We present a new approach for iris recognition based on stochastic autoregressive models with exogenous input (ARX). Iris recognition is a method to identify persons, based on the analysis of the eye iris.
A typical iris recognition system is composed of four phases: image acquisition and preprocessing, iris localization and extraction, iris features characterization, and comparison and matching. The main contribution in this work is given in the step of characterization of iris features by using ARX models. In our work every iris in database is represented by an ARX model learned from data. In the comparison and matching step, data taken from iris sample are substituted into every ARX model and residuals are generated. A decision of accept or reject is taken based on residuals and on a threshold calculated experimentally. We conduct experiments with two different databases. Under certain conditions, we found a rate of successful identifications in the order of 99.7 % for one database and 100 % for the other.Applications in Artificial Intelligence - ApplicationsRed de Universidades con Carreras en Informática (RedUNCI

Garza Castañon, Luis E.

Montes de Oca, Saúl

Morales Menéndez, Rubén

SEDICI - Repositorio de la UNLP

An Application of ARX Stochastic
Models to Iris Recognition
Luis E. Garza Castan˜o´n1, Sau´l Montes de Oca2, and
Rube´n Morales-Mene´ndez1
1 Department of Mechatronics and Automation, ITESM Monterrey Campus,
{legarza,rmm}@itesm.mx
2 Automation Graduate Program Student, ITESM Monterrey Campus,
saul montesdeoca@yahoo.com.mx
Av. Eugenio Garza Sada Sur No. 2501
Monterrey, N.L. 64849 Me´xico
Abstract. We present a new approach for iris recognition based on sto-
chastic autoregressive models with exogenous input (ARX). Iris recogni-
tion is a method to identify persons, based on the analysis of the eye iris.
A typical iris recognition system is composed of four phases: image ac-
quisition and preprocessing, iris localization and extraction, iris features
characterization, and comparison and matching. The main contribution
in this work is given in the step of characterization of iris features by
using ARX models. In our work every iris in database is represented
by an ARX model learned from data. In the comparison and matching
step, data taken from iris sample are substituted into every ARX model
and residuals are generated. A decision of accept or reject is taken based
on residuals and on a threshold calculated experimentally. We conduct
experiments with two different databases. Under certain conditions, we
found a rate of successful identifications in the order of 99.7 % for one
database and 100 % for the other.
1 Introduction
Iris recognition is related to the area of biometrics. The main intention of bio-
metrics is to provide reliable automatic recognition of individuals based on the
measuring of a physical or behavioral characteristic of persons. Biometrics can
be used for access control to restricted areas, such as airports or military instal-
lations, access to personal equipments such as laptops and cellular phones, and
public applications, such as banking operations [13]. A wide variety of biomet-
rics systems have been deployed and resulting systems include different human
features such as: face, fingerprint, hand shape, palmprint, signature, voice and
iris [8]. The last one may provide the best solution by offering a much more dis-
criminating power than the others biometrics. Specific characteristics of iris such
as a data-rich structure, genetic independence, stability over time and physical
protection, makes the use of iris as biometric well recognized.
