Int. J. Communications, Network and System Sciences, 2010, 3, 722-729
doi:10.4236/ijcns.2010.39096 Published Online September 2010 (http://www.SciRP.org/journal/ijcns)
A New Method for Improving Robustness of Registered
Fingerprint Data Using the Fractional Fourier Transform
Reiko Iwai, Hiroyuki Yoshimura
Graduate School of Engineering, Chiba University, Chiba, Japan
Received June 21, 2010; revised July 29, 2010; accepted August 30, 2010
Abstract
Inspired by related studies, a new data processing method in fingerprint authentication using the fractional
Fourier transform (FRT) was proposed for registered fingerprint data. In this proposal, protection of personal
information was also taken into account. We applied the FRT instead of the conventional Fourier transform
(FT) which has been used as one of the representative fingerprint authentication algorithm. Our method
solved the problem of current registration method and the robustness was verified. In this study, a modeled
fingerprint image instead of the original raw fingerprint images was analyzed in detail to make the character-
istic clear. As one dimensional (1D) modeled fingerprint image, we used the finite rectangular wave which is
regarded as the simplification of the grayscale distribution in an arbitrary scanned line of the raw fingerprint
images. As a result, it was clarified that the data processed by the FRT provide higher safety than those proc-
essed by the FT, because it is difficult to specify the orders from the intensity distribution of FRTs (the in-
tensity FRTs) when the combination of the various FRT’s order at every scanned line is used.
Keywords: Fractional Fourier Transform, Fingerprint Authentication, Biometrics, Personal Information
Protection
1. Introduction
The fingerprint images are indispensable and easy to use
the information to identify individuals. It is often-used for
logging into a PC, access control, as well as diligence and
indolence management in an office. Registration methods
of the fingerprint images are classified into three major
categories. One method is to register the whole fingerprints
images as two dimensional (2D) data without modification.
The others include to register a priori extracted features of
the images, such as minutiae templates [1], and to register
spatial frequency data transformed from 1D data extracted
from the original 2D image in a specific direction [2]. In
the former two methods, there exists the problem that un-
fair use is possible when the information leaks out. For the
security reasons, the third method might be more preferable.
However, it also has following problems: 1) the registered
data can easily be decoded to the original 1D data by the
inverse FT (IFT), when the registered data are generated
using the conventional FT; 2) additional processing time is
necessary for any trials to solve the issue 1).
We focused on the application of the FRT [3] by genera-
lizing the FT to solve these problems, where the FRT’s
order can be set arbitrarily. Because of this unique feature
of the FRT, it has capability of the encryption if the FRT’s
order is not revealed. The FRT of the 1D image out of a 2D
fingerprint image includes such features as less computa-
tional complexity to retrieve the original, if needed, and
impossibility to be decoded to the corresponding raw data
by unauthorized persons who have no knowledge on the
FRT’s order used. Therefore, even if the registered finger-
print info rmation leak ed out fro m an iden tification s ystem,
security would be guaranteed. Since the FRT will be
processed by an optical system incorporating a lens and a
laser light source to conduct the necessary process physi-
cally from scanning the fingerprint images to generating
the FRT image [4]. Therefore, it could drastically reduce
the time needed for individual identification.
In the present study, taking into account such future
application, the intensity FRTs are considered in terms of
any possibility of unlawful reveal of the hidden FRT’s or-
ders used to register the information corresponding to the
fingerprint images. Namely, we compare the intensity dis-
tribution data obtained by the FT and FRT. In the com-
parison, the actual data are simplified to be the finite-length
rectangular waveforms because th e analytical results could
be explained easily. The purpose of this analysis is to
demonstrate that unauthorized third persons cannot retrieve

R. IWAI ET AL.
723
the original data from the registered data in FRT. In this
paper, we analyze from the following two perspectives: 1)
behavior of the peak value of the cross-correlation function
between the finite rectangular wave and the intensity FRT,
and 2) behavior of the peak value of the cross-correlation
function between the finite rectangular wave and the inten-
sity inverse FRT (IFRT). Th ese analyses allow us to show
the difference between the intensity FRT, intensity IFRT
and the original image quantitatively. This fact means that
we cannot identify the original fingerprint image as the
difference becomes greater and greater. In addition, we
verify the robustness from the following two viewpoints: 1)
behavior of the second-maximum value of auto-correlation
function of the intensity FRT, and 2) behavior of the peak
value of the cross-correlation function between the finite
rectangular wave and the intensity IFRT (the FRT’s and
IFRT’s orders are different cases). First, we show that the
FRT’s order cannot be identified from the characteristics of
the intens ity F RT. Next, we s how th e inten sit y FRT canno t
be decoded to the corresponding original data by using the
conventional IFT by unauthorized persons who have no
knowledge on the FRT’s order used.
2. A New Data Processing Method in
Fingerprint Authentication by Use of the
Intensity FRT
2.1. Definition of the Fractional Fourier
Transform (FRT)
The FRT is the generalization of a conventional FT. The
FRT of 1D input data u(x) is defined [3, 5] as


