An effective method for extracting singular points in fingerprint images

Int. J. Electron. Commun. (AEÜ) 60 (2006) 671 – 676 www.elsevier.de/aeue

LETTER

An effective method for extracting singular points in fingerprint images Lin Wanga,∗ , Mo Daib a Image Laboratory, Department of Mathematics and Computer Science, Guizhou University for Nationalities, 550025 Guiyang, China b Image Laboratory, Institute EGID-Bordeaux 3, University of Bordeaux 3, 1 Allee Daguin, 33607 Pessac cedex, France

Received 2 August 2005

Abstract In most algorithms of fingerprint identification and fingerprint classification, extracting the number and the precise location of singular points (SPs) is of great importance. In this paper, a new algorithm based on the distribution of Gaussian–Hermite moments is presented for detection of SPs. All other SPs extraction methods can only extract two types of SPs (core and delta). They often ignore a true pair of core–delta that are close to each other. Our algorithm can detect not only the core and the delta, but also a pair of core–delta. It is shown how very accurate detection of the SPs. These estimates can for instance be used for fingerprint classification and accurate registration of two fingerprints in a fingerprint verification system. 䉷 2006 Elsevier GmbH. All rights reserved. Keywords: Fingerprint image; Core; Delta; A pair of core–delta; Gaussian–Hermite moments

1. Introduction In recent years, fingerprints are most widely used for personal identification. Fingerprint images are directionoriented patterns formed by ridges and valleys. The singular point (SP) area is defined as a region where the ridge curvature is higher than normal and where the direction of the ridge changes rapidly [1,2] (see Fig. 1). In most fingerprint identification algorithms and fingerprint classification algorithms, extracting the number and the precise location of SPs is of great importance [3]. Most of the approaches to detect SPs are known in the literatures. Examples of these algorithms are based on sliding neural networks [4], local energy of the direction field [5], directional mask [6]. A common technique to extract SPs in fingerprints is to use the Poincare index introduced

∗ Corresponding author. Tel.: +86 851 3610156.

E-mail address: [email protected] (L. Wang). 1434-8411/$ - see front matter 䉷 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2006.01.002

by Kawagoe and Tojo [7]. The Poincare index takes the values 1/2, −1/2, and 0 for a core point, a delta point, and an ordinary point, respectively. This technique has been used in [3,8–10] to define and extract SPs. However, these algorithms can only extract two types of SPs: core and delta. They cannot detect a true pair of core–delta that are close to each other. In this paper, we propose a new algorithm for extracting SPs in fingerprint images, which is based on the behavior of Gaussian–Hermite moments (GHMs). The proposed algorithm is able to locate SPs in fingerprint with high accuracy. Unlike all other SPs extraction methods, besides core and delta, our algorithm can detect a pair of core–delta. We denoted core, delta and a pair of core–delta by SC , SD and SCD , respectively. This paper is organized as follows: Section 2 introduces the GHMs and their behaviors in the fingerprint image. Section 3 presents the algorithm based on GHMs for extracting SPs, and Section 4 presents some experimental results. Finally, Section 5 concludes this paper.

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L. Wang, M. Dai / Int. J. Electron. Commun. (AEÜ) 60 (2006) 671 – 676

Fig. 1. Singular points in fingerprint images: ◦ – core,
– delta, and × – a pair of core–delta.

2. Gaussian–Hermite moments (GHMs) and their behaviors in fingerprint image 2.1. Gaussian–Hermite moments Moments, such as geometric moments and orthogonal moments, are widely used in pattern recognition and image processing. In order to better represent local characteristics of images, smoothed orthogonal GHMs were proposed [11,12]. Given a Gaussian smoothing function g(t,
) with 2 −1/2

g(t,
) = (2
)

exp(−x /2
), 2

2

(1)

the nth-order smoothed GHMs Mn (x, S(x)) of the signal S(x) is defined as
∞ Mn (x, S(x)) = Bn (t)S(x + t) dt, −∞

n = 0, 1, 2, . . .

(2)

with Bn (t) = g(t,
)Pn (t),

(3)

where Pn (t) is a scaled Hermite polynomial function of order n, defined as Pn (t) = Hn (t/
)

(4)

with Hn (t) = (−1)n exp(t 2 )(dn /dt n ) exp(−t 2 ).

and in particular M0 (x, S(x)) = g(x,
) ∗ S(x), M1 (x, S(x)) = 2
d[g(x,
) ∗ S(x)]/dx,

(9) (10)

where S (m) (x) = dm S(x)/dx m , S (0) (x) = S(x) and ∗ denotes the convolution operator. Two-dimensional (2D) orthogonal GHMs of order (p, q) of an input image I (x, y) can be defined similarly

Mp,q (x, y) =

∞ −∞

G(t, v,
)Hp,q (t/
, v/
)

× I (x + t, y + v) dt dv,

(11)

where G(t, v,
) is the 2D Gaussian function, and Hp,q (t/
, v/
) is the scaled 2D Hermite polynomial of order (p, q) with

(5) Hp,q (t/
, v/
) = Hp (t/
)Hq (v/
).

