Seeded Ising Model and Statistical Natures of Human Iris Templates
Song-Hwa Kwon∗
Department of Mathematics, Catholic University of Korea,
43 Jibong-ro, Bucheon-si, Gyeonggi-do 14662, Republic of Korea

Hyeong In Choi
Department of Mathematics, Seoul National University,
1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea

Sung Jin Lee

arXiv:1802.02223v1 [stat.AP] 3 Jan 2018

Department of Mathematics, Daejin University, 1007 Hoguk-ro,
Pocheon-si, Gyeonggi-do 11159, Republic of Korea

Nam-Sook Wee
Division of Smart Management Engineering, Hansung University,
116 Samseongyo-ro 16-gil, Seongbuk-gu, Seoul 02876, Republic of Korea
(Dated: February 8, 2018)
We propose a variant of Ising model, called the Seeded Ising Model, to model probabilistic nature
of human iris templates. This model is an Ising model in which the values at certain lattice points
are held fixed throughout Ising model evolution. Using this we show how to reconstruct the full iris
template from partial information, and we show that about 1/6 of the given template is needed to
recover almost all information content of the original one in the sense that the resulting Hamming
distance is well within the range to assert correctly the identity of the subject. This leads us to
propose the concept of effective statistical degree of freedom of iris templates and show it is about
1/6 of the total number of bits. In particular, for a template of 2048 bits, its effective statistical
degree of freedom is about 342 bits, which coincides very well with the degree of freedom computed
by the completely different method proposed by Daugman.

I.

INTRODUCTION

Human iris texture exhibits very intricate, even random or chaotic patterns. It is known that no two human
beings, even identical twins, have the same iris pattern.
Exploiting this fact, Daugman invented an iris recognition method which is widely used as a means of identifying human individuals [1, 2].
Daugman’s method, roughly put, creates a twodimensional binary array called template or iris code
from the annular iris region using Gabor transform; and
the templates are matched using the Hamming distance.
Figure 2 and 3 show a template in which the black pixels are the bits whose binary values are 0 and the white
ones with binary value 1. One can easily observe that 0s
and 1s tend to cluster together in somewhat randomly
alternating fashion. The clusters represent folding patterns of the 3-dimensional shapes which are the result of
stochastic or possibly chaotic development in utero whose
detailed morphogenesis depends on initial conditions in
the embryonic mesoderm.
In this paper, we study a probabilistic model which we
hope will elucidate this phenomenon of random mixing
with clustering. The model we propose is what we call the

Seeded Ising Model. It is an Ising model [3] in which bits
in certain locations are held fixed throughout Ising model
dynamic evolution. This way, our Seeded Ising Model
reconstructs a template from a fraction of the information
of the whole template.
It turns out that this model has a few remarkable properties. First of all, the reconstructed template exhibits
similar visual quality when compared qualitatively with
the original one. (See Figure 4.) However, more remarkable is the fact that the reconstructed templates actu-

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∗

Corresponding author. skwon@catholic.ac.kr; This work was
supported by the Catholic University of Korea, Research Fund,
2016.

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500

600

FIG. 1. A sample of eye image (from ICE2005)

2

FIG. 2. Iris template from the image in Fig. 1.

FIG. 3. Real (top) and imaginary (bottom) parts of the iris
template in Fig. 2.