2In last years, there have been different implementations of iris recognition sys-
tems. Daugman’s system [2] used multiscale quadrature wavelets (Gabor filters)
to extract texture phase structure information of the iris to generate a 2,048-bit
iris code and compared the difference between a pair of iris representations by
their Hamming distance. In [11] iris features are extracted by applying a dyadic
wavelet transform with null intersections. To characterize the texture of the iris,
Boles and Boashash [1] calculated a one dimension wavelet transform at various
resolution levels of a concentric circle on an iris image. In this case the iris match-
ing step was based on two dissimilarity functions. Wildes [15] represented the
iris texture with a Laplacian pyramid constructed with four different resolution
levels and used the normalized correlation to determine whether the input image
and the model image are from the same class. A Similar method to Daugman’s
is reported in [10], but using edge detection approach to localize the iris, and
techniques to deal with illumination variations, such as histogram equalization
and feature characterization by average absolute deviation. In [7] a new method
is presented to remove noise in iris images, such as eyelashes, pupil, eyelids and
reflections. The approach is based on the fusion of edge and region information.
In [3] an iris recognition approach based on mutual information is developed. In
that work, couples of iris samples are geometrically aligned by maximizing their
mutual information and subsequently recognized.
In our work we apply standard techniques as integro-differential operators to
locate the iris, and histogram equalization over extracted iris area to compen-
sate for illumination variations. The main contribution in this work is given in
the step of characterization of iris features by using stochastic ARX models,
commonly used by the automatic control community. In our work every iris in
database is represented by an ARX model learned from data. In the comparison
and matching step, data taken from an arriving iris sample are substituted in
every ARX model and residuals are generated. A decision to accept or reject the
sample is taken based on the residuals and on a threshold calculated experimen-
tally. The architecture of the proposed method is shown in Fig. 1.
We conduct experiments with UBIRIS database [14] and MILES database [12].
Under certain conditions we found a rate of successful identifications in the order
of 99.7 % and 100 % respectively.
2 THE PROPOSED APPROACH
The implementation of our approach relies on the use of colored eyes images
from UBIRIS and MILES databases. Eyes images include samples where iris
is free from any occlusion, and others with moderate obstruction from eyelids
and eyelashes. Noisy samples from UBIRIS database are shown in Fig. 2. We
transform the images color representation to just grey level pixels, because this
process is sufficient to reveal the relevant features of iris.
Our iris recognition system consists of four steps: iris localization and extraction,
iris features characterization, and comparison and matching.
3Fig. 1. Architecture of the iris recognition system based on ARX models
Fig. 2. Eyes samples with noise (moderate obstruction)
42.1 Iris Localization
The search of limbic and pupilar limits is achieved with the use of the integro-
differential operator shown in eqn 1.
max
(r, x0, y0)=
∣∣∣∣ ∂∂rG(r) ∗
∮
r,xc,yc
I(x, y)
2pir
ds
∣∣∣∣ (1)
where I(x, y) is an image containing an eye.
The operator behaves as an iterative circular edge detector, and searches over
the image domain (x, y) for the maximum in the partial derivative with respect
