Improved Authentication Accuracy by Individually Set Orders of the Fractional Fourier Transform and Effects of Damage of Fingerprint Image on Authentication Accuracy

doi:10.4236/ijcns.2011.432095

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Int. J. Communications, Network and System Sciences, 2011, 4, 770-777

doi:10.4236/ijcns.2011.432095 Published Online December 2011 (http://www.SciRP.org/journal/ijcns)

Improved Authentication Accuracy by Individually

Set Orders of the Fractional Fourier Transform and

Effects of Damage of Fingerprint Image on

Authentication Accuracy

Reiko Iwai, Hiroyuki Yoshimura

Graduate School of Engineering, Chiba University, Chiba, Japan

E-mail: [email protected], [email protected]

Received Octoberr 21, 2011; revised November 29, 2011; accepted December 5, 2011

Abstract

Recently, ubiquitous personal devices with a fingerprint authentication function have been increasing. In

such devices, there is almost no possibility of the authentication by impostors unless they are lost or stolen.

However, for example, in the management of entering and leaving a building, not only the fingerprint au-

thentication device but also the other authentication measures, such as an IC card, a key, etc., are generally

used. In our previous studies, we have analyzed the authentication accuracy of the fingerprint authentication

devices for personal possessions where other authentication measures are not needed. As a result, we made

clear that the authentication accuracy in our method has extremely high compared with that in the marketed

compact fingerprint authentication products, even if dirt, sebum, etc., are attached to the fingertip and there

are scratches. In this study, we analyze the damage ratio of the fingerprint image where the genuine authen-

tication can be conducted without problems, because the fingertip is easily got large cuts. Moreover, we

analyze the impostor authentication of the fingerprint authentication devices for public possessions in the

two cases of without and with other authentication measures. As a result, it is found that clearer impostor

authentication can be achieved in the case of with other authentication measures. In addition, it is found that

the damage ratio of the fingerprint image to conduct clearer genuine authentication without the image cor-

rection is less than 14.3%.

Keywords: Fractional Fourier Transform, Fingerprint Authentication, Biometrics, Personal Information

Protection

1. Introduction

The authentication of personal identities by fingerprints

has been much researched until now. In particular, it has

recently been paid attention to because of ease, cheap

price and being used everywhere. In the use of a finger-

print authentication system, the users do not have to re-

member the fingerprint itself and there is no worry to

lose it like a password, once the relevant information is

registered as a fingerprint template in the system. The

fingerprint, however, cannot be changed like a password

if the information leaks out from the system. Moreover,

the fingerprint identifier does not always have the same

condition so that there is a possibility of false authentica-

tion. In our previous studies [1-4], in order to solve these

problems, we proposed a new data processing method

using the fractional Fourier transform (FRT) [5-9] to

generate the fingerprint templates. Recently, the resear-

ches related to the fingerprint authentication using the

FRT have been conducted to deal with the fake finger-

prints, to consider the fingerprint encryption and decryp-

tion algorithm for enhanced security, to consider the size

of the encrypted fingerprint images for secure transmis-

sion on digital communication networks, and so on [10-

12].

In our previous studies [1-4], we have assumed the

fingerprint authentication devices for personal posses-

sions such as cell phones, where other authentication

measures such as an IC card, a key, etc., are not needed.

As a result, we made clear that the authentication accu-

R. IWAI ET AL.

771

racy in our method has extremely high compared with

that in the marketed compact fingerprint authentication

products, even if dirt, sebum, etc., are attached to the

fingertip and there are scratches. In this study, we ana-

lyze the damage ratio of the fingerprint image where the

genuine authentication can be conducted without prob-

lems, because the fingertip is easily got large cuts.

Moreover, we assume the fingerprint authentication de-

vices for public possessions such as ATMs in a bank,

diligence and indolence management in an office, etc.,

and analyze the impostor authentication in the two cases

of without and with other authentication measures. Spe-

cifically, 1) we compare the impostor authentication ac-

curacy in the case that the FRT’s orders are fixed in each

fingerprint authentication device with that in the case

that the FRT’s orders are changed in each fingerprint

identifier; 2) we analyze the effects of the damage, such

as large cuts, in the fingerprint image on the genuine

authentication accuracy.