ally match the original ones rather quite accurately when
computed with Hamming distance. This means that the
reconstructed templates retain quite a bit of information
content of the original ones as far as iris recognition is
concerned.
It is better to elaborate a bit more on seeds. First of
all, it is true that the morphogenetic initial conditions
relating to the embryonic development of mesoderm and
ectoderm are important factors for the full biological development of iris, which again is responsible for the fractal or chaotic patterns of iris. However, it is very hard
to pin down biologically these initial conditions. One
can only guess they must be somehow random in nature.
Similarly, in our model seeds are chosen randomly, which
is meant to be a mathematical abstraction of the random
nature of biological initial conditions.
The clustering phenomenon of iris template indicates
that not all binary bits can be independent. Then a question arises as to what is its degree of freedom. Daugman
looked at this question by examining the impostor distribution. By approximating it with a binomial distribution, he claims that the degree of freedom must be about
12.16% (249 bits out of 2048 bits) [2]. In this paper we
use somewhat different template generation and matching algorithm [4]. The impostor distribution gotten by
the algorithm we use is somewhat narrower, which implies that the degree of freedom computed à la Daugman
is about 17.19% (352 bits out of 2048 bits). (For details,
see Fig. 11.)
We also look at this problem of degree of freedom from
a completely different angle. We examine the genuine,
not impostor, matching and find that our Seeded Ising
Model needs 342 bits as seeds out of 2048 bits in order
to recover quite faithfully the information content of the
original template. It translates into about 16.70% (342
bits out of 2048 bits). It is remarkable to notice that the
degrees of freedom gotten by two completely different
approaches coincide so well.
The degree of freedom computed this way is only statistical in nature and furthermore, it only relates to the
information content from the view point of iris recognition. Because of this reason, we call it the Effective
Statistical Degree Of Freedom.
As far as we are aware, this kind of Seeded Ising
Model was not so far studied anywhere. However, it

exhibits some remarkable physical property resembling
re-normalization phenomenon. We hope to be able to
publish this finding elsewhere.
First, in Section II, we describe the proposed Seeded
Ising Model, and present a method of reconstructing iris
templates from partial information. It is a sampling
method based on the Metropolis algorithm adapted to
our model. We then examine how well our model reflects
the nature of human iris templates by conducting, in Section III, a series of statistical experiments based on the
proposed model. Subsequently we discuss the concept
of effective statistical degree of freedom, and finally give
concluding remarks in Section IV.

II.

SEEDED ISING MODEL

A real part or imaginary part of the human iris template is modeled by a binary random field x on an m × n
regular lattice. As we said above, each bit of iris template is binary with value 0 or 1, but when it comes to
the Ising model presentation, we use the convention that
each bit has value −1 or 1, instead. This convention
is purely for the sake of simplicity of notation of Ising
model. So according to this convention, 0 in iris template is replaced with −1 for the Ising model, and vice
versa. With this notational convention, which should be
understood in context throughout this paper, the space
of all iris templates is denoted by

T = {x|xi,j ∈ {−1, 1} for 1 ≤ i ≤ m, 1 ≤ j ≤ n}.
In the following description, we use the univariate indexing to denote the position in an m × n regular lattice by utilizing a mapping, for example, (i, j) 7→ k =
i + (j − 1) × m. With the univariate indexing scheme,
the space of iris templates can be simply written by
T = {x|xk ∈ {−1, 1} for 1 ≤ k ≤ mn}.
For a given subset I ⊆ {1, 2, · · · , mn}, a map s : I →
{−1, 1} is regarded as a ‘partial template data’ specifying
the value of the template at positions in I. So, the set of
templates which have the same partial template data as
s is denoted by T (s) = {x|xk = s(k) for each k ∈ I}.
All the templates in T (s) have the same value at positions of s, and the values at other positions of a template
in T (s) may be regarded as having grown from the fixed
‘seed’ s. Our way of looking at this “growth” is in fact
picking the most ‘likely’ sample from T (s) under some
probability distribution.
For a given seed s, we model the probability distribu-

3

FIG. 4. Reconstructed iris template

tion P (x) on the space of T (s) by


X
1
P (x) = exp 
Ji,j xi xj  ,
Z
i∼j
where i ∼ j means the positions i and j are adjacent
to each other, thus, the sum is done over all adjacent
positions, Ji,j is a constant parameter for the adjacent
positions i and j, and Z is the partition function given
by


X
X
Z=
exp 
Ji,j xi xj  .
Since we use different Ji,j depending on whether i and
j are horizontally or vertically adjacent to each other, we
say the relation i ∼v j means that two positions i and
j are vertically adjacent to each other, and the relation
i ∼h j that two positions i and j are horizontally adjacent to each other with circular-end conditions employed
for each row of a template. Note that we think of the
first column and the last column in a regular lattice are
adjacent to each other with circular-end conditions employed. In this paper, we set Ji,j = Jv when two positions
i and j are adjacent vertically, and Ji,j = Jh when two
positions i and j are adjacent horizontally. With these
conventions, P (x) can be written as


X
X
1
P (x) = exp Jv
xi xj + J h
xi xj  .
Z
i∼ j
i∼ j
v

h

We call this probabilistic model of the space of templates
T (s) with seed s the Seeded Ising Model.
A.