to an increasing radius r, of the normalized contour integral of I(x, y) along a
circular arc ds of radius r and center coordinates (x0, y0). The symbol ∗ denotes
convolution and Gσ(r) is a smoothimg function, tipically a Gaussian of scale σ.
The result of this localization operator is shown in Fig. 3.
Fig. 3. localization of limbic and pupilar limits with integro-differential operators
This operator behaves well in most cases with moderate noise conditions, but
requires some fine tuning of parameters, in order to deal with pupil reflections,
obscure eyes and excess of illumination. Heavy occlusion of iris by eyelashes or
eyelids needs to be handled by other methods. In our work, eye images with
heavy occlusion were discarded.
The extracted iris image has to be normalized to compensate for pupil dilation
and contraction under illumination variations. This process is achieved by a
transformation from cartesian to polar coordinates, using equations 2 and 3.
The output of this transformation is a rectangular image strip , shown in Fig. 4.
x(r, θ) = (1− r)xp(θ) + rxs(θ) (2)
y(r, θ) = (1− r)yp(θ) + rys(θ) (3)
where x(r, θ) and y(r, θ) are defined as a linear combination of pupil limits
(xp(θ), yp(θ)) and limbic limits (xs(θ), ys(θ)). r is defined in the interval [0, 1],
and θ in the interval [0, 2pi].
5Fig. 4. iris strip image
2.2 Feature Characterization by ARX Models
We propose the representation of iris image by an stochastic ARX model. An
ARX model represents the behavior of a dynamic system in discrete time [5],
where the output Vk depends on the input Uk and past values of both variables.
To represent iris image by an ARX model, we first divide the iris strip image in
a rectangular grid, and define output Vk as the mean grey level value of every
subarea in the grid. The input Uk is defined as the corresponding row number
of subarea on the grid.
In discrete time the ARX model is defined as follows:
vk = a1vk−1 + a2vk−2 + · · ·+ anavk−na + b1uk−1−nd + · · ·+ bnbuk−nb−nd (4)
Where a1, a2, · · · ana and b1, b2, · · · bnb are the model coefficients to be learned
by a least squares (LS) algorithm, nd is an integer number representing the
number of times steps that output Vk takes to show the effect of a given input
Uk. This term usually is called dead time. In our case we assume nd = 0. Former
model can be represented as a discrete transfer function in the complex z domain,
and is expressed as:
V (z)
U(z)
=
b1z
−1 + b2z−2 + · · ·+ bnbz−nb
1− a1z−1 − a2z−2 − · · · − anaz−na z
−nd (5)
The coefficients a1, a2, · · · ana and b1, b2, · · · bnb are learned by a least squares
(LS) algorithm. This method minimizes an index based on differences between
the real data and the model. We define first the following vectors:
ΨT = [vk−1 vk−2 · · · vk−na uk−1−nd · · · uk−nb−nd] (6)
θN = [a1 a2 · · · ana b1 · · · bnb] (7)
and then we can rewrite eqn. 4 as follows:
vk = ΨTk θ (8)
LS algorithm find the coefficients vector θ that makes the best estimate of
output vk, defined as:
6v̂k = ΨTk θ + ek (9)
where k = nm,nm+ a, ..., N and nm = max(na, nb+ nd).
difference between real data and model is given by:
eN = VN − ΨNθN (10)
The performance index that LS algorithm minimizes is given by:
J =
N∑
k=nm
e2k = e
T
NeN (11)
Iris database then is composed by an ARX model for every iris. When an iris
arrives for recognition, the comparison is made by using eqn. (11), and obtaining
the error in eqn. (10), where VN and θN belong to iris in database (IT ) and
matrix ΨN belongs to iris simple (IS), this lead us to following equations:
eN = V I
T
N − Ψ I
S
N θ
IT
N (12)
where eN , V I
T
N , Ψ
IS
N y θ
IT
N are defined as:
eN =