GHMs can be recursively calculated as follows: Mn (x, S (m) (x)) = 2(n − 1)Mn−2 (x, S (m) (x)) + 2
Mn−1 (x, S (m+1) (x)) for m
0 and n
2

Fig. 2. 2D base functions of GHMs.

(6)

with M0 (x, S (m) (x)) = g(x,
) ∗ S (m) (x) for m
0,

(7)

M1 (x, S (m) (x)) = 2
d[g(x,
)]/dx ∗ S (m) (x) for m
0

(8)

(12)

Obviously, 2D orthogonal GHMs are separable, so the recursive algorithm in 1D cases can be applied for their calculation. Fig. 2 shows the spatial responses of the bidimensional GHMs kernels of different orders. In fact, GHMs are linear combinations of different order derivatives of the signal filtered by a Gaussian filter. As it is well known, the derivatives have been extensively used for image representation in pattern recognition. For a fingerprint image, the SP can be regarded a mode. That is why we use GHMs to better extract the SP.

L. Wang, M. Dai / Int. J. Electron. Commun. (AEÜ) 60 (2006) 671 – 676

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3. Behavior of Gaussian–Hermite moments in fingerprint image In order to characterize the fingerprint image, we use four GHMs of different order: M0,1 , M1,0 , M0,3 and M3,0 . The number and the order of GHMs required were empirically determined. For detection of SPs, using four moments resulted in a satisfied performance. A further increase in the number of moments did not provide any increase in the performance. We define Mu (x, y) = aM 1,0 (x, y) + (1 − a)M3,0 (x, y), Mv (x, y) = aM 0,1 (x, y) + (1 − a)M0,3 (x, y),

(13)

where a (0< a <1) is the weight associated with the GHMs of different orders of the fingerprint image (in our experiment, a =0.5). For each pixel (x, y) of the fingerprint image, we thus obtain a characteristic vector [Mu , Mv ]T by Eq. (13). Fig. 3 shows the distribution of [Mu , Mv ]T in an ordinary area, a SP area and a blank area, respectively. As is shown in Fig. 3, in an ordinary area, the distribution of [Mu , Mv ]T is along the direction orthogonal to the local ridge orientation, while in a SP area, the distribution of [Mu , Mv ]T is almost a uniform distribution over all directions. So using these behaviors of the distribution of [Mu , Mv ]T , we can identify a SP. We use the principal component analysis (PCA) to analyze the distribution of [Mu , Mv ]T . The estimate of the covariance matrix CM of the vectors [Mu , Mv ]T is given by   Muu Muv CM = (14) Muv Mvv

Fig. 3. Distribution of [Mu , Mv ]T in an ordinary area (a), a SPs area (b), and a blank area (c).

with Muu =



(Mu − mu )2 ,

W

Muv =



(Mu − mu )(Mv − mv ),

Fig. 4. Images function of C in different areas.

W

Mvv =



(Mv − mv )2 ,

W

1  Mu , n×n W 1  mv = Mv , n×n mu =

W

where n × n is the size of the window W. Let 1
2 be two eigenvalues of the covariance matrix of the vectors [Mu , Mv ]T . In an ordinary area, the distribution of [Mu , Mv ]T will be mainly along the long axis, i.e., in the direction orthogonal to the orientation of ridges, so 1 ?2 . On the contrary, in a SP area, 2 will be close to 1 . Therefore, we can define the coherence C as follows:  2 (Muu − Mvv )2 + 4Muv 1 − 2 C= = . (15) 1 + 2 Muu + Mvv

Fig. 4 shows a fingerprint image and its C image. In C image, white indicates 1 and black indicates 0. It can be clearly seen that C is close to 1 in the ordinary areas (see Fig. 4(c)), while in a SP it is close to 0 and it is a local minimum. The image function of C in a SP area approximates a taper (see Fig. 4(d)). Although C in a noisy area also is very low, it is not local minimum (see Fig. 4(e)). Using these behaviors of C image, we can well extract the SPs in fingerprint images.

4. Extraction of singular points We have analyzed the relation between the SP and the distribution of GHMs. A SP corresponds to a local minimum of C. In order to reduce the computational cost and the influence of noise, we only compute C in an area where the ridge curvature is highest. We can find such areas by the orientation field.

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L. Wang, M. Dai / Int. J. Electron. Commun. (AEÜ) 60 (2006) 671 – 676

Fig. 5. Extraction of SC and SD : in P+ + P− , black indicates −1/2, white indicates 1/2, and gray indicates 0; in C image, black indicates 0, and white indicates 1; ◦ − SC point, and
− SD point.