Sampling via Metropolis Algorithm

For a given seed s : I → {−1, 1}, we sample templates
in T (s) according to the distribution defined by P (x) via
the Metropolis algorithm [5]. First note that, under the
Seeded Ising Model, the probability P (x) is proportional
to


X
X
exp Jv
xi xj + J h
xi xj 
i∼v j

x = (x1 , x2 , · · · , xk−1 , xk , xk+1 , · · · , xmn ).
Then, the Metropolis algorithm modified for our context
would have the following steps:
1. Start with an initial template x ∈ T (s).
2. Select randomly a non-seed index
k ∈ {1, 2, · · · , mn} \ I.
3. Propose a new template x0 as
x0 = (x1 , x2 , · · · , xk−1 , −xk , xk+1 , · · · , xmn ).

i∼j

x∈T (s)

Let a template x ∈ T (s) be represented by a vector

i∼h j


= exp Jv (mn − n − 2dvx ) + Jh (mn − 2dhx ) ,
where dvx denotes the number of disagreeing vertical edges
in template x and dhx denotes the number of disagreeing
horizontal edges. Thus, P (x) is also proportional to the
un-normalized probability π(x) = exp(−2Jv dvx − 2Jh dhx ).

4. Define the proposal probability of t1 → t2 , moving
from a template t1 to a template t2 by

1

, if t1 , t2 differ at exactly


 mn − |I|
one non-seed index.
Q(t1 → t2 ) =




0,
otherwise.
Then, accept x0 with probability A(x → x0 ),


π(x0 )Q(x0 → x)
0
A(x → x ) = min 1,
,
π(x)Q(x → x0 )
where π(x) = exp(−2Jv dvx − 2Jh dhx ) is the unnormalized probability of x. Since x, x0 differ in exactly one non-seed index k by the construction of
x0 from x, Q(x → x0 ) = Q(x0 → x) > 0, and thus
we get


π(x0 )
0
A(x → x ) = min 1,
.
π(x)
The ratio in A(x → x0 ) is
exp(−2Jv (dvx0 − dvx ) − 2Jh (dhx0 − dhx ))
= exp(2Jv (dvx,k − avx,k ) + 2Jh (dhx,k − ahx,k )),
where dvx,k is the number of disagreeing vertical
edges between the index k and its vertically adjacent indices in template x, avx,k is the number of
agreeing vertical edges for x at k. dhx,k and ahx,k are
defined similarly.
5. Generate a uniform random number u ∈ (0, 1) and
accept x0 as the current template if u < A(x → x0 ).
Otherwise, keep x as the current template and go
to Step 2.

4
B.

Reconstruction of Iris Templates

By the Metropolis algorithm for the Seeded Ising
Model, we can sample a template x ∈ T (s) for a given
seed s according to the distribution defined by P (x).
However, since the space T (s) is so huge, it is highly
unlikely that one sampled template can serve as a representative template in T (s). For this reason, we produce
a reconstructed template by using the idea of bagging
(bootstrap aggregating) [6] in machine learning. The
procedure of obtaining a reconstructed template is as follows: Let t0 be an initial template for Metropolis algorithm, and tn be the template obtained after n iterations
by the Metropolis algorithm. Then, for predetermined
positive integers L and nj ’s for j = 1, 2, · · · , L with the
condition that 0 ≤ n1 < n2 < · · · < nL , we define the
reconstructed template rn whose value rin at position i
is given by the following:

L
X

n

 1,
if
ti j ≥ 0


j=1
(1)
rin =
L
X


nj


ti < 0,
 −1, if
j=1

where n denotes the vector of n = (n1 , · · · , nL ).
Fig. 4 shows an example of template rn reconstructed
from a seed s that is extracted from a full human iris
template in Fig. 2. When two templates in Fig. 2 and in
Fig. 4 are compared, the visual qualities look quite similar even though the reconstructed template rn is generated by utilizing only the partial information in Fig.
2. This may be taken as a supporting evidence that our
Seeded Ising Model explains reasonably well the statistical nature of human iris templates, at least visually.
III.