enm
enm+1
...
eN
 , V ITN =

vI
T
nm
vI
T
nm+1
...
vI
T
N
 ,
Ψ I
S
N =

vI
S
nm−1 · · · vI
S
nm−na u
IS
nm−1−nd · · · uI
S
nm−nb−nd
vI
S
nm · · · vI
S
nm−na+1 u
IS
nm−nd · · · uI
S
nm−nb−nd+1
...
. . .
...
...
. . .
...
vI
S
N−1 · · · vI
S
N−na u
IS
N−1−nd · · · uI
T
N−nb−nd
 , θITN =

aI
T
1
aI
T
2
...
aI
T
na
bI
T
1
...
bI
T
nb

3 Experiments
Experiments were ran for UBIRIS and MILES databases. Images with too much
occlusion and noise were discarded, because the difficulty to locate the iris region
with integro-differential operators. Then, our UBIRIS experimental database was
built with 1013 samples coming from 173 users, and MILES database consists of
7DB Size (%) # of iris # Threshold samples # Test samples
100 1013 52 788
90 912 46 693
80 811 42 596
70 710 36 501
50 507 26 308
Table 1. UBIRIS Databases used for experiments with 173 users. First column refers
to the percent of database used. For instance, 90 % means that 10 % of worst user
samples were discarded. Second column refers to total number of iris samples, third
column refers to the number of samples used to calculate the decision threshold, and
fourth column refers to the total number of samples used for testing.
DB Size (%) # of iris # Threshold samples # Test samples
100 36 10 167
75 36 8 116
Table 2. MILES Databases used for experiments with 119 users.
100 % 90 % 80 % 70 % 60 % 50 %
grid size DB DB DB DB DB DB
4 × 4 84.64 85.39 93.42 96.65 97.93 99.7
5 × 5 86.31 87.82 91.69 94.23 97.93 98.8
8 × 8 83.21 84.98 86.68 92.36 94.25 95.51
10 × 10 80.12 80.38 83.54 88.64 86.67 90.12
Table 3. Results of experiments for UBIRIS database (in % of accuracy) with different
grid size. Best results are highlighted
100 % 75 %
grid size DB DB
4 × 4 91.53 100.0
5 × 5 94.35 100.0
8 × 8 93.41 99.19
10 × 10 93.79 99.19
Table 4. Results of experiments for MILES database (in % of accuracy) with different
grid size. Best results are highlighted
8213 samples from 199 users. With these databases, we perform some experiments
with 100 % of samples and others experiments where worst user samples were
discarded. Tables 1 and 2 shows the different databases used.
The order of ARX model (number of coeffficients ai,bi) was determined em-
pirically by doing differents experiments, and best results were obtained for
na = 5 and nb = 5 in both, UBIRIS and MILES databases. In tables 3 and 4
these results are shown. We can see that best results were obtained for cleaner
databases.
In Fig.5 we can see the ROC curves for UBIRIS databases used in experiments.
Databases with cleaner iris samples reflects better results. In Fig. 6 we can see
the authentic-impostor distribution curves for two UBIRIS databases used. The
Overlapping between distribution curves in Fig. 6 (a) leads to worst results.
Similar ROC and authentic-impostor curves were obtained for MILES database
experiments and are not shown.
Fig. 5. Receiver operating characteristic (ROC) curve for experiments with different
databases.
4 Comparison to Previous Work
Daugman’s system [2] has been tested thoroughly with databases containing
thousands of samples, and reports of 100 % of accuracy have been given. In [11],
the experimental results given are in the order of 97.9 %, by working with a
database of 100 samples from 10 persons. Boles and Boashash [1] report best
results in the order of 100 % but working with very small sets of images. Wildes
[15] report results in the order of 100 % by working with a database of 600 sam-
ples coming from 40 individuals. In [10], a report is given about a performance
of 99.09 % in experiments with a database of 500 iris images from 25 individuals.
In [7] the results are between 98 % and 99 % by working with CASIA database
9(a) (b)
Fig. 6. authentic-impostor distribution for ARX based system. (a) database with 100%
of data. (b) database with 50% of data.
(2255 samples from 213 subjects). In [3] best results are in the order of 99.05 %
with a database of 384 images from 64 persons.
None of these works specify the quality of databases, so a direct comparison is
not possible. What we can say is that we are competitive with most methods
when our method work with clean databases, which means eye images with no
obstruction and noise. Our best results were 99.7 % for UBIRIS database and
100 % for MILES database obtained with cleanest databases.
5 Conclusions and Future Works
A new approach for iris recognition has been presented. The novel contribution
relies on the feature characterization of iris by the use of stochastic ARX models.
Although experimental results show better results for databases with cleaner
eyes images, we are looking forward to improve the methodology by combining
statistical sampling methods and stochastic models. We believe the combination
of best aspects of both approaches will lead us to a more robust and accurate
iris identification system.
References
1. W. Boles and B. Boashash, ”Iris Recognition for Biometric Identification using
dyadic wavelet transform zero-crossing”, IEEE Transactions on Signal Processing,
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Systems for Video Technology, Vol. 14, No. 1, 2004, pp. 21-30.
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English