We prepare three kinds of real fingerprint images: the

genuine fingerprint images; the impostor fingerprint im-

ages; the genuine fingerprint images with various dam-

ages such as large cuts, and focus on the peak value of

the two-dimensional (2D) normalized cross-correlation

function (NCF) between the intensity distributions of the

FRTs (i.e., the intensity FRTs). The intensity FRT is ob-

tained by extracting 256 × 256 pixels at the center part of

the 2D original fingerprint image and conducting the

FRTs with the random FRT’s orders in the different lines

of the extracted image. Specifically, we obtain the peak

values of the NCFs in the following: 1) the intensity FRT

of the extracted genuine fingerprint image is registered

(we call it the fingerprint template); 2) the intensity FRT

of the newly scanned and extracted fingerprint image is

obtained (we call it the impostor intensity FRT or the

damaged genuine intensity FRT); 3) the peak value of

the NCF derived from 1) and 2) is obtained. Finally, we

obtain the minimum error rate (MER) by a value satis-

fied with the condition the false acceptance rate (FAR)

and the false rejection rate (FRR) take the same value

[13]. The MER is derived from the properties of the

NCFs. The authentication threshold is also decided.

In Section 2, first, the generation method of the fin-

gerprint templates is explained. Next, to conduct clearer

impostor authentication, we compare the properties of

the peak values of the NCFs between the fingerprint

templates and the impostor intensity FRTs in cases of 1)

the FRT’s orders to obtain the impostor FRT intensity

are the same as those used to generate the fingerprint

template and 2) the FRT’s orders to obtain the impostor

FRT intensity are different from those used to generate

the fingerprint template. In Section 3, the properties of

the peak values of the NCFs between the fingerprint

templates and the intensity FRTs of the damaged genuine

fingerprint images caused by large cuts (i.e., the dam-

aged genuine intensity FRTs) are obtained. In Section 4,

we evaluate the authentication accuracy based on the

MER derived from the results obtained in Sections 2 and

3. Finally, in Section 5, conclusions in our study and

future study are described.

2. Properties of the Peak Values of the NCFs

between the Fingerprint Templates and

the Impostor Intensity FRTs

In this Section, first, we explain the generation method of

the fingerprint templates. Next, we obtain the peak value

of the NCF between the fingerprint template and the

impostor intensity FRT when the FRT’s orders used to

obtain the impostor intensity FRT are the same as those

used to obtain the fingerprint template, i.e., the impostor

authentication accuracy in the case that the FRT’s orders

are fixed in each fingerprint authentication device. We

also obtain the peak value when the FRT’s orders used to

obtain the impostor intensity FRT are different from

those used to obtain the fingerprint template, i.e., the

impostor authentication accuracy in the case that the

FRT’s orders are changed in each fingerprint identifier.

Finally, we analyze the mean values and the standard

deviations of the NCFs in cases of the same FRT’s

orders and the different ones.

2.1. Generation Method of the Fingerprint

Templates

In this subsection, the generation method of the finger-

print templates is explained. Fingerprint images provided

by the Biometric System Laboratory [14] were used as

original fingerprint images. The provided fingerprint

images were 880 from 110 fingertips. For each fingertip,

there were 8 fingerprint images. We selected one finger-

print image for each fingertip. Therefore, we used 110

original fingerprint images which were not affected by

abrasion and distortion. As an example, Figure 1 visual-

izes the data in the TIF format with 480 vertical and 640

horizontal pixels. In our previous study, it was clear that

the fingerprint authentication accuracy is scarcely af-

fected by extracted size [3]. Therefore, in this study, as

depicted in Figure 2, we analyzed using the fingerprint

images with 256 vertical and 256 horizontal pixels ex-

tracted from the center of Figure 1. We call this image

the genuine fingerprint image. The real size corresponds

to 13.0 mm by 13.0 mm. In this study, as shown in Fig-

ure 2, height and width of the images are called line and

column, respectively.

The FRT is the generalization of a conventional Fourier

772 R. IWAI ET AL.

Column

Line

Figure 1. An example of the original fingerprint image.