STATISTICAL EXPERIMENTS

To examine how good the proposed Seeded Ising Model
is, we performed statistical experiments. In the experiments, we used the algorithm developed by Lee et al.
[4] for iris template generation and matching. The size
of iris template generated by this algorithm is 8 × 256
which is of the same size as the template in Fig. 2. Since
the template consists of real part and imaginary part, the
size of real (or imaginary) part of templates as in Fig. 3
is 8 × 128.
The dataset used in this paper for statistical experiments consists of reasonably good images selected from
ICE2005 Dataset which was used for Iris Challenge Evaluation 2005 [7]. Table I shows basic statistics of ICE2005
Dataset and Table II shows basic statistics of the dataset
used in this paper.
Fig. 5 shows the matching results of the dataset used
in this paper when no rotation is considered in matching,
and Fig 6 shows the matching results of the same dataset
with rotation applied. From the distribution, we may

TABLE I. ICE2005 Dataset used (with error correcteda )
Position
Left
Right
Total
a

# of Images
1527
1426
2953

# of Subjects
119
124
132

[7]

TABLE II. Dataset used in this paper
Position
Right

# of Images
# of Subjects
948
120
# of Genuine matchings # of Impostor matchings
5,953
442,925

observe that most of genuine matching (matching two iris
templates from the same person) has Hamming distance
less than 0.4, and most of impostor matching (matching
two iris templates from two other persons) has Hamming
distance more than 0.4. So, roughly speaking, the best
threshold that distinguishes whether a given pair of iris
templates comes from the same person or not would be
around 0.4.
A.

The parameter J

To verify our Seeded Ising Model explains well the statistical nature of iris templates, we first need to find the
parameter J = (Jv , Jh ). To determine the best parameter, we first performed statistical experiments for a range
of parameter values. For the experiment, we first fixed
one iris template x = (xR , xI ) where xR and xI denote
the real and imaginary part of template x, respectively,
and for each
J = (Jv , Jh ) ∈ {(0.1 ∗ i, 0.1 ∗ j)| 1 ≤ i, j ≤ 10 },
100 templates are reconstructed using random seeds
s = (sR , sI ), where sR and sI denote the seed for the
real and imaginary part of templates, respectively, and
matched with the original template x. To obtain each reconstructed template, we first randomly chose an index
set I ⊆ {1, 2, · · · , mn}, where mn is the size of real or
imaginary part of template x, which is 1024(= 8×128) in
our case. In this experiment, we chose I so that |I| = 256,
i.e., the ratio of information in seed s to that in x is 1/4.
After obtaining the index set I, the seed sR : I → {−1, 1}
is naturally given from template xR by the relation that
I
sR (k) = xR
k for k ∈ I. The seed s for the imaginary
part of template is also determined similarly from xI .
Once the seed sR is determined, the real part of a reconstructed template is computed by Equation (1) with
n = 104 (1, 2, · · · , 100). The imaginary part of a reconstructed template is also generated from the seed sI similarly, and these two parts are combined to produce the
final reconstructed template.

5

Dist

16

genuine
impostor

14

0.34

12

0.33
0.32

mean dist

Density

10
8

0.31
0.3

6

0.29

4

0.28
1
0.8

2

1
0.8

0.6
0.6

0.4

0

0.2

Jh

0

0.2

0.4

0.6

0.4
0.2

0.8

FIG. 5.
Genuine/Impostor Distribution of Hamming distance without rotation (5,953 genuine matchings, 442,925 impostor matchings)

FIG. 7. Minimum mean distance (0.2846) at J = (0.2, 0.3).

Dist

35
30

Dist
25

25

genuine
impostor

20

20

Jv

rec(1/5)
rec(1/6)
rec(1/7)
init(1/5)
init(1/6)
init(1/7)

Density

15
15

10
10

5
0
0.2

5

0.25

0.3

0.35

0.4

0.45

0.5

0
0

0.2

0.4

0.6

0.8

FIG. 6.
Genuine/Impostor Distribution of Hamming distance with rotation (5,953 genuine matchings, 442,925 impostor matchings)

The average of 100 Hamming distances for each value
of J = (Jv , Jh ) is shown in Figure 7. The minimum
of 0.2846 of mean Hamming distance is obtained when
J = (Jv , Jh ) = (0.2, 0.3). In the subsequent experiments,
we use this values for the parameter J unless mentioned
otherwise.