()2 2 2
()()()exp[ ()/tan]
p
pp p
uxFuxuxix xs

 

2
exp[ 2/sin],
p
ixxs dx   (1)
where a constant factor has been dropped; /2p



,
where p is the FRT’s order; s is a constant. In particular,
in the optical FRT, s is called a scale parameter expressed
in terms of
s
s
f

where

is the wavelength and
s
f
is an arbitrarily fixed focal length. In this paper, the
value of s was fixe d at 1 .0.
When p takes a value of 4n+1, n be ing any integ er, the
FRT corresponds to the conventional FT. The intensity
distribution of the FRT, Ip(xp), is obtained by calculating
|up(xp)|2. In addition, up(xp) can be decoded to u(x) by the
IFRT with the order –p as follows:
()
()( ).
ppp
uxFu x


(2)
In our study, we call p in Equation (2) the IFRT’s or-
der. “Disfrft.m” [6] was used in our numerical calcula-
tion of the FRT.
2.2. Mo deling W ave form P atte rn of t he
Fingerprint
Although the grayscale levels are composed of interme-
diate values between 0 and 255 at the actual scanned
lines in the case of 2D black and white image of 8 bits, in
order to highlight the FRT as our new method together
with its feasibility, a finite rectangular wave is assumed
to be the simplification of the grayscale distribution of
the fingerprint image as shown in Figure 1. Horizontal
axis is intentionally made of 1024 (210) pixels for the
results of the FT and the FRT to be smoothly illustrated.
In the present paper we show the result only at the one
scanned line especially. However, we premise the appli-
cation of the FRT to the 2D original fingerprint image
which has multiple lines with random FRT’s orders. In
addition, the FRT’s orders can be used as arbitrary real
numbers.
2.3. Application of the FRT
The algorithm of the FRT has been intensively studied
[7-9]. Alternatively, the FRT was also applied to the fake
finger detectio n [1 0].
In the present paper we apply the FRT to the 1D finite
rectangular wave data shown in Figure 1 as a model of
fingerprint images. Basically, the FRT with the order p is
applied to the finite rectangular wave in Equation (1).
The FRT with the order p can be decoded to the finite
rectangular wave by the IFRT with the same order p as
already explained in Equation (2).
Figure 2 demonstrates the results of the FRTs in com-
parison with the conventional FT (i.e., the FRT with
p=1.0). Namely, Figure 2(a) shows the result of the FT
as the amplitude distribution at the upper portion and the
phase distribution at the lower portion. Figures 2(b) and
Figure 1. The finite rectangular wave as a modeled finger-
print image.

R. IWAI ET AL.
724
(a)
(b)
(c)
Figure 2. Examples of the amplitude and phase distribu-
tions of the FRTs applied to the finite rectangular wave,
when ps = (a) 1.0, (b) 0.9 and (c) 0.8.
2(c) are the results of the FRTs with ps = 0.9 and 0.8,
respectively. As a result, the peak values of the ampli-
tude distributions in Figures 2(a), 2(b) and 2(c) are
4.04

103, 6.59

102 and 5.80102, respectively.
It is found that the peak value of the amplitude distri-
bution falls remarkably and the width of spread increases
when the value of the FRT’s order p decreases. It is also
found that there is little difference in phase distributions
between Figures 2(b) and 2(c). In the case of FT shown
in Figure 2(a), the order p can be identified through the
waveform of the amplitude and phase distributions.
However, in the case of FRT, the order p might not be
identified through them. In particular, it is difficult to
identify the FRT’s orders ps through the waveforms of
the phase distributions shown in Figures 2(b) and 2(c).
Therefore, this fact led us the new method safer than the
conventional method using the FT, because the FRT’s
order has highly-confidential in the applied FRT condi-
tion.
In this way, we focused on the intensity distribu tion of
the FRT from a viewpoint of the security of individual
information, because the intensity FRT may not be com-
pletely decoded to the original fingerprint image by the
IFRT. Figure 3 depicts the intensity FT of Figure 2(a)
and the intensity FRT of Figure 2(b). The peak values of
the intensity distributions in Figures 3(a) and 3(b) are
1.63