Line

FRT’s O

der

p255

p256

Column

Figure 2. Genuine fingerprint image with 256 lines and 256

columns extracted from the center of the image shown in

Figure 1.

transform (FT). The FRT of the one-dimensional input

data is defined as [15,16]



222

exp πtan

exp2 π/sind,

pp p

uxuxix x s

ixx sx















 (1)

where a constant factor has been dropped; π2p





where p is the FRT’s order; s is a constant. In particular,

in the optical FRT, s is called a scale factor expressed in

terms of



 where



is the wavelength and

is an arbitrarily fixed focal length. In this study, the

value of s was fixed at 1.0. In particular, p takes a value

of 4n + 1, n being any integer, the FRT corresponds to

the conventional FT. The intensity FRT,



x, is ob-

tained by calculating



In this study, the grayscale distribution in one line of a

genuine fingerprint image shown in Figure 2 could be

regarded as a wave pattern. The FRT was performed

using Equation (1) in each line of the image by changing

the FRT’s order randomly and the intensity FRT distri-

butions with the different FRT’s orders in different lines

were obtained. In Figure 2, the FRT’s orders are ex-

pressed in terms of p1, p2, , p256 and take values from

0.1 to 1.9 because this range has been thought to be the

best performance in our processing method [4].

ux .



Figure 3 shows the intensity FRT of the genuine fin-

gerprint image shown in Figure 2. In this study, this im-

age is called the fingerprint template. The peak value of

the fingerprint template is 1.18 × 106. We prepared 110

fingerprint templates.

2.2. Peak Value of the NCF between the

Fingerprint Template and the Impostor

Intensity FRT in Case of the Same FRT’s

Orders

In this subsection, we obtain the peak value of the NCF

between the fingerprint template and the impostor inten-

sity FRT, when the FRT’s orders used to obtain the im-

postor intensity FRT are the same as those used to obtain

the fingerprint template. Figure 4 illustrates an example

of the impostor fingerprint image with 256 lines and 256

columns. Figure 5 shows the impostor intensity FRT of

Figure 4. The peak value is 3.23 × 106.

Figure 6 illustrates the NCF between the fingerprint

template shown in Figure 3 and the imposter intensity

FRT shown in Figure 5. The peak value is 0.703 and this

means that the identification between a genuine person

and an impostor person can be conducted correctly

Figure 3. Fingerprint template of the genuine fingerprint

image shown in Figure 2.

Column

Line

Figure 4. Impostor finger print image with 256 lines and 256

columns.

R. IWAI ET AL.

773

Figure 5. Impostor intensity FRT of Figure 4. The FRT’s

orders used to obtain this figure are the same as those used

in generating the fingerprint template shown in Figure 3.

Figure 6. NCF between the fingerprint template shown in

Figure 3 and the impostor intensity FRT shown in Figure 5.

from the viewpoint of the result obtained from our pre-

svious study [4]. In our analysis, we used 110 original

fingerprint images so that the number of the peak values

of the NCFs was 5995 (110C2).

2.3. Peak Value of the NCF between the

Fingerprint Template and the Impostor

Intensity FRT in Case of the Different

FRT’s Orders

In this subsection, we obtain the peak value of the NCF

between the fingerprint template and the impostor inten-

sity FRT, when the FRT's orders used to obtain the im-

postor intensity FRT are different from those used to

obtain the fingerprint template. Figure 7 shows the im-

postor intensity FRT of Figure 4. The peak value is 3.16

× 106 and comparable with that in Figure 5 (3.23 × 106).

Figure 8 illustrates the NCF between the finger-

Figure 7. Impostor intensity FRT of Figure 4. The FRT’s

orders used to obtain this figure are different from those used

in generating the fingerprint template shown in Figure 3.

Figure 8. NCF between the fingerprint template shown in

Figure 3 and the impostor intensity FRT shown in Figure 7.

print template shown in Figure 3 and the imposter inten-

sity FRT shown in Figure 7. The peak value is 0.267 and

very low. This means that the identification between a

genuine person and an impostor person can be conducted

more correctly. The number of the peak values of the

NCFs was the same as that in subsection 2.2, i.e., 5995.