B.

Hamming Distances of Reconstructed
Templates

To compare the original iris template and the reconstructed template, we selected 100 images from ICE2005
Dataset, and generated 100 reconstructed templates for

FIG. 8.

each iris template and measured Hamming distance between the original template and the reconstructed template. For reconstruction, we used the same procedure
as described in Section III A with several different values
of seed size, i.e., 1/5, 1/6, 1/7 of template size.
In what follows, an initial template is the one randomly
selected from T (s) as in Step 1 of the Metropolis algorithm. In practice, initial template is made by randomly
assigning values at non-seed positions, while using the
fixed values of s at the positions of seed s.
Figure 8 shows the distributions of Hamming distances. In Figure 8, the dotted curves represent the
distributions of Hamming distances between the original
templates and the initial templates in the reconstruction
procedure, and the solid curves represent the distributions of Hamming distances between the original templates and the reconstructed templates given by Equation (1). The numbers 1/5, 1/6, 1/7 in the legend represent the proportion of the seed size in the full template.

6
TABLE III. Means and Standard Deviations of distances
Initial Template
mean
std
0.3998
0.0126
0.4164
0.0128
0.4281
0.0130

Reconstructed Template
mean
std
0.3123
0.0179
0.3362
0.0179
0.3547
0.0182

Table III shows the mean distance and standard deviation for each case. Compared to the mean Hamming distance of initial templates, the mean Hamming distance
of reconstructed templates is significantly small. For instance, in the case of 1/6 seed size, the mean distance of
0.3362 is more than 3.5σ away from the threshold value
of 0.4, which is a relatively good threshold determining
whether two iris templates come from the same person or
not as discussed in the beginning of this section. Even in
the case of 1/7 seed size, the mean distance is more than
2.4σ away from the threshold. This implies that the reconstructed template is statistically similar to, or essentially have the same information content as, the original
iris template. In other words, the reconstructed template
can be safely deemed to have come from the same person
to whom the original template belongs. This, we believe,
is a good evidence that the proposed Seeded Ising Model
is a reasonably good model for probabilistic model of iris
templates.

C.

Effective statistical degree of freedom

Once the Seeded Ising Model has been shown as a good
model for iris templates, the next question would be how
big the seed size is in order to reliably reconstruct human
iris templates. For that, we use the cumulative distribution function of Hamming distance.
For a given collection of M matching distances

1

Match Rate

0.8

1/5
1/6
1/7
genuine

0.6
0.4
0.2
0
0.1

0.2

0.3

threshold

FIG. 9.

0.4

0.5

1

0.995

Match Rate

size of seeds
1/5
1/6
1/7

1/5
1/6
1/7
genuine

0.99

0.985

0.98
0.375 0.38 0.385 0.39 0.395

0.4

0.405 0.41

threshold

FIG. 10.

(d1 , d2 , · · · , dM ), match rate m(d) is defined as the ratio of the number of matchings with Hamming distance
less than or equal to d,

m(d) =

|{i|di ≤ d}|
.
M

Figure 9 shows match rate curves for several different
collections of matching distances. The dotted curve represents the matching rate for genuine matchings of 948
images. (See Table II for the details.) The solid curves
in blue, red, yellow represent the matching rates of Hamming distances between the original templates and reconstructed templates from the seeds of size 1/5, 1/6, 1/7,
respectively.
Figure 10 is the zoomed figure of Figure 9 around Hamming distance 0.395. From these figures, we may say the
matching rate of the reconstructed templates from the
seeds of size 1/6 is comparable to, with slightly better
performance, the matching rate of the genuine matchings of real iris templates. This analysis gives approximate size of seeds for the real iris templates. Roughly
speaking, only about 1/6 ≈ 16.7% of template information is needed to account for the information content of
the whole iris templates through our Seeded Ising Model.
From this, we may conclude that the proposed Seeded
Ising Model needs 342 bits as seeds out of 2048 bits in
order to recover quite faithfully the information content
of the original template. It translate into about 16.70%
(342 bits out of 2048 bits).
As we alluded in the beginning, the clustering phenomenon of iris templates indicates that not all binary
bits can be independent, and the above result suggests
that the degree of freedom, the size of seeds, is about
16.7%. The degree of freedom computed this way is only
statistical in nature and furthermore, it only relates to
the information content from the view point of iris recognition. Because of this reason, we call it the Effective
Statistical Degree Of Freedom.