An application of ARX stochastic models to iris recognition

Servicio de Difusión de la Creación Intelectual

An Application of ARX StochasticModels to Iris RecognitionLuis E. Garza Castan˜o´n1, Sau´l Montes de Oca2, andRube´n Morales-Mene´ndez11 Department of Mechatronics and Automation, ITESM Monterrey Campus,{legarza,rmm}@itesm.mx2 Automation Graduate Program Student, ITESM Monterrey Campus,saul montesdeoca@yahoo.com.mxAv. Eugenio Garza Sada Sur No. 2501Monterrey, N.L. 64849 Me´xicoAbstract. We present a new approach for iris recognition based on sto-chastic autoregressive models with exogenous input (ARX). Iris recogni-tion is a method to identify persons, based on the analysis of the eye iris.A typical iris recognition system is composed of four phases: image ac-quisition and preprocessing, iris localization and extraction, iris featurescharacterization, and comparison and matching. The main contributionin this work is given in the step of characterization of iris features byusing ARX models. In our work every iris in database is representedby an ARX model learned from data. In the comparison and matchingstep, data taken from iris sample are substituted into every ARX modeland residuals are generated. A decision of accept or reject is taken basedon residuals and on a threshold calculated experimentally. We conductexperiments with two different databases. Under certain conditions, wefound a rate of successful identifications in the order of 99.7 % for onedatabase and 100 % for the other.1 IntroductionIris recognition is related to the area of biometrics. The main intention of bio-metrics is to provide reliable automatic recognition of individuals based on themeasuring of a physical or behavioral characteristic of persons. Biometrics canbe used for access control to restricted areas, such as airports or military instal-lations, access to personal equipments such as laptops and cellular phones, andpublic applications, such as banking operations [13]. A wide variety of biomet-rics systems have been deployed and resulting systems include different humanfeatures such as: face, fingerprint, hand shape, palmprint, signature, voice andiris [8]. The last one may provide the best solution by offering a much more dis-criminating power than the others biometrics. Specific characteristics of iris suchas a data-rich structure, genetic independence, stability over time and physicalprotection, makes the use of iris as biometric well recognized.2In last years, there have been different implementations of iris recognition sys-tems. Daugman’s system [2] used multiscale quadrature wavelets (Gabor filters)to extract texture phase structure information of the iris to generate a 2,048-bitiris code and compared the difference between a pair of iris representations bytheir Hamming distance. In [11] iris features are extracted by applying a dyadicwavelet transform with null intersections. To characterize the texture of the iris,Boles and Boashash [1] calculated a one dimension wavelet transform at variousresolution levels of a concentric circle on an iris image. In this case the iris match-ing step was based on two dissimilarity functions. Wildes [15] represented theiris texture with a Laplacian pyramid constructed with four different resolutionlevels and used the normalized correlation to determine whether the input imageand the model image are from the same class. A Similar method to Daugman’sis reported in [10], but using edge detection approach to localize the iris, andtechniques to deal with illumination variations, such as histogram equalizationand feature characterization by average absolute deviation. In [7] a new methodis presented to remove noise in iris images, such as eyelashes, pupil, eyelids andreflections. The approach is based on the fusion of edge and region information.In [3] an iris recognition approach based on mutual information is developed. Inthat work, couples of iris samples are geometrically aligned by maximizing theirmutual information and subsequently recognized.In our work we apply standard techniques as integro-differential operators tolocate the iris, and histogram equalization over extracted iris area to compen-sate for illumination variations. The main contribution in this work is