2.4. Mean Values and Standard Deviations of the

Peak Values of the NCFs in Cases of the

Same FRT’s Orders and the Different Ones

The mean values and the standard deviations of the peak

values of the NCFs are summarized in Table 1. In this

table, the mean value and the standard deviation take

values of 0.761 and 0.108, respectively, in case of the

same FRT’s orders. On the other hand, they take values

of 0.287 and 0.0484, respectively, in case of the different

FRT’s orders. These data were calculated from 5995

774 R. IWAI ET AL.

Table 1. Mean values and standard deviations in cases of

the same FRT’s orders and the different ones.

Same FRT’s orders Different FRT’s orders

Mean value 0.761 0.287

Standard deviation 0.108 0.0484

peak values of the NCFs. We can understand from Table

1 that in case of the different FRT’s orders the mean

value decreases by less than 40% and the standard devia-

tion becomes smaller more than 45% in comparison with

those in case of the same FRT’s orders. Therefore, we

can say that clearer authentication is possible in case of

the different FRT’s orders.

3. Properties of the Peak Values of the NCFs

between the Fingerprint Templates and

the Damaged Genuine Intensity FRTs

In this Section, we calculate the mean values and the

standard deviations of the peak values of the NCFs be-

tween the fingerprint templates and the damaged genuine

intensity FRTs. The damaged genuine intensity FRT was

derived from the FRT of a genuine fingerprint images

with the damage caused by large cuts. Specifically, the

damage was given by the image deletion from the genu-

ine fingerprint image. The damaged areas were changed

by the variation of the deleted numbers of lines and

columns from 1 to 19 one by one. In addition, for the

fixed deleted number of the lines and the columns, 50

damaged genuine fingerprint images were generated by

changing the damaged position randomly. Since there

were 110 genuine fingerprint images, 5500 peak values

of the NCFs between the fingerprint templates and the

damaged genuine intensity FRTs were obtained. The

whole number of pixels in a genuine fingerprint image is

256 × 256 = 65,536. In the case that the damaged area is

composed of 1 line and 1 column, the damage ratio be-

comes (256 × 2 – 1)/(256 × 256) × 100 = 0.780%. In the

case that the damaged area is composed of 10 lines and

10 columns, the damage ratio becomes (256 × 20 – 10 ×

10)/(256 × 256) × 100 = 7.66%. As an example, Figure

9 denotes the damaged genuine fingerprint image with

7.66% damage ratio. Figure 10 shows the intensity FRT

of Figure 9. The peak value is 1.28 × 106 and compara-

ble with that in Figure 3 (1.18 × 106). Figure 11 shows

the NCF between the fingerprint template shown in Fig-

ure 3 and the damaged genuine intensity FRT shown in

Figure 10. The peak value is 0.768.

The mean values and the standard deviations for sev-

eral damage ratios are summarized in Table 2. In the

table, damage ratios are 0.780% (1 line & 1 column),

3.87% (5 lines & 5 columns), 7.66% (10 lines & 10

Column

Line

Figure 9. An example of the 7.66% damaged genuine fin-

gerprint image shown in Figure 2.

Figure 10. Damaged genuine intensity FRT shown in Figure

Figure 11. NCF between the fingerprint template shown in

Figure 3 and the damaged genuine intensity FRT shown in

Figure 10.

columns), 10.6% (14 lines & 14 columns) and 14.3% (19

lines & 19 columns). Then, the mean values take values

from 0.888 to 0.992 and the standard deviations take

values from 0.0261 to 0.0931. We can understand from

the mean values in Table 2 that the authentication accu-

R. IWAI ET AL.

775

Table 2. Mean values and standard deviations for various damage ratios. ( ): the numbers of lines and columns corresponding

to the damage.

Damage ratio 0.780%

(1 line & 1 colum) 3.87%

(5 lines & 5 columns)7.66%

(10 lines & 10 columns)10.6%

(14 lines & 14 columns) 14.3%

(19 lines & 19 columns)

Mean value 0.992 0.962 0.934 0.908 0.888

Standard deviation 0.0261 0.0540 0.0700 0.0854 0.0931

racy judged as a genuine person becomes worse with an

increase in the damage ratio. However, it is found that

there is no problem in the identification between a genu-

ine person and an impostor person, because the mean

value for every damage ratio is fully higher than 0.287 in

cases of the different FRT’s orders as shown in Table 1.