7
tion (2), we got p = 0.4947 and N = 352. This implies
that the degree of freedom calculated from the impostor
distribution as suggested by Daugman is about 17.19%
(352 bits out of 2048 bits), which is very similar to 16.7%,
our result obtained above through the genuine matching
via template reconstruction from seeds.
It is remarkable to notice that the our definition of degree of freedom coincides so well with that of Daugman,
although the two approached to degree of freedom are
completely different.

16
14
12

Density

10
8
6
4

IV.

CONCLUSION

2
0
0

0.2

0.4

0.6

0.8

1

Hamming Distance

FIG. 11. Distribution of Hamming Distances from 442, 925
impostor comparisons. The histogram is fitted by the binomial distribution with p = 0.4947 and N = 352 degrees-offreedom in solid curve (2).

Our result has a parallel counterpart coming from different angle. Concerning the question of what is the degree of freedom of iris templates, Daugman examined
the impostor distribution. (Ours is through the genuine
distribution.) By approximating it with a binomial distribution given by f (x),

f (x) =

N!
pm (1 − p)N −m ,
m!(N − m)!

(2)

where x = m/N is the outcome fraction of N Bernoulli
trials with probability of success p, he claims that the
degree of freedom must be about 12.16% (249 bits out of
2048 bits) [2].
To replicate Daugman’s finding in our context, we need
to exercise some caution. First, the algorithms we use for
template generation and matching [4] is different from
the one he uses, and we are also using different image
dataset in this paper. The impostor distribution plot
in Figure 11 looks somewhat different from that in [2]
in that our impostor distribution is somewhat narrower.
When we calculate, using our algorithm, the binomial
distribution approximation as defined by f (x) in Equa-

[1] J. G. Daugman, IEEE Transactions on Pattern Analysis
and Machine Intelligence 15, 1148 (1993).
[2] J. Daugman, Circuits and Systems for Video Technology,
IEEE Transactions on 14, 21 (2004).
[3] B. A. Cipra, Am. Math. Monthly 94, 937 (1987).
[4] Y. Lee, R. J. Micheals, P. J. Phillips, and J. J. Filliben,

In this paper, we proposed the Seeded Ising Model,
a probabilistic model of human iris templates. Inspired
by the biological processes how human iris texture patterns develop, we devised a probabilistic model by introducing ‘seeds’ that is a mathematical abstraction of
initial conditions in the embryonic mesoderm and ectoderm from which the chaotic iris texture develops. Also,
we found the best parameter value of J for the proposed
Seeded Ising Model from the iris recognition point of
view. We also provided statistical evidences that the proposed model is a reasonably good model capable of explaining the probabilistic nature of human iris templates.
In fact, it turns out that artificially generated iris templates based on the proposed model with the best parameter value of J share many probabilistic natures of real
human iris templates in that (1) The reconstructed templates are visually similar to real iris templates including
clustering phenomenon; (2) The reconstructed templates
from the partial template information are much more
closer to the original iris templates in terms of Hamming
distance.
We also found approximate size of seeds for the real iris
templates. Roughly speaking, our Seeded Ising Model
implies that only about 1/6 ≈ 16.7% of template information is needed to account for the information content of the whole iris templates. Based on this finding.
we propose the concept of Effective Statistical Degree Of
Freedom of human iris templates. Surprisingly, this estimated value coincides very well with the degree of freedom computed by the completely different method proposed by Daugman.
Many evidences presented in this paper suggest that
we may conclude the proposed model reflects well the
real nature of human iris templates.

Journal of Research of NIST 118, 218 (2013).
[5] M. Newman and G. Barkema, Monte Carlo Methods in
Statistical Physics (Clarendon Press, 1999).
[6] L. Breiman, Machine Learning 24, 123 (1996).
[7] P. J. Phillips, K. W. Bowyer, P. J. Flynn, X. Liu, and
W. T. Scruggs, 2nd IEEE International Conference on Biometrics: Theory, Applications and Systems , 1 (2008).

