SINGULAR POINTS WITH FOUR-LEVEL BREAKDOWN OF DISTANCE (IN PIXELS). Fourier Series


Core Point [0 10] 169 300 (10 20] 401 424 (20 40] 308 188 >40 32 6

Delta point [0 10] 286 326 (10 20] 58 49 (20 40] 14 3 >40 16 1


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of singular points.

C. Fingerprint Matching

As aforementioned, the orientation field plays an important role in fingerprint image enhancement. As a way to

indirectly measure the performance of orientation modeling, we have carried out matching experiment using the NIST

fingerprint software [34] for minutia detection and matching. Fig. 9 plots the Receiver Operating Curve (ROC) based

on the FVC 2000 DB1a (Row 1) and the FVC 2004 DB1a (Row 2) dataset, where the variation method takes into

account only the kinetic energy. As the proposed method converges very fast, the performance when both energy terms

are considered is very similar. The NIST results (without orientation modeling) are also given for reference. From Fig.

9, it can be seen that the performance between our proposed (solid line) and the Fourier series modeling method (dotted

line) [19] is very close in FVC 2000 DB1a but our proposed method is slightly better in FVC 2004 Db1a. Both

outperform the NIST results where no orientation modeling has been carried out.

As seen in the above section, our proposed method turns out to solve two partial diffusion equations (PDE). On the

other hand, PDE-based image processing has been a hot topic since the pioneering work of anisotropic diffusion by

Perona and Malik in 1990 [35]. Anisotropic diffusion reduces image noise without damaging salient image features by

Fig. 9. ROC on FVC 2000 DB1a (top) and 2004 DB1a (bottom), where the solid line represents the proposed orientation modeling method, the dotted line the Fourier Series modeling method and the dash line the NIST method without orientation modeling.


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a filtering process adaptive to the local content. For application to fingerprint enhancement, the difference of

conventional PDE method and the proposed method lies in that the former is implemented in the image domain directly

and the latter in the orientation domain. The combination of these two concepts would lead to a method for fingerprint

enhancement which adapts the filter in anisotropic diffusion using the orientation information from the process of

orientation modeling. This directed anisotropic diffusion is the idea of coherence enhancement anisotropic diffusion

[36]. Fig. 10 compares the matching performance between these two enhancement methods, where the solid, red curve

stands for the original anisotropic diffusion and the dashed, blue curve the directed anisotropic diffusion. It can be seen

that for both datasets the matching performance is improved with the added information.

V. CONCLUSION

In this study we introduced a framework for fingerprint orientation modeling which is based on the variational

principle. Different from existing popular methods to approximate the orientation field using some function, the

variational method needs no explicit form of the approximated function and the solution is derived implicitly from a

functional space, where the desired features for the solution are modeled. The proposed framework is advantageous in

terms of having less number of parameters and more freedom to preserve singularities in original fingerprints. An

Fig. 10. ROC on FVC 2000 DB1a (top) and 2004 DB1a (bottom), where the solid red line represents anisotropic diffusion with the orientation derived from the proposed orientation modeling method, the dashed blue line the directed anisotropic diffusion.


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illustration based on this framework has been provided. The performance has been validated using the FVC datasets

and been compared with the Fourier series expansion method. It turns out that the proposed method is accurate and

stable, and converges fast. The performance is at least comparable with the state-of-the-art in terms of singularity

preservation and fingerprint matching. It should be pointed out that the model for singularity preservation as illustrated

here is a very simple model. Even though the feasibility of this simple model has been observed, more intelligent design

is still necessary and would be the future direction of this framework.

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Zujun Hou (M’06) received a degree of B.S. from Dept. of Physics, Beijing Normal University in 1991, a degree of

M.S. from Dept. of Physics, National University of Singapore in 1999 and a PhD degree from Dept. of Computational

Science, National University of Singapore in 2003.

He was a Research Associate with the Centre of Advanced Numerical Engineering Simulations, Nanyang

Technological University from 2001 to 2003. After that, he joined the Biomedical Imaging Lab, Singapore BioImaging

Consortium as a Research Scientist. Since 2006, he has been with the Institute for Infocomm Research, A*STAR,

Singapore, as a Senior Research Fellow. His research interest is primarily on image analysis and he has published

more than 40 papers (including 18 in journal) in the area. He is a member of Editorial Board of KSII Transactions on

Internet and Information Systems.

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