107 and 4.34

105, respectively. It is found from
the comparison between Figures 2 and 3 that the peak
value of the wave pattern of the intensity distribution is
very high. In the next section, we compare the registered
information of the inten sity FT and the intensity FRT by
changing the FRT’s order p.
3. Cross-Correlation Properties between the
Intensity FRT and the Finite Rectangular
Wave
First, we focused on and analyzed the peak value of the
normalized cross-correlation function between the finite
rectangular wave and the intensity FRT by changing the
FRT’s order p. The peak value quantitatively indicates
the waveform difference between the two.
Figure 4 depicts the normalized cross-correlation
function between the finite rectangular wave shown in
Figure 1 and the intensity FRT shown in Figure 3. The
peak values of the normalized cross-correlation functions
in Figures 4(a) and 4(b) are 0.0522 and 0.312, respec-
tively. It is found that, in the case of p = 1.0, the wave-
form is similar to the finite rectangular wave, though the
peak value is very small. On the other hand, in the case
of p = 0.9 , the waveform is not similar to the finite rectan-
gular wave, though the peak value is higher than that in
the case of p = 1.0. However significant peak cannot be
seen in both of the cases.

R. IWAI ET AL.
725
(a)
(b)
Figure 3. The intensity distributions of the FRTs of the fi-
nite rectangular wave shown in Figure 1, when ps = (a) 1.0
and (b) 0.9.
(a)
(b)
Figure 4. The normalized cross-correlation functions be-
tween the finite rectangular wave shown in Figure 1 and the
intensity FRT shown in Figure 3, when ps = (a) 1.0 and (b)
0.9.
Figure 5 illustrates the peak value of the normalized
cross-correlation function by changing the FRT’s order p
from 0.1 to 1.0 by 0.1. As a result, it is understood that
the waveform difference between the finite rectangular
wave and the registered intensity d istribution of the FRT
becomes larger with increasing FRT’s order p. In par-
ticular, the intensity FRT with p=0.1 is proximate to the
finite rectangular wave. However, basically the original
fingerprint image is 2D and consisted of multiple scanned
lines. Therefore, it is not a problem in our method because
Figure 5. Behavior of the peak value of the cross-correlation
function between the in tensity FRT and th e finite rectangular
wave on the FRT’s order p.
the combination of the various FRT’s orders ps would be
applied in the 2D fingerprint image and the values of the
low FRT’s order p would not be use d.
4. Cross-Correlation Properties between the
Intensity IFRT and the Finite
Rectangular Wave
Next, we focused on and analyzed the peak value of
normalized cross-correlation function between the fi-
nite rectangular wave shown in Figure 1 and the intensity
distribution of the IFRT (the intensity IFRT) of inten-
sity FRT shown in Figure 3, in order to investigate the
difficult y to r etr ieve the o rig inal f ingerp rint image.
Figure 6 depicts examples of the intensity IFRT
when the FRT’s and IFRT’s orders ps are 1.0 and 0.9,
respectively. From this figure, it is understood that the
intensity FRT cannot be decoded to the finite rectan-
gular wave Figure 7 shows the normalize cross-
correlation function between the finite rectangular
wave shown in Figure 1 and the intensity IFRT shown
in Figure 6. We numerically calculated the normalized
cross-correlation function to analyze the difference
between them quantitatively. As a result, the peak val-
ues of the normalized cross-correlation functions in
Figures 7(a) and 7(b) are 0.869 and 0.197, respec-
tively, and the peak value when the order p = 0.9 is
much smaller than that when the order p = 1.0.
The analytical result of the every FRT’s order is
shown in Figure 8. It is found that the intensity FRT
can be decoded to closely the finite rectangular wave in
the cases of FRT’s orders ps = 0.1 and 1.0. The peak