given inthe step of characterization of iris features by using stochastic ARX models,commonly used by the automatic control community. In our work every iris indatabase is represented by an ARX model learned from data. In the comparisonand matching step, data taken from an arriving iris sample are substituted inevery ARX model and residuals are generated. A decision to accept or reject thesample is taken based on the residuals and on a threshold calculated experimen-tally. The architecture of the proposed method is shown in Fig. 1.We conduct experiments with UBIRIS database [14] and MILES database [12].Under certain conditions we found a rate of successful identifications in the orderof 99.7 % and 100 % respectively.2 THE PROPOSED APPROACHThe implementation of our approach relies on the use of colored eyes imagesfrom UBIRIS and MILES databases. Eyes images include samples where irisis free from any occlusion, and others with moderate obstruction from eyelidsand eyelashes. Noisy samples from UBIRIS database are shown in Fig. 2. Wetransform the images color representation to just grey level pixels, because thisprocess is sufficient to reveal the relevant features of iris.Our iris recognition system consists of four steps: iris localization and extraction,iris features characterization, and comparison and matching.3Fig. 1. Architecture of the iris recognition system based on ARX modelsFig. 2. Eyes samples with noise (moderate obstruction)42.1 Iris LocalizationThe search of limbic and pupilar limits is achieved with the use of the integro-differential operator shown in eqn 1.max(r, x0, y0)=∣∣∣∣ ∂∂rG(r) ∗∮r,xc,ycI(x, y)2pirds∣∣∣∣ (1)where I(x, y) is an image containing an eye.The operator behaves as an iterative circular edge detector, and searches overthe image domain (x, y) for the maximum in the partial derivative with respectto an increasing radius r, of the normalized contour integral of I(x, y) along acircular arc ds of radius r and center coordinates (x0, y0). The symbol ∗ denotesconvolution and Gσ(r) is a smoothimg function, tipically a Gaussian of scale σ.The result of this localization operator is shown in Fig. 3.Fig. 3. localization of limbic and pupilar limits with integro-differential operatorsThis operator behaves well in most cases with moderate noise conditions, butrequires some fine tuning of parameters, in order to deal with pupil reflections,obscure eyes and excess of illumination. Heavy occlusion of iris by eyelashes oreyelids needs to be handled by other methods. In our work, eye images withheavy occlusion were discarded.The extracted iris image has to be normalized to compensate for pupil dilationand contraction under illumination variations. This process is achieved by atransformation from cartesian to polar coordinates, using equations 2 and 3.The output of this transformation is a rectangular image strip , shown in Fig. 4.x(r, θ) = (1− r)xp(θ) + rxs(θ) (2)y(r, θ) = (1− r)yp(θ) + rys(θ) (3)where x(r, θ) and y(r, θ) are defined as a linear combination of pupil limits(xp(θ), yp(θ)) and limbic limits (xs(θ), ys(θ)). r is defined in the interval [0, 1],and θ in the interval [0, 2pi].5Fig. 4. iris strip image2.2 Feature Characterization by ARX ModelsWe propose the representation of iris image by an stochastic ARX model. AnARX model represents the behavior of a dynamic system in discrete time [5],where the output Vk depends on the input Uk and past values of both variables.To represent iris image by an ARX model, we first divide the iris strip image ina rectangular grid, and define output Vk as the mean grey level value of everysubarea in the grid. The input Uk is defined as the corresponding row numberof subarea on the grid.In discrete time the ARX model is defined as follows:vk = a1vk−1 + a2vk−2 + · · ·+ anavk−na + b1uk−1−nd + · · ·+ bnbuk−nb−nd (4)Where a1, a2, · · · ana and b1, b2, · · · bnb are the model coefficients to be learnedby a least squares (LS) algorithm, nd is an integer number representing thenumber of times steps that output Vk takes to show the effect of a given inputUk. This term usually is called dead time. In our case we assume nd = 0. Formermodel can be represented as a discrete transfer function in the complex z domain,and is expressed as:V (z)U(z)=b1z−1 + b2z−2 + · · ·+ bnbz−nb1− a1z−1 − a2z−2 − · · · − anaz−na z−nd (5)The coefficients a1, a2, · · · ana and b1, b2, · · · bnb are learned by a least squares(LS) algorithm. This method minimizes an index based on differences betweenthe real data and the model. We define first the following vectors:ΨT = [vk−1 vk−2 · · · vk−na uk−1−nd · · · uk−nb−nd] (6)θN = [a1 a2 · · · ana b1 · · · bnb] (7)and then we can rewrite eqn. 4 as follows:vk = ΨTk θ (8)LS algorithm find the coefficients vector θ that makes the best estimate ofoutput vk, defined as:6v̂k = ΨTk θ + ek (9)where k = nm,nm+ a, ..., N and nm = max(na, nb+ nd).difference between real data and model is given by:eN = VN − ΨNθN (10)The performance index that LS algorithm minimizes is given by:J =N∑k=nme2k = eTNeN (11)Iris database then is composed by an ARX model for every iris. When an irisarrives for recognition, the comparison is made by using eqn. (11), and obtainingthe error in eqn. (10), where VN and θN belong to iris in database (IT ) andmatrix ΨN belongs to iris simple (IS), this lead us to following equations:eN = V ITN − Ψ ISN θITN (12)where eN , V ITN , ΨISN y θITN are defined as:eN =enmenm+1...eN , V ITN =vITnmvITnm+1...vITN ,Ψ ISN =vISnm−1 · · · vISnm−na uISnm−1−nd · · · uISnm−nb−ndvISnm · · · vISnm−na+1 uISnm−nd · · · uISnm−nb−nd+1.... . ........ . ....vISN−1 · · · vISN−na uISN−1−nd · · · uITN−nb−nd , θITN =aIT1aIT2...aITnabIT1...bITnb3 ExperimentsExperiments were ran for UBIRIS and MILES databases. Images with too muchocclusion and noise were discarded, because the difficulty to locate the iris regionwith integro-differential operators. Then, our UBIRIS experimental database wasbuilt with 1013 samples coming from 173 users, and MILES database consists of7DB Size (%) # of iris # Threshold samples # Test samples100 1013 52 78890 912 46 69380 811 42 59670 710 36 50150 507 26 308Table 1. UBIRIS Databases used for experiments with 173 users. First column refersto the percent of database used. For instance, 90 % means that 10 % of worst usersamples were discarded. Second column refers to total number of iris samples, thirdcolumn refers to the number of samples used to calculate the decision threshold, andfourth column refers to the total number of samples used for testing.DB Size (%) # of iris # Threshold samples # Test samples100 36 10 16775 36 8 116Table 2. MILES Databases used for experiments with 119 users.100 % 90 % 80 % 70 % 60 % 50 %grid size DB DB DB DB DB DB4 × 4 84.64 85.39 93.42 96.65 97.93 99.75 × 5 86.31 87.82 91.69 94.23 97.93 98.88 × 8 83.21 84.98 86.68 92.36 94.25 95.5110 × 10 80.12 80.38 83.54 88.64 86.67 90.12Table 3. Results of experiments for UBIRIS database (in % of accuracy) with differentgrid size. Best results are highlighted100 % 75 %grid size DB DB4 × 4 91.53 100.05 × 5 94.35 100.08 × 8 93.41 99.1910 × 10 93.79 99.19Table 4. Results of experiments for MILES database (in % of accuracy) with differentgrid size. Best results are highlighted8213 samples from 199 users. With these databases, we perform some experimentswith 100 % of samples and others experiments where worst user samples werediscarded. Tables 1 and 2 shows the different databases used.The order of ARX model (number of coeffficients ai,bi) was determined em-pirically by doing differents experiments, and best results were obtained forna = 5 and nb = 5 in both, UBIRIS and MILES databases. In tables 3 and 4these results are shown. We can see that best results were obtained for cleanerdatabases.In Fig.5 we can see the ROC curves for UBIRIS databases used in experiments.Databases with cleaner iris samples reflects better results. In Fig. 6 we can seethe authentic-impostor distribution curves for two UBIRIS databases used. TheOverlapping between distribution curves in Fig. 6 (a) leads to worst results.Similar ROC and authentic-impostor curves were obtained for MILES databaseexperiments and are not shown.Fig. 5. Receiver operating characteristic (ROC) curve for experiments with differentdatabases.4 Comparison to Previous WorkDaugman’s system [2] has been tested thoroughly with databases containingthousands of samples, and reports of 100 % of accuracy have been given. In [11],the experimental results given are in the order of 97.9 %, by working with adatabase of 100 samples from 10 persons. Boles and Boashash [1] report bestresults in the order of 100 % but working with very small sets of images. Wildes[15] report results in the order of 100 % by working with a database of 600 sam-ples coming from 40 individuals. In [10], a report is given about a performanceof 99.09 % in experiments with a database of 500 iris images from 25 individuals.In [7] the results are between 98 % and 99 % by working with CASIA database9(a) (b)Fig. 6. authentic-impostor distribution for ARX based system. (a) database with 100%of data. (b) database with 50% of data.