4. Comparison of Authentication Accuracy

between Our Method and the

Conventional Fingerprint Authentication

Device

In this Section, we evaluate the authentication accuracy

based on the MER which is calculated from the results

obtained in Sections 2 and 3. In particular, as for the re-

sults obtained in Section 2, the one in case of the differ-

ent FRT’s orders is used, because better MER could be

obtained. Figure 12 is the result showing a set of histo-

grams of the peak values of the NCFs obtained in Sec-

tions 2 and 3. In Figure 12, the left side one corresponds

to the impostor distribution and the right side one does to

the genuine distribution. The left side one was obtained

in case of the different FRT’s orders in Section 2 and the

right side one was obtained in the case that the damage

ratio was 7.66% in Section 3. In Figure 12, the MER is

2.03 × 10–6 and the authentication threshold is 0.552.

Figure 12. A set of histograms obtained from the results in

Sections 2 and 3 when the damage ratio is 7.66%.

print authentication systems in the market. This fact means

that our method can sufficiently achieve the authentica-

tion accuracy with the normal security level shown in

Table 4 when the genuine fingerprint images have ap-

proximately less than 14.3% damage ratio as understood

from Table 3. In addition, our method is equivalent to

the authentication accuracy with the high security level

when the genuine fingerprint images have approximately

less than 10.6% damage ratio. Also the other smaller

damage ratios are satisfied with the authentication accu-

racy with the high security level. Moreover, in our pre-

vious study [3], the mean value of the MERs of the vari-

ously extracted fingerprint images was about 1.10 × 10–3.

Therefore, it was also found that the authentication ac-

curacy is considerably improved when the damage ratio

is 10.6% or less as understood from Table 3.

MERs for various damage ratios are summarized in

Table 3. In the table, the MERs take values from 1.47 ×

10–19 to 1.09 × 10–3 and the thresholds take values from

0.492 to 0.745. The data indicated by red letters corre-

spond to the result obtained from Figure 12. For com-

parison, the most recent specifications of FARs and

FRRs of the marketed products based on the frequency

analysis method are indicated in Table 4 [17]. In Table 4,

the authentication accuracy is approximately less than

from 0.0001% to 0.01% in the FAR and less than 0.1%

in the FRR. The FAR and the FRR take different values,

because the marketed products focus on the FAR. In or-

der to decrease the FAR in our method, we may move

the threshold to the right side in Figure 12. In this paper,

however, we directly compare the MERs for various

damage ratios in Table 3 with the FARs in Table 4.

5. Conclusions

In this paper, the impostor authentication accuracy in the

case that the FRT’s orders are changed to each finger-

print identifier was compared with that in the case that

they are fixed to each fingerprint authentication device.s

As a result, we found that our method has higher au-

thentication accuracy in comparison with those shown in

the recent available specification sheet of major finger-

776 R. IWAI ET AL.

Table 3. MERs for various damage ratios.

Damage ratio 0.780% 3.87% 7.66% 10.6% 14.3%

MER (%) 1.47 × 10–19 2.04 × 10–9 2.03 × 10–6 1.69 × 10–4 1.09 × 10–3

Threshold 0.745 0.606 0.552 0.511 0.492

Table 4. Specifications of the marketed products based on

the frequency analysis method.

Security level FAR FRR

Low <0.01% N/A

Normal <0.001% <0.1%

High <0.0001% N/A

As a result, we found that clearer impostor authentication

can be achieved in the case that they are changed to each

fingerprint identifier. Moreover, the effects of damage of

the 2D real fingerprints on the genuine authentication

accuracy were analyzed. As a result, it was found that

higher genuine authentication can be achieved in our

method in comparison with that in the conventional fin-

gerprint authentication device based on the frequency

analysis method, even if the damage ratio is 14.3% and

the image correction is not conducted.

As a further study, we would analyze the effects of the

misalignment of the scanned fingerprint image with that

used in generating the fingerprint template on the au-

thentication accuracy.

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