R. IWAI ET AL.
726
(a)
(b)
Figure 6. Examples of the intensity IFRT of the intensity
FRT shown in Figure 3, when the FRT’s and IFRT’s orders
ps = (a) 1.0 and (b) 0.9.
(a)
(b)
Figure 7. The normalized cross-correlation function be-
tween the finite rectangular wave shown in Figure 1 and the
intensity IFRT shown in Figure 6, when the FRT’s and
IFRT’s orders ps = (a) 1.0 and (b) 0.9.
value of the normalized cross-correlation function
when the order p=1.0 becomes drastically high as
shown in Figure 8. It is understood from Figure 6 (a)
that the intensity FR T in the cas e of FRT’s order p=1.0
can be decoded to closely the finite rectangular wave.
However, it may not be decoded to the finite rectangu-
lar wave in the cases of the other FRT’s orders ps. As
Figure 8. Behavior of the peak value of the normalized
cross-correlation functions between the intensity IFRT and
the finite rectangular wave when the IFRT’s order p is the
same as the FRT’s order p.
described previously in Section 3, in our method, the
combination of the various FRT’s orders ps would be
applied and the FRT’s order p = 1.0 and the low FRT’s
order p would not be used. Therefore, it is clarified that
the data processed by the FRT would provide higher
safety than the cases processed only by the FT.
5. Verification of Robustness
Finally, we verify the robustness of our proposed
method from the following two viewpoints: 1) behavior
of the second-maximum value of auto-correlation func-
tion of the intensity FRT, and 2) behavior of the peak
value of the cross-correlation function between the
finite rectangular wave and the intensity IFRT (the
FRT’s and IFRT’s orders are different cases).
First, we show the FRT’s order cannot be identified
from the characteristics of the intensity FRT. Next, we
show the intensity FRT cannot be decoded to the finite
rectangular wave by using the conventional IFT by
unauthorized persons who have no knowledge on the
FRT’s orders used.
5.1. The Order Dependency on the Second-
Maximum Values of the Auto-Correlation
Function of the Intensity FRT
First, we focused on and analyzed the second-maximum
value of the auto-correlation function of the intensity
distribution of the FRT by changing the order p, be-
cause the peak values of the auto-correlation functions
are all 1 and are not distinguishable.
Figure 9 depicts examples of the auto-correlation
function of the intensity distribution of the FRT when
the FRT’s orders ps = 1.0 and 0.9 shown in Figure 3.
Order p

R. IWAI ET AL.
727
As a result, the second-maximum values in Figures
9(a) and 9(b) are 0.501 and 0.390, respectively.
Figure 10 illustrates the FRT’s order dependency on
the second-maximum value of the auto-correlation
function by changing the order p. The second maxi-
mum peak values for the FRT’s orders from 0.1 to 1.0
by 0.1 are 0.180, 0.223, 0.191, 0.179, 0.145, 0.225,
0.299, 0.359, 0.390 and 0.501, respectively. In par-
ticular, the second-maximum peak value is the smallest
when the FRT’s order p = 0.5. The reason can be con-
sidered that the intensity FRT equally includes both of
the characteristics of the spatial information and spatial
frequency information of the finite rectangular wave
when the FRT’s order p = 0.5.
As a result, it is shown that the FRT’s order cannot
be found from the characteristics of the intensity FRT,
because we cannot see noticeable difference in the
second-maximum values in any cases of FRT’s order p.
(a)
(b)
Figure 9. The normalized auto-correlation functions of the
intensity FRTs shown in Figure 3, when ps = (a) 1.0 and (b)
0.9.
Figure 10. Dependence of the second-maximum value of the
normalized auto-correlation functions on th e FRT’s order p.
5.2. Cross-Correlation Properties between the
Intensity IFRT and the Finite Rectangular
Wave (The FRT’s and the IFRT’s Orders
are Different Cases)
Next, we focused on and analyzed the peak values of
normalized cross-correlation function between the finite
rectangular wave and the intensity IFRT of intensity
FRT, when the FRT’s and IFRT’s orders are different
from each other. In particular, we fixed the IFRT’s
order at 1.0 which corresponds to the conventional
IFT.
Figure 11 depicts examples of the intensity IFRT of
intensity FRT when the FRT’s and IFRT’s orders ps are
(a) 0.9 and 1.0 and (b) 0.8 and 1.0, respectively. As a
result, it is understood that the waveforms are completely
different from the finite rectangular wave. Figure 12
shows the normalized cross-correlation function between
the finite rectangular wave shown in Figure 1 and the
intensity IFRT shown in Figure 11. As a result, the peak
values in Figures 11(a) and 11( b) are 0.135 and 0.0579,
respectively, and the peak values are very small for both
of the FRT’s orders ps=0.9 and 0.8.
The analytical result of every FRT’s order is shown in
Figure 13. The peak values of the FRT’s orders from 0.1
to 0.9 by 0.1 are 0.0563, 0.0592, 0.0587, 0.0624, 0.0579,
0.0690, 0.0884, 0.107 and 0.135, respectively, and fully
small. The peak value decreases with a decrease in the
FRT’s order p when p takes a value from 0.9 to 0.5 by
0.1. However, the peak value is almost the same value
when p takes a value from 0.5 to 0.1 by 0.1. The reason
can be considered that the retrieving effect to the finite
rectangular wave by the FT is very small when p=0.5 or
less because the intensity FRT has little characteristics of
the spatial frequency information of the finite rectangular
wave. As a result, it is found that the intensity IFRT
Order p
Position of pixel

