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
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
r
der
p1
p2
p3
p4
p5
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]

ux



222
2
exp πtan
exp2 π/sind,
pp p
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
s
s
f

 where

is the wavelength and
s
f
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,

p
p
I
x, is ob-
tained by calculating

2
pp
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
9.
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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