(2255 samples from 213 subjects). In [3] best results are in the order of 99.05 %with a database of 384 images from 64 persons.None of these works specify the quality of databases, so a direct comparison isnot possible. What we can say is that we are competitive with most methodswhen our method work with clean databases, which means eye images with noobstruction and noise. Our best results were 99.7 % for UBIRIS database and100 % for MILES database obtained with cleanest databases.5 Conclusions and Future WorksA new approach for iris recognition has been presented. The novel contributionrelies on the feature characterization of iris by the use of stochastic ARX models.Although experimental results show better results for databases with cleanereyes images, we are looking forward to improve the methodology by combiningstatistical sampling methods and stochastic models. We believe the combinationof best aspects of both approaches will lead us to a more robust and accurateiris identification system.References1. W. Boles and B. Boashash, ”Iris Recognition for Biometric Identification usingdyadic wavelet transform zero-crossing”, IEEE Transactions on Signal Processing,Vol. 46, No. 4, 1998, pp. 1185-1188.2. J. Daugman, ”How Iris Recognition Works”, IEEE Transactions on Circuits andSystems for Video Technology, Vol. 14, No. 1, 2004, pp. 21-30.3. M. Dobes, L. Machala, P. Tichasvky, and J. Pospisil, ”Human Eye Iris RecognitionUsing The Mutual Information”, Optik, No. 9, 2004, pp. 399-404.4. A. Efros, and T. Leung, ”Texture Synthesis by Non-Parametric Sampling”, in Pro-ceedings of the 7th IEEE International Conference on Computer Vision, September1999, Vol. 2, pp. 1033-1038.105. G. Franklin, J. Powell, and M. Workman, Digital Control of Dynamic Systems,Addison-Wesley, 3a edition, 19976. J. Hammersley, ”Monte Carlo Methods for Solving Multivariate Problems”, Annalsof New York Academy of Science, 1960, No. 86, pp. 844-874.7. J. Huang, Y. Wang, T. Tan, and J. Cui, ”A New Iris Segmentation Method forIris Recognition System”, In Proceedings of the 17th International Conference onPattern Recognition, 2004, pp. 554-557.8. A. Jain, A. Ross, A. Prabhakar, ”An Introduction to Biometric Recognition”, IEEETransactions on Circuits and Systems for Video Technology, Vol. 14, No. 1, 2004,pp. 4-20.9. L. Liang, C. Liu, Y. Xu, B. Guo, and H. Shum, ”Real-time Texture Synthesis byPatch-based Sampling”, ACM Transactions on Graphics, Vol. 20, No. 3, July 2001,pp. 127-150.10. L. Ma, Y. Wang, T. Tan, and D. Zhang, ”Personal Identification Based on Iris Tex-ture Analysis”, IEEE Transactions on Pattern Analysis and Machine Intelligence,Vol. 25, No. 12, 2003, pp. 1519 - 1533.11. D. de Martin-Roche, C. Sanchez-Avila, and R. Sanchez-Reillo, ”Iris Recognitionfor Biometric Identification using dyadic wavelet transform zero-crossing”, In Pro-ceedings of the IEEE 35th International Conference on Security Technology, 2001,pp. 272-277.12. Miles Research. Sample iris Pictures. http://www.milesresearch.com/13. M. Negin, Chmielewski T., Salganicoff M., Camus T., Cahn U., Venetianer P., andZhang G. ”An Iris Biometric System for Public and Personal Use ”, Computer,Vol. 33, No. 2, 2000, pp. 70-75.14. H. Proenca, and L. Alexandre,”UBIRIS: A Noisy Iris Image Database”, in Pro-ceedings of the International Conference on Image Analysis and Processing 2005,Vol. 1, pp. 970-977.15. R. Wildes, ”Iris Recognition: An Emerging Biometric Technology”, Proceedings ofthe IEEE, Vol. 85, No. 9, 1997, pp. 1348-1363.

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An application of ARX stochastic models to iris recognition

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