R. IWAI ET AL.
728
(a)
(b)
Figure 11. Examples of the intensity IFRT of the intensity
FRT shown in Figure 3, when the FRT’s and IFRT’s orders
ps = (a) 0.9 and 1.0 and (b) 0.8 and 1.0, respectively.
(a)
(b)
Figure 12. The normalized cross-correlation function be-
tween the finite rectangular wave shown in Figure 1 and the
intensity IFRT shown in Figure 11, when the FRT’s and
IFRT’s orders ps = (a) 0.9 and 1.0 and (b) 0.8 and 1.0, re-
spectively.
cannot be decoded to the finite rectangular wave in all
cases of the FRT’s orders ps.
Moreover, it is understood by comparison between
Figures 8 and 13 that the intensity FRT may not be de-
coded to the finite rectangular wave for all FRT’s orders
ps when the FRT’s and IFRT’s orders ps are
different. Therefore, it is clarified that the data processed
by the FRT would provide higher safety than the cases
processed only by the FT.
Figure 13. Behavior of the peak value of the normalized
cross-correlation function between the finite rectangular
wave and the intensity IFRT, when the IFRT’s order is
fixed at 1.0 and the FRT’s orders ps are different from 1.0.
From the above-mentioned results about the robust-
ness of image processed by our method and the results
shown in Figure 5 in Section 3 and Figure 8 in Section
4, we can say that th e most suitab le single valu e of the
FRT’s order p is 0.9, though the combination of the
various p at every scanned line of the raw fingerprint
image is significant in our method.
6. Conclusions
In the present study, we proposed a new data processing
method for registering the fingerprint image by using the
FRT. Moreover, the robustness was examined. In this
study, our new method was analyzed in detail by using a
modeled fingerprint image instead of the original raw
fingerprint image to make the characteristic clear. As a
result, it was found that our proposed method can register
the fingerprint related data that cannot be easily de-
coded to the corresponding original fingerprint data by
unauthorized persons. In addition, our new method was
performed by simple numerical calculation. Furthermore,
it was understood that our new method has high robust-
ness and security in the combination of the various
FRT’s orders. As a further study, we would analyze the
accuracy of our method for dust and sebum regarded as
noise and hurts regarded as the lost part.
7. References
[1] D. Maltoni, D. Maio, A. K. Ja in and S. Prabhakar, “Hand-
book of Fingerprint Recognition,” Springer, New York,
2003.
[2] H. Takeuchi, T. Umezaki, N. Matsumoto and K. Hiraba-
yashi, “Evaluation of Low-Quality Images and Imaging
Enhancement Methods for Fingerprint Verification,”
Order p

R. IWAI ET AL.
729
Electronics and Communications in Japan, Vol. 90, No.
10, 2007, pp. 40-53.
[3] H. M. Ozakta s, Z. Zalevsky and M. A. Kutay, “The Frac-
tional Fourier Transform,” John Wiley & Sons, New Jer-
sey, 2001.
[4] A. W. Lohmann, “Image Rotation, Wigner Rotation, and
the Fractional Fourier Transform,” Journal of the Optical
Society of America A, Vol. 10, No. 10, 1993, pp. 2181- 2186.
[5] A Bultheel and H. E. Martinez Sulbaran, “Computation
of the Fractional Fourier Transform,” Applied and Compu-
tational Harmonic Analysis, Vol. 16, No. 3, 2004, pp.
182-202.
[6] A.Bultheel and H. M. Sulbaran, “Computation of the
Fractional Fourier Transform”. http://nalag.cs. kuleuven.
be/research/software/FRFT/
[7] F. J. Marinho and L. M. Bernardo, “Numerical Calcula-
tion of Fractional Fourier Transforms with a Single Fast-
Fourier-Transform Algorithm,” Journal of Optical Society
of America A, Vol. 15, No. 8, 1998, pp. 2111-2116.
[8] X. Yang, Q. Tan, X. Wei, Y. Xiang, Y. Yan and G. Jin,
“Improved Fast Fractional-Fourier Transform Algorithm,”
Journal of Optical Society of America A, Vol. 21, No. 9,
2004, pp. 1677-1681.
[9] D. H. Bailey and P. N. Swarztrauber, “The Fractional
Fourier Transform and Applications,” SIAM Review, Vol.
33, No. 3, 1991, pp. 389-404.
[10] H. Lee, H. Maeng and Y. Bae, “Fake Finger Detection
Using the Fractional Fourier Transform,” Lecture Notes
in Computer Science, Heidelberg, Vol. 5707, 2009, pp.
318-324.