We use the method, which is presented by Zhang [8], to compute and to smooth the orientation fields of fingerprints. For each point (x, y) in the orientation field, which is enclosed by a digital curve (with Np points), a positive cumulative change and a negative cumulative change of orientation can be computed as follows: Np −1

P+ (x, y) = (1/2)




+ (k)

(16)


− (k)

(17)

i=0

and Np −1

P− (x, y) = (1/2)

 i=0

with 
+ (k) =

(k) if 0 (k) < /2, (k) +  if (k) − /2, 0 otherwise

(18)

(k) if 0 > (k) > − /2, (k) −  if (k)
/2, 0 otherwise,

(19)

and 
− (k) =

Fig. 6. Extraction of a SCD : in P+ , black indicates 0, and white indicates maximal absolute value; in C image black indicates 0, and white indicates 1; × – singular point SCD .

where (k)=(x(k+1) mod Np , y(k+1) mod Np )−(xk , yk ), and 0 (x, y) <  is the direction of point (x, y). The digital curve moves in a counter-clockwise direction from 0 to Np , and Np is selected as 4. The main steps of our algorithm for detection of SPs are as follows: (i) If P+ (xk , yk ) + P− (xk , yk ) = 1/2 (respectively, P+ (xk , yk ) + P− (xk , yk ) = −1/2), a possible SC point (respectively, a SD ) is detected. Its position is extracted as follows: We compute C by Eq. (15) in a square of 6 × 6 centered on point (xk , yk ), where  denotes the average fingerprint ridge period. The local minimum of C is only extracted in this square. The local minimum is estimated as: If C(x ∗ , y ∗ ) < C(x, y) + T for any point (x, y) in r 2 < (x − x ∗ )2 + (y − y ∗ )2 < (r + 1)2 , then C(x ∗ , y ∗ ) is a local minimum. T is a threshold. T and r were empirically determined. In our experiment, r = , and T = 10 were used. If the number of local minimum is greater than 1, we take the point that the value of C is minimal. If C(x ∗ , y ∗ ) is a local minimum, a SC point (respectively, a SD ) is found at point (x ∗ , y ∗ ). If the local minimum does not exist, this is not a SP.

L. Wang, M. Dai / Int. J. Electron. Commun. (AEÜ) 60 (2006) 671 – 676

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Table 1. Result of proposed algorithm Type

A T L R W Total

N

100 100 100 100 100 500

False SPs

Missed SPs

SC

SD

SCD

SC

SD

SCD

0 1 2 1 1

1 2 2 3 1 16

2 0 0 0 0

0 6 4 4 3

0 7 6 5 4 42

0 3 0 0 0

N – number, A – Arch, T – Tented Arch, L – Left Loop, R – Right Loop, W – Whorl.

Table 2. Result of algorithm based on Poincare index Type

A T L R W Total

N

100 100 100 100 100 500

False SPs

Missed SPs

SC

SD

SCD

SC

SD

SCD

1 2 2 3 1

2 3 3 4 2 23

0 0 0 0 0

0 5 3 4 2

0 7 5 5 3 62

0 23 3 2 0

The extraction of SC and SD is shown in Fig. 5. In this figure, all SPs (a SC point and a SD point) are exactly detected. (ii) If for any point (x, y) in the orientation field, P+ (x, y)+ P− (x, y) = 0, i.e. no SC point and no SD point, we find the point which P+ is maximal. Let (xk , yk ) be such a point. We compute C by Eq. (15) in a square of centered on point (xk , yk ). The local minimum of C is only extracted in this square. If C(x ∗ , y ∗ ) is a local minimum, a SCD point is found at point (x*,y*). If the local minimum does not exist, there is not any SP in the fingerprint image. The Fig. 6 illustrates the extraction of SCD . In this figure, a SCD is exactly detected.

5. Experimental results In this section, we present the experimental results of our SPs detection algorithm and a comparison with the algorithm based on Poincare index used in [3,7–10]. In order to validate the performance of our algorithm, it was tested on the special fingerprint database NIST-4 [13]. We randomly selected 500 fingerprint images in NIST-4. Due to the fact that true position of the SPs in the fingerprint are not known, we were obliged to do a visual inspection of the positions of the estimated SPs for each fingerprint in the database. The results of proposed algorithm are shown in Table 1, and the results of the algorithm based on Poincare index are shown

in Table 2. With our algorithm, the number of false SPs is 16 and the number of missed SPs is 42. With the algorithm based on Poincare index, the number of false SPs is 23 and the number of missed SPs is 62. From these results, our algorithm has good performance.

6. Conclusion In this paper, a new method based on Gaussian–Hermite moments (GHMs) is presented for extracting SPs. It can extract not only normal SPs (core and delta), but also a pair of core–delta. Human inspection on experimental results for real fingerprint images shows that the proposed method provides accurate results, which would facilitate fingerprint identification and fingerprint classification afterwards. The comparison between our method and the method based on Poincare index used in a lot of literatures for SPs detection shows that our method has a good performance of SPs detection.

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[13] Watson CI, Wilson CL. Nist special database 4, fingerprint database. National Institute of Standards and Technology, March, 1992. Lin Wang is an Associate Professor of the Department of Mathematics and Computer Science, Guizhou University for Nationalities, China. He received his Ph.D. in 2005 from the University of Bordeaux-3, France. He is the author/co-author of more than 10 publications in image processing. His research interests include pattern recognition and image processing for personnel identification. Mo Dai is an Associate Professor of University of Bordeaux-3, France. He received his Ph.D. in 1986 from the University of Bordeaux-1, France. He is the author/co-author of more than 30 publications in image processing. His recent research interests include image processing and